Econometrics I: Multivariate Time Series Econometrics (1)

Multivariate Time Series Econometrics (1) Dean Fantazzini Dipartimento di Economia Politica e Metodi Quantitativi ... df is degrees of freedom for (ap...

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Econometrics I: Multivariate Time Series Econometrics (1) Dean Fantazzini

Dipartimento di Economia Politica e Metodi Quantitativi University of Pavia

Overview of the Lecture

1st EViews Session XIII: VAR residual diagnostics

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

2

Overview of the Lecture

1st EViews Session XIII: VAR residual diagnostics 2nd EViews Session XIV: Estimate and forecast VAR

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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Overview of the Lecture

1st EViews Session XIII: VAR residual diagnostics 2nd EViews Session XIV: Estimate and forecast VAR 3rd EViews Session XV: VAR lag order selection

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XIII: VAR residual diagnostics −→ The following program computes residual diagnostics from a VAR. For the residual correlogram (autocorrelation), EViews only provides the asymptotic standard error which only depends on the sample size. ’ VAR residual tests ’ replicates Lutkepohl (1991, pp.148-158) ’ 1/10/2000 h last checked 3/25/2004 ’change path to program path %path = @runpath cd %path ’load workfile load lut1 ’ estimate VAR smpl 1960:1 1978:4 Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XIII: VAR residual diagnostics var1.ls 1 2 y1 y2 y3 @ c ’ residual correlograms (Fig 4.2, p.149) freeze(fig42) var1.correl(12,graph) show fig42 ’ portmanteau test (p.152) freeze(tab p152) var1.qstats(12,name=qstat) show tab p152 ’ normality test (p.158) freeze(tab p158) var1.jbera(factor=chol,name=jbera) show tab p158 You should get the following results: Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XIII: VAR residual diagnostics Figure 1: Autocorrelation Graphs. Autocorrelations with 2 Std.Err. Bounds Cor(Y1,Y1(-i))

Cor(Y1,Y2(-i))

Cor(Y1,Y3(-i))

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Cor(Y2,Y1(-i))

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Cor(Y2,Y3(-i))

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Multivariate Time Series Econometrics (1) Dean Fantazzini

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EViews Session XIII: VAR residual diagnostics Table 1: Table p. 152 VAR Residual Portmanteau Tests for Autocorrelations H0: no residual autocorrelations up to lag h Date: 04/21/03 Time: 23:39 Sample: 1960:1 1978:4 Included observations: 73 Lags

Q-Stat

Prob.

Adj Q-Stat

Prob.

df

1

0.920768

NA*

0.933556

NA*

NA*

2

2.044941

NA*

2.089396

NA*

NA*

3

9.328680

0.4075

9.685295

0.3766

9

4

21.03897

0.2775

22.07444

0.2287

18

5

26.38946

0.4971

27.81836

0.4204

27

6

30.77054

0.7154

32.59177

0.6315

36

7

35.57594

0.8416

37.90683

0.7642

45

8

44.83454

0.8085

48.30495

0.6928

54

9

48.27351

0.9147

52.22752

0.8315

63

10

56.81194

0.9051

62.12126

0.7904

72

11

66.09500

0.8846

73.05132

0.7235

81

12

73.51723

0.8966

81.93365

0.7157

90

*The test is valid only for lags larger than the VAR lag order. df is degrees of freedom for (approximate) chi-square distribution

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XIII: VAR residual diagnostics Table 2: Table p. 158 VAR Residual Normality Tests Orthogonalization: Cholesky (Lutkepohl) H0: residuals are multivariate normal Component

Skewness

Chi-sq

df

Prob.

1

0.119351

0.173310

1

0.6772

2

-0.383159

1.786194

1

0.1814

3

-0.312723

1.189845

1

0.2754

3.149350

3

0.3692

Joint

Component

Kurtosis

Chi-sq

df

Prob.

1

3.933079

2.648186

1

0.1037

2

3.739590

1.663770

1

0.1971

3

2.648386

0.376049

1

0.5397

4.688005

3

0.1961

Joint

Component

Jarque-Bera

df

Prob.

1

2.821496

2

0.2440

2

3.449965

2

0.1782

3

1.565894

2

0.4571

Joint

7.837355

6

0.2503

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XIV: Estimate and forecast VAR −→ The following program estimates an unrestricted VAR, creates a model object out the estimated VAR, and obtains dynamic forecasts from the VAR by solving the model object. ’ estimate VAR and forecast ’ replicates example in Lutkepohl (1991) pp.70-73, pp.89-91 ’ 1/7/2000 h last checked 3/25/2004 ’change path to program path %path = @runpath cd %path ’ load workfile load lut1 ’ estimate VAR smpl 1960:1 1978:4 var1.ls 1 2 y1 y2 y3 @ c Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XIV: Estimate and forecast VAR ’ replicates p.72, (3.2.22) & (3.2.24) ’ note that the variables are ordered differently freeze(out1) var1.output show out1 ’ make model out of estimated VAR var1.makemodel(mod1) ’ change sample to forecast period smpl 1979:1 1980:1 ’ solve model to obtain dynamic forecasts mod1.solve ’ plot actual and forecasts smpl 1975:1 1980:1

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XIV: Estimate and forecast VAR for !i=1 to [email protected] group gtmp y!i y!i 0 freeze(gra!i) gtmp.line %gname = %gname + ‘‘gra’’ + @str(!i) + ‘‘ ’’ next ’ merge all graphs into one freeze(gfcst) %gname gfcst.options size(8,2) gfcst.align(1,0.1, 0.5) gfcst.legend position(0.1,0.1) ’gfcst.scale(left)+zeroline gfcst.draw( dashline,left,rgb(155,155,155) ) 0.0 show gfcst Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XIV: Estimate and forecast VAR

You should get the following results:

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XIV: Estimate and forecast VAR Table 3: Estimation results Vector Autoregression Estimates Date: 04/21/03 Time: 23:56 Sample(adjusted): 1960:4 1978:4 Included observations: 73 after adjusting endpoints Standard errors in ( ) & t-statistics in [ ]

Y1(-1)

Y1(-2)

Y2(-1)

Y2(-2)

Y1

Y2

Y3

-0.319631

0.043931

-0.002423

(0.12546)

(0.03186)

(0.02568)

[-2.54774]

[ 1.37891]

[-0.09435]

-0.160551

0.050031

0.033880

(0.12491)

(0.03172)

(0.02556)

[-1.28537]

[ 1.57728]

[ 1.32533]

0.145989

-0.152732

0.224813

(0.54567)

(0.13857)

(0.11168)

[ 0.26754]

[-1.10220]

[ 2.01305]

0.114605

0.019166

0.354912

(0.53457)

(0.13575)

(0.10941)

[ 0.21439]

[ 0.14118]

[ 3.24398]

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XIV: Estimate and forecast VAR Y3(-1)

0.961219

0.288502

-0.263968

(0.66431)

(0.16870)

(0.13596)

[ 1.44694]

[ 1.71015]

[-1.94151]

0.934394

-0.010205

-0.022230

(0.66510)

(0.16890)

(0.13612)

[ 1.40490]

[-0.06042]

[-0.16331]

-0.016722

0.015767

0.012926

(0.01723)

(0.00437)

(0.00353)

[-0.97072]

[ 3.60427]

[ 3.66629]

R-squared

0.128562

0.114194

0.251282

Adj. R-squared

0.049340

0.033666

0.183217

Sum sq. resids

0.140556

0.009064

0.005887

S.E. equation

0.046148

0.011719

0.009445

F-statistic

1.622807

1.418070

3.691778

Log likelihood

124.6378

224.6938

240.4444

Akaike AIC

-3.222954

-5.964214

-6.395737

Schwarz SC

-3.003321

-5.744581

-6.176104

Mean dependent

0.018229

0.020283

0.019802

S.D. dependent

0.047330

0.011922

0.010451

Y3(-2)

C

Determinant Residual Covariance

1.66E-11

Log Likelihood (d.f. adjusted)

595.2689

Akaike Information Criteria

-15.73339

Schwarz Criteria

-15.07449

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XIV: Estimate and forecast VAR Figure 2: VAR forecasts. .12 Y1

Y1 (Baseline)

.08 .04 .00 -.04 1975Q1

1975Q3

1976Q1

1976Q3

1977Q1

1977Q3

1978Q1

1978Q3

1979Q1

1979Q3

1980Q1

1977Q1

1977Q3

1978Q1

1978Q3

1979Q1

1979Q3

1980Q1

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1978Q1

1978Q3

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.05 .04

Y2

Y2 (Baseline)

.03 .02 .01 .00 -.01 1975Q1

1975Q3

1976Q1

1976Q3

.04 .03

Y3

Y3 (Baseline)

.02 .01 .00 -.01 1975Q1

1975Q3

1976Q1

1976Q3

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XV: VAR lag order selection −→ This program computes various criteria to select the lag order of a VAR. The results from EViews do not quite match those reported in Lutkepohl (1991, Tables 4.4 and 4.5). While Table 4.4 reports the standard LR statistics, EViews reports the modified statistics as explained in the EViews 5 User’s Guide. The program computes the unmodified LR statistics that exactly replicate those reported in Table 4.5 by using the log likelihood values stored in the output matrix returned from the “mname=” option in the laglen command. Note that the stored log likelihood values do not make a degrees of freedom adjustment to the residual covariance matrix and will not match those reported in the estimation output. The information criteria reported in Table 4.5 do not appear to include the constant term in the log likelihood. However, even after correcting for the constant term, we are not able to replicate the values for AIC, HQ, and SC in Table 4.5. (There appears to be a typo in Table 4.5. The HQ and SC values for lag order 0 are unlikely to be the same.)

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XV: VAR lag order selection ’ VAR lag order selection ’ replicates Lutkepohl (1991) Table 4.4 (p.127) and Table 4.5 (p.130) ’ 1/10/2000 h ’ last checked 3/25/2004 ’change path to program path %path = @runpath cd %path ’ load workfile load lut1 ’ estimate VAR smpl 1960:1 1978:4 var1.ls 1 2 y1 y2 y3 @ c ’ lag length criteria freeze(tab45) var1.laglen(4,vname=vlag,mname=mlag) Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XV: VAR lag order selection show tab45 ’ unmodified LR test (exactly replicate Table 4.4, p.127) ’ 1st column:

(unmodified) LR statistic

’ 2nd column:

p-value

!m = @rows(mlag)-2 matrix(!m,2) tab44 !df = [email protected] * [email protected] ’ degrees of freedom of test for !r=!m to 1 step -1 tab44(!r,1) = 2*(mlag(!r+1,1) - mlag(!r,1)) tab44(!r,2) = 1 - @cchisq(tab44(!r,1),!df) next show tab44 Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XV: VAR lag order selection

Table 4: Table 4.5 VAR Lag Order Selection Criteria Endogenous variables: Y1 Y2 Y3 Exogenous variables: C Date: 04/22/03 Time: 00:22 Sample: 1960:1 1978:4 Included observations: 71 Lag

LogL

LR

FPE

AIC

SC

HQ

0

564.7842

NA

2.69E-11

-15.82491

-15.72930*

-15.78689*

1

576.4087

21.93905

2.50E-11

-15.89884

-15.51641

-15.74676

2

588.8591

22.44588*

2.27E-11*

-15.99603*

-15.32679

-15.72989

3

591.2373

4.086484

2.75E-11

-15.80950

-14.85344

-15.42931

4

598.4565

11.79471

2.91E-11

-15.75934

-14.51646

-15.26508

* indicates lag order selected by the criterion LR: sequential modified LR test statistic (each test at 5% level) FPE: Final prediction error AIC: Akaike information criterion SC: Schwarz information criterion HQ: Hannan-Quinn information criterion

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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EViews Session XV: VAR lag order selection

Table 5: Table 4.4 C1

C2

R1

23.24884

0.005661

R2

24.90090

0.003083

R3

4.756399

0.855006

R4

14.43835

0.107564

Multivariate Time Series Econometrics (1) Dean Fantazzini

July 2007

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