econometrics i: multivariate time series econometrics...

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



2nd EViews Session XIV: Estimate and forecast VAR


Dean Fantazzini July 2007 2-a



2nd EViews Session XIV: Estimate and forecast VAR

3rd EViews Session XV: VAR lag order selection


Dean Fantazzini July 2007 2-b

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




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:




Figure 1: Autocorrelation Graphs.

-.3

-.2

-.1

.0

.1

.2

.3

1 2 3 4 5 6 7 8 9 10 11 12

Cor(Y1,Y1(-i))

-.3

-.2

-.1

.0

.1

.2

.3

1 2 3 4 5 6 7 8 9 10 11 12

Cor(Y1,Y2(-i))

-.3

-.2

-.1

.0

.1

.2

.3

1 2 3 4 5 6 7 8 9 10 11 12

Cor(Y1,Y3(-i))

-.3

-.2

-.1

.0

.1

.2

.3

1 2 3 4 5 6 7 8 9 10 11 12

Cor(Y2,Y1(-i))

-.3

-.2

-.1

.0

.1

.2

.3

1 2 3 4 5 6 7 8 9 10 11 12

Cor(Y2,Y2(-i))

-.3

-.2

-.1

.0

.1

.2

.3

1 2 3 4 5 6 7 8 9 10 11 12

Cor(Y2,Y3(-i))

-.3

-.2

-.1

.0

.1

.2

.3

1 2 3 4 5 6 7 8 9 10 11 12

Cor(Y3,Y1(-i))

-.3

-.2

-.1

.0

.1

.2

.3

1 2 3 4 5 6 7 8 9 10 11 12

Cor(Y3,Y2(-i))

-.3

-.2

-.1

.0

.1

.2

.3

1 2 3 4 5 6 7 8 9 10 11 12

Cor(Y3,Y3(-i))

Autocorrelations with 2 Std.Err. Bounds




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




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

Joint 3.149350 3 0.3692

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

Joint 4.688005 3 0.1961

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



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


%path = @runpath

cd %path

’ load workfile load lut1

’ estimate VAR

smpl 1960:1 1978:4

var1.ls 1 2 y1 y2 y3 @ c




’ 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




for !i=1 to var1.@neqn

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




You should get the following results:




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 Y2 Y3

Y1(-1) -0.319631 0.043931 -0.002423

(0.12546) (0.03186) (0.02568)

[-2.54774] [ 1.37891] [-0.09435]

Y1(-2) -0.160551 0.050031 0.033880

(0.12491) (0.03172) (0.02556)

[-1.28537] [ 1.57728] [ 1.32533]

Y2(-1) 0.145989 -0.152732 0.224813

(0.54567) (0.13857) (0.11168)

[ 0.26754] [-1.10220] [ 2.01305]

Y2(-2) 0.114605 0.019166 0.354912

(0.53457) (0.13575) (0.10941)

[ 0.21439] [ 0.14118] [ 3.24398]




Y3(-1) 0.961219 0.288502 -0.263968

(0.66431) (0.16870) (0.13596)

[ 1.44694] [ 1.71015] [-1.94151]

Y3(-2) 0.934394 -0.010205 -0.022230

(0.66510) (0.16890) (0.13612)

[ 1.40490] [-0.06042] [-0.16331]

C -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

Determinant Residual Covariance 1.66E-11

Log Likelihood (d.f. adjusted) 595.2689

Akaike Information Criteria -15.73339

Schwarz Criteria -15.07449




Figure 2: VAR forecasts.

-.04

.00

.04

.08

.12

1975Q1 1975Q3 1976Q1 1976Q3 1977Q1 1977Q3 1978Q1 1978Q3 1979Q1 1979Q3 1980Q1

Y1 Y1 (Baseline)

-.01

.00

.01

.02

.03

.04

.05

1975Q1 1975Q3 1976Q1 1976Q3 1977Q1 1977Q3 1978Q1 1978Q3 1979Q1 1979Q3 1980Q1

Y2 Y2 (Baseline)

-.01

.00

.01

.02

.03

.04

1975Q1 1975Q3 1976Q1 1976Q3 1977Q1 1977Q3 1978Q1 1978Q3 1979Q1 1979Q3 1980Q1

Y3 Y3 (Baseline)



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.)




’ 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


%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)




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 = var1.@neqn * var1.@neqn ’ 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




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




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



econometrics i: multivariate time series econometrics...

Documents