Applied Financial Econometrics Slides Rolf Tschernig — Florian Brezina University of Regensburg

Version: 18 July 20121

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c Rolf Tschernig. I very much thank Joachim Schnurbus for his important corrections and suggestions.

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Contents

1 Introduction 1.1 Themes . . . . . . . 1.2 Some basics . . . . . 1.3 Stochastic processes 1.4 R Code . . . . . . . 2 The 2.1 2.2 2.3 2.4 2.5 2.6

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basics of time series modeling Autoregressive processes . . . . . Moving average processes . . . . . ARMA processes . . . . . . . . . Trajectory examples . . . . . . . . Estimation . . . . . . . . . . . . R Code . . . . . . . . . . . . . .

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15 15 29 35 38 41 62

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3 Forecasting (financial) time series 3.1 Some general remarks and definitions . 3.2 Decomposition of prediction errors . . 3.3 AR Model Specification . . . . . . . 3.4 Prediction with AR models . . . . . . 3.5 Evaluation of forecasts . . . . . . . . 3.6 R Code . . . . . . . . . . . . . . . . 4 More on modeling time series 4.1 Unit root tests . . . . . . . . . . . . 4.2 Model Checking . . . . . . . . . . . . 4.3 Estimating dynamic regression models 4.4 R Code . . . . . . . . . . . . . . . .

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5 Modeling volatility dynamics 5.1 Standard conditional volatility models . . . . . . . . . 5.2 Maximum Likelihood Estimation . . . . . . . . . . . . 5.3 Estimation of GARCH(m, n) models . . . . . . . . . . 5.4 Asymmetry and leverage effects . . . . . . . . . . . . 5.5 Testing for the presence of conditional heteroskedasticity 5.6 Model selection . . . . . . . . . . . . . . . . . . . . . iii

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117 119 125 131 140 142 144

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5.7 Prediction of conditional volatility . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 R Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Long-run forecasting 151 6.1 Estimating/predicting unconditional means . . . . . . . . . . . . . . . . . . . . . . 151 6.2 Predicting long-term wealth: the role of arithmetic and geometric means . . . . . . 160 6.3 Are long-term returns predictable? . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7 Explaining returns and estimating factor models 7.1 The basics of the theory of finance . . . . . . . . 7.2 Is the asset pricing equation empirically relevant? 7.3 Factor-pricing models . . . . . . . . . . . . . . . 7.4 Regression based tests of linear factor models . . 7.5 Supplement: Organisation of an empirical project

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Applied Financial Econometrics — General Information — U Regensburg — July 2012

General Information

Schedule and locations see http://www-wiwi.uni-regensburg.de/Institute/VWL/Tschernig/Lehre/Applied.html.en

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Applied Financial Econometrics — General Information — U Regensburg — July 2012

Contact Rolf Tschernig

Florian Brezina

Building

RW(L), 5th floor, Room 515

RW(L), 5th floor, Room 517

Tel.

(+49) 941/943 2737

(+49) 941/943 2739

Fax

(+49) 941/943 4917

(+49) 941/943 4917

[email protected] [email protected]

Homepage http://www-wiwi.uni-regensburg.de/Institute/VWL/Tschernig/Team/index.html.en

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Applied Financial Econometrics — General Information — U Regensburg — July 2012

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Grading The grade for the course will be based on the written exam, the graded presentations of the exercises (up to a maximum of 15 points), and a mid-term exam. Details are provided at: http://www-wiwi.uni-regensburg.de/images/institute/vwl/tschernig/lehre/Hinweise_Kurse_Noten_ab_ SS2010.pdf.

Final Exam early in August: precise date and time will be announced.

Prerequisites ¨ ¨ Okonometrie I or even better Methoden der Okonometrie or comparable courses.

Literature • Basic textbook(s): – Cochrane, J.H. (2005). Asset Pricing, rev. ed., Princeton University Press.

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Applied Financial Econometrics — General Information — U Regensburg — July 2012

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– Kirchg¨assner, G. and Wolters, J. (2008, 2007). Introduction to modern time series analysis, Springer, Berlin. (In the campus network full text available) – L¨utkepohl, Helmut und Kr¨atzig, Markus (2004, 2008). Applied Time Series Econometrics, Cambridge University Press. • Introduction to software R: – Kleiber, C. and Zeileis, A. (2008). Applied econometrics with R, Springer, New York. (In the campus network full text available) – Ligges, U. (2008). Programmieren mit R, Springer, Berlin. (In the campus network full text available) Additional Reading: • Introductory level: – Brooks, C. (2008). Introductory econometrics for finance, 2nd ed., Cambridge University Press. – Diebold, F.X. (2007). Elements of forecasting, 4. ed., Thomson/South-Western. – Wooldridge, J.M. (2009). Introductory Econometrics. A Modern Approach, Thomson South-Western.

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Applied Financial Econometrics — General Information — U Regensburg — July 2012

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• Graduate level: – Campbell, J.Y., A.W. Lo, and A.C. MacKinlay (1997). The Econometrics of Financial Markets, Princeton University Press. – Enders, W. (2010). Applied econometric time series, Wiley. – Franke, J., H¨ardle, W., and Hafner, C. (2011). Statistics of financial markets. An introduction, Springer, (Advanced, old edition in German available) – Tsay, R.S. (2010). Analysis of financial time series, Wiley. • German Reading: – Franke, J., H¨ardle, W., and Hafner, C. (2004). Einf¨ uhrung in die Statistik der Finanzm¨arkte, 2. ed., Springer. (Advanced, newer English edition available) – Kreiß, J.-P. and Neuhaus, G. (2006). Einf¨ uhrung in die Zeitreihenanalyse, Springer. (Advanced, In the campus network full text available) – Neusser, K. (2009). Zeitreihenanalyse in den Wirtschaftswissenschaften, 2. Auflage, Teubner. (In the campus network full text available, many copies of the first edition (2006) available for lending)

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Applied Financial Econometrics — 1 Introduction — U Regensburg — July 2012

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1 Introduction

1.1 Themes • How to measure returns and risks of financial assets? • Are asset returns predictable? In the short run - in the long run? −→ requires command of time series econometrics • Does the risk of an asset vary with time? Is this important? How can one model time-varying risk? • Is the equity premium (excess returns of stocks over bonds) really that high? • How can one explain variations in stock returns across various stocks? For outline of the course see Contents

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Applied Financial Econometrics — 1.1 Themes — U Regensburg — July 2012

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This course provides an introduction to the basics of financial econometrics, mainly to analyzing financial time series. There are many more topics in financial econometrics that cannot be covered by this course but are treated in advanced textbooks such as Franke et al. (2011) or Tsay (2010). A selection of advanced topics not treated here is: • Statistics of extreme risks • Credit risk management and probability of default • Interest rate models and term structure models • Analyzing high-frequency data and modeling market microstructure • Analyzing and estimating models for options • Multivariate time series models • Technical methods such as state-space models and the Kalman filter, principal components and factor models, copulae, nonparametric methods, ....

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Applied Financial Econometrics — 1.2 Some basics — U Regensburg — July 2012

1.2 Some basics • Return Rt (or gross return) Rt = • Net return

Pt + Dt Pt−1

(Pt − Pt−1) + Dt = Rt − 1 Pt−1

• Log returns rt or continuously compounded returns – Recall: ln(1) = 0,

∂ ln(x) ∂x

= x1 . Taking a Taylor expansion of degree 1 at x0 delivers

ln x ≈ ln x0 +

∂ ln(x) 1 (x − x0) = ln x0 + (x − x0) ∂x |x0 x0

Thus, expanding at x0 = 1, one has for x close to 1 ln x ≈ x − 1 – Replacing x by Rt gives rt = log(Rt) ≈ Rt − 1

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Applied Financial Econometrics — 1.2 Some basics — U Regensburg — July 2012

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• Real prices Pt CP It Note that if real prices should be given in prices of year s, then one has to compute CP Is real pricet(s) = Pt CP It real pricet(t) =

• Real return real returnt =

CP It−1 Pt/CP It + Dt/CP It Pt + Dt CP It−1 = = Rt Pt−1/CP It−1 Pt−1 CP It CP It

• Log real returns CP It−1 logged real returnt = log Rt CP It = log(Rt) + log CP It−1 − log CP It = rt + log CP It−1 − log CP It • Excess log returns of asset A over asset B excess log returnt = log(RtA) − log(RtB ) = rtA − rtB = rtA + log CP It−1 − log CP It − rtB + log CP It−1 − log CP It = excess log real returnt

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Applied Financial Econometrics — 1.2 Some basics — U Regensburg — July 2012

A first look at data: S&P 500 Composite Index Real prices of the S&P 500 Composite and real earnings, January 1871 – March 2012, Source: homepage of Robert Shiller: www.econ.yale.edu/~shiller/ 2500

450

2000

350 300

1500 250

Price 200 1000

150 100

500

50

Earnings 0

1870

0

1890

1910

1930

1950

1970

Year

Are real prices RPt predictable? 10

1990

2010

Real S&P Composite Earnings

Real S&P 500 Stock Price Index

400

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Applied Financial Econometrics — 1.2 Some basics — U Regensburg — July 2012

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• Estimating a simple model RPt = α0 + α1RPt−1 + ut, Call: lm(formula = RP.t ~ 1 + RP.tm1) Residuals: Min 1Q -251.036 -5.128

Median 0.356

3Q 6.094

Max 155.825

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.707306 0.742315 0.953 0.341 RP.tm1 1.000171 0.001299 769.665 <2e-16 *** --Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 21.56 on 1693 degrees of freedom Multiple R-squared: 0.9972,Adjusted R-squared: 0.9971 F-statistic: 5.924e+05 on 1 and 1693 DF, p-value: < 2.2e-16

What do you learn from that? Potential pitfalls?

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t = 1, . . . , 1695

(1.1)

Applied Financial Econometrics — 1.3 Stochastic processes — U Regensburg — July 2012

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Potential issues when estimating a model like (1.1): – lagged endogeneous regressors, – nonstationarity of regressor, – heteroskedastic errors, – lag length. Models where the regressors are exclusively lagged endogenous variables are called autoregressive models, see section 2.1.

1.3 Stochastic processes First, some more terms: Stochastic process or random process: A stochastic process {. . . , y−2, y−1, y0, y1, y2, . . . , } is a collection of random variables yt’s with their indices ordered with respect to time. An observed realization of a stochastic process is called trajectory or sample path.

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Applied Financial Econometrics — 1.3 Stochastic processes — U Regensburg — July 2012

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The simplest example of a stochastic process is: White Noise (WN): A collection of random variables {ut}∞ t=−∞ is called white noise if • the unconditional mean is zero E[ut] = 0 for all t, and • the variance is identical for all t, i.e. V ar(ut) = σ 2 for all t, and • the random variables are uncorrelated over time Cov(ut, us) = 0 for all t, s and t 6= s. Note that there may be dependence in the higher order moments (moments of order three or more). E.g., one has E[u2t u2s ] 6= E[u2t ]E[u2s ]). The latter case is excluded if the ut and us are stochastically independent and identically (i.i.d.) distributed. The i.i.d. condition is automatically fulfilled if the random variables are normally distributed and uncorrelated. Then one has a stronger version of white noise: Gaussian white noise. In case stock returns are white noise, what does this imply?

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Applied Financial Econometrics — 1.4 R Code — U Regensburg — July 2012

1.4 R Code Code for estimating (1.1): # Estimating AR(1) model for real_price data series from Shiller’s dataset ie_data.xls # RP is vector containing real_price data RP.0to1 <- embed(RP, 2)

# matrix containing contemporaneous and (once) lagged data

RP.t RP.tm1

# contemporaneous series # lagged series

<- RP.0to1[,1] <- RP.0to1[,2]

est.ar1 <- lm(RP.t ~ 1 + RP.tm1) summary(est.ar1)

# ols regression with constant # summary output of linear regression

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Applied Financial Econometrics — 2 The basics of time series modeling — U Regensburg — July 2012

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2 The basics of time series modeling

2.1 Autoregressive processes 2.1.1 Autoregressive processes of order one (AR(1) processes) A stochastic process {yt}t∈T that is generated by the following (stochastic) difference equation yt = α0 + α1yt−1 + ut,

t∈T

where the ut’s are white noise and T = {0, 1, 2, . . . , } or T = Z = {. . . , −2, −1, 0, 1, 2, . . . , } is called autoregressive process of order one (AR(1) process). If T = Z we say that the process started a long time age. The random variable ut is called error, disturbance, shock, or innovation.

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Applied Financial Econometrics — 2.1.1 AR(1) processes — U Regensburg — July 2012

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Statistical properties of an AR(1) process • For α1 = 0 the stochastic process yt − α0 is white noise. • Expressing yt as a weighted sum of past and present white noise terms (plus starting value) yt = α0 + α1 (α0 + α1yt−2 + ut−1) + ut = α0 + α1α0 + α12yt−2 + α1ut−1 + ut ... j−1 2 = α0 1 + α1 + α1 + · · · + α1 + α1j yt−j + α1j−1ut−(j−1) + · · · + α12ut−2 + α1ut−1 + ut = α0

j−1 X

α1k + α1j yt−j +

k=0

j−1 X

α1k ut−k .

k=0

– For |α1| < 1 and j → ∞ (process has run since ever) one has j 2 1 + α1 + α1 + · · · + α1 + . . . = 1/(1 − α1) α1j y0 → 0 and therefore yt = α0/(1 − α1) +

∞ X

α1j ut−j .

(2.1)

j=0

The random variable yt is a weighted infinite sum of past values and the present value of the

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Applied Financial Econometrics — 2.1.1 AR(1) processes — U Regensburg — July 2012

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white noise process {ut}. The importance of shocks declines quickly. E.g. for α1 = 0.9 one has 0.910 = 0.349 and 0.950 = 0.05. – For α1 = 1 one cannot let j → ∞. Typically one chooses j = t and obtains yt = tα0 + y0 + ut + ut−1 + · · · + u1,

t = 1, 2, . . . .

∗ If α0 = 0, the stochastic process {yt}∞ t=1 is called random walk because yt is an unweighted sum of random increments. Shocks keep their importance. One also says that yt has long memory (even perfect memory). ∗ For α0 6= 0, it will be seen below that E[yt] = α0t + E[y0]. Thus, one has a combination of a (pure) random walk with a deterministic trend or drift. In this case {yt} is called a random walk with drift. – For α1 > 1 the influence of a shock increases with its distance to the present observation. One has an explosive autoregressive model. • The conditional expectation E[yt|yt−1] = α0 + α1yt−1 is in general different from zero.

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Applied Financial Econometrics — 2.1.1 AR(1) processes — U Regensburg — July 2012

• The unconditional expectation for period t is α0/(1 − α1) µt ≡ E[yt] = α0(1 + α1 + . . . + α1t−1) + α1t E[y0] tα0 + E[y0]

if |α1| < 1 and t ∈ Z — independent of t, if t = 1, 2, . . . — dependent on t, if α1 = 1 and t = 1, 2, . . . — dependent on t.

Knowing µt one can rewrite the autoregressive process as yt − µt = α1(yt−1 − µt−1) + ut. Check! • Unconditional variance: V ar(yt) ≡ E[(yt − µt)2] 2 = E (α1(yt−1 − µt−1) + ut) = E α12(yt−1 − µt−1)2 + 2E[α1(yt−1 − µt−1)ut] + E[u2t ] = α12V ar(yt−1) + 2 · 0 + σ 2 = α12V ar(yt−1) + σ 2. Again, further results depend on α1:

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Applied Financial Econometrics — 2.1.1 AR(1) processes — U Regensburg — July 2012

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Inserting iteratively delivers 2 2 ) if |α1| < 1 and t ∈ Z — independent of t, σ /(1 − α 1 P V ar(yt) = σ 2 t−1 α12j if V ar(y0) = 0 and t = 1, 2, . . . — dependent on t, j=0 2 σ t if α1 = 1 and V ar(y0) = 0 and t = 1, 2, . . . — dependent on t. • Definition of the autocovariance function: Cov(yt, ys) ≡ E[(yt − µt)(ys − µs)] Computation: Take for simplicity s = t − 1: Cov(yt, yt−1) = E [(α1(yt−1 − µt−1) + ut) (yt−1 − µt−1)] 2 = E α1(yt−1 − µt−1) + ut(yt−1 − µt−1) = α1E[(yt−1 − µt−1)2] + E[ut(yt−1 − µt−1)] = α1V ar(yt−1). Thus, with the results for the unconditional variance we obtain 2 − α12) if |α1| < 1 and t ∈ Z — independent of t, α1σ /(1 P Cov(yt, yt−1) = α1V ar(yt−1) = α1σ 2 t−2 α12j if V ar(y0) = 0 and t = 1, 2, . . . — dependent on t, j=0 (t − 1)σ 2 if α1 = 1, V ar(y0) = 0, t = 1, 2, . . . — dependent on t.

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Applied Financial Econometrics — 2.1.1 AR(1) processes — U Regensburg — July 2012

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Similarly one can show that k 2 − α12) α1 σ /(1 Pt−1−k 2j Cov(yt, yt−k ) = α1k V ar(yt−k ) = α1σ 2 j=0 α1 (t − k)σ 2

if |α1| < 1 and t ∈ Z — independent of t, if V ar(y0) = 0 and t = 1, 2, . . . — dependent on t, if α1 = 1 and V ar(y0) = 0, t = 1, 2, . . . — dep. on t.

• Weak stationarity or Covariance stationarity: A stochastic process {yt} is called weakly stationary if the first and second unconditional moment are independent of the time index t: – E[yt] = µt = µ for all t ∈ T, – Cov(yt, yt−k ) = γk for all t ∈ T. • Thus, if |α1| < 1 and it has run since ever, the AR(1) process is weakly stationary. – Stochastic processes for which lim E(yt) = µ,

t→∞

lim Cov(yt, yt−k ) = γk ,

t→∞

are called asymptotically stationary. Of course any stationary process is also asymptotically stationary. – If a process is not asymptotically stationary, it is nonstationary. 20

Applied Financial Econometrics — 2.1.1 AR(1) processes — U Regensburg — July 2012

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– Get a feeling for the different behavior of the AR(1) processes by using the Excel file AR1 simulation.xlsx. – There is another concept of stationarity, frequently called strict stationarity. A stochastic process is called strictly stationary if the joint distribution of (yt1 , yt2 , . . . , ytm ) is the same as for (yt1+h, yt2+h, . . . , ytm+h) for any t1, . . . , tm and for all m ∈ N, h ∈ Z. Is a weakly stationary AR(1) process also strictly stationary? If not, which additional assumptions do you need? • Properties of a (weakly) stationary AR(1) process: – The autocovariances are always different from zero albeit they may be very small. – The autocovariances converge to zero exponentially fast if |α1| < 1: γk = αk γ0. Therefore, stationary AR(1) processes are called stochastic processes with short memory. The effect of a shock ut in time t has a negligible effect on yt+h for h large. The opposite holds for a random walk where the effect of ut stays the same for any yt+h in the future! Therefore, random walks are said to have long memory. • Further remark: If {yt} is weakly stationary, it can be represented as a weighted sum of past and present white noise terms plus a constant. This result holds for all stationary stochastic processes and is known as the Wold decomposition, see (2.1). 21

Applied Financial Econometrics — 2.1.2 Autoregressive processes of higher order — U Regensburg — July 2012

• If one replaces in (2.1) α1j by φj , then one obtains a so called moving average process yt = µ +

∞ X

φj ut−j ,

t ∈ Z,

j=0

see also section 2.2.

2.1.2 Autoregressive processes of higher order • Some more notation: – Backshift or lag operator Lut ≡ ut−1. Thus, L2ut = L(Lut) = Lut−1 = ut−2 and Lk ut = ut−k . – Differencing operator ∆xt = (1 − L)xt = xt − Lxt = xt − xt−1. • A stochastic process {yt} is called autoregressive process of order p (AR(p) process) if yt = α0 + α1yt−1 + α2yt−2 + · · · + αpyt−p + ut,

t∈T

where ut is white noise. The index set can be T = Z or T = {p, p + 1, p + 2, . . .}. 22

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Applied Financial Econometrics — 2.1.2 Autoregressive processes of higher order — U Regensburg — July 2012

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Alternative representations with lag operator yt − α1yt−1 − · · · − αpyt−p = α0 + ut 1 − α1Lyt − · · · − αpLpyt = α0 + ut (1 − α1L − · · · − αpLp)yt = α0 + ut α(L)yt = α0 + ut where α(L) ≡ (1 − α1L − · · · − αpLp) is called an AR(p) lag polynomial. • Relationship of an AR(2) process with AR(1) processes: Example: Let {wt} and {ut} be a weakly stationary AR(1) process and white noise, respectively. Then wt = λ1wt−1 + ut,

t = . . . , −2, −1, 0, 1, 2, . . . ,

(1 − λ1L)wt = ut. Now consider the process {yt} yt = λ2yt−1 + wt,

t = . . . , −2, −1, 0, 1, 2, . . . ,

(1 − λ2L)yt = wt – Is the stochastic process {yt} weakly stationary? – Can the stochastic process {yt} be represented as an AR(p) process? 23

Applied Financial Econometrics — 2.1.2 Autoregressive processes of higher order — U Regensburg — July 2012

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Answer: ut (1 − λ1L) (1 − λ2L)(1 − λ1L)yt = ut (1 − λ2L)yt =

does this work?

2 ((1 −λ L − λ L + λ λ L 1 2 1 2 | {z } | {z })yt = ut −α1 L

−α2 L2

yt − α1yt−1 − α2yt−2 = ut yt = α1yt−1 + α2yt−2 + ut with α1 = λ1 + λ2,

α2 = −λ1λ2.

• In general it holds that one can factor an AR(p) process as 1 − α1z − · · · − αpz p = (1 − λ1z) · · · (1 − λpz) where the values λi can be complex numbers. • Complex numbers C: √ – Define i = −1. Thus i2 = −1. i is called an imaginary number. Any number obtained by multiplying i with a scalar produces again an imaginary number, e.g. 5i. – A complex number consists of a real part and imaginary part z = a + bi 24

Applied Financial Econometrics — 2.1.2 Autoregressive processes of higher order — U Regensburg — July 2012

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where a and b are real scalars. – Examples: λ1 = a + bi, λ2 = a − bi: −α1 = −(a + bi) − (a − bi) = −a − bi − a + bi = −2a real number −α2 = (a + bi)(a − bi) = a2 − abi + abi − b2i2 = a2 − b2(−1) = a2 + b2 real number (a + bi)(a + bi) = a2 + abi + abi + b2i2 = a2 + b2(−1) + 2abi = a2 − b2 + 2abi complex number – For more information on complex numbers see the article on Wikipedia or Neusser (2009, Appendix A). • The solutions z ∗ to the so-called characteristic equation (1 − λ1z) · · · (1 − λpz) = 0 are called its roots. • Each factor (1 − λiz) can be viewed as AR(1) polynomial which produces a weakly stationary process if its root is outside the unit circle, that is |zi| > 1 for all i = 1, . . . , k. Equivalently |λi| < 1 for all i = 1, . . . , k.

25

(2.2)

Applied Financial Econometrics — 2.1.2 Autoregressive processes of higher order — U Regensburg — July 2012

26

Thus, (2.2) is called stability condition of an AR(p) process. One also says: All roots of the AR(p) polynomial α(L) are outside the unit circle. Frequently (2.2) is also called stationarity condition. This, however, is not entirely correct because the process is only asymyptotically stationary if it starts in t = 0 even if the stability condition holds. Why? • Is for exactly one factor the root z = 1 (let’s say for λ1 = 1) and all others fulfil the stability condition, then one has ∗ z p−1) (1 − z)(1 − λ2z) · · · (1 − λpz) = (1 − z)(1 − α1∗z − · · · − αp−1

α(L) = (1 − L)α∗(L) where α∗(L) fulfils the stability condition (2.2) and the AR(p) process contains a random walk component. One also says that it contains a unit root or is integrated of order 1, yt ∼ I(1). • In general: An AR(p) process {yt} is said to be integrated of order d, if α(L) = (1 − L)dα∗(L) and α∗(L) fulfils the stability condition (2.2). The integration parameter d may take real values → long memory models (Tschernig 1994, Chapter 3).

26

Applied Financial Econometrics — 2.1.2 Autoregressive processes of higher order — U Regensburg — July 2012

27

• Moments of a stationary AR(p) process: – Mean: E(yt) = µ,

for all t.

It holds that α0 = α(L)µ = α(1)µ = µ(1 − α1 − α2 − · · · − αp) since from α(L)yt = α0 + ut one obtains α(L)(yt − µ) = −α(L)µ + α0 + ut by subtracting α(L)µ on both sides. If µ represents the mean, then −α(L)µ + α0 must be zero. – Variance and autocovariance function: cf. Hamilton (1994, p. 59, eq. (3.4.36)) α γ + α γ + · · · + α γ + σ 2 for k = 0 1 1 2 2 p p γk = α1γk−1 + α2γk−2 + · · · + αpγk−p for k = 1, 2, . . . – Partial autocorrelation function Corr(yt, yt−k |yt−1, . . . , yt−k+1): In an AR(p) process it holds that ak = Corr(yt, yt−k |yt−1, . . . , yt−k+1). Thus, all partial autocorrelations for k > p are zero since ak = αk = 0 for k > p.

27

(2.3)

Applied Financial Econometrics — 2.1.2 Autoregressive processes of higher order — U Regensburg — July 2012

28

• It can be shown that the autocovariances of a stationary AR(p) process converge exponentially fast towards zero (Hamilton 1994, p. 59), (Kirchg¨assner & Wolters 2008, Example 2.4). For the AR(1) process this was shown in section 2.1.1. • A stationary AR(p) process exhibits the following representation yt = φ(L)ut

(2.4)

yt = ut + φ1ut−1 + φ2ut−2 + . . . + φiut−i + . . . where the lag polynomial φ(L) is determined by the following equations 1 = φ(L) α(L) α(L)φ(L) = 1

(2.5) (2.6)

The parameters of φ(L) can be determined by comparing coefficients (method of undetermined coefficients (Kirchg¨assner & Wolters 2008, Section 2.1.2)): φj =

j X

φj−iαi,

j = 1, 2, . . . ,

α0 = 1, αi = 0 for i > p

i=1

The representation (2.4) is an example of an MA(∞) process, see section 2.2.3.

28

Applied Financial Econometrics — 2.2 Moving average processes — U Regensburg — July 2012

29

2.2 Moving average processes 2.2.1 MA(1) processes • A stochastic process {yt} is called moving average process of order 1 (MA(1) process) if it fulfils the following equation yt = ut + m1ut−1,

t = . . . , −2, −1, −, 1, 2, . . .

yt = (1 + m1L)ut where {ut} is white noise. • Properties: – Mean: E(yt) = 0 – (Auto)covariance function: Cov(yt, yt−k ) = E(ytyt−k ) = E((ut + m1ut−1)(ut−k + m1ut−k−1)) = E(utut−k ) + m1E(utut−k−1) + m1E(ut−1ut−k ) + m21E(ut−1ut−k−1)

29

Applied Financial Econometrics — 2.2.2 MA(q) processes — U Regensburg — July 2012

∗ Lag 0, k = 0: V ar(yt) = E(u2t ) + m1E(utut−1) + m1E(ut−1ut) + m21E(ut−1ut−1) = σ 2 + m21σ 2 = γ0 ∗ Lags -1,1, k = 1 oder k = −1: Cov(yt, yt−1) = E(utut−1) + m1E(utut−2) + m1E(u2t−1) + m21E(ut−1ut−2) = m1σ 2 = γ1 ∗ For all lagged variables with |k| ≥ 2 holds that Cov(yt, yt−k ) = 0 = γk

2.2.2 MA(q) processes • A stochastic process {yt} is called MA(q) process if it has the following representation yt = ut + m1ut−1 + · · · + mq ut−q , where {ut} is white noise. 30

t = . . . , −2, −1, −, 1, 2, . . .

30

Applied Financial Econometrics — 2.2.2 MA(q) processes — U Regensburg — July 2012

• Short notation using lag polynomials: yt = (m0 + m1L + · · · + mq Lq )ut,

m0 = 1

yt = m(L)ut • Properties: – Mean: E(yt) = 0 – (Auto)covariance function: Similarly to above we have ∗ Lag 0, k = 0: V ar(yt) = γ0 = σ 2 + m21σ 2 + · · · m2q σ 2 ! q q X X = σ2 1 + m2i = σ 2 m2i i=1

i=0

∗ Lag k, −q ≤ 0 < k ≤ q: Cov(yt, yt−k ) = γk = σ 2

q−k X

mimk+i

i=0

∗ For all lags |k| > q: Cov(yt, yt−k ) = γk = 0 31

31

Applied Financial Econometrics — 2.2.3 MA(∞) processes — U Regensburg — July 2012

32

– Remarks: ∗ All autocovariances for lags larger than q are 0! Put differently: A shock that occurred q or more periods before does not influence the stochastic behavior of yt ! ∗ All partial autocorrelations are unequal zero for any lag, see later.

2.2.3 Moving Average of infinite order (MA(∞) processes) • A stochastic process {yt} is called MA(∞) process if it has the following representation yt = ut + m1ut−1 + · · · + mq ut−q + · · · , yt = (1 + m1L + · · · + mq Lq + · · · )ut yt = m(L)ut where {ut} is white noise. • Properties: – Mean: E(yt) = 0 – (Auto)covariance function:

32

Applied Financial Econometrics — 2.2.3 MA(∞) processes — U Regensburg — July 2012

33

∗ Variance V ar(yt) = σ

2

∞ X

m2i

i=0

Remark: P∞ 2 The variance of an MA(∞) process only exists if the infinite sum i=0 mi converges to a finite number. This a necessary condition for weak stationarity. Why? (Why is it not a necessary condition for strict stationarity?) In other words: Weak stationarity requires that the influence of a shock ut−k decreases fast enough if the number of lags k increases. ∗ Autocovariance function: Cov(yt, yt−k ) = γk = σ 2

∞ X

! mimk+i

i=1

– A MA(∞) process for which the MA polynomial m(z) has all roots outside the unit circle is called invertible and has an AR(∞) or even AR(p) representation. • Remarks: – In practice one cannot estimate an MA(∞) process since one cannot estimate the infinitely many parameters mi. However, one can approximate an MA(∞) process by an MA(q) process 33

Applied Financial Econometrics — 2.2.3 MA(∞) processes — U Regensburg — July 2012

34

with high order q. This only makes sense if the DGP that generated the observed time series does not have autocovariances for lags larger q. – If small but non-zero autocovariances occur for lags larger than q, then using a stationary AR(p) model can be more appropriate since any stationary AR(p) process can be represented as an MA(∞) process. See (2.1) for an AR(1) process and (2.4) for an AR(p) process. – Summary: ∗ AR(p) process: γk 6= 0 for k > p, ak = 0 for k > p ∗ MA(q) process: γk = 0 for k > q, ak 6= 0 for k > q. These properties can be used to select between AR and MA models in practice if one has reliable estimators for the autocorrelation and partial autocorrelation function. This is an integral part of the Box-Jenkins model specification strategy. – Sometimes, neither an AR(p) nor an MA(q) process is appropriate, then a combination of both may do it: the ARMA(p, q) process.

34

Applied Financial Econometrics — 2.3 ARMA processes — U Regensburg — July 2012

35

2.3 ARMA processes • A stochastic process {yt} is called autoregressive moving average process (ARMA(p, q) process) with AR order p and MA order q if it follows the equation α(L)yt = m(L)ut and {ut} is white noise and if {yt} is weakly stationary. • Properties: – Mean: E(yt) = 0 for all periods t – Autocovariance function: ∗ For |k| ≥ max(p, q) holds γk = α1γk−1 + α2γk−2 + · · · + αpγk−p. For large lags the ARMA process performs like an AR process. ∗ For |k| < max(p, q +1) the computation is somewhat more complicated than in the AR(p) case, see e.g. Hamilton (1994, Section 3.6). – How does the partial autocorrelation function behave? – The stationarity condition for an ARMA(p, q) process and an AR(p) process are identical. 35

Applied Financial Econometrics — 2.3 ARMA processes — U Regensburg — July 2012

36

– Therefore, a stationary ARMA process exhibits an MA(∞) representation that can be obtained via comparing coefficients like (2.5). – Is the MA polynomial invertible, then the ARMA(p, q) process can be written as an AR(∞) process. Invertibility requires: 1 + m1z + · · · + mq z q 6= 0 for |z| ≤ 1 where z can be complex. – Attention: If the AR polynomial α(L) and the MA polynomial m(L) have common roots, then some or all parameters of these polynomials are not identified, (see examples) and cannot be estimated. Therefore one needs a reliable determination of the AR and MA order. More on that later when model selection criteria are discussed. Examples: ∗ yt = ut (1 − α1L)yt = (1 − α1L)ut so that m1 = −α1.

36

Applied Financial Econometrics — 2.3 ARMA processes — U Regensburg — July 2012

∗ (1 − λ1L)(1 − λ2L)yt = (1 − λ1L)(1 + λ3L)ut 2 λ1λ3 L2)ut (1 − (λ1 + λ2) L + λ 1 λ2 L )yt = (1 + (−λ1 + λ3 ) L + |{z} |{z} | {z } | {z } α1

−α2

m1

(1 − |{z} λ2 L)yt = (1 + |{z} λ3 L)ut α10

m01

37

m2

37

Applied Financial Econometrics — 2.4 Trajectory examples — U Regensburg — July 2012

38

2.4 Trajectory examples yt = 0.2yt−1 + ut, where ut i.i.d.N (0, 1) and y0 = 0.

−2

−1

0

y

1

2

AR(1) with alpha = 0.2

0

50

100

150

38

200

250

Applied Financial Econometrics — 2.4 Trajectory examples — U Regensburg — July 2012

39

yt = 10 + 0.2yt−1 + ut, where ut i.i.d.N (0, 1) and y0 = 0.

9

10

11

12

y

13

14

15

AR(1) with alpha = 0.2

0

50

100

150

39

200

250

Applied Financial Econometrics — 2.4 Trajectory examples — U Regensburg — July 2012

40

yt = yt−1 + ut, where ut i.i.d.N (0, 1) and y0 = 0.

0

5

10

y

15

20

25

30

AR(1) with alpha = 1

0

50

100

150

40

200

250

Applied Financial Econometrics — 2.5 Estimation — U Regensburg — July 2012

41

2.5 Estimation 2.5.1 OLS Estimation of AR(p) Models AR(p) process yt = α0 + α1yt−1 + α2yt−2 + · · · + αpyt−p + ut with {ut} being white noise • Some more notation and convention: – One has p presample values y−p+1, . . . , y0 and a sample with T observations y1, . . . , yT . Hence, for estimating the parameters of an AR(p) process we need T + p observations.

41

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

– Define the vectors and matrices α 1 0 α y 1 t−1 α = α2 Yt−1 = yt−2 . . .. .. αp yt−p y1 1 y0 y−1 y 1 y y0 1 2 y = . , X = . . ... .. .. .. yT 1 yT −1 yT −2

· · · y1−p

· · · yT −p

uT

y = Xα + u • OLS estimator 0

−1

0

α ˆ = (X X) X y =

t=1

42

u1 u · · · y2−p 2 , u = . ... ... ..

Thus, one can write

T X

!−1 0 Yt−1Yt−1

T X t=1

Yt−1yt

42

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

43

Example: If p = 1, one has yt = αyt−1 + ut and PT PT y y t−1 t t=1 yt−1 ut ˆ = Pt=1 = α + α P |{z} T T 2 2 y t−1 t=1 t=1 yt−1 conditions? or PT yt−1ut ˆ − α = Pt=1 α T 2 t=1 yt−1 All the important properties of the OLS estimator can be illustrated for a simple AR(1) model. • Statistical properties in finite samples: – Consider

0

−1

0

ˆ = α + E (X X) X u E [α]

= α + E E (X 0X)−1X 0u|X = α + E (X 0X)−1X 0 E [u|X] | {z } {z } | 6=0

6=0

The reason why the conditional expectation is not zero is that X is not exogenous in the present case because it consists of lagged yt’s that are weighted sums of ut’s. Thus, the ut’s are already contained in the condition. This is different from the simple linear model for cross-section data.

43

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

44

Therefore, the OLS estimator α ˆ is biased in finite samples. ˆ 6= α E[α] ˆ is not exactly known even if the ut’s are normally distributed. – The probability distribution of α Note that in this case also the yt’s follow a normal distribution. Why? Consider the OLS estimator of a simple AR(1) model PT yt−1yt ˆ = Pt=1 α T 2 t=1 yt−1 As one can see, the same normal random variables appear in the numerator and denominator. Thus, the OLS estimator is a highly nonlinear function of normal random variables for which the distribution is generally unknown.

44

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

45

– Example: Distribution of α, ˆ when the process is yt = 0.8yt−1 + ut and ut ∼ IN (0, 1):

OLS estimator Normal distribution

Sample Size = 30 Mean = 0.750576362436378 Variance = 0.0170138496172623

0.0

0.5

1.0

1.5

Density

2.0

2.5

3.0

3.5

Density of OLS estimator (true alpha = 0.8) and normal density

0.0

0.2

0.4

0.6

0.8

1.0

alpha_hat

Note: finite sample distribution is not centered around the true value 0.8 but around 0.75; 45

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

deviates considerably from normal distribution.

46

46

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

• Statistical properties in very large samples — asymptotic properties: – Consider the OLS estimator of the AR(1) model first: PT PT 1 yt−1yt t=1 yt−1 yt T ˆ = Pt=1 = α P T T 1 2 2 y t−1 t=1 t=1 yt−1 T Note that if the AR(1) process is stationary it holds that (see above) Cov(yt, yt−1) = αV ar(yt) and thus α= Note also that

Cov(yt, yt−1) . V ar(yt)

T 1X yt−1yt T t=1

is an estimator of Cov(yt, yt−1) and T 1X 2 y T t=1 t−1

is an estimator of V ar(yt−1) = V ar(yt).

47

47

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

48

– In order to check the properties of the OLS estimator for large samples, one may check the behavior of the covariance and the variance estimator. If for an arbitrary > 0 ! T 1 X 2 yt−1 − V ar(yt) < = 1 (+) lim P T →∞ T t=1 P 2 converges in then the estimation error disappears. One says that the estimator T1 Tt=1 yt−1 probability to the true value, which is V ar(yt) in the present case. A short hand notation is T 1X 2 plim yt−1 = V ar(yt). T T →∞ t=1

An estimator for which the estimation error diminishes with increasing sample size is also called consistent. It can be shown that the variance and the covariance estimator are both consistent if the autoregressive process is stationary. Moreover, for two arbitrary sequences of random numbers zT and wT , for which plim zT = z plim wT = w it holds that plim zT wT = plim zT plim wT = zw. Note that this property does not hold for expected values!

48

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

49

Thus, if the model is correctly specified and thus Cov(yt, yt−1) = αV ar(yt) holds, one has P plim T1 Tt=1 yt−1yt = αV ar(yt)/V ar(yt) = α. plim α ˆ= PT 2 1 plim T t=1 yt−1 The OLS estimator is consistent. The following figure (animated) shows the distribution of the OLS estimator of the AR(1) model when the true data generating process is yt = 0.8yt−1 + ut where ut ∼ IN (0, 1). Sample sizes vary from 20 to 1000.

49

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

20

Density of OLS estimator (true alpha = 0.8)

10 0

5

Density

15

sample size: 20

0.0

0.2

0.4

0.6 alpha_hat

Note: variance diminishes with larger sample size

50

0.8

1.0

50

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

51

– Using a central limit theorem one can also show that √ d ˆ − α) −→ N (0, q) T (α with some asymptotic variance q. In order to state the asymptotic properties of the OLS estimator for AR(p) models we state the conditions that the unknown but true AR(p) process has to fulfil: – Assumption A: 1. The AR(p) process is stationary and correctly specified. This implies that the order p is correctly chosen and the errors are not serially correlated. 2. The errors are homoskedastic V ar(ut) = σ 2 for all periods t = 1, . . . , T . 3. E[u4t ] < ∞ for all periods t = 1, . . . , T .

51

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

Under Assumption A the following holds for the OLS estimator α: ˆ ∗ It is consistent, i.e. lim P (|α ˆ − α| < ) → 1

T →∞

for arbitrary > 0 or in short hand notation ˆ = α. plimT →∞α ∗ The OLS estimator is asymptotically normally distributed: √ d ˆ − α) −→ N (0, Q) T (α where Q denotes the asymptotic covariance matrix 2

Q = σ plimT →∞ T

−1

0

XX

−1

= σ 2plimT →∞ T −1

T X

!−1 0 Yt−1Yt−1

.

t=1

A less precise but more intuitive way to write this result for reasonably large T is ! −1 T X −1 2 0 0 2 . ˆ ≈ N α, σ (X X) ˆ ≈ N α, σ Yt−1Yt−1 α or α t=1

– Remarks

52

52

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

53

∗ Cf. L¨utkepohl & Kr¨atzig (2008, Section 2.4.1), for proofs see Brockwell & Davis (1991, Sections 8.7 and 8.8) who require ut ∼ IID(0, σ 2) instead of the weaker white noise assumptions. ∗ Weaker conditions that only require stability of the AR polynomial and some regularity of the error process are possible, see e.g. Davidson (2000, Chapter 6). ∗ OLS estimation is used in EViews if the lags of a variable Y are specified by Y(-1), Y(-2), etc. and is automatically used in the software JMulTi. In R, function ar() uses a different estimation technique by default (using the Yule-Walker equations). You need to add the option method = "ols" to get the least squares estimator. ∗ The OLS estimator is even asymptotically normally distributed if {yt} contains one unit root and the AR order is at least 2. However, in this case the asymptotic covariance matrix Q is singular (which, for example, implies that F tests do not work). This is due to the faster convergence rate of the unit root compared to the stationary roots.

53

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

54

• Asymptotic distribution of the OLS estimator in case of a simple random walk – Note that a simple random walk process yt = yt−1 + ut with {ut} being white noise does not fulfil Assumption A. What are then the asymptotic P 2 . Does it converge with properties? To see that things change, consider again T1 Tt=1 yt−1 increasing sample size T to a finite quantity? Remember that in case of a random walk V ar(yt) = tσ 2. – The asymptotic distribution of the OLS estimator for the autoregression parameter α in case of a random walk as DGP is PT 1 1 (X − 1) t=1 yt−1 ut d T −→ 2 T (α ˆ − 1) = 1 PT 2 Z t=1 yt−1 T2 R1 2 2 where X = [W (1)] ∼ χ (1) and Z ∼ 0 [W (r)]2dr and X and Z are not independent. For specialists: The W (r) denotes Brownian motion which is the continuous time version of a simple discrete time random walk. Thus: In case of p = 1 and α1 = 1 one obtains a ∗ completely different asymptotic distribution (graphic shows distribution of T (α ˆ − 1):

54

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

55

Density of standardized and scaled OLS estimator (true alpha = 1) and normal density

standardized and scaled OLS estimator Normal distribution

0.00

0.05

0.10

Density

0.15

0.20

Sample Size = 5000 Mean = 0.998251038269106 Variance = 9.76669126968408e−06

−10

−5

0

5

10

alpha_hat

∗ the OLS estimator converges much faster to the true value α = 1. It is called super consistent. The variance of the OLS estimator converges much faster towards zero than in the stationary case. Here, the rate is T −1 instead of T −1/2 in the stationary case. The reason is that with an increasing number of observations the variation of the regressor and the regressand increase — in contrast to the stationary case:

55

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

56

250

Density of OLS estimator (random walk)

0

50

100

Density

150

200

sample size: 20

0.0

0.2

0.4

0.6

0.8

1.0

alpha_hat

ˆ is much smaller than in the stationary case Note: even for T = 20 the variance of α ∗ For conducting valid inference one has to check which distribution applies!! This can be done with an appropriate unit root test, see section 4.1. ∗ If the model contains deterministic terms (constant, trend, seasonal dummies, 56

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012

57

structural breaks, etc.), the asymptotic distribution changes, see section 4.1. ∗ The result holds for ut being white noise or Gaussian white noise. ∗ If the {ut} process is not white noise but correlated, then the asymptotic distribution changes −→ Phillips-Perron-Test ∗ In order to avoid correlated ut, one can also estimate a more general model, e.g. an AR(p) model. However, for testing purposes it has to be rewritten in a specific way −→ Augmented Dickey-Fuller-Test, see section 4.1.

57

Applied Financial Econometrics — 2.5.1 OLS Estimation of AR(p) Models — U Regensburg — July 2012 0 ˆ • Estimation of the error variance using the OLS residuals uˆt = yt − Yt−1 α

– OLS estimator (for the model including a constant term) T

X 1 2 uˆ2t σˆ = T − p − 1 t=1 – Maximum Likelihood estimator

T X 1 σ˜ 2 = uˆ2t T t=1

– Asymptotic properties: under suitable conditions both estimators are consistent plimT →∞σˆ T2 = σ 2 plimT →∞σ˜ T2 = σ 2

58

58

Applied Financial Econometrics — 2.5.2 Alternative Estimators for AR(p) models — U Regensburg — July 2012

59

2.5.2 Alternative Estimators for AR(p) models • Nonlinear least squares: A stationary AR(1) process can alternatively be written as (similarly for an AR(p) process) yt = µ + vt

(2.7a)

vt = α1vt−1 + ut.

(2.7b)

yt = µ(1 − α1) + α1yt−1 + ut.

(2.8)

Inserting (2.7a) into (2.7b) delivers

This equation is nonlinear in the parameters µ and α1 and can be estimated by nonlinear least squares. This is done in EViews if lagged endogenous variables are specified by AR(1), AR(2), etc. Be aware that in this case in the EViews output C refers to the estimated mean µ, not the estimated α0. • Both, the OLS and nonlinear LS estimator require that the first p observations of a sample are needed as starting values, thus reducing the number of observations by p. This can be avoided if one can assume ut ∼ N ID(0, σ 2). Then, one can apply maximum likelihood (ML) estimation. See section 5.2 for a brief introduction to maximum likelihood estimation of stationary AR(p) models and e.g. Hamilton (1994, Sections 5.2 and 5.3) for details.

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2.5.3 Estimation of MA(q) and ARMA(p, q) models The estimation of MA(q) and ARMA(p, q) models cannot be done by OLS directly. Mainly three procedures are used: • 2-step procedure using OLS techniques: 1. step: if the MA or ARMA process is invertible, it has an infinite AR representation. This can be approximated by an AR(p∗) model with p∗ rather large. So fit an AR(p∗) model with ∗0 ∗ ˆ , t = 1, . . . , T α p∗ >> max(p, q) and calculate residuals uˆt(p∗) = yt − Yt−1 2. step: Estimate ARMA(p, q) model with residuals uˆt(p∗) yt = ν + α1yt−1 + · · · + αpyt−p + ut + m1uˆt−1(p∗) + · · · + mq uˆt−q (p∗) using the OLS estimator. Under suitable conditions asymptotically normally distributed. • Maximum likelihood (ML) estimation. See section 5.2 for an introduction to ML estimation ¨ or Kapitel 5 in Fortgeschrittene Okonometrie. This estimator is highly nonlinear in the ARMA parameters, requires nonlinear optimization techniques and starting values. The latter can be obtained via the first method. Under suitable conditions both estimators are asymptotically normally distributed.

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A detailed treatment can be found in Hamilton (1994, Chapter 5). Technical details for algorithms and proofs can be found in Brockwell & Davis (1991, Chapter 8). • Nonlinear least squares with backcasting techniques (EViews 5, User Guide, Chapter 17; EViews 6, User guide, Chapter 26)

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2.6 R Code Three different ways to simulate an AR(2) process: # Simulating an AR(2) process ################## # using a ’for’ loop T <- 100 # sample length u <- rnorm(T, mean = 0, sd = 1) # draw innovations from standard normal distributioni y <- rep(0, T) # initialize vector y (all values set to zero) for(t in 3:T){ # for t = 3,...,T recursively calculate values in y y[t] = 0.8 * y[t-1] - 0.3 * y[t-2] + u[t] } ################## # using filter() T <- 100 # sample length u <- rnorm(T, mean = 0, sd = 1) # draw innovations from standard normal distributioni y <- filter(u, c(0.8, -0.3), method = "recursive") ################### # using arima.sim() y <- arima.sim(model = list( ar = c(0.8, -0.3), order = c(2,0,0)), n = 100, rand.gen = rnorm)

Why are the simulated series y not identical?

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Simulate an ARMA(2, 1) process: # Simulating an ARMA(2,1) process ################## # using a ’for’ loop T <- 100 # sample length u <- rnorm(T, mean = 0, sd = 1) # draw innovations from standard normal distributioni y <- rep(0, T) # initialize vector y (all values set to zero) for(t in 3:T){ # for t = 3,...,T recursively calculate values in y y[t] = 0.8 * y[t-1] - 0.3 * y[t-2] + u[t] - 0.1 * u[t-1] } ################## # using filter() T <- 100 # sample length u <- rnorm(T, mean = 0, sd = 1) # draw innovations from standard normal distributioni e2 <- embed(u,2) %*% c(1, -0.1) # apply MA filter e <- filter(u, c(1,-0.1) , method = "convolution", sides = 1) y <- filter(e, c(0.8, -0.3), method = "recursive") # apply AR filter ################### # using arima.sim() y <- arima.sim(model = list( ar = c(0.8, -0.3), ma = c(-0.1), order = c(2,0,1)), n = 100, rand.gen = rnorm)

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Estimate (fit) an AR(p) process: # Estimating parameters of an AR(p) model # y denotes the data series ################## # choose lag order p <- 2 # Order of AR model to be estimated ################## # using lm() M <- embed(y, p+1) # matrix of contemporaneous and lagged y y.t <- M[,1] # y_t vector Y.tm1.tmp <- M[,-1] # y_{t-1}, y_{t-2},...,y_{t-p} estimate <- lm(y.t ~ -1 +Y.tm1.tmp) summary(estimate) # nice output ################## # using ar.ols() estimate <- ar.ols(y, order.max = p, aic = FALSE, demean = FALSE) estimate$asy.se.coef # asymptotic standard errors

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3 Forecasting (financial) time series

3.1 Some general remarks and definitions • To begin with, one may call prediction a general statement about the dependent variable given some conditioning information, e.g. predicting the wage of an individual knowing his/her age, experience, education etc. • Forecasting is a particular type of prediction of a future value of the dependent variable or a value of it that is outside the current sample. • A forecasting rule is any systematic operational procedure for making statements about future events. • Before we continue, we need to define the conditioning information set for which we introduce the general notation It. Intuitively, It contains all sets for which a probability can be assigned. In

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case of lagged endogenous variables, for example in case of AR(p) processes, these sets contain all combinations of intervals in which past values could have been observed. In practice, one often writes for the given example It−1 = {yt−1, yt−2, . . .}. Then in case of an AR(1) process yt = αyt−1 + ut,

ut ∼ i.i.d.(0, σ 2)

one has E[yt|It−1] = αyt−1 6= E[yt] = 0. Thus, knowing the condition helps to predict yt. Predictability is not only limited to the (conditional) mean. One may state this even more general w.r.t. to densities. • A random variable is unpredictable with respect to a given information set It−1 if the conditional probability distribution F (yt|It−1) is identical to the unconditional/marginal probability distribution F (yt) F (yt|It−1) = F (yt). • Any prediction is very likely to be false. Let yT +h|T denote a predictor for yT +h based on some information set up to time T . Then the prediction error is eT +h|T = yT +h − yT +h|T . To evaluate the prediction error one may e.g. miminize the – mean squared error of prediction (MSEP) M SEP (yT +h|T ) ≡ E[(yT +h − yT +h|T )2] 66

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– mean absolute error of prediction(MAEP) M AEP (yT +h|T ) ≡ E[|yT +h − yT +h|T |] The function of which the mean is taken is called loss function, in case of the M SEP it is the squared prediction error. Note that it does not make much sense to minimize the squared 2 error of prediction yT +h − yT +h|T alone since the optimal predictor may vary from sample to sample even if the information set stays the same. One therefore takes the mean of all potential samples. • By the law of iterated expectations, the MSEP (equivalently for the MAEP) can be written as 2 M SEP (yT +h|T ) = E E[(yT +h − yT +h|T ) IT ] , (3.1) 2 where IT denotes the information set used to compute yT +h|T . Note that E[(yT +h − yT +h|T ) IT ] measures the mean of squared prediction errors for a given path up to time T due to the conditioning on IT so that yT +h|T is the same while yT +h varies randomly. Some authors call (more precisely) E[(yT +h − yT +h|T )2 IT ] the mean squared error of prediction and denote the expression E E[(yT +h − yT +h|T )2 IT ] as mean integrated squared error of prediction since it ’integrates’ out the impact of history up to time T . Both are the same in case of ARMA processes so that we follow the standard textbook usage and call (3.1) MSEP (L¨utkepohl 2005, Section 2.2.2), (Hamilton 1994, Chapter 4). These concepts in general differ in case of nonlinear time series processes. • The minimal mean squared prediction error is obtained by using the conditional expectation

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E[yT +h|IT ] (does not minimize the MAEP), i.e. use yT +h|T = E[yT +h|IT ]. Thus, forecasting by using conditional expectations implicitly means that one aims at minimizing the mean squared prediction error. • Possible approaches for predicting time series data: – using exclusively past/current observations (univariate/multivariate time series models — example: AR(p)/VAR(p) model) – using regression models including conditioning/explanatory variables yt = β1 + β2xt2 + · · · + βk xt,k + ut where ut is white noise and the xtj ’s are observable at time t. Having found a regression model explaining yt is not sufficient for predictions since the xtj ’s are unknown in future periods. One then needs an auxiliary model for predicting all explanatory variables. – a combination of the pure time series and the pure regression approach: dynamic econometric models Example: yt = β0 + α1yt−1 + β1xt1 + β2xt−1,1 + β3xt2 + ut

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• Important issues: – How to choose the best suited approach for a given prediction problem? – Which model framework should be chosen? univariate vs. multivariate models, linear vs. non-linear models,...) – Is it necessary for obtaining good forecasts to have a model with good explanatory power (’good’ model)? – Alternatives to econometric/statistical forecasts that are based on models: judgement, expert opinions,... – Forecasting horizon • Further issues — not treated here – Combination of forecasts and encompassing forecasts – Interval forecasts – Forecasting the complete conditional density

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3.2 Decomposition of prediction errors • Some notation – Optimal predictor if MSEP has to be minimized: conditional expectation yT +h|T ≡ E[yT +h|IT ] Note that the optimal predictor yT +h|T is in general unfeasible since the conditional expected value of yT +h given the information set is unknown. The conditional expected value (and the conditional density) would be known if the DGP were known. Supplement (not applied): Proof: 1. Let pT be any predictor that uses all information given by IT . Without loss of generality one can set pT = E[yT +h |IT ] + aT where aT denotes the deviation of the predictor pT from the conditional expectation. 2. By the law of iterated expectations one has for the MSEP E (yT +h − pT )2 = E E (yT +h − pT )2 |IT We first focus on the conditional expectation E [(yT +h − pT )2 |IT ].

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3. Next we show that the conditional expectation E [(yT +h − pT )2 |IT ] is minimized for aT = 0: E (yT +h − pT )2 |IT = E (yT +h − (E[yT +h |IT ] + aT ))2 |IT = E (yT +h − E[yT +h |IT ] − aT )2 |IT = E ((yT +h − E[yT +h |IT ]) − aT )2 |IT = E (yT +h − E[yT +h |IT ])2 |IT − 2E [aT (yT +h − E[yT +h |IT ])|IT ] + E a2T |IT Now we have 3 terms: the first one is independent of aT and can thus be ignored for finding the minimum. Let us look at the second term E [aT (yT +h − E[yT +h |IT ])|IT ]. Since aT is known given IT , it can be taken out of the conditional expectation. Then one can solve the conditional expectation and one obtains E [aT (yT +h − E[yT +h |IT ])|IT ] = aT (E[yT +h |IT ] − E [E [yT +h |IT ] |IT ]) = aT (E[yT +h |IT ] − E[yT +h |IT ]) = 0 Finally, the third conditional expectation contains a square and is thus minimized if aT = 0. 4. Since the above step holds for any realization, that is for any history IT , it holds also for the MSEP. More formally E (yT +h − pT )2 = E E (yT +h − E[yT +h |IT ])2 |IT − 2E [aT 0] + E E a2T |IT = E (yT +h − E[yT +h |IT ])2 + E a2T which again is minimized for aT = 0.

q.e.d.

5. We conclude that the MSEP is minimized if we use the predictor pT = E[yT +h |IT ], i.e. we use the conditional expectation.

– In order to estimate the conditional expectation one has in general to select/specify a model that contains a set of conditional expectations that vary with respect to a parameter vector θ, say an AR(3) model. 71

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– In general one has a set of model candidates, say all AR(p) models with p = 0, 1, . . . , pmax where the maximum order pmax is chosen by the researcher. Let Mj denote the j-th model of the available set of models. The predictor based on model Mj with parameter vector θj is E[yT +h|IT , Mj , θj ]. At this point the parameter vector θj is unspecified. The optimal predictor in the MSEP sense given model Mj is obtained by minimizing min E (yT +h − E[yT +h|IT , Mj , θj ])2 θj ∈Mj

with respect to all θj included in model Mj . This minimizing parameter vector θj0 is frequently called pseudo-true parameter vector. Note that the pseudo-true parameter vector can change if one minimizes another criterion such as e.g. MAEP. – A feasible predictor of yT +h given information set IT and model Mj is obtained by estimating the model parameters θj and inserting them into the conditional expectation of model Mj yˆT +h|T ≡ E[yT +h|IT , Mj , θˆj ]. Feasible means that the predictor can be computed once a sample with observations up to time T is available. 72

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• During the process described a number of wrong decisions can be made that all contribute to the total prediction error. • Classification of prediction errors eˆT +h|T = yT +h − yˆT +h|T = yT +h − E[yT +h|IT ] {z } | unavoidable error + E[yT +h|IT ] − E[yT +h|IT , Mj , θj0] | {z } approximation error/model misspecification + E[yT +h|IT , Mj , θj0] − E[yT +h|IT , Mj , θˆj ] | {z } estimation error/parameter uncertainty • Example: DGP: AR(2) process yt = α1yt−1 + α2yt−2 + ut, selected model: AR(1) model yt = ayt−1 + vt: yT +1|T = E[yT +1|yT , yT −1, . . . , y0] = α1yT + α2yT −1 E[yT +1|yT , yT −1, . . . , y0, AR(1), a] = ayT yˆT +1|T = E[yT +1|yT , yT −1, . . . , y0, AR(1), aˆ] = aˆyT – Unavoidable error: uT +1 73

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– Approximation error: α1yT + α2yT −1 − a0yT where a0 denotes the parameter value that minimizes the mean squared approximation error. – Estimation error: (a0 − aˆ)yT • Notes: – One cannot avoid the genuine prediction error eT +h|T = yT +h − E[yT +h|IT ]. – The approximation error is zero if the DGP is in the chosen model Mj , e.g. data are generated by an AR(p) process and we use an AR(q) model for prediction, where p ≤ q. – In order to reduce a possible approximation error one may have to select a model that is ’closer’ to the correct model (includes DGP). It may have more parameters. – If the correct model has many parameters, there is a tradeoff between facing large parameter uncertainty and/or a considerable approximation error. Thus, for prediction a ’wrong’ model may be superior to the correct model that contains the DGP. – Model selection procedures are designed to optimize tradeoff between approximation and estimation error by miminizing the mean of the squared approximation and estimation error, see the following section.

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• Further potential sources for prediction errors: – Parameter instability: the parameters of the model may change over time. In this case the stationarity assumption for the model is violated. – Variable mis-measurement: data is not exactly measured or is not yet completely known (e.g.: preliminary GNP data) – Initial condition uncertainty: the starting value y0 in a dynamic model is also random. – Incorrect exogeneity assumptions. – Policy changes: the model may change completely due to policy changes.

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3.3 AR Model Specification See L¨utkepohl & Kr¨atzig (2008, Section 2.5) • (Economic) theory usually does not give many hints about the possible AR order p. Therefore p has to be determined with statistical methods. • Model specification procedures are also useful to identify a model for which the assumptions are fulfilled. • There are three possible procedures: – Model selection criteria – Sequential testing: Start with a large lag order p = pmax and test for significance of the parameter for the pmaxth lag. If it is significantly different from zero, you are done, if not, reduce the lag order to p = pmax − 1 and repeat the testing procedure. Continue until you are able to reject the null hypothesis. – Box-Jenkins method: The Box-Jenkins method requires to define and estimate the partial autocorrelation function. Since this procedure is not so popular any more, we do not discuss it here.

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Model selection criteria • General structure: Criterion(p) = ln σ˜ 2(p) + with

cT φ(p) | {z } penalty term

T 1X 2 σ˜ (p) = uˆ (p) T t=1 t 2

where the uˆt(p)’s denote the residuals of a fitted AR(p) model. – One chooses a maximum order pmax and computes Criterion(p) for p = 0, 1, . . . , pmax. The selected order pˆ is the order for which the selection criterion achieves its minimum. –

∂φ(p) ∂p

> 0, i.e. adding lags increases the penalty term.

T – cT can be seen as the weight of φ(p) in the criterion function. ∂c ∂T < 0. i.e. adding lags for short time series (T small) increases the penalty term by a larger amount than for long time series.

– σ˜ 2(p) ≤ σ˜ 2(p − 1) when AR models are fitted by OLS. – Note that the maximum likelihood estimator for the error variance is used, i.e. there is no correction for degrees of freedom.

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– For p = 0, 1, . . . , pmax the sample size used for fitting the AR(p) model should be kept fixed, see Ng & Perron (2005) for a simulation study. • Standard order selection criteria: (all have φ(p) = p) – Akaike Information Criterion (AIC) AIC(p) = ln σ˜ 2(p) +

2 p T

– Hannan-Quinn Criterion (HQ) HQ(p) = ln σ˜ 2(p) +

2 ln ln T p T

– Schwarz (and Rissanen) Criterion (SC, SIC or BIC) SC(p) = ln σ˜ 2(p) +

ln T p T

• Asymptotic properties – Condition I: true order p0 < ∞ and pmax ≥ p0, , pˆT denotes the estimated order, the roots of the AR polynomial are all outside or on the unit circle (unit root and stationary processes!) – AIC: P (ˆ pT > p0) > 0 – HQ: plimT →∞pˆT = p0 78

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a.s.

– SC: pˆT −→ p0 – T ≥ 16:

pˆ(SC) ≤ pˆ(HQ) ≤ pˆ(AIC)

– Condition II: true order p0 infinite: AIC is the best

3.4 Prediction with AR models See L¨utkepohl & Kr¨atzig (2008, Section 2.8) • One-step ahead predictions for T + 1 based on an AR(p) model (1 − α1L − · · · αpLp)yt = ut with known AR parameters: yT +1|T = E[yT +1|yT , yT −1, . . . , y1, . . . , y−p+1] = α1yT + α2yT −1 + · · · + αpyT −p+1 • One can show that the prediction based on the conditional expectation has the smallest mean squared error of prediction (MSEP) if ut ∼ i.i.d.(0, σ 2). 79

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The MSEP, often also abbreviated by MSE, is given for the present case: h 2 i = E[u2T +1] = σ 2 M SE = E yT +1 − yT +1|T • Predictions for several time periods ahead, h > 1, usually abbreviated as h-step predictions, can be calculated recursively yT +h|T = α1yT +h−1|T + α2yT +h−2|T + · · · + αpyT +h−p|T , with yT +j|T = yT +j for j ≤ 0.. (3.2) • Prediction error: A stationary AR(p) process α(L)yt = ut can be represented as yt = φ(L)ut = ut + φ1ut−1 + φ2ut−2 + · · · . This representation is an example of an infinite Moving Average Process (MA(∞) process). It is possible to compute the moving average parameters φj , j = 1, 2, . . . , by solving α(L)φ(L) = Ps 1 leading to φs = j=1 φs−j αj , s = 1, 2, . . . with φ0 = 1 and αj = 0 for j > p (see e.g. (2.4) in section 2.1.2). One then obtains for the conditional expectation yT +h|T = E [yT +h|yT , yT −1, · · · ] = E [uT +h + φ1uT +h−1 + · · · + φh−1uT +1 + φhuT + φh+1uT −1 + · · · |yT , yT −1, · · · ] = φhuT + φh+1uT −1 + · · ·

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and the prediction error is given by yT +h − yT +h|T = uT +h + φ1uT +h−1 + · · · + φh−1uT +1 + φhuT + φh+1uT −1 + · · · − (φhuT + φh+1uT −1 + · · · ) = uT +h + φ1uT +h−1 + · · · + φh−1uT +1. • The mean squared error of prediction (MSEP) of an h-step prediction is therefore h−1 h X 2 i 2 2 σy (h) = E yT +h − yT +j|T =σ φ2j , j=0

since the errors ut are assumed to be i.i.d. Note that if ut ∼ i.i.d.(0, σ 2), then the h-step predictor yT +h|T is optimal, that is it exhibits among all possible prediction methods the smallest MSEP. • If the errors ut are just white noise, then equation (3.2) delivers the best linear prediction (where then the parameters depend on the goal function and are then pseudo-true values). • Since the variance σy2 of a stationary process is given by σy2

=σ

2

∞ X

φ2j

j=0

(see Neusser (2009, Abschnitt 2.4, Theorem 2.3)), the variance of the prediction error σy2(h) = Ph−1 2 2 2 σ j=0 φj of an h-step prediction approaches σy with the number of steps h increasing. 81

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• If the errors ut are Gaussian white noise, then the prediction errors are also i.i.d. normally distributed and one can compute a prediction interval for a given confidence level (1 − γ)100% [yT +h|T − c1−γ/2σy (h), yT +h|T + c1−γ/2σy (h)] where c1−γ/2 denotes the 1 − γ/2 quantile of the standard normal distribution. • One can use the predictor (3.2) also for autoregressive processes with a unit root. As a technical remark note that in this case the parameters φj do not represent the parameters of an MA(∞) process, simply because the latter does not exist. Observe that the prediction variance increases in this case with the number of prediction periods h towards infinity since the variance of a random walk also increases unboundedly with t. • In empirical work the parameters of the data generating process are unknown and have to be estimated. One therefore replaces in (3.2) all unknown parameters by their estimates and obtains the feasible predictor yˆT +h|T = α ˆ 1yˆT +h−1|T + α ˆ 2yˆT +h−2|T + · · · + α ˆ pyˆT +h−p|T . This leads to the prediction errors yT +h − yˆT +h|T = [yT +h − yT +h|T ] + [yT +h|T − yˆT +h|T ] =

h−1 X

φj uT +h−j + [yT +h|T − yˆT +h|T ].

j=0

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(3.3)

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The second term on the right hand side through which the estimation uncertainty is captured approaches zero with increasing sample size if the parameters are estimated consistently (correct order, etc.). The variance of the prediction error with estimated parameters is given by h 2 i 2 σyb(h) = E yT +h − yˆT +h|T = σy2(h) + ’something that appraoches zero for T → ∞’. Note that in small samples the additional variance due to the estimation uncertainty is nonnegligible and should be included when computing prediction intervals. Details can be found e.g. in L¨utkepohl (2005, Section 3.5).

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3.5 Evaluation of forecasts • Properties of optimal forecasts – The optimal predictor (=use model that includes DGP with the true parameter values) is unbiased (by definition). – Prediction errors based on optimal 1-step ahead predictions are white noise. – Prediction errors based on optimal h-step ahead predictions may be correlated. (If so, they follow a Moving Average Process of order h − 1 or less). – The variance of the prediction errors of h-step predictions converges to the unconditional variance (in case the DGP is stationary). Thus: Residuals from good forecasts must be unpredictable! This property can be used for checking the quality of a forecasting procedure by testing e.g. in eˆt+1|t = β1 +

k X

βixti + errort,

t = 1, . . . , T,

i=1

whether βi = 0, i = 2, . . . , k. If some xti has significant explaining power, then we should incorporate this variable into our forecasting procedure.

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• Quantities to compare forecasts – Forecast error: eˆT +h|T = yT +h − yˆT +h|T . – Forecast percent error:

yT +h −ˆ yT +h|T . yT +h

• Measures to compare forecast errors: – Mean error of prediction (MEP)

M EP (h) = E eˆT +h|T

– Mean squared error of prediction (MSEP) h i 2 M SEP (h) = E eˆT +h|T – Root mean squared error of prediction (RMSEP) r h i 2 RM SEP (h) = E eˆT +h|T – Mean absolute error of prediction (MAEP) M AEP (h) = E eˆT +h|T .

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Notes: – These measures can also be defined for the forecast percent error. – The RMSEP and the MAEP have the same scale as the prediction error. – The MAEP is not minimized by using the conditional expectation as predictor. – There also exist measures that weight positive and negative prediction errors differently. – All measures are generally not observable and have to be estimated. A popular tool for estimation are out-of-sample forecasts.

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• Out-of-sample forecasts: – Split the sample into 2 parts with the first T1 observations in subsample 1 and the second T − T1 observations in subsample 2. – Estimate the model on basis of subsample 1 and predict yT1+h where h ≤ T − T1. Denote the prediction error by eˆT1+h|T1 = yT1+h − yˆT1+h|T1 – Re-estimate the model using all data from t = 1, . . . , T1 + 1, predict yT1+1+h and save the prediction error. – Repeat the previous step until you predict yT . – At the end you estimate the mean of your measure by averaging over all out-of-sample predictions. For example, the MSEP is estimated by T −h

X 1 \ eˆ2j+h|j . M SEP = T − (T1 − 1) j=T1

• Advanced literature to testing and comparing predictive quality of various models: Diebold & Mariano (1995), West (1996), Giacomini & White (2006).

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3.6 R Code Automatic lag length selection for an AR model using the Akaike Information Criterion (AIC) and h-step forecasting using a fitted AR model: # Model selection by AIC # y denotes data series p.max <- 10 estimate p.hat

# maximum order

<- ar.ols(y, order.max = p.max, aic = TRUE, demean = TRUE) <- estimate$order # chosen lag length

# h-step forecasting h <- 10 predict(estimate, n.ahead = h)

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Applied Financial Econometrics — 4 More on modeling time series — U Regensburg — July 2012

4 More on modeling time series

4.1 Unit root tests 4.1.1 Dickey-Fuller test • The simplest case – Consider the estimation of the simple AR(1) model yt = αyt−1 + ut – Null and alternative hypothesis: H0 : α = 1 versus H1 : α < 1

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– t-statistic tT = with σˆ α2ˆ

= PT

σˆ 2

2 t=1 yt−1

90

ˆ−1 α σˆ αˆ T

,

1 X σˆ = (yt − αy ˆ t−1)2 T − 1 t=1 2

– Asymptotic distribution of the t-statistic in case of a random walk as DGP 2 1 2 [W (1)] − 1 d tT −→ hR i1/2 1 2 0 [W (r)] dr A table of selected values of this distribution is given below. • Dickey-Fuller tests for various hypotheses A: No deterministic trend 1. So far: H0 : random walk versus H1 : stationary AR(1) without constant (Case 1 in Hamilton (1994, Section 17.4))

90

(I)

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2. More relevant in empirical work: H0 : random walk versus H1 : stationary AR(1) with constant

(II)

(Case 2 in Hamilton (1994, Section 17.4)) ∗ Remark: Two representations of an AR(1) with constant yt − µ0 = α(yt−1 − µ0) + ut, yt = ν + αyt−1 + ut,

with ν = µ0(1 − α).

Thus, in case of the null hypothesis α = 1 the constant ν = µ0(1 − 1) = 0 drops out! ∗ The asymptotic distribution changes if a constant is allowed under the alternative. One then has: R 2 1 [W (1)] − 1 − W (1) W (r)dr 2 d tT −→ hR 2i1/2 , R 1 2 W (r)dr 0 [W (r)] dr − see the table below and e.g. Hamilton (1994, Section 17.4) for mathematical details B: Linear deterministic trend in DGP 3. a) H0: Stochastic and deterministic trend (random walk with drift) yt = µ0 + yt−1 + xt,

∆yt = µ0 + xt, 91

xt stationary AR(1)

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b) H1: Deterministic trend only (difference stationary) yt = µ0 + µ1t + xt,

xt stationary AR(1)

Representation of the null and alternative hypothesis yt = µ0 + µ1t + xt,

xt = αxt−1 + ut

H0 : α = 1 versus H1 : α < 1 Inserting delivers: yt − µ0 − µ1t = α (yt−1 − µ0 − µ1(t − 1)) + ut yt = µ0(1 − α) + µ1t(1 − α) + αµ1 + αyt−1 + ut H0 :

yt = µ1 + yt−1 + ut random walk with drift

H1 :

yt = ν + δt + αyt−1 + ut AR(1) with linear trend.

Summary H0 : random walk with drift versus H1 : AR(1) with linear trend

(III)

(Case 4 in Hamilton (1994, Section 17.4)) For this case the asymptotic distribution of the Dickey-Fuller test turns out to be different again (see e.g. Hamilton (1994, Equation 17.4.55)) and the table below. 92

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4. If a linear trend is excluded from the alternative and the pair of hypotheses is the following H0 : random walk with drift versus H1 : AR(1) with constant (Case 3 in Hamilton (1994, Section 17.4)), one finds that the t-statistic is asymptotically normally distributed. This case is empirically not very relevant since a trend must be of stochastic nature due to the particular choice of the alternative hypothesis. • Asymptotic critical values for the Dickey-Fuller unit root tests Quantile 1% 2.5% 5% 10% 97.5% t-statistic Pair of hypotheses (I)

-2.56 -2.23 -1.94 -1.62 1.62

Pair of hypotheses (II) -3.43 -3.12 -2.86 -2.57 0.24 Pair of hypotheses (III) -3.96 -3.66 -3.41 -3.13 -0.66 T (α ˆ − 1) Pair of hypotheses (I)

-13.7 -10.4 -8.0

-5.7

1.6

Pair of hypotheses (II) -20.6 -16.9 -14.1 -11.2 0.4 Pair of hypotheses (III) -29.4 -25.1 -21.7 -18.2 -1.8 Quelle: Davidson & MacKinnon (1993, Table 20.1, p.708)

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• How to proceed in empirical work – If a trend is expected in the data, one chooses the pair of hypotheses (III). – If the data should not contain a trend for economic reasons, e.g. interest rate date, one chooses the pair of hypotheses (II). – The fewer parameters have to be estimated, that is the smaller the number of the pair of hypothesis (I)-(III), the larger is the power of the test. – There is the possibility to test the null hypothesis α = 1 und δ = 0 jointly with an F test. The corresponding F statistic has also an asymptotic nonstandard distribution, see e.g. Hamilton (1994, Section 17.4). If this hypothesis is rejected, there is empirical evidence for a deterministic linear trend.

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4.1.2 Unit Root Tests in the Presence of Autocorrelation Augmented Dickey-Fuller Test • In general, the DGP may have a larger order than 1 and be an AR(p) process (1 − α1L − α2L2 − · · · − αpLp)xt = ut α(L)xt = ut with ut being white noise. – The asymptotic distributions of the Dickey-Fuller statistics were derived for null hypotheses based on the AR(1) case, e.g. xt = αxt−1 + ut, H0 : α = 1, H1 : α < 1 . If the DGP is in fact an AR(p) process, then the asymptotic distributions all change! Then the asymptotic distribution contains unknown parameters that depend on the correlation structure of the DGP! One solution is to estimate these additional parameters nonparametrically and adapt the test statistic accordingly −→ Phillips-Perron test (This test will not be discussed in this course. Details are found in Hamilton (1994, Section 17.6)) – Alternative: one includes the additional lags in the estimation equation −→ Augmented Dickey-Fuller test • In order to obtain the (Augmented) Dickey-Fuller test statistic for the general AR(p) case, one 95

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has to use the following decomposition ∗ xt = (α1 + α2 + · · · + αp)xt−1 + α1∗∆xt−1 + · · · + αp−1 ∆xt−(p−1) + ut

or, after subtracting xt−1 on both sides and writing φ = (α1 + α2 + · · · + αp) − 1 = −α(1), ∆xt = φxt−1 +

p−1 X

αj∗∆xt−j + ut.

(4.1)

j=1 ∗ Lp−1) ∆xt = φxt−1 + ut (1 − α1∗L − · · · − αp−1 | {z } α∗ (L)

Try this decomposition for an AR(2) process! Note that – the order of the AR model (4.1) in first differences is exactly p − 1; – under H0 and H1 the stationarity condition holds for the α∗(z) polynomial, that is, all roots lie outside the unit circle. Thus, the dependent variable ∆xt is both under the null and the alternative hypothesis stationary (H0: xt contains random walk component, H1: xt stationary). The same holds for the lagged differences ∆xt−j , j = 1, . . . , p − 1; – the process {xt} contains exactly one unit root if φ = 0, and thus α(1) = 0; – this version of the Dicky-Fuller test that is augmented by lagged differences is called Augmented Dickey-Fuller test (ADF test); 96

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– Notice that under H0 the regressor xt−1 on the right hand side is non-stationary. Since its variance increases with increasing sample size, one observes that the estimate φˆ converges √ particularly fast towards the true value 0. This occurs with rate T (instead of T in the stationary case). One therefore calls the OLS estimator for φ in this case superconsistent. What is the rate under H1? • The pairs of hypotheses: A: No trend H0 : unit root AR(p) versus H1 : stationary AR(p) without constant

(I)

H0 : φ = 0 versus H1 : φ < 0 H0 : unit root AR(p) versus H1 : stationary AR(p) with constant

(II)

H0 : φ = 0 versus H1 : φ < 0 B: With trend yt = µ0 + µ1 t + xt ,

∆xt = φxt−1 +

p−1 X

αj∗∆xt−j + ut

j=1

H0 : unit root AR(p) with drift versus H1 : stationary AR(p) with linear trend H0 : φ = 0 versus H1 : φ < 0

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(III)

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• Asymptotic distributions – φ: The asymptotic distribution of T φˆ ∗ ˆ 1∗ − · · · − α ˆ p−1 1−α ˆ − 1) for the corresponding pair corresponds under H0 with the asymptotic distribution of T (α of hypotheses in the AR(1) case, see previous table for critical values. – t-statistic for φ: The asymptotic distribution is given by the asymptotic distribution of the AR(1) case for the corresponding pair of hypotheses, see previous table for critical values. – The reason for this property is the superconsistency of the OLS estimator for φ under H0. Further details including derivations can be found in Hamilton (1994, Section 17.7) or Davidson (2000, Chapter 14) – αj∗ and corresponding t statistics: like in the stationary AR model – All the results mentioned so far only hold if the order p is not chosen too small that is ∗ ut is white noise or ∗ ut is approximately white noise. This means that given the sample size one makes the residuals resemble white noise as much as possible. One can achieve this by letting the 98

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order p increase with sample size T such that the residual process becomes ’whiter’. This case occurs if the true order is infinite. It is important that the p for the estimated model does neither grow too fast nor too slow (Too fast: not enough observations to estimate parameters with small enough variance; too slow: residuals do not become white enough). • In sum: to obtain a valid asymptotic distribution, the order p has to be large enough. However, a too large order leads to a loss in power (since the variance of parameter estimates decreases not fast enough). Luckily, it is possible to determine an appropriate order by using model selection criteria like in the stationary case: The Hannan-Quinn HQ(n) and the Schwarz criterium SC(n) are consistent if the time series is generated by an AR(p) process with finite p and pmax ≥ p. • In empirical work it may happen that the degree of integration is larger than I(1). One then can use the Pantula principle to determine the appropriate order of integration, see e.g. L¨utkepohl & Kr¨atzig (2008, Section 2.7.1).

4.1.3 Other Unit Root Tests • Phillips-Perron Test: see comments above and for an overview e.g. Kirchg¨assner & Wolters (2008, Section 5.3.2). A

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detailed derivation can be found in Hamilton (1994, Section 17.6). • KPSS Test: The null hypothesis of all tests discussed so far is that yt ∼ I(1). Kwiatkowski et al. (1992) have developed a test that allows to test stationarity directly using the pair of hypotheses: H0 : yt ∼ I(0)

versus

H1 : yt ∼ I(1).

– See e.g. Kirchg¨assner & Wolters (2008, Section 5.3.4) or L¨utkepohl & Kr¨atzig (2008, Section 2.7.4) for an overview. – This test is based on the Beveridge-Nelson decomposition (not covered in this course but you ¨ may see the slides of Fortgeschrittene Dynamische Okonometrie (available on request) that states that an autoregressive process with a unit root can be decomposed into a random walk and a stationary component yt = zt + ηt zt = zt−1 + vt,

vt ∼ iid(0, σv2).

– The pair of hypotheses can now be stated more precisely as H0 : σv2 = 0 versus H1 : σv2 > 0. If H0 holds, then zt is a constant and yt is stationary. 100

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– The test statistic is: 1 KP SS = 2 T

PT hPt t=1

j=1 (yj

− y¯)

2

101

i

2 σˆ η,∞

2 denotes the so-called long-run variance of the stationary process {ηt}. where ση,∞

If yt is stationary (H0 holds), then the numerator converges. If yt is non-stationary (H1 2 < ∞, holds), then the numerater diverges to plus infinity! Since for both hypotheses ση,∞ one rejects the null hypothesis if the numerator and thus the test statistic is too large. – The asymptotic distribution of the KPSS test is also non-standard and is given in the tables of Kwiatkowski et al. – If there is a linear trend under the null hypothesis (trend stationarity), then one considers yt = µ1t + zt + ηt. Further details can be found e.g. in L¨utkepohl & Kr¨atzig (2008, Section 2.7.4) • If one uses the ADF or the Phillips-Perron test jointly with the KPSS test, it may happen that both tests reject or do not reject. This happens if one of the two tests does not have enough power or if the alternative does not contain the true data generating process. • A structural break in the data in general reduces the power of the ADF or the Phillips-Perron test (meaning it is less likely to reject the unit root hypothesis if it is not true). The reason is 101

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that the standard alternative hypothesis cannot capture the structural break well. Thus, if one suspects a structural break, one has to include it in the alternative hypothesis. If one does not know the break point exactly, one may estimate it. This can be done with the freely available menu-driven software JMulTi, see L¨utkepohl & Kr¨atzig (2008, Section 2.7.3) for a description. • There are a number of other tests, see e.g. other tests in R package urca. • Besides the unit root discussed so far it may also happen that the data driven process exhibits seasonal roots. If a data generating process contains all seasonal roots, then one can obtain a stationary process by applying the operator (1 − LS ) to the data. Depending on the number of seasons S one can decompose this filter differently. Example S = 4: (1 − L4) = (1 − L)(1 + L)(1 + L2). One notices that the (seasonal) unit roots occur at 1, −1, i. With the HEGY test one can test which seasonal unit roots cannot be rejected for a given time series. This test is e.g. implemented in JMulTi and described in L¨utkepohl & Kr¨atzig (2008, Section 2.7.5).

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4.2 Model Checking 4.2.1 Descriptive Analysis of the Residuals • Outliers or structural breaks frequently show up in the plotted time series of the residuals. =⇒ Plot the standardized residuals uˆst: uˆt − u¯ˆt s uˆt = σ˜ u

with u¯ˆt = T −1

T X

uˆt and

t=1

σ˜ u2

=T

−1

T X

uˆt − u¯ˆt

2

t=1

(Technical note: the OLS residuals do not sum to 0 if there is no constant in the model; therefore the mean of the residuals has to be subtracted.) If the residuals are standard normally distributed (e.g. in case the DGP is Gaussian white noise with variance 1), about 95% of the residuals should be within the interval [−2, 2]. If the residuals are approximately normally distributed (e.g. if the DGP has Gaussian white noise errors), then the rule can be applied approximately for identifying outliers and/or structural breaks. • Analysis of the squared residuals in order to check the homoskedasticity assumption. If the errors are homosekdastic, the squared residuals should not exhibit very large peaks. • Analysis of the autocorrelation structure in the residuals: 103

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Let

Cov(ut, ut−i) ρu,i = p V ar(ut)V ar(ut−i) denote the i-th autocorrelation of the error process {ut}. An estimator of the autocorrelation coefficient ρu,i is P T −1 Tt=i+1 uˆt − u¯ˆt uˆt−i − u¯ˆt ρˆu,i = σ˜ u2ˆ T X = T −1 uˆstuˆst−i. t=i+1

If the error process is white noise, the estimated autocorrelations of the residuals should be in the √ √ interval [−2/ T , 2/ T ] with roughly 95% probability. In small samples one should correct for the sample size, see e.g. L¨utkepohl (2005, Proposition 4.6). √ √ Observing ’too many’ autocorrelations outside the interval [−2/ T , 2/ T ] is an indication that relevant lags have been missed in the specified model. • For the spectral estimation of the residuals, you are referred e.g. to L¨utkepohl & Kr¨atzig (2008, Section 2.2.2).

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4.2.2 Diagnostic tests • Tests for autocorrelation in the residuals In order to test whether the error process underlying an estimated model is a white noise process it is in principle necessary to check whether all its autocorrelations are 0. In empirical work one restricts oneself to test whether the first h autocorrelations are jointly 0. Since the errors are unobservable, the test must be based on the residuals uˆt. – Portmanteau test for residual autocorrelation The Portmanteau (=luggage) test checks whether the first h autocorrelations are jointly 0: H0 : ρu,1 = · · · = ρu,h = 0 versus H1 : ρu,i 6= 0 for at least one i = 1, . . . , h. Portmanteau test statistic (Box-Pierce statistic): Qh = T

h X

ρˆ2u,j

(4.2)

j=1

Under H0 and for T → ∞ it holds that Qh follows asymptotically a χ2(h−p−q)-distribution. Note: Test can only be performed for h > p + q. Problem: The test is not very reliable in small samples. One therefore uses in practice a modified version described next. 105

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– Modified Portmanteau test statistic (Ljung-Box statistic) QLB h

h X ρˆ2u,j = T (T + 2) T −j j=1

(4.3)

This test also has an asymptotic χ2(h − p − q)-distribution under H0. – Lagrange multiplier (LM) test for autocorrelation in the residuals: ∗ The Lagrange multiplier test is a general testing principle that is derived in the setting of maximum likelihood estimation that will be discussed later during this course. The test statistic of the Lagrange multiplier test varies with the specific model and null hypothesis under consideration. ∗ The basis for the LM test for autocorrelation in the residuals of AR models is an autoregressive model for the errors ut = β1ut−1 + · · · + βhut−h + errort that allows to test the null hypothesis H0 : β1 = · · · = βh = 0 versus H1 : β1 6= 0 or · · · or βh 6= 0. This test is not feasible since the errors are not observable. To make the test feasible one uses the residuals uˆt of the fitted model instead. 106

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∗ The computation of the test statistic is quite easy: LMh = T R2 where the coefficient of determination R2 is obtained (in case of a fitted AR(p) model) from the following auxiliary regression uˆt = ν + α1yt−1 + · · · + αpyt−p + β1uˆt−1 + · · · + βhuˆt−h + et. The Lagrange multiplier test statistic follows asymptotically a χ2-distribution with h degrees of freedom, thus, d LMh = T R2 → χ2(h). ∗ In small samples, the F version of the LM statistic frequently delivers a better approximation of the finite sample distribution R2 T − p − h − 1 ≈ F (h, T − p − h − 1). F LMh = 1 − R2 h ∗ The LM test is frequently called the Breusch-Godfrey test or the Godfrey test. • Test for nonnormal errors: Lomnicki-Jarcqe-Bera test – If the assumption of normally and identically distributed errors is not violated, then the following holds for the third and fourth moment of the standardized errors ust = ut/σ: E (ust)3 = 0, E (ust)4 = 3. 107

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The Lomnicki-Jarcqe-Bera test checks whether these two moments correspond to the values implied by a normal distribution: s 4 s 3 H0 : E (ut ) = 0 and E (ut ) = 3 H1 : E (ust)3 6= 0 or E (ust)4 6= 3 – The test statistic is "

LJB =

T T −1 6

T X

(ˆ ust)3

#2

"

+

t=1

T T −1 24

T X

(ˆ ust)4 − 3

#2 d

−→ χ2(2).

t=1

– Remark: If H0 is not rejected, then this does not imply that the errors are normally distributed but merely that their first four moments are compatible with a normal distribution.

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• Testing for (conditional) heteroskedasticity in the error process – Important definitions: A stochastic process {yt}, t = . . . , −2, −1, 0, 1, 2, . . . , is called ∗ homoskedastic if V ar(yt) = σy2 for all t, ∗ (unconditionally) heteroskedastic if 2 2 6= V ar(ys) = σy,s for some t 6= s, V ar(yt) = σy,t

∗ conditionally heteroskedastic if V ar(yt|yt−1, yt−2, . . . , ) 6= V ar(yt) for some t. ∗ These definitions apply also to the noise process {ut} of a stochastic process {yt}. – To test for the presence of unconditional heteroskedasticitiy in the noise process one can use tests for structural breaks. To test for the presence of conditional heteroskedasticity in the noise process one can use the ARCH-LM test that is described in section 5.5.

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4.3 Estimating dynamic regression models • The dynamic regression model yt = Xt0β + ut with

Xt0

= xt1 xt2 · · · xtk ,

(4.4) β1 β 2 β=. .. βk

and Xt0 belongs to the information set It. The following assumptions are made about the stochastic processes of the error term and the regressors: ¨ 1. Assumption B (compare to assumptions (C1) to (C4) in Methoden der Okonometrie): a) The conditional expectation of the error term ut given the information set up to time t is zero E[ut|It] = 0. b) The conditional variance of the error term ut given the information set up to time t is 110

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constant (= the error term ut is homoskedastic) E[u2t |It] = σ 2. c) The regressor matrix behaves nicely in large samples, that is the empirical moment matrix converges asymptotically to a fixed matrix T T 1X 1X 0 plim T →∞ XtXt = lim E [XtXt0] = MXX T →∞ T T t=1 t=1

with MXX being positive definite. d) Strict stationarity of all variables or E |λ0Xtut|2+δ ≤ B < ∞,

δ > 0,

for all t, for all λ with λ0λ = 1.

Then it can be shown that the OLS estimator is asymptotically normally distributed: !−1 T T X X √ 1 1 d 0 2 −1 ˆ √ Xt Xt T β−β = Xtut −→ N 0, σ MXX T t=1 T t=1 • Notes to the assumptions: – Note that Assumption A for estimating AR(p) models, see subsection 2.5.1, guarantees that Assumption B holds. Thus, the dynamic regression model is a generalization of AR(p) 111

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models by allowing also for further explanatory variables that are not lagged endogenous variables. This implies that parameters associated with lagged endogenous variables have to fulfil the stationarity conditions known from the AR models! This has to be checked in the model analysis! – The main issue in B(c) is that it is excluded that ∗ asymptotically the moment matrix MXX becomes singular (Example: xt1 = 1, xt2 = 1/t) ∗ the empirical second moments do not converge such that MXX does not exist (Example: all regressors contain a random walk). If all regressors are stationary, Assumption B(c) is fulfilled. It basically restricts the memory of the regressors (remember that a random walk has perfect/infinite memory and therefore violates the requirement of restricted memory). – As conditioning/explanatory variables one may include ∗ deterministic variables (but watch out: deterministic trends can violate B(c), see below.) ∗ lagged variables ∗ current dated (=contemporaneous variables) that are weakly exogenous for β and σ 2. This

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requires for example that they are uncorrelated with ut. – The homoskedasticity assumption in B(b) can be relaxed to heteroskedastic errors. This, however, requires to use a heteroskedasticity-robust estimator for the covariance matrix of the parameter estimates (or the use of FGLS methods, which may be difficult). – Assumption B(d) generalizes Assumption A. This assumption is needed such that the vari√ PT ance of 1/ T t=1 xtiut asymptotically exists. Among other things, this requires that the variances of ut or xti are not allowed to increase without bound. It also avoids that the error or the regressor variable have too fat tails. • A consistent estimator for the covariance matrix is T

σˆ2 (X 0X)

−1

,

1 X 2 2 uˆ . σˆ = T − k t=1 t

• Because the OLS parameters are asymptotically normally distributed, it can be shown that the standard t and F tests can be applied, however, only asymptotically (for more details on this, see ¨ Methoden der Okonometrie). • Some further (important) notes: – If dummy variables are used, there must be a nonzero fraction of 1’s or 0’s in any sample. Thus, there is no asymptotic distribution for a dummy that is 1 for a single (or a finite number 113

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of periods) and 0 otherwise in case the sample size increases. Tests for such dummies must be based on other grounds. Including a dummy for a single period means asymptotically that this observation is dropped from the sample. – A variable is called trend-stationary if regressing it on t delivers stationary residuals. If one includes variables that are trend-stationary, then the parameter estimates corresponding to √ these variables do not have the T rate of convergence and thus, the standard errors are not correct. Solution: Include a deterministic trend xti = t as an additional regressor. However: the trend variable has now a different (faster) convergence rate, see also unit root tests. – Random walks and cointegration. Will not be discussed here.

4.4 R Code Testing time series y for a unit root # y denotes data series install.packages("urca") library(urca)

# only necessary when used for the first time

# ADF test with up to 5 lags (chosen by AIC) against the alternative of # a stationary process around a (non-zero) constant (pair of hypotheses (II)) y.adf <- ur.df(y, type = "drift", lags = 5, selectlags = "AIC") summary(y.adf)

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# KPSS test with number of lags depending on T against the alternative of # a unit root process without deterministic trend y.kpss <- ur.kpss(y, type = "mu", lags = "long") summary(y.kpss) # See ur.pp() for the Phillips-Perron Test ?ur.pp # Other unit root tests in package urca that have not been discussed so far ?ur.ers ?ur.sp ?ur.za

Diagnostic tests: serial correlation and non-normality # Analysing residuals from AR(4) regression ar.est <- ar(y, demean = FALSE, aic = FALSE, order.max = 4) ar.resid <- ar.est$resid # contains NA values for first 4 observations # Autocorrelation structure of residuals (ignoring NA values) acf(ar.resid, na.action = na.pass) # Portmanteau test # up to order h h <- 7 Box.test(ar.resid, # modified version Box.test(ar.resid,

for serial correlation in residuals

lag = h, type = "Box-Pierce", fitdf = 4) of Portmanteau test lag = h, type = "Ljung-Box", fitdf = 4)

# Test for nonnormal errors ar.resid <- na.omit(ar.resid)

# getting rid of NA values

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Applied Financial Econometrics — 4.4 R Code — U Regensburg — July 2012 T <- length(ar.resid) ar.resid.mean <- mean(ar.resid) ar.resid.demean <- ar.resid - ar.resid.mean ar.resid.var <- mean( ar.resid.demean^2 ) ar.resid.stndrd <- ar.resid.demean / sqrt(ar.resid.var) # standardized residuals ar.resid.skew <- mean( ar.resid.stndrd^3 ) # skewness ar.resid.kurt <- mean( ar.resid.stndrd^4 ) - 3 # (excess) kurtosis LJB <- T/6 * (ar.resid.skew^2) + T/24 * (ar.resid.kurt^2) # LJB test statistic 1- pchisq(LJB, df = 2) # p-value

Dynamic regression # Dynamic regressions are easily performed using the interface provided # by the dynlm package library(dynlm) ?dynlm

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5 Modeling volatility dynamics A brief introduction: • Consider the very simple conditionally heteroskedastic stochastic process {yt} yt = ut where the noise process is conditionally heteroskedastic V ar(ut|ut−1, ut−2, . . .) 6= V ar(ut). Note that in the special case considered at the moment V ar(ut|yt−1, yt−2, . . .) = V ar(ut|ut−1, ut−2, . . .) since the ut−j ’s are known if and only if the yt−j ’s are known for all j = 1, 2, . . .. Note that E[ut|yt−1, yt−2, . . .] = 0 as in the independent white noise case.

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(5.1)

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• A more general conditionally heteroskedastic process is the conditionally heteroskedastic autoregressive process of order p (1 − α1L − α2L2 − · · · αpLp)yt = ut where the noise process exhibits conditional heteroskedasticity as given by (5.1). If the autoregressive parameters are known, then V ar(ut|yt−1, yt−2, . . .) = V ar(ut|ut−1, ut−2, . . .). If the autoregressive parameters are not known, then the two conditioning sets are different since the errors ut are unobservable and have to be estimated. • In the following we present models for conditional heteroskedastic noise and for simplicity we assume that the errors are observable. Later on this assumption will be relaxed. • In order to save notation we frequently write in the following σt2 = V ar(ut|ut−1, ut−2, . . .). • In all what follows we assume that ut = σtξt,

ξt ∼ IID(0, 1)

and ξt and us are stochastically independent for t > s. The distribution for ξt may vary, depending on the specific application. 118

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5.1 Standard conditional volatility models • Autoregressive Conditionally Heteroskedastic Models (ARCH(m) Models) – Simplest case: ARCH(1) process σt2 = V ar(ut|ut−1, ut−2, . . .) = V ar(ut|ut−1) = γ0 + γ1u2t−1. The conditional variance only depends on one lagged error. Properties: ∗ Sufficient conditions for positive variance: γ0 > 0 and γ1 > 0. ∗ How can one generate a realization of an ARCH(1) process with conditionally normally distributed innovations? ut = σtξt, ξt ∼ i.i.d.N (0, 1) with σt2 given above. Try the R code in section 5.8 for generating ARCH(1) realizations. ∗ Unconditional mean E[ut] = 0 if the unconditional variance exists since q E[ut] = E γ0 + γ1u2t−1 E[ξt] = D 0. |{z} | {z } =0 by assumption D

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Thus, D has to be finite which can be shown to be the case if the unconditional variance exists. ∗ Unconditional variance V ar(ut) = E[u2t ]

law of iterated expectations

=

E[E[u2t |ut−1]]

= E[γ0 + γ1u2t−1] = γ0 + γ1E[u2t−1] V ar(ut) = γ0 + γ1V ar(ut−1) Under the assumption of stationarity V ar(ut) = V ar(ut−1) = σ 2 we get γ0 V ar(ut) = 1 − γ1 Note the unconditional variance only exists if γ1 < 1. ∗ Conditional fourth moment E[u4t |ut−1] = (γ0 + γ1u2t−1)2E[ξt4] ∗ Unconditional fourth moment (only exists if fourth moment stationary). Then one has E[u4t ] = E[u4t−1] = m4 and one can derive under the assumption of normally distributed ξt 3γ02(1 + γ1) m4 = . (1 − γ1)(1 − 3γ12) 120

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Note that the unconditional fourth moment only exists if the unconditional variance exists and γ12 < 1/3. ∗ Unconditional kurtosis in case of normally distributed ξt: m4 1 − γ12 E[u4t ] κ= 2 = 2 = 3 2 > 3. 2 1 − 3γ (V ar(ut)) (σ ) 1 Thus, an ARCH(1) process is leptokurtic: the tail distribution of ut is heavier than that of a normal distribution. Put differently, compared with a normal distribution, ’outliers’ are more likely. – ARCH(m) process: σt2 = V ar(ut|ut−1, ut−2, . . . , ut−m) = γ0 + γ1u2t−1 + · · · + γmu2t−m The conditional variance depends on m lagged errors. The ARCH(m) model was proposed by the nobel prize winner Robert Engle in Engle (1982) Properties: ∗ Sufficient conditions for positive variance: γ0 > 0 and γj > 0, j = 1, . . . , m. ∗ Unconditional variance: The unconditional variance V ar(ut) = σ 2 =

121

γ0 1 − γ1 − · · · − γm

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exists if 1 − γ1 − · · · − γm > 0. ∗ The ARCH(m) process corresponds to a conditionally heteroskedastic AR(m) process in the squared errors u2t = σt2ξt2 u2t = σt2 + σt2(ξt2 − 1) | {z } ηt

u2t = γ0 + γ1u2t−1 + · · · + γmu2t−m + ηt where E[ηt] = E[E[ηt|ut−1, . . . , ut−m]] = E[σt2E[(ξt2 − 1)]] = E[σt2 0] = 0, since E[ξt2] = 1 by assumption. ∗ Other moments and conditions can also be derived but are messier to present. Drawbacks of ARCH(m) models: ∗ Positive and negative shocks have the same impact on volatility. In practice one observes frequently an asymmetric behavior. ∗ The parameter restrictions are quite severe if one requires (unconditional) fourth moments to be stationary. ∗ The model may not be very parsimonious if large lags have an important impact (m large).

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• Generalized Autoregressive Conditionally Heteroskedastic Processes (GARCH(m, n) process) – To solve the problem of having too many parameters in an ARCH(m) model, Taylor (2008) and Bollerslev (1986) independently suggested the GARCH(m, n) model that uses lagged conditional variances as explanatory variables in addition to the lagged squared shocks 2 2 σt2 = V ar(ut|ut−1, ut−2, . . .) = γ0 + γ1u2t−1 + · · · + γmu2t−m + β1σt−1 + · · · + βnσt−n .

– A special case is the so-called integrated generalized autoregressive conditionally heteroskedastic (IGARCH(1,1)) process for which γ1 + β1 = 1. – Properties: ∗ Unconditional variance of ut: The unconditional variance is V ar(ut) =

γ0 1 − γ1 − . . . − γm − β1 − . . . − βn

and exists if γ1 + . . . + γm + β1 + . . . + βn < 1. ∗ Conditional fourth moment: E[u4t ]|ut−1, ut−2, . . .]

= γ0 +

γ1u2t−1

+ ··· +

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γmu2t−m

+

2 β1σt−1

+ ··· +

2 2 βnσt−n E

4 ξt

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∗ Unconditional fourth moment for a GARCH(1,1) process: If ξt ∼ IN (0, 1), then E[ξt4] = 3 and then one can show that E[u4t ]

3γ02(1 + γ1 + β1) = . (1 − γ1 − β1)(1 − β12 − 2γ1β1 − 3γ12)

The fourth moment exists if (β12 + 2γ1β1 + 3γ1)2 < 1. Note that this condition is stronger than the one for a stationary variance. ∗ Kurtosis for a GARCH(1,1) process: If ξt ∼ IN (0, 1), then E[ξt4] = 3 and then one can show that κ=

E[u4t ] (E[u2t ])

2

=

3(1 − γ1 − β1)(1 + γ1 + β1) > 3. 1 − β12 − 2γ1β1 − 3γ1

Thus, a conditionally normally distributed GARCH(1,1) process is leptokurtic, meaning that the tails of the unconditional probability density of ut are thicker than those of a normal density that has a kurtosis of 3: large shocks are more likely to occur than in case of a normally distribution. Note that the type of the distribution is not normal even if ξt is IN(0,1). ∗ Although the unconditional distribution does not correspond to a known standard distribution, Nelson (1990) has shown that a GARCH(1,1) process is strictly stationary and ergodic if E log β1 + γ1ξt2 < 0. 124

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This condition means that one should not observe too often values of β1 + γ1ξt2 that are way beyond 1 and thus getting too much impact. Note that this condition is weaker than the stationarity condition for the unconditional variance since an IGARCH(1,1) process is strictly stationary but not covariance stationary. • There exist a number of extensions of ARCH and GARCH processes. They will be discussed after the presentation of estimation procedures.

5.2 Maximum Likelihood Estimation • Introductory Example: Consider having thrown a coin 10 times with the results of 9 heads and 1 tail. Do you think that this is a ’fair’ coin (a coin for which the probability for observing head is 0.5)? – Note that the probability to obtain k times head out of n throws is given by the binomial distribution n! pk (1 − p)(n−k) (5.2) P (’k heads of out n trials’|p) = (n − k)!k! where p denotes the probability for getting a head in one throw.

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– Thus, the probability to observe the outcome stated above is for various p: p = 1/2

−→ P (’9 heads of out 10 trials’|p = 1/2) ≈ 0.01

p = 3/4

−→ P (’9 heads of out 10 trials’|p = 3/4) ≈ 0.19

p = 9/10

−→ P (’9 heads of out 10 trials’|p = 9/10) ≈ 0.39

Hence, the probability of observing the given result is only about 1% if the coin is ’fair’. It seems more likely that the probability of getting head in one throw is far larger than 0.5. – One may now change the use of the probability function (5.2) and use it to assign a given event a probability based on a chosen value for p. In this interpretation one calls (5.2) a likelihood function in order to distinguish it from the use of a probability function. One writes for the present case n! pk (1 − p)(n−k) (5.3) L(p|’k heads of out n trials’) = (n − k)!k! – Since one can compute the likelihood L(p|’k heads of out n trials’) for a given outcome, e.g. ’9 heads out of 10 throws’, for any p, one can maximize the likelihood L(p|’k heads of out n trials’) with respect to p. One then obtains an estimate pˆ for p that maximizes the likelihood for which the observed outcome may be observed. Therefore, this estimator is called maximum likelihood (ML) estimator. – In the current case, one can easily obtain the ML estimator pˆ by setting the first derivative 126

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of (5.3) to zero and solving for p. – Very often it is more convenient to maximize the likelihood function after taking logarithms. That does not change the maximum likelihood estimator since taking logarithms is a strictly monotone transformation. However, it makes the analytical or numerical optimization much easier. The log-likelihood function in the present case is l(p|’k heads of out n trials’) = ln L(p|’k heads of out n trials’) n! = ln + k ln p + (n − k) log(1 − p). (n − k)!k! The first derivative is ∂ ln L(p|·) k n − k ! = − = 0. ∂p p 1−p The ML estimate for p is therefore pˆ = k/n = 9/10. (For completeness one has to check whether the extremum is a maximum. This requires the second derivative to be negative around pˆ.) • Maximum likelihood estimation in the case of continuous random variables: – For a continuous random variable Y it holds that the probability ’Y takes the value y’ is zero, that is P (Y = y) = 0. This is because there are an infinite number of possible values and a sum of infinitely many positive probabilities cannot sum to 1.

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Instead one considers an interval for Y , e.g. [a, b] or frequently [−∞, y]. For the latter interval one obtains the probability distribution function F (Y < y) = P (Y < y) which must be nonzero for some intervals. Thus one can also analyze the change of the probability if one increases the interval by an marginal amount δ > 0. This delivers the absolute change in probability P (Y < y + δ) − P (Y < y) and the relative change in probability P (Y < y + δ) − P (Y < y) . δ Letting the change in the interval length δ go to 0, one obtains the probability density function P (Y < y + δ) − P (Y < y) f (y) = lim δ→0 δ that must be nonzero at some y since otherwise the probability would not change if the interval is increased. Since P (y ≤ Y < y + δ) = P (Y < y + δ) − P (Y < y)

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one obtains, loosely speaking, P (y ≤ Y < y + δ) ≈ f (y)δ. One can approximate the probability that a realization of Y is observed to be in the interval [y, y+δ) by the density multiplied with the interval length. Of course, this approximation is the better, the smaller δ. The density is approximately proportional to the probability of Y being observed in a very short interval around y. – Equivalently, if the probability and density function depend on a parameter θ one has P (y ≤ Y < y + δ|θ) ≈ f (y|θ)δ. Maximizing the likelihood for observing Y (to be in a extremely tiny interval around y) thus can be done by maximizing the density with respect to θ. For continuous random variables the density therefore has the interpretation of the likelihood function L(θ|y) = f (y|θ). The ML estimator θˆ for θ is then given by max L(θ|y) (= max f (y|θ)). θ

θ

– Thus, for deriving the ML estimator for a specific problem one has to choose an appropriate parameterized density function. 129

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– For a sample of T observations y1, y2, . . . , yT the likelihood function is the joint density with respect to θ L(θ|y1, y2, . . . , yT ) = f (y1, y2, . . . , yT |θ) – The joint density for T i.i.d. observations is the product of T marginal densities. Thus, the likelihood is given by L(θ|y1, y2, . . . , yT ) = f (y1, y2, . . . , yT |θ) = f (y1|θ) · · · f (yT |θ) and the log-likelihood is the sum of the log-likelihood for each observation l(θ|y1, y2, . . . , yT ) = ln f (y1, y2, . . . , yT |θ) =

T X

ln f (yt|θ).

t=1

This property is very convenient for maximizing the (log)-likelihood! – In case the observations are not i.i.d. one can use the following decomposition L(θ|y1, y2, . . . , yT ) = f (y1, y2, . . . , yT |θ) = f (yT |yT −1, . . . , y1; θ)f (yT −1, yT −2, . . . , y1|θ) = f (yT |yT −1, . . . , y1; θ)f (yT −1|yT −2, . . . , y1|θ)f (yT −2, . . . , y1|θ) = f (yT |yT −1, . . . , y1; θ)f (yT −1|yT −2, . . . , y1|θ) · · · f (y2|y1; θ)f (y1|θ)

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Taking logarithms one obtains the sum l(θ|y1, y2, . . . , yT ) = ln f (y1, y2, . . . , yT |θ) =

T X

ln f (yt|yt−1, . . . , y1; θ) + ln f (y1|θ).

t=1

– If the term ln f (y1|θ) is ignored one obtains the conditional likelihood function that is conditional on y1. Its maximization delivers the conditional maximum likelihood estimator.

5.3 Estimation of GARCH(m, n) models • GARCH models are usually estimated with the (conditional) maximum likelihood estimator. This requires to assume a density for the errors ξt. • Assume independent normally distributed errors ξt ∼ N (0, 1) with density 1 1 f (ξt) = √ exp − ξt2 . 2 2π 131

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Since by definition

ut σt

ξt = one has

ut|ut−1, . . . ∼ N (0, σt2) and thus by the properties of the normal density 1 u2t f (ut|ut−1, . . . , u1; γ0, . . . , γm, β1, . . . , βn) = p exp − 2 2 σt 2πσt2 1

The log-likelihood function of an GARCH(m, n) model is then given by l(γ0, . . . , γm, β1, . . . , βn|uT , uT −1, . . . , u1) =

T X

ln f (ut|ut−1, . . . ; γ0, . . . , γm, β1, . . . , βn)

t=1

=

T X t=1

( ln

) 2 1u 1 p exp − t2 2 σt 2πσt2 T

T

1 X u2t T 1X 2 ln σt − = − ln(2π) − 2 2 t=1 2 t=1 σt2 which after inserting σt2 becomes T

T

1X T 1X u2t 2 2 − ln(2π) − ln γ0 + γ1ut−1 + · · · + βnσt−n − 2 2 2 t=1 2 t=1 γ0 + γ1u2t−1 + · · · + βnσt−n 132

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This log-likelihood function has to be maximized for obtaining the ML estimates γˆ0, . . . , βˆn based on the assumption of conditionally normally distributed errors. • Frequently residuals of GARCH models of financial time series are leptokurtic (they have fatter tails than normally distributed residuals). In this case the assumption of conditionally normally distributed errors is wrong and such a model is misspecified. Solutions: – Use other error distribution, e.g. t-distribution. – Use quasi-maximum likelihood estimator. • Assume conditionally t-distributed errors. The density of a t-distributed error variable u with ν degrees of freedom and variance σ 2 is − ν+1 ν+1 2 Γ 2 u2 1+ f (u; ν) = √ ν vπ Γ ν2 R∞ where Γ(h) denotes the gamma function Γ(h) = 0 xh−1 exp(−x)dx, h > 0. The moments are – E[u] = 0 if ν ≥ 2 – V ar(u) =

ν ν−2

if ν ≥ 3 133

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– all odd moments are zero – r-th moment, see Johnson et al. (1995), 1

µr (u) = ν 2 r

1 · 3 · · · (r − 1) (ν − r)(ν − r + 2) · · · (ν − 2)

ν V ar(u) – E[u4] = 3 ν−4 ν−2 – kurtosis: κ = 3 ν−4

• Assume conditional generalized error distribution (GED) !ν/2 1/2 Γ ν3 νΓ ν3 ν f (u; ν) = exp −|u| 1 1 3/2 Γ ν 2Γ ν where ν > 0. For ν = 2 one obtains a normal distribution N (0, σ 2) and for ν < 2 one has a leptokurtic distribution. • Transformation of random variables: How can one obtain the density for a random variable X if one knows the density fZ (z) of Z = h(X) and the function h(·)? Then the density of X is obtained by fX (x) = fZ (h(x))|h0(x)|, see e.g. Davidson & MacKinnon (2004, p. 438-439) for a derivation. 134

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• Setting U = Zs with u = h(z) = random variable the density

z s

135

and h0(z) = 1/s one obtains e.g. in case of a t-distributed ν+1 2

Γ f (z; ν, s) = √ s vπ Γ

z2 1+ 2 ν sν 2

− ν+1 2 .

• Numerical optimization There exists a number of algorithms for the optimization of nonlinear functions. For maximizing the log-likelihood function a particularly suited algorithm was developed by Berndt et al. (1974) (BHHH algorithm). In the i-iteration you update the estimator θˆi by !−1 T T X X ∂l ∂l ∂l t t t θˆi = θˆi−1 + φ 0 ∂θ ∂θ ∂θ ˆ θ=θi−1 t=1 i=1

θ=θˆi−1

where φ denotes the step length that also can be automatically chosen. For running these iterations you have to choose – starting values θ0, – a maximum number of iterations imax, – a stopping rule, e.g. |θˆi − θˆi−1| < where the precision is a priori chosen, 135

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– a numerical method or the analytical derivatives to compute the various derivatives, – possibly an algorithm to determine the step length. Attention: the choice of the starting values matters whatever numerical algorithm one chooses. Badly chosen starting values may result in a local instead of a global maximum! Thus, it is often useful to try several starting values, e.g. even by drawing random numbers / vectors. If possible, one may get the starting values from an auxiliary model (e.g. for estimating an ARCH model from the parameter estimates of the ARCH-LM test, see below.) • Under the regularity conditions, see below, the ML estimator is asymptotically normally distributed √ d ˆ T θ − θ −→ N 0, S −1 with T ∂lt ∂lt 1X E S = lim T T ∂θ ∂θ0 t=1 1 ∂l ∂l = lim E T T ∂θ ∂θ0 1 ∂l = lim V ar T T ∂θ denoting the asymptotic Fisher information matrix. The Fisher information matrix or the ∂l t t covariance matrix of the gradient of the log-likelihood function is given by E ∂l ∂θ ∂θ0 . 136

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Note that the expectation of the score (=first derivative of the log-likelihood function) computed at the true parameter vector is zero, " # ∂l E =0 ∂θ θ=θtrue value ¨ if the model is correctly specified! For details see e.g. Methoden der Okonometrie. • Let It denote the information set that is available for computing the log-likelihood function at time t. In case of an AR(p) model, the information set is given by all past observations of the endogenous variable. The regularity conditions are – E[ut|It] = 0 The conditional mean of the error is zero (standard assumption) – E[u2t |It] = σt2 Conditional heteroskedasticity is correctly modeled. P P 0 0 = M < ∞ and M is invertible. – plim T1 Tt=1 Yt−1Yt−1 = lim T1 Tt=1 Yt−1Yt−1 This means that even for very large samples the regressors are well behaved, e.g. not becoming perfectly linearly dependent. – E |ut|4+δ < ∞ 137

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The fourth moments of the errors exist. • In case no error distribution is found to be appropriate, one may use quasi-maximum likelihood estimation (QML estimation) : one uses the maximum likelihood estimator based on a normal error density although the true errors may follow a different distribution. Properties: – The QML estimator is consistent. – The QML estimator requires the computation of a more complicated covariance matrix of the parameter estimates. Denote the limit of the negative expectation of the Hessian matrix (matrix of all second partial derivatives) divided by T by 2 ∂ l 1 . D = − lim E T T ∂θ∂θ0 Then the covariance matrix of the QML estimator is D−1SD−1. Thus, the QML estimator is not efficient since in general the covariance matrix is ’larger’ than the covariance matrix based on the correct error density. – The QML estimator is asymptotically normally distributed √ d T θˆ − θ −→ N 0, D−1SD−1 .

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– If the information matrix equality D−1 = S holds asymptotically, then the asymptotic distribution of the QML estimator is identical with that of the ML estimator. Note, however, that the matrices S and D have to be estimated ˆ −1SˆD ˆ −1 6= Sˆ−1 and one should use the ML estimator if the in practice. Thus, in general D errors are indeed normal. ¨ For details see e.g. Methoden der Okonometrie. Warnings: – The QML principle only works with the normal distribution. Using another ’wrong’ distribution does not lead to a consistently and asymptotically normally distributed estimator. – If one has estimated the covariance matrix by QML, one cannot produce forecasting intervals since the error distribution is still unknown. • Tests based on ML estimators in case of nonnormal errors are misleading since the standard errors of the parameter estimates are incorrectly estimated! • Estimation of conditional mean and conditional variance functions: – The information matrix is block diagonal if the covariances between all parameter estimators for the conditional mean function and all parameter estimators for the conditional variance 139

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function are zero. – Sequential estimation is possible if the information matrix is block diagonal. One then can consistently estimate in the first step the conditional mean function (e.g. an AR(p) model) and in the second step based on the residuals from the first step the parameters of the conditional variance function (e.g. GARCH(1,1) parameters). Nevertheless, both the conditional mean and the conditional variance have to be specified correctly for obtaining correct standard errors for the parameters of the conditional mean estimates. Otherwise, the information matrix equality is violated as well. Models with a block diagonal information matrix: ARCH, GARCH, TGARCH with symmetric errors (see next section) – Models without a block diagonal information matrix: EGARCH, GARCH-in-mean and TGARCH models with skewed (=asymmetric) errors (see next section).

5.4 Asymmetry and leverage effects In many cases it has been observed that negative shocks have a larger impact on the volatility than a positive shock of the same magnitude. Models that allow for such asymmetric effects are e.g.:

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• Threshold GARCH (TGARCH model) 2 σt2 = γ0 + γ1u2t−1 + γ1−I(ut−1 < 0)u2t−1 + β1σt−1

where I(A) is the indicator function, taking the value 1 if the argument A is true and the value 0 otherwise. Properties: – if the parameter γ1− has a positive sign, then negative shocks increase volatility more than positive shocks. One also calls this phenomenon leverage effect. • Exponential GARCH (EGARCH model) 2 log σt2 = γ˜0 + γ˜1 (|ξt−1| − E [|ξt−1|]) + γ˜1−ξt−1 + β˜1 log σt−1

Properties: – If ξt ∼ N (0, 1), then E [|ξt|] =

p

2/π

– γ˜1 > 0 causes volatility clustering γ˜1− < 0 causes a leverage effect – By construction one always has a positive variance since log σt2 implies σt2 > 0. – In practice it was found that outliers get too much impact on the estimation.

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• ARCH-in-mean model It may happen that the conditional risk also influences the return itself, e.g. if there is a nonlinear relationship between risk and return. A potentially useful model is an AR model that also has an ARCH effect in the conditional mean function. yt = α1yt−1 + · · · + αpyt−p + ρσt−1 + ut,

ut = σtξt σt2 = γ0 + γ1u2t−1 + · · · + γmu2t−m

5.5 Testing for the presence of conditional heteroskedasticity • Before one enters into the stage of modeling conditional heteroskedasticity, one should test for the presence of (conditional) heteroskedasticity. A good test is the ARCH-LM test. • ARCH-LM test: One considers the following autoregression model for the squared errors u2t = β0 + β1u2t−1 + · · · + βq u2t−q + errorst In case of homoskedastic errors all βj = 0, j = 1, 2, . . . , q. Thus, one has the pair of hypotheses H0 : β1 = β2 = · · · = βq = 0 versus H1 : at least one βj 6= 0, j = 1, 2, . . . , q 142

Applied Financial Econometrics — 5.5 Testing for the presence of conditional heteroskedasticity — U Regensburg — July 143 2012

Since the errors are not directly observable, they are replaced the residuals uˆt. One then estimates uˆ2t = β0 + β1uˆ2t−1 + · · · + βq uˆ2t−q + errorst and computes the resulting R2. Then the test statistic and its asymptotic distribution are given by d ARCHLM (q) = T R2 −→ χ2(q). Choice of q: In long time series one easily should choose q large enough. However, in short time series testing jointly too many parameters may lead to a loss of power since the estimated variances of the parameter estimates are quite large. Very often, choosing q small is already sufficient to reject the null of homoskedasticity. • Inspection of autocorrelation of squared residuals If there are many estimated autocorrelations of the squared residuals outside the 95% confidence interval, one definitely should conduct the ARCH-LM test of adequate order. • Important: If the null hypothesis is rejected, the asymptotic distribution of the OLS estimator for the conditional mean function is invalid! Solutions: – Use heteroskedasticity consistent standard errors, e.g. the White procedure – Specify model for conditional heteroskedasticity, e.g. ARCH, GARCH etc. 143

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5.6 Model selection In general the type and order of the conditional mean and the conditional volatility function are unknown and have to be estimated as well. The typical procedure is: 1. test for the presence of stochastic or deterministic trends or other kinds of nonstationarity (e.g. using ADF tests) 2. specify the conditional mean function (e.g. select order and terms in AR model) 3. check residuals of conditional mean model for remaining autocorrelation and potential (conditional) heteroskedasticity (e.g. using the LM test for autocorrelation and the ARCH-LM test). Again: ignoring (conditional) heteroskedasticity implies false inference! 4. specify a conditional heteroskedasticity model (e.g. ARCH or TGARCH) and choose a ML or the QML estimator 5. check standardized residuals

uˆt ξˆt = σˆ t of full model for remaining autocorrelation and check choice of error distribution in the previous step (problem: asymptotic distribution of ARCH-LM test not known except in the case of H0: “no remaining ARCH); visual inspection of density estimate (e.g. histogram) of standardized residuals

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5.7 Prediction of conditional volatility For simplicity assume a simple GARCH(1,1) process 2 , σt2 = V ar(ut|ut−1, ut−2, . . .) = γ0 + γ1u2t−1 + β1σt−1

ξt ∼ i.i.d.(0, 1)

Derivation of the h-step ahead forecast of σT2 +h given information up to time T : • Rewriting the GARCH(1,1) for various periods T + j, j = 1, 2, 3 one obtains σT2 +1 = γ0 + γ1u2T + β1σT2 σT2 +2 = γ0 + γ1u2T +1 + β1σT2 +1 σT2 +3 = γ0 + γ1u2T +2 + β1σT2 +2 • It is well known that the conditional expectation is the best predictor for minimizing the mean squared error of prediction (here for σT2 +h). Therefore, the optimal predictor given information up to time T is σT2 +h|T ≡ E[σT2 +h|IT ] where IT denotes the available information up to time T . • Note that by the law of iterated expectations one has E[σT2 +h|IT ] = E[E[σT2 +h|IT +h−1]|IT ] 145

Applied Financial Econometrics — 5.7 Prediction of conditional volatility — U Regensburg — July 2012

Applying to h = 2 one obtains σT2 +2|T = E[E[σT2 +2|IT +1]|IT ] = E[γ0 + γ1u2T +1 + β1σT2 +1|IT ] = γ0 + γ1E[u2T +1|IT ] + β1E[σT2 +1|IT ] Since u2t = σt2ξt2 and therefore E[u2T +1|IT ] = E[σT2 +1|IT ] E[ξT2 +1|IT ] | {z } | {z } =σT2 +1 =1 since ξt iid one obtains σT2 +2|T = γ0 + γ1σT2 +1 + β1σT2 +1 = γ0 + (γ1 + β1)σT2 +1 • Similarly one can compute the optimal predictor for h = 3: σT2 +3|T = E[E[σT2 +3|IT +2]|IT ] = E[γ0 + γ1u2T +2 + β1σT2 +2|IT ] = γ0 + γ1E[u2T +2|IT ] + β1E[σT2 +2|IT ] = γ0 + (γ1 + β1)σT2 +2|T

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Inserting σT2 +2|T delivers σT2 +3|T

2 = γ0 + (γ1 + β1) γ0 + (γ1 + β1)σT +1 = γ0 [1 + (γ1 + β1)] + (γ1 + β1)2σT2 +1

• One can continue in this way in order to derive the h-step ahead predictor, h ≥ 2, σT2 +h|T

h−1 X = γ0 (γ1 + β1)i−1 + (γ1 + β1)h−1σT2 +1 i=1

• In practice the optimal predictor is usually not feasible since the parameters are unknown. They have then to be replaced by consistent estimators. • In R it is possible to compute these forecasts by applying method predict() to an object of class garch. An object of class garch is returned by function garch(), which estimates a GARCH(m,n) model by maximum likelihood. This function is part of package tseries.

5.8 R Code ################# # ARCH simulation

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Applied Financial Econometrics — 5.8 R Code — U Regensburg — July 2012 T <- 1000 gamma.0 <- 2 gamma.1 <- 0.5 # unconditional variance (under the assumption of stationarity) sigma.sq <- gamma.0 / (1 - gamma.1) sigma.t.sq <- numeric(T) xi <- rnorm(T) u <- numeric(T) # starting value of process u (variance set to unconditional variance) u.0 <- sqrt( sigma.sq ) * rnorm(1) # computing conditional variance and realization of u for period 1 sigma.t.sq[1] <- gamma.0 + gamma.1 * u.0^2 u[1] <- sqrt( sigma.t.sq[1] ) * xi[1] # computing conditional variance and u for all following periods for(t in 2:T){ sigma.t.sq[t] <- gamma.0 + gamma.1 * u[t-1]^2 u[t] <- sqrt( sigma.t.sq[t] ) * xi[t] } # plot u and its conditional variance par( mfrow =c(2,1)) ts.plot(u) ts.plot(sigma.t.sq) ##################################################################################### # Fitting the mean and variance equation for the log return on a stock price series # price denotes vector of stock prices

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Applied Financial Econometrics — 5.8 R Code — U Regensburg — July 2012 # Definition of return (in %) r <- 100* diff( log(price) ) # Look at sample mean, variance and histogram of returns mean(r) var(r) hist(r, breaks = 100) # serial correlation acf(r) pacf(r) # formal test - Ljung Box version of Portmanteau test for(h in c(5,10,15,20) ){ ACtest <- Box.test(r, lag = h, type = "Ljung-Box", fitdf = 0) print(ACtest) } # result: all tests reject the null of no serial correlation in returns # A suitable model is the ARMA(1,3) specification (chosen by AIC) # using arima() allows to estimate ARMA models via ML estimation arma13 <- arima(r, order = c(1,0,3), include.mean = TRUE) # Looking at the AIC we see that an ARMA(1,3) model is suggested # Checking for remaining autocorrelation resid13 <- arma13$residuals for(h in c(5,10,15,20) ){ ACtest <- Box.test(resid13, lag = h, type = "Ljung-Box", fitdf = 4) print(ACtest) } # at least for 5 and 10 lags there is no significant autocorrelation left # for now, this is going to be our model for the mean

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# serial correlation in squared residuals acf(resid13^2) pacf(resid13^2) # testing for conditional heteroskedasticity # using Portmanteau test for(h in c(5,10,15,20) ){ CHtest <- Box.test(resid13^2, lag = h, type = "Ljung-Box", fitdf = 0) print(CHtest) } # using ARCH-LM test for(h in c(5,10,15,20) ){ res.2 <- resid13^2 ARCHLM <- dynlm(res.2 ~ 1+ L(res.2, 1:h)) teststat <- summary(ARCHLM)$r.squared * length(ARCHLM$residuals) pvalue <- 1 - pchisq(teststat, df = h) print(pvalue) } # indicates conditional heteroskedasticity # Estimating GARCH(1,1) model for variance and ARMA(1,3) for mean with assumption of normally distributed innovations library(rugarch) # this package contains a lot of useful functions for fitting GARCH-type models # Try a parsimonious specification for GARCH model, here GARCH(1,1) # Look at the help page ?ugarchspec spec1 <- ugarchspec(variance.model = list(garchOrder = c(1,1)) ,mean.model = list(armaOrder = c(1,3), include.mean = TRUE)) # also have a look at ?ugarchfit fit1 <- ugarchfit( spec1, r) # Analyse the output, which contains tests for autocorrelation and conditional heteroskedasticity (and more) of model residuals # Forecasting ugarchforecast(fit1)

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6 Long-run forecasting

6.1 Estimating/predicting unconditional means • Time series and (dynamic) regression models are normally used to compute conditional predictions. • For long-term investment decisions reliable forecasts of the unconditional mean of stock returns, interest rates, etc. are important as well: for large h, the h-step ahead forecast for a stationary time series is the unconditional mean of the series. • Consider a stationary process {yt} (of returns) that has mean µ and variance σy2, yt ∼ (µ, σy2). Note that we did not assume that yt is i.i.d., thus the covariances Cov(yt, ys) may be non-zero.

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The traditional mean estimator of a realization with T observations is T 1X µˆ = yt T t=1

Properties: – Expected Value T T 1X 1X E(yt) = µ=µ E(ˆ µ) = T t=1 T t=1

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– Variance !2

T X 1 µ) = E yt − µ V ar(ˆ T t=1 !2 !2 T T X X 1 1 1 (yt − µ) =E yt − T µ = 2 E T t=1 T T t=1 " T T # X X 1 = 2E (yt − µ)(ys − µ) T t=1 s=1 T T T 1 X X 1 X = 2 V ar(yt) + 2 Cov(yt, ys) T t=1 T t=1 s=1,s6=t

T T 1 X X 1 2 Cov(yt, ys) = σy + 2 T T t=1 s=1,s6=t

This expression simplifies if {yt} is uncorrelated. Then σy2 V ar(ˆ µ) = . T – Asymptotic normality √

d

µ − µ) −→ N T (ˆ 153

0, σy2

+ γ∞

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with γ∞

T T 1 X X Cov(yt, ys). = lim 2 T →∞ T t=1 s=1,s6=t

Note that γ∞ = 0 if yt is uncorrelated. The term γ∞ converges if yt is a stationary autoregressive process. • Estimating annual returns – Given a time series of annual observations one has in case of uncorrelated returns the estimaσy2 tion variance T .

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– Does it help to collect a time series of daily observations of daily returns? Let s denote the day of the year and t the year with 250 working days per year. Then 250 Y

Rt =

Rts

s=1

and in log returns rt ≈

250 X

rts.

s=1

For simplicity, assume i.i.d. daily returns with µday = µyear /250 and σday

√ = σyear / 250.

Then E[rt] = E

" 250 X

# rts = 250µday = µyear

s=1

V ar(rt) = V ar

250 X s=1

=

! rts

2 250 X σyear iid 2 = V ar(rts) = 250σday = 250 √ 250 s=1

2 σyear

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∗ Thus, one has to multiply the daily mean by 250 in order to get the annual mean! ∗ And the estimation variance for the annual mean stays the same since ! T X 250 X 1 µday ) = V ar 250 rts V ar(250ˆ 250T t=1 s=1 ! T X 250 X 1 = V ar rts T t=1 s=1 T 250 1 XX = V ar(rts) T t=1 s=1

iid

2 2 = 250σday = σyear .

Therefore, sampling at higher frequencies does not help to reduce the estimation variance! ∗ This conclusion remains qualitatively unchanged if returns are correlated. • Estimating long-horizon annualized returns – If one is interested in estimating the 20-year return, then the same argument holds as above: sampling at a yearly frequency does not help to reduce the 20-year return variance. – However, if one is interested in the annualized h-year return? Let Rh−year denote the

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h-year return and rh−year the corresponding log return. The annualized return is then p 1 Rh−year,annualized = h Rh−year = (Rh−year ) h and the annualized log return 1 rh−year,annualized = rh−year . h Therefore, µh−year,annualized = µh−year /h and 2 2 σh−year,annualized = σh−year /h2.

Note that one has here the same scaling effect as for yearly and daily returns. Thus, the estimation variance for estimating h-year returns cannot be reduced by this trick. – To make it completely clear: If one is interested in the 100-year overall return and has a sample 2 of 100 annual returns, then the estimation variance is just 100σyear (while for estimating the 2 /100). 1-year return it is σyear • Don’t look only at the estimated mean alone! It always pays to compute a confidence interval or even estimate the probability distribution of the returns. The latter can be done by looking at the – histogram or 157

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– quantiles of the empirical distribution. • Overlapping observations If one observes only T /h observations for the h-year mean return but one has yearly observations, then one may compute overlapping h-year returns: r1 + · · · + rh, r2 + · · · + rh+1, . . ., rT −h+1 + · · · + rT . Note that computing a histogram/quantiles with overlapping observations does not deliver much more information since each observation is used h times! Nevertheless doing this delivers nicer pictures. And is done in practice. In the book of Dimson, E., Marsh, P. and Staunton, M. (2002), Triumph of the Optimists, Princeton University Press one finds a large collection of results on the performance of various assets in OECD countries over the last 100 years. The following graph from this book shows the quantiles for the estimated 20-year return minus the risk free rate using overlapping observations:

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Applied Financial Econometrics — 6.2 Predicting long-term wealth: the role of arithmetic and geometric means — U 160 Regensburg — July 2012

6.2 Predicting long-term wealth: the role of arithmetic and geometric means Timing schedule for discrete payments / returns on investments Time

period 0 begin end begin

Payments Wealth

W0

period 1

period 2 end begin

period 3 end begin

W0(R1 − 1)

W0(R2 − 1)

W | 0{z· R}1

W {z1 · R}2 | 0·R

W1

W2

¯ h over h periods if received payments are reinvested: • Average of a given flow of discrete returns R Wh = W0 · R1 · · · Rh ¯ ···R ¯ = W0 · R ¯ h. = W0 · (R) =⇒ Average of a given flow of discrete return is given by the geometric mean of returns 1 ¯ = [R1 · · · Rh] h R

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• Uncertain flow of discrete returns: Consider the expected value of the uncertain final wealth Wh " # h Y E [Wh] = E W0 (Rt)

(6.1)

t=1

– If discrete returns are independently distributed, then the expectation of the returns is the product of the expectations and one obtains indep.

µW ≡ E [Wh] = W0

h Y

E [Rt]

(6.2)

t=1

= W0

h Y

E[Rt].

(6.3)

t=1

If the means of the Rt’s are known, then the expected final wealth is easily computed. – If discrete returns are independently and identically distributed, then this simplifies with µR = E[Rt] to µW = W0(µR )h.

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• Additional estimation risk: If the means are unknown, one faces in addition estimation risk. Now one has to be careful how to estimate E[Wh]! For simplicity, we continue to assume i.i.d. returns Rt = µR + ε t ,

εt ∼ i.i.d.(0, σR2 ).

It is well known from statistics that one then can estimate an expected value by the sample mean, thus estimate T 1X µˆ R = Rt. T t=1 This estimator for µR is unbiased since T 1X E[µR + εt] = µR = E[Rt]. E[ˆ µR ] = T t=1

In order to estimate µW µ W = W0

h Y j=1

one has to replace µR by some estimate:

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µR = W0(µR )h

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– Arithmetic mean: Replacing in µW the unknown µR by its arithmetic mean estimate delivers h µˆ W

T 1 X = W0 Rt T . t=1 | {z } =ˆ µR

However, µˆ W is biased since (for simplicity consider h = 2) µW ] = W0E [(ˆ µR )(ˆ µR )] E [ˆ = W0E µˆ 2R

h i 2 = W0 E (µR + ε¯) | {z } =E [µ2R +2µR ε¯+¯ ε2 ] 1 = W0 µ2R + σ 2 T > W0 µ2R = µW . Thus, using the arithmetic mean estimator results in overestimating the expected final wealth. Exception: For very large T , the arithmetic mean is fine because the nonlinearity does no 163

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longer matter. Therefore, the arithmetic mean estimator is consistent. – Geometric mean: Replacing in µW the unknown µR by its geometric mean !1/T T Y µ˜ R = R1 · · · RT t=1

delivers T Y

µ˜ W = W0

!h/T R1 · · · RT

.

t=1

Thus, the expected value is µ˜ W = W0E

T Y

!h/T R1 · · · RT

t=1

h i h i iid h/T h/T = W0 E R 1 · · · E RT . If h = T and E[Rt] = µR , one obtains µ˜ W = W0µhR . Only in this special case, h = T , the geometric mean estimator delivers an unbiased wealth estimate because there is no averaging effect.

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Since both mean estimators are unbiased only in extreme situations, recently a combination of both was suggested to obtain an unbiased predictor (Eric Jacquier, Alex Kane, and Alan J. Marcus (2003). ”Geometric or Arithmetic Mean: A Reconsideration”, Financial Analysts Journal, 59, 46-53): µ˘ W = W0 (Arithmetic average × (1 − h/T ) + Geometric average × h/T )h (see Bodie, Kane & Marcus (2005), Investments, McGraw-Hill, 8th edition, p. 866). – Note that the correction is horizon dependent! – The correction can be large if the difference between the arithmetic and the geometric mean is large! – Note that while the above estimator delivers an unbiased estimator of the annualized rate of return over h periods, this estimator does not exhibit the smallest mean squared error of prediction. For obtaining this, yet another estimator for µR has to be used, see Bodie, Kane & Marcus (2002). Optimal Forecasts of Long-Term Returns and Asset Allocation: Geometric, Arithmetic, or Other Means?, Working Paper or a paper of the same authors in Financial Analysts Journal, 2003. – This case is a nice example that one has to consider the goal function explicitly in case of nonlinear functions.

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6.2.1 Log or continuously compounding returns • Using the definition of continuously compounding returns rt ≡ log(Rt) ≈ Rt − 1 and therefore ert = Rt one can rewrite the wealth equation Wh = W0 · R1 · · · Rh as Wh = W0er1 · · · erh = W0er1+···+rh Ph

j=1 rj

= W0 e

= W0eh¯rh where

h

1X r¯h = rj h j=1 defines the arithmetic mean of the continuously compounding returns.

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• Summary: h

Ph Wh Y = (Rj ) = e j=1 rj W0 j=1

and h1 h Y 1 Ph r ¯ = e h j=1 j ≡ er¯h . Rh ≡ Rj j=1

The geometric mean of discrete returns translates into the arithmetic mean of log returns. • In case (log) returns are uncertain, one obtains – h

Ph

E [Wh] = W0E e

j=1 rj

i

– If stock returns are independent, then this equation simplifies to E [Wh] = W0

h Y

E [erj ]

j=1

– If stock returns are i.i.d., this equation further simplifies to E [Wh] = W0 (E [erj ])h

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(6.4)

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– To check the equality of both expectations indicates whether the returns are i.i.d.! This property can be used for testing the i.i.d. hypothesis. – In order to compute the expected values one has to know the probability distribution for Rj . If it is unknown, in many cases a reasonable assumption is that Rj is i.i.d.lognormally distributed. This means assuming that rt = µ + εt,

εt ∼ i.i.d.N (0, σ 2)

and applying statistical rules to Rt = ert so that 1

1 2

E[Rt] = E [ert ] = eE[rt]+ 2 V ar(rt) = eµ+ 2 σ . Thus, if the discrete returns are i.i.d.lognormally distributed, then the expected value of final wealth is h Y 1 2 µ+ 12 σ 2 E[Wh] = W0 e = W0eh(µ+ 2 σ ) 6= W0ehE[rt]. j=1

The inequality sign should not surprise since the expected value cannot be switched with nonlinear functions! – If the mean µ and the variance σ 2 of the log returns rt have to be estimated, one faces the same problems as for estimating the means of the Rj ’s. One cannot simply insert estimators 168

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169

for µ in the exponent. This leads to a biased estimate. The reason for these difficulties is the same as above, the nonlinearity of the exponential function.

6.3 Are long-term returns predictable? • This is an ongoing debate in current research. Some recent literature is: – Campbell, J.Y., A.W. Lo, and A.C. MacKinlay (1997). The Econometrics of Financial Markets, Princeton University Press – Campbell, J.Y. and Shiller, R.J. (2004). Valuation ratios and the long-run stock market outlook: an update, in: Barberis, N. und Thaler, R. (eds.). Advances in Behavioral Finance, Volume II, Russel-Sage Foundation. Working paper can be downloaded. – Cochrane, J.H. (2006). The dog that did not bark: a defense of return predictability, University of Chicago. – Goyal, A. and Welch, I. (2003), Predicting the Equity Premium with Dividend Ratios, Management Science, 49, 639-654. – Goyal, A. and Welch, I. (2006), A Comprehensive Look at the Empirical Performance of Equity Premium Prediction (January

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11, 2006). Yale ICF Working Paper No. 04-11 Available at SSRN: – Pesaran, M.H. und Timmermann, A.G. (2000). A recursive modelling approach to predicting UK stock returns, Economic Journal, 110, 159-191.

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7 Explaining returns and estimating factor models

7.1 The basics of the theory of finance • Literature: – Cochrane, J. (2005). Asset Pricing, Princeton University Press (book is used in graduate courses and succeeds in providing a unifying approach to finance using stochastic discount factors) – Bodie, Kane, Marcus (2005). Investments, MacGrawHill (an undergraduate book, technically much less demanding with a very broad focus and very up-to-date with empirical and theoretical develepments) • The basic problem of an investor – The setup is made as simple as possible. There are many, more complicated extensions that do

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not alter the basic message here. The presentation follows largely Cochrane (2001), Section 1.1-1.4. – Assumptions: ∗ The investor is exclusively interested in consumption. Additional consumption is viewed less important if the consumption level is high −→ utility function is concave du(ct) declines in ct. dct ∗ Future consumption is uncertain and future expected consumption is discounted using the discount factor β. ∗ There is one investor who lives 2 periods having utility U (ct, ct+1) = u(ct) + βE[u(ct+1)]. ∗ In each period the investor receives a fixed endowment et and et+1. ∗ The investor can invest some of his endowment in period t into an asset with price pt and risky payoff xt+1 in period t + 1 by buying ξ items of the asset. ct = et − ptξ ct+1 = et+1 + xt+1ξ.

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173

Thus, his control variable is ξ which allows her to maximize her utility max U (ct, ct+1) = max u(ct) + βE[u(ct+1)|It]. ξ

ξ

∗ There are no buying or selling restrictions. ∗ There are no trading costs, taxes, etc.... – Solving the model: Insert the budget constraints into the utility equation U (ct, ct+1) = u(et − ptξ) + βE[u(et+1 + xt+1ξ)|It] and maximize with respect to ξ. The first-order condition (FOC) is ∂U (ct, ct+1) ! = −u0(ct)pt + βE [u0(ct+1)xt+1|It] = 0. ∂ξ Note that taking the derivative of the expectation works because Z Z ∂E[g(x; α)] ∂g(x; α) ∂ ∂ = g(x; α)f (x)dx = g(x; α)f (x)dx = E ∂α ∂α ∂α ∂α which can be applied here as well by defining g(xt+1; ξ) ≡ u(c(xt+1, ξ)),

173

c(xt+1, ξ) = et+1 + xt+1ξ.

(7.1)

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174

Rearranging the FOC and using the notation Et[·] ≡ E[·|It] delivers the central asset pricing formula: • Central asset pricing formula pt = Et

0

βu (ct+1) xt+1 u0(ct)

(7.2)

– Note that ct and ct+1 depend on ξ. Thus finding the optimal ξ requires the nonlinear solution of the asset pricing equation. – The theory of asset pricing can be viewed as providing specializations and manipulations of the asset pricing equation. – Interpretation: The investor is willing to buy an extra unit of the asset at price pt if the marginal loss in utility due to less consumption is lower than the expected discounted marginal gain in future utility. The marginal gain in future utility is just the payoff times the marginal utility in consumption. – The term in front of the future payoff has a special function and is called stochastic discount factor. It will be discussed next.

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• Stochastic discount factor: – Note that the random variable

βu0(ct+1) mt+1 ≡ u0(ct) plays the function of a discount factor of the future payoff xt+1. Since it is random, it is called stochastic discount factor.

– Since in general β < 1 one has in general also mt+1 < 1 except if ct+1 is much smaller than ct due to the decreasing marginal utility of consumption. In other words, given current consumption ct, the stochastic discount factor is the smaller, the larger future consumption ct+1 is, implying larger discounting! – Thus, discounting depends on the uncertain level of future consumption. Therefore, the discount factor must be stochastic. – Small example: Imagine that there are 3 different states of the world in period 2 that imply 3 different payoffs. The larger the payoff, the larger the consumption in period 2, discounting is largest in the state with the largest payoff. Since in period 1 it is unknown which state will realize in period 2, the investor takes expectations. Note that by using the stochastic discount factor, the investor already takes care about her future utility of utility/consumption. – Thus, the stochastic discount factor does not only take into account the risk with respect to

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future payoff but directly the risk with respect to future consumption. – The stochastic discount factor is independent of a specific asset. Its value for a current state only depends on the consumption level of that state. The function of the stochastic discount factor is the same for all assets! – However, the correlation between the values of the stochastic discount factor and the payoffs can be different among assets. It therefore provides an asset-specific risk correction w.r.t. utility maximization, the ultimate goal of investment. – The specific function of the stochastic discount factor changes if the utility function is changed. – Note that future payoff is linearly correlated with future marginal utility of consumption. In contrast, the dependence of future payoff with future consumption is in general highly nonlinear and invokes all moments!

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– Special cases: ∗ No uncertainty: There is only one future state and 1 xt+1. Rf Thus, in a risk-free world the stochastic discount factor corresponds to the inverse of the risk-free rate Rf . pt = mt+1xt+1 =

∗ Uncertainty w.r.t. pay-off: relationship between future consumption and payoff is ignored 1 pt = i Et[xt+1]. R Here, the risk-adjustment is done by using the asset-specific discount factor 1/Ri for asset i. ∗ Uncertainty w.r.t. consumption: relationship between future consumption and payoff taken into account → asset pricing equation pt = Et[mt+1xt+1] – Other names for the stochastic discount factor: marginal rate of substitution, pricing kernel, change of measure, state-price density.

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• Relating payoff risk and utility/consumption risk – Since Cov(X, Y ) = E[XY ] − E[X]E[Y ] the asset pricing equation can be rewritten as pt = Et[mt+1]Et[xt+1] + Covt(mt+1, xt+1). – Assume for the moment that future consumption is stochastically independent with future payoff. Then Covt(mt+1, xt+1) = 0. Therefore, future payoff cannot help to control future consumption but it still may improve future consumption on average. In this case one can obtain the risk-free rate 1 Rf = E[mt+1] as the inverse of expected stochastic discount factor. The price of such assets is pt =

Et[xt+1] . Rf

– Interpretation of the risk adjustment term Covt(mt+1, xt+1): The asset price increases if an increase in payoff is associated with an increase in the stochastic discount factor ∼ increase in future marginal utility ∼ decrease future consumption and vice versa. Thus, a positive covariance helps to reduce the variation between current and future consumption (or in the level of marginal utility). A positive covariance between the stochastic discount factor and payoff smoothes the consumption stream. In

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contrast, if the covariance is negative, the volatility of the consumption stream is increased and the investor wants to be payed for this by a higher return. • Systematic and idiosyncratic risk If Cov(mt+1, xt+1) 6= 0, then there is correlation between the fluctuations in utility and in payoff. This joint risk is called systematic risk while the risk in payoff that is without impact on future consumption/utility is called idiosyncratic. From the asset pricing formula it follows that only the systematic risk influences the asset price. The idiosyncratic risk does not contribute to smoothing/aggrevating the consumption stream and is therefore not priced. • Important special cases of the asset pricing equation: – The asset pricing formula (7.2) can be applied to a wide range of assets, e.g. to returns. This delivers the mean-variance frontier. – Be aware that the formulas resulting from the asset pricing formula (7.2) hold for an investor that optimized her portfolio in period 1 by buying the optimal number ξ of the asset. – A ’return’ of an asset i costs in period t one unit and pays in period t + 1 Ri units. Thus, dropping time indices, 1 = E[mRi] E[Ri] i + Cov(m, R ). 1= Rf 179

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Rewriting delivers an explanation for expected excess returns = −Rf Cov(m, Ri) . E[Ri] − Rf | {z } | {z } expected excess return risk adjustment

(7.3)

Note that the stochastic discount factor m depends on ct, ct+1 and thus on ξ. Interestingly, m is the quantity that is influenced by the investor by choosing ξ. Thus, by choosing ξ one influences the expected value or the covariance in the equation above. • Expected return-beta representation and the mean-variance frontier – Rewriting equation (7.3) for expected excess returns delivers the expected return-beta representation or beta-pricing model Cov(m, Ri) V ar(m) (−1) E[R ] − R = E[m] V ar(m) | {z } | {z } i

f

λm

E[Ri] − Rf = λmβi,m.

βi,m

(7.4)

In the representation (7.4) the coefficient λm represents the price of risk that is independent of the specific return considered and depends exclusively on the type and volatility of the stochastic discount factor. The beta-parameter βi,m denotes the quantity of systematic risk associated with return i. – Rewriting equation (7.3) for expected excess returns in a slightly different way, using Cov(X, Y ) = 180

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Corr(X, Y )σ(X)σ(Y ) with σ(X) =

p

181

V ar(X), delivers

Corr(m, Ri)σ(m)σ(Ri) E[R ] − R = − E[m] E[Ri] − Rf i σ(m) = −Corr(m, R ) . σ(Ri) E[m] i

f

The expression E[Ri] − Rf σ(Ri) is called Sharpe ratio. Since |Corr(m, Ri)| ≤ 1, one has E[Ri] − Rf σ(m) σ(Ri) ≤ E[m] . The Sharpe ratio that is the slope of the mean-volatility line of any return i cannot be larger in absolute value than the coefficient of variation σ(m) E[m] of the stochastic discount factor m. The slope of the coefficient of variation is said to provide the mean-variance frontier for all possible returns. The mean-variance frontier is also called capital market line. Note that in the present context the capital market line depends on the decision of the investor through ξ. Properties of the mean-variance frontier:

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∗ all returns on the frontier are perfectly correlated with the stochastic discount factor since |Corr(m, Ri)| = 1. ∗ all returns on the upper frontier are maximally risky since consumption is highest when payoffs are highest and thus there is the least possible smoothing of consumption. Therefore the investor wants to be paid for taking this risk and the expected excess return is highest. ∗ all returns on the lower frontier are minimally risky since consumption is lowest when payoffs are highest and thus there is the most possible smoothing of consumption. Thus, the expected excess return is the lowest. ∗ any pair of two frontier returns are perfectly correlated and therefore spans/synthesizes any frontier return. ∗ any mean-variance efficient return, abbreviated by Rmv carries all pricing information since Corr(m, Rmv ) = −1 and Rmv = a + bm. Then Cov(m, Rmv ) = −σ(m)σ(Rmv ) = −bσ(m)σ(m). For the expected excess return of a mean-variance efficient return the beta pricing model delivers bσ 2(m) mv f = bλm. E[R ] − R = λm 2 σ (m)

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Inserting E[Rmv ] − Rf λm = b into the expected return-beta representation (7.4) for return i finally produces E[Rmv ] − Rf E[R ] − R = βi,m b i f or E[R ] − R = βi,mv E[Rmv ] − Rf i

since βi,mv

f

Cov(Rmv , Ri) b Cov(m, Ri) Cov(Ri, m) βi,m = 2 = ≡ . ≡ V ar(Rmv ) b V ar(m) bV ar(m) b

• Example: power utility function – Power utility: 1 1−γ c 1−γ u0(c) = c−γ . u(c) =

– Stochastic discount factor:

mt+1 = β

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ct+1 ct

−γ

(7.5)

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– Case I: no uncertainty f Rt+1

1 1 = = mt+1 β

ct+1 ct

γ

Real interest rates are high if ∗ people are impatient (=β low) ∗ consumption growth is high ∗ the desire for smooth consumption is large (γ large) – Case II: uncertainty Assumption: consumption growth is log-normally distributed ct+1 ln = µt + ut, ut ∼ N (0, σt2). ct The risk-free rate is f Rt+1 =

184

1 . E[mt+1]

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Using the assumption of log-normal consumption growth one has, e−δ ≡ β, −γ −γ ct+1 = e−δ eln ct+1−ln ct = e−δ e−γ(∆ ln ct+1) β ct " −γ # h i 2 ct+1 −δ −γ(∆ ln ct+1 ) −δ −γµt + γ2 σt2 Et β =e e = Et e e ct 2 2 −1 γ f Rt+1 = e−δ e−γµt+ 2 σt . Real interest rates are high if ∗ impatience δ is high ∗ mean consumption growth µt is high ∗ more smoothing γ needed and ∗ consumption variance σt2 is small. Note that the parameter γ in the power utility function controls ∗ the degree of intertemporal substitution (=aversion to a consumption stream that varies over time)

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∗ risk aversion (=aversion to a consumption stream that varies over states of nature) ∗ the degree of precautionary savings (σt2γ 2/2 term). These links can be broken up by using more flexible utility functions. – Using the power utility function one can also express the beta pricing model directly in terms of changing consumption. Applying Taylor expansions to λm and βi,m in (7.4), one obtains E[Ri] − Rf ≈ βi,∆c γV ar(∆ct+1) . | {z } λ∆c

Thus, the price of risk increases with ∗ an increase in the variance of consumption growth ∗ a larger risk aversion/degree of intertemporal substitution. • The equity premium puzzle – If one computes the mean-variance frontier for the power utility function, then the Sharpe ratio of a mean-variance efficient return can be approximated by E[Rmv ] − Rf σ(Rmv ) ≈ γσ(∆ ln ct+1) – Over the last 100 years the real stock return of the stock market index in the US was about 9% with standard deviation 16% while the real return of treasury bills was 1%. This delivers 186

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a Sharpe ratio of about 0.5. For the same period consumption growth exhibited mean 1% and standard deviation 1%. Thus the risk aversion parameter γ has to be 50! This number is way too large given empirical evidence on risk aversion from economic experiments. Thus the expected excess return of the market portfolio was too large given the risk-aversion of people. This is the equity premium puzzle. • Note that equation (7.5) is almost the capital asset pricing model. The capital asset pricing model (CAPM) states that the mean-variance efficient return is given by return on the market or total wealth portfolio i f W f E[R ] − R = βi,RW E[R ] − R . (7.6) Note that going from (7.5) to (7.6) requires additional assumptions. One of the following set of assumptions is enough, see e.g. Cochrane (2001), section 9.1, – quadratic utility function u(c) = (c − c∗)2, c∗ fixed, and no labor income or – exponential utility u(c) = −e−ac, a fixed, and normally distributed returns • Random Walks and Efficient Markets – A wide spread hypothesis: Financial markets are efficient if prices follow a random walk pt = Et[pt+1] pt+1 = pt + ut,

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u ∼ (0, σ 2).

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This says: prices tomorrow are unpredictable given prices and other information until today. – A generalization of a random walk is a martingale yt+1 = yt + ut,

ut ∼ (0, σt2)

where the variance of the error term may be (conditionally) heteroskedastic. – Consider the asset pricing formula: pt = Et[mt+1xt+1]. This relationship only reduces to a random walk if mt+1 = 1 and xt+1 = pt+1. This implies that ∗ one considers a short time horizon such that β ≈ 1, ∗ investors are risk neutral ⇔ u(·) linear or σc2 = 0 and ∗ there is no dividend payment in t + 1. – How to view the asset pricing formula as a random walk? Prices follow a random walk after scaling and adjusting by discounted marginal utility. – Since changes in marginal utility may not matter in the very short run, the random walk hypothesis is likely to hold in the very short run. But in the long run?

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Reconsider the beta-pricing model (7.4) with power utility that was used to explain the equity premium puzzle Et[Rmv ] − Rf ≈ γtσt(∆ ln ct+1)σt(Rmv ). A time index was added to each quantity since the asset pricing formula was derived conditional on information up to time t. The expected return can vary if ∗ the risk aversion γt changes, ∗ the conditional variance of the mean-variance return changes, ∗ the conditional variance of consumption growth changes Note that all quantities may vary in the long-run but not in the short-run. Thus, prices should only be predictable in the long-run if at all! • Infinite horizon models – Imagine an investor with infinite horizon that can purchase a stream of dividends {dt+j } at price pt. The asset pricing formula continues to hold pt = Et[mt+1(pt+1 + dt+1)]

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with xt+1 = pt+1 + dt+1 being the payoff in period t + 1. It may also be written as ∞ 0 X u (ct+j ) j pt = Et mt,t+j dt+j , mt,t+j = β 0 u (ct) j=0 by iterating the asset pricing formula forward (=iteratively inserting) and using the assumption (=transversatility condition) limj→∞ Et[mt+j pt+j ] = 0. Thus, asset prices are expected future dividends stochastically discounted. • Important remarks: The asset pricing formula 1. is no equilibrium condition, 2. does not require complete markets or a representative investor, 3. does not require specific assumptions on future prices/returns, 4. does not require a specific utility function, 5. does not exclude other income sources, 6. allows for stochastic discount factors that vary across individuals, 7. makes a statement about optimal investment for a marginal investment for an individual

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investor given current price, 8. states the price given the joint distribution of the (subjective) stochastic discount factor (or consumption) and the asset payoff or 9. states current consumption u0(ct) given the joint distribution of the (subjective) stochastic discount factor (or consumption) and the asset payoff and today’s price 10. does not specify truly exogenous stochastic processes that drive consumption and payoffs. See Cochrane (2001, Section 2.1). In the following some of the items above are treated in more detail. • Finding the individual willingness to pay for an asset using the asset pricing equation – This amounts to finding the price for investing an extra marginal amount ξ into an asset. This increases the investor utility from u(ct) + βEt[u(ct+1)] to u(ct − ptξ) + βEt[u(ct+1 + xt+1ξ)].

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This implies the loss :u(ct − ptξ) − u(ct) gain :βEt[u(ct+1 + xt+1ξ) − u(ct+1)]. Using first-order Taylor expansions taken at ct and ct+1 one obtains approximatively loss :u(ct) + u0(ct)ptξ − u(ct) = u0(ct)ptξ gain :βE[u(ct+1) + u0(ct+1)xt+1ξ − u(ct+1)] = βE[u0(ct+1)xt+1ξ]. Comparing the loss and the gain one obtains 0 > βE[u0(c )x ξ] u (ct+1) t+1 t+1 =E β 0 xt+1 ≡ vt. pt = = 0 (c )ξ u u (c ) t t < On the left hand side is the market price pt. On the right hand side one has the private valuation vt of an extra unit of the asset given everything else. Now the investor faces three situations ∗ vt > pt (private valuation larger than market valuation) the investor buys more of the asset. This increases future consumption on average leading to an average decline of future marginal utility and thus a decrease in the private valuation of additional units of the asset. The investor stops buying once vt = pt. 192

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∗ vt = pt nothing happens. The investor is in post-trade or in the equilibrium situation. In the optimum for the investor loss equals gain. ∗ vt < pt (private valuation smaller than market valuation) the investor sells more of the asset. This decreases future consumption on average leading to an average increase of future marginal utility and thus an incrase in the private valuation of additional units of the asset. The investor stops selling once vt = pt. • Complete markets, the stochastic discount factor, and risk-neutral probabilities – The stochastic discount factor can be related to the very important concept of complete markets. First we need a class of specific assets: – Contingent claims: Consider a simple world with finitely many states in the future, let’s say S states. A contingent claim for state s is an asset that pays exactly 1 unit in state s and nothing in any other state. Thus, by buying a contingent claim for all S states one can ensure to always obtain 1 unit in the future period. This certain payoff costs simply the sum of the prices of all contingent claims. Note that this insurance type of investment only works if there exists a contingent claim for every state. In this case, a market is called complete. Otherwise, a market is called

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incomplete. In reality, markets are in general incomplete. For a market to be complete it is not necessary that contingent claims themselves exist. It is sufficient that there exist enough assets such that a contingent claim for each state can be constructed by choosing an appropriate portfolio of available assets. One says the assets span/synthezise all contingent claims. – Relating complete markets with the stochastic discount factor ∗ Price of contingent claim for state s: pc(s) ∗ Recall that the price of a payment that pays x(1), . . . , x(S) units in states 1 to S in the next period is S X p(x) = x(s)pc(s). s=1

In order to write this as an expected value write S X

pc(s) π(s) π(s) s=1 pc(s) = E x(s) π(s)

p(x) =

x(s)

where π(s) denotes the probability that state s occurs in the next period.

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∗ Now observe that the ratio of contingent claims prices and probabilities play the role of the stochastic discount factor! Thus, define the stochastic discount factor for state s m(s) ≡

pc(s) , π(s)

s = 1, . . . , S

and obtain the asset pricing equation p(x) = E [x(s)m(s)] . ∗ Thus, if markets are complete, a stochastic discount factor exists. If the state space is continuous a similar result can be shown but this is more difficult. ∗ Now consider the price for a certain payout of 1 unit S X

pc(s) =

s=1

S X pc(s)

π(s) s=1 | {z }

π(s) = E [m(s)] .

m(s)

Thus, the risk-free rate of return is Rf = PS

1

s=1 pc(s)

as shown before. – Risk-neutral probabilities 195

=

1 E [m(s)]

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∗ The concept of risk-neutral probabilities is very important for pricing options or computing term structure models for interest rates. It can be very nicely explained using the asset pricing equation. ∗ Note that if the investor were risk-neutral, then marginal utility would be constant and the asset pricing equation would simplify to p(x) = βE[x(s)]. ∗ Now we expand the asset pricing equation once more. This time by E[m(s)] which leads to m(s) E[m(s)] p(x) = E x(s) E[m(s)] S X m(s) x(s) = E[m(s)] π(s) . E[m(s)] s=1 | {z } π ∗ (s)

Note that π ∗(s) are also probabilities since they add up to 1 (because

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m(s) E[m(s)]

is a weight

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function that sums to 1). Thus, one can write in short p(x) = E[m(s)]

S X

x(s)π ∗(s)

s=1

=

1 ∗ E [x(s)] Rf

where E ∗[·] denotes the expectation using the probabilities π ∗(s). The probabilities π ∗(s) are called risk-neutral probabilities since now the price corresponds to the formula of a risk-neutral investor. Thus, one was able to move the risk-aversion into the probabilities by changing them. ∗ A few remarks: · Agents are risk-neutral with respect to the probability distribution π ∗. · Note that π ∗ gives a greater weight to states with higher than average future marginal utility u0(c(s)), see example below. · Risk aversion is equivalent to paying more attention to unpleasant states relative to their actual probability of occurrence. · The combination of π×m is the most important piece of information for many decisions!

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· There is also technical jargon (not needed in this course but for later reference) π ∗(s) =

m(s) π(s). E[m(s)] | {z } change of measure

m(s) is called change of measure or derivative since it allows The expression E[m(s)] to switch from the measure (=probability distribution) π to the risk-neutral measure (=probability distribution) π ∗.

– Example ∗ Power utility: u(c) =

1 1−γ 1−γ c

with γ = 12 . Thus, marginal utility is u0(c) = c−1/2.

∗ For simplicity, the discount rate β = 1. ∗ Consumption in period t is already chosen and ct = 9. Thus, current marginal utility is u0(ct) = 13 . ∗ Endowment in period t + 1 is independent of the state and is et+1 = 6. The only other income in period t + 1 is the payoff xt+1(s) of the asset. ∗ In the table below the payoff for each of the S = 3 states is specified and the resulting future marginal utility, the discount factor for each state, the risk-neutral probability and the price of a contingent claim that pays 1 unit in one state. 198

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state probability payoff

future

future marginal

consumption

utility

stochastic

risk-neutral

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price of

discount factor probability contingent claims

s

π(s)

xt+1(s)

ct+1(s)

u0(ct+1(s))

mt+1(s)

π ∗(s)

pc(s)

1

1 2 1 3 1 6

3

9

1

10

16

−2

4

1 3 1 4 1 2

1 2 1 4 1 4

1 2 1 4 1 4

2 3

3 4 3 2

Notice that the risk-neutral probability for the state with negative payoff is much higher than the probability itself, indicating the risk aversion of the investor! ∗ Price of asset with payoff xt+1F pt = E [mt+1xt+1] =

3 X

1 30 1 1 7 mt+1(s)xt+1(s)π(s) = 3 + −3 = . 2 43 6 2 s=1

Price of asset using risk-neutral probabilities ∗

pt = E [xt+1] =

3 X

1 1 1 7 xt+1(s)π ∗(s) = 3 + 10 − 2 = . 2 4 4 2 s=1

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7.2 Is the asset pricing equation empirically relevant? • In order to check the asset pricing equation empirically, one has to assume something about the stochastic discount factor and to model its joint distribution with future payoff. Moreover one also has to assume that there exists something like an average utility function or a representative agent. – Simplest case: power utility function, see above. This assumption is not very well supported by the data. – Alternatives: ∗ Other utility functions: variables driving utility; separating intertemporal substitution and risk aversion, etc. ∗ General equilibrium models: link consumption to other variables e.g. income, interest rates; model covariance in beta explicitly ∗ Factor pricing models: model stochastic discount factor (ratio of marginal utilities) in terms of other variables mt+1 = a + bA

A B ft+1 +bB ft+1 + ··· , |{z} factor A

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Special cases: · Capital Asset Pricing Model (CAPM) W mt+1 = a + bRt+1 W where Rt+1 is the rate of return on a claim to total wealth.

· Arbitrage Pricing Theory (APT) · Intertemporal Capital Asset Pricing Model (ICAPM) · Term structure models ∗ Arbitrage or near-arbitrage pricing: use the no-arbitrage condition to determine the price of one payoff in terms of the prices of other payoffs (most famous: Black-Scholes option prices). Works because of the existence of the asset pricing equation and positive marginal utility.

7.3 Factor-pricing models • Idea: link stochastic discount factor to other (observable) data Easiest: use a link that linearly combines several factors: linear factor pricing models. This 201

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is mostly used in empirical work, see below. First, we discuss its theoretical properties. • Factor pricing models – replace consumption-based expression for marginal utility growth version of stochastic discount factor with a linear model mt+1 = a + b0ft+1 where ft+1 is a vector of factors that have to be determined. – are equivalent to the multivariate/multiple version of the expected return-beta representation or beta-pricing model i E[Rt+1 ] = γ + β 0λ

where λ is a vector that captures the various prices of risks that influence expected returns. – Main issue: which factors to choose? ∗ If one starts again from the simple consumption-based model, then the factors must proxy aggregate marginal utility growth u0(ct+1) ≈ a + b0ft+1 β 0 u (ct) ∗ Why is this view useful?

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“the essence of asset pricing is that there are special states of the world in which investors are especially concerned that their portfolios not do badly. They are willing to trade off some overall performance—average return—to make sure that portfolios do not do badly in those particular states of nature. The factors are variables that indicate that these “bad states” have occured.” Cochrane (2005a, p.149) ∗ One needs factors that · measure the state of the economy: u0(ct) · forecast the state of the economy: u0(ct+1) ∗ Factors that measure the state of the economy: · returns on broad-based portfolios · interest rates · growth in GDP · investment · other macroeconomic variables · returns to real production processes

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· etc. ∗ Factors that forecast the state of the economy must be related to news · variables that are correlated with “changes in the investment opportunity set” · term premium · dividend price ratio · stock returns · etc. – Should factors be predictable over time? ∗ Simple case: assume constant real interest rate Rf =

1 . Et[mt+1]

Then using the consumption based stochastic discount factor mt+1 one obtains 1 = Rf βEt

0

u0(ct+1) =β 0 u (ct)

u (ct+1) , u0(ct) 204

Et

0

u (ct+1) 1 = . u0(ct) βRf

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Thus

1 u0(ct+1) = + εt+1, u0(ct) βRf

205

E[εt+1] = 0

∗ In reality, the real interest rate varies, however, not too much. Thus, marginal utility growth cannot be expected to be highly predictable! This carries over to all factors that should proxy marginal utility growth! ∗ Thus, one may choose factors that represent changes themselves: GNP growth, portfolio returns, price-dividend ratios, etc. • Capital Asset Pricing Model (CAPM) – Most famous asset pricing model – A one factor model with the factor return on the “wealth portfolio” W mt+1 = a + bRt+1 .

– how to proxy the wealth portfolio: in practice S&P500, value- or equally-weighted NYSE, DAX(?) • One simple derivation based on two-period quadratic utility: Assumptions:

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– quadratic utility u(c) = − 12 (ct − c∗)2. – one investor, living two periods, maximizing the utility 1 1 ∗ 2 ∗ 2 U (ct, ct+1) = − (ct − c ) − βE (ct+1 − c ) . 2 2 – Initial endowment/wealth Wt exogenously given at the beginning of period 1 – No labor income – Investment opportunities: N assets with price pit for asset i with payoff xit+1. Thus, asset i i has return Rt+1 . The optimization problem is to maximize intertemporal utility given the budget constraint by selecting the portfolio weights wi and implicitly ct, ct+1: ct+1 = Wt+1 W Wt+1 = Rt+1 (Wt − ct) W Rt+1 =

N X

i wiRt+1 ,

N X

wi = 1

i=1

i=1

W i where Rt+1 and Rt+1 denote the rate of return on total wealth and asset i, respectively. Note

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that shortselling is allowed since wi maybe negative. mt+1

W (Wt − ct) − c∗ Rt+1 =β ct − c∗ βc∗ β (Wt − ct) W = ∗ − R c| {z − c}t | c∗ {z − ct } t+1 at

bt

W mt+1 = at − btRt+1

Note that the denominator was rearranged such that at and bt are non-negative for ct ≤ c∗ which is the relevant part of the utility function. Note also that the parameters at and bt are W time-dependent since they depend on the situation at time t. Inserting mt+1 = at − btRt+1 into the expected returns - beta representation results in a time-varying price of risk λW,t and beta

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βW,i,t i i ) V art(mt+1) −Covt(mt+1, Rt+1 f Et Rt+1 − Rt+1 = Et[mt+1] V art(mt+1) W W i V art(Rt+1 )b2t (−1)2btCovt(Rt+1 , Rt+1 ) = W ] W )b2 at − btEt[Rt+1 V art(Rt+1 t | {z } | {z } λW,t

βW,i,t

W f − Rt+1 = λW,t · 1 Et Rt+1 λW,t

= and therefore W f = Et Rt+1 − Rt+1

• Other factor models are the Intertemporal Capital Asset Pricing Model (ICAPM) or the Arbitrage Pricing Theory (APT), see e.g. Cochrane (2005a, Sections 9.2 - 9.5)

7.4 Regression based tests of linear factor models • Using time series regressions – Recall the linear single factor model using the notation of excess returns Rei = Ri − Rf and

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ReW = RW − Rf : E[Rtei] = βi,W λ – Note that the excess return-beta representation is also obtained by taking expectations of Rtei = βi,W RteW + εit,

E[εit] = 0,

i = 1, . . . , N, t = 1, . . . , T.

– This model is nested in the standard time series regression model (including a constant) Rtei = αi + βi,W RteW + εit. In practice, all pricing errors αi should be zero. • Estimation procedure – Estimate price of risk λ by

1 T

PT

eW t=1 Rt

– Run N OLS regressions of (7.7) and test for each H0 : αi = 0 versus H1 : αi 6= 0.

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(7.7)

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Example: Estimation of a CAPM model – Data: monthly data January 2003 to June 2012 of DAX, Bayer, Daimler, E.ON, Deutsche Post, Deutsche Lufthansa, SAP, Siemens and short-term interest rates; see file capm.RData. – Computation of excess returns of stock i Rtei = ln Pt − ln Pt−1 − Rtf /(12 ∗ 100), since the three-month interest rates are given annualized in percent. – Estimation of time series regressions in R Regressing excess returns for the Bayer stock on the excess returns of the DAX yields Call: lm(formula = excess.stock ~ +1 + excess.market) Residuals: Min 1Q Median -0.217861 -0.032947 -0.001235

3Q 0.030595

Max 0.125766

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.005474 0.005143 1.065 0.289 excess.market 0.992579 0.086037 11.537 <2e-16 *** --Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.05443 on 111 degrees of freedom

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Hence, the constant term is not significant and beta is close to unity. • Conclusion: Do you think that the CAPM captures most of the variation of the equity premia for the considered period and stocks? In general, adding additional factors play a role in explaining equity premia such as the difference between returns of big and small companies or the difference in returns between companies with a high book-to-market ratio and those with a low ratio. For a recent survey on the CAPM and its extensions, see e.g. Fama, Eugene F., French, Kenneth R. (2004). The Capital Asset Pricing Model: Theory and Evidence, Journal of Economic Perspectives 18, 25 – 4 Perold, Andr F. (2004). The Capital Asset Pricing Model Source, Journal of Economic Perspectives 18, 3 – 24

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7.5 Supplement: Organisation of an empirical project • Purpose of project and data definition – goal of project / model – definition of original data what is measured, how is measured (e.g. construction of financial indices/redefinitions, etc.) – period, sampling frequency – data source – specific data choice, e.g. opening/closing price... – data transformations (taking logarithms, etc.) • Descriptive statistics and first analysis of data properties – plot of original and transformed data – mean – median – max/min

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– skewness – kurtosis – normality – correlations – unit root testing: ∗ choice of lags ∗ choice of deterministic components (trend, constant, seasonal dummies) ∗ choice of test(s) and required assumptions ∗ level, differences – seasonal components – structural breaks, etc. – Summary of findings and guess of model class(es) • Model choice and diagnostics – choice of model class: e.g. AR or ARMA

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– choice of lag orders – choice of deterministic components: constant, time trend, seasonal dummies – choice of estimator and required assumptions – residual diagnostics ∗ residual (auto)correlation ∗ (conditional) heteroskedasticity ∗ normality ∗ stationarity – if necessary: model modification: e.g.: ARCH, GARCH, TGARCH ∗ model choice ∗ lag choice ∗ residual check, see above plus correspondence of assumption about error distribution with residual distribution • Statistical results and interpretation

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• Forecasting or other uses – point forecasts – interval forecasts – take into account prior data transformations • Check spelling

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• Offizielle Beschreibungen der ’DAXe’ von der Deutschen B¨orse – DAX Aktienindex, der die Wertentwicklung der 30 nach Marktkapitalisierung gr¨oßten und umsatzst¨arksten deutschen Aktien im Prime Standard der FWB Frankfurter Wertpapierb¨orse abbildet. Der DAX-Index (Deutscher Aktienindex) wird von der Deutschen B¨orse aus den Kursen der 30 umsatzst¨arksten deutschen Aktien berechnet und ist der meist beachtete Indikator f¨ur die Entwicklung des deutschen Marktes. Die DAX-Werte notieren im Prime Standard. Kriterien f¨ur die Gewichtung der Aktien in DAX sind B¨orsenumsatz und Marktkapitalisierung des Streubesitzes. DAX wird als Kurs- und Performance-Index aus Xetra-Kursen sek¨undlich berechnet und aktualisiert. Kursdaten zu DAX und den enthaltenen Werten sowie die Termine der Neuzusammensetzung finden Sie auf boerse-frankfurt.de/indizes. – MDAX Index, der die Wertentwicklung der 50 gr¨oßten auf die DAX-Werte folgenden Unternehmen der klassischen Branchen im Prime Standard abbildet. MDAX wird seit dem 19. Januar 1996 berechnet. Der Index enth¨alt die 50 nach Mark216

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tkapitalisierung und B¨orsenumsatz gr¨oßten Unternehmen der klassischen Branchen im Prime Standard unterhalb der DAX-Werte. Basis der Berechnung ist der 30. Dezember 1987 mit einem Wert von 1.000 Punkten. Die Indexzusammensetzung wird u¨blicherweise halbj¨ahrlich u¨berpr¨uft und mit Wirkung zum M¨arz und September angepasst. Kriterien f¨ur die Gewichtung der Aktien in MDAX sind B¨orsenumsatz und Marktkapitalisierung auf Basis des Streubesitzes (Freefloats). Ein Unternehmen kann außerhalb der ordentlichen u¨berpr¨ufungstermine aus dem Index genommen werden, wenn es beim Kriterium Marktkapitalisierung oder B¨orsenumsatz nicht mehr zu den 75 gr¨oßten Unternehmen z¨ahlt, bzw. in den Index aufgenommen werden, wenn es bei den Kriterien Marktkapitalisierung und B¨orsenumsatz eines der 40 gr¨oßten Unternehmen ist. Der Austausch findet zum n¨achsten Verkettungstermin statt. u¨ber Ver¨anderungen in MDAX entscheidet der Vorstand der Deutsche B¨orse AG. Er wird dabei vom Arbeitskreis Aktienindizes beraten. – SDAX Index der 50 gr¨oßten auf die MDAX-Werte folgenden Unternehmen der klassischen Branchen des Prime Standard. SDAX startete am 21. Juni 1999. Er umfasst die 50 nach Marktkapitalisierung und B¨orsenumsatz gr¨oßten Unternehmen der klassischen Branchen unterhalb der MDAX-

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Werte. Der Index wird als Kurs- und Performance-Index von der Deutschen B¨orse berechnet. Basis der Berechnung ist der 30. Dezember 1987 mit einem Wert von 1.000 Punkten. Die Indexzusammensetzung wird u¨blicherweise halbj¨ahrlich u¨berpr¨uft und mit Wirkung zum M¨arz und September angepasst. Kriterien f¨ur die Gewichtung der Aktien in SDAX sind: B¨orsenumsatz und Marktkapitalisierung auf Basis des Streubesitzes sowie Branchenrepr¨asentativit¨at. u¨ber Ver¨anderungen in SDAX entscheidet der Vorstand der Deutschen B¨orse. Er wird dabei vom Arbeitskreis Aktienindizes beraten. – TecDAX Index f¨ur die Wertentwicklung der 30 gr¨oßten Technologieaktien im Prime Standard unterhalb der DAX-Titel. TecDAX startete am 24. M¨arz 2003. Er umfasst die 30 nach Marktkapitalisierung und B¨orsenumsatz gr¨oßten Unternehmen der Technologiebranchen im Prime Standard unterhalb des Leitindex DAX. Der Index wird als Kurs- und als Performance-Index berechnet. Basis der Berechnung ist der 30. Dezember 1997 mit einem Wert von 1.000 Punkten. Die Indexzusammensetzung wird u¨blicherweise halbj¨ahrlich u¨berpr¨uft und mit Wirkung zum M¨arz und September angepasst. Kriterien f¨ur die Gewichtung der Aktien in

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TecDAX sind: B¨orsenumsatz und Marktkapitalisierung auf Basis des Streubesitzes (Freefloats). Ein Unternehmen kann außerhalb der ordentlichen u¨berpr¨ufungstermine aus dem Index genommen werden, wenn es beim Kriterium Marktkapitalisierung oder B¨orsenumsatz nicht mehr zu den 45 gr¨oßten Unternehmen z¨ahlt, bzw. in den Index aufgenommen werden, wenn es bei den Kriterien Marktkapitalisierung und B¨orsenumsatz eines der 25 gr¨oßten Unternehmen ist. Ein Austausch findet zum n¨achsten Verkettungstermin statt. u¨ber Ver¨anderungen in TecDAX entscheidet der Vorstand der Deutsche B¨orse AG. Er wird dabei beraten vom Arbeitskreis Aktienindizes.

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