SAS/ETS 9.22 User''''s Guide 222 docx

2202 F Chapter 32: The VARMAX ProcedureOutput 32.1.11 shows the innovation covariance matrix estimates, the various information criteria results, and the tests for white noise residuals.

2202 F Chapter 32: The VARMAX Procedure

Output 32.1.11 shows the innovation covariance matrix estimates, the various information criteria results, and the tests for white noise residuals The residuals have significant correlations at lag 2 and

3 The Portmanteau test results into significant These results show that a VECM(3) model might be better fit than the VECM(2) model is.

Output 32.1.11 Diagnostic Checks

Covariances of Innovations

Information Criteria

AICC -40.6284 HQC -40.4343 AIC -40.6452 SBC -40.1262 FPEC 2.23E-18

Schematic Representation of Cross Correlations of Residuals Variable/

+ is > 2*std error, - is < -2*std error, is between

Portmanteau Test for Cross Correlations of Residuals

Up To

Output 32.1.12 describes how well each univariate equation fits the data The residuals for y3 and y4 are off from the normality Except the residuals for y3, there are no AR effects on other residuals Except the residuals for y4, there are no ARCH effects on other residuals.

Output 32.1.12 Diagnostic Checks Continued

Univariate Model ANOVA Diagnostics

Standard Variable R-Square Deviation F Value Pr > F

Univariate Model White Noise Diagnostics

Variable Watson Chi-Square Pr > ChiSq F Value Pr > F

Univariate Model AR Diagnostics

Variable F Value Pr > F F Value Pr > F F Value Pr > F F Value Pr > F

2204 F Chapter 32: The VARMAX Procedure

The PRINT=(IARR) option provides the VAR(2) representation in Output 32.1.13

Output 32.1.13 Infinite Order AR Representation

Infinite Order AR Representation

Output 32.1.14 shows whether each variable is the weak exogeneity of other variables The variable y1 is not the weak exogeneity of other variables, y2, y3, and y4; the variable y2 is not the weak exogeneity of other variables, y1, y3, and y4; the variable y3 and y4 are the weak exogeneity of other variables.

Output 32.1.14 Weak Exogeneity Test

Testing Weak Exogeneity of Each Variables

Variable DF Chi-Square Pr > ChiSq

Example 32.2: Analysis of German Economic Variables

This example considers a three-dimensional VAR(2) model The model contains the logarithms of a quarterly, seasonally adjusted West German fixed investment, disposable income, and consumption expenditures The data used are in Lütkepohl (1993, Table E.1).

title 'Analysis of German Economic Variables';

data west;

date = intnx( 'qtr', '01jan60'd, _n_-1 );

format date yyq ;

input y1 y2 y3 @@;

y1 = log(y1);

y2 = log(y2);

y3 = log(y3);

label y1 = 'logarithm of investment'

y2 = 'logarithm of income' y3 = 'logarithm of consumption';

datalines;

180 451 415 179 465 421 185 485 434 192 493 448

more lines

data use;

set west;

where date < '01jan79'd;

keep date y1 y2 y3;

run;

proc varmax data=use;

id date interval=qtr;

model y1-y3 / p=2 dify=(1)

print=(decompose(6) impulse=(stderr) estimates diagnose) printform=both lagmax=3;

causal group1=(y1) group2=(y2 y3);

output lead=5;

run;

First, the differenced data is modeled as a VAR(2) with the following result:

yt D

0

@

0:01672 0:01577 0:01293

1

A C 0

@

0:31963 0:14599 0:96122 0:04393 0:15273 0:28850 0:00242 0:22481 0:26397

1

A yt 1

C 0

@

0:16055 0:11460 0:93439 0:05003 0:01917 0:01020 0:03388 0:35491 0:02223

1

A yt 2C t

The parameter estimates AR1_1_2, AR1_1_3, AR2_1_2, and AR2_1_3 are insignificant, and the VARX model is fitted in the next step.

2206 F Chapter 32: The VARMAX Procedure

The detailed output is shown in Output 32.2.1 through Output 32.2.8

Output 32.2.1 shows the descriptive statistics.

Output 32.2.1 Descriptive Statistics

Analysis of German Economic Variables

The VARMAX Procedure

Observation(s) eliminated by differencing 1

Simple Summary Statistics

Standard

Simple Summary Statistics

Variable Difference Label

Output 32.2.2 shows that a VAR(2) model is fit to the data.

Output 32.2.2 Parameter Estimates

Analysis of German Economic Variables

The VARMAX Procedure

Estimation Method Least Squares Estimation

Constant

Variable Constant

AR

2208 F Chapter 32: The VARMAX Procedure

Output 32.2.3 shows the parameter estimates and their significance.

Output 32.2.3 Parameter Estimates Continued

Schematic Representation

Variable/

+ is > 2*std error,

-is < -2*std error,

is between, * is N/A

Model Parameter Estimates

Standard Equation Parameter Estimate Error t Value Pr > |t| Variable

AR1_1_1 -0.31963 0.12546 -2.55 0.0132 y1(t-1) AR1_1_2 0.14599 0.54567 0.27 0.7899 y2(t-1) AR1_1_3 0.96122 0.66431 1.45 0.1526 y3(t-1) AR2_1_1 -0.16055 0.12491 -1.29 0.2032 y1(t-2) AR2_1_2 0.11460 0.53457 0.21 0.8309 y2(t-2) AR2_1_3 0.93439 0.66510 1.40 0.1647 y3(t-2)

AR1_2_1 0.04393 0.03186 1.38 0.1726 y1(t-1) AR1_2_2 -0.15273 0.13857 -1.10 0.2744 y2(t-1) AR1_2_3 0.28850 0.16870 1.71 0.0919 y3(t-1) AR2_2_1 0.05003 0.03172 1.58 0.1195 y1(t-2) AR2_2_2 0.01917 0.13575 0.14 0.8882 y2(t-2) AR2_2_3 -0.01020 0.16890 -0.06 0.9520 y3(t-2)

AR1_3_1 -0.00242 0.02568 -0.09 0.9251 y1(t-1) AR1_3_2 0.22481 0.11168 2.01 0.0482 y2(t-1) AR1_3_3 -0.26397 0.13596 -1.94 0.0565 y3(t-1) AR2_3_1 0.03388 0.02556 1.33 0.1896 y1(t-2) AR2_3_2 0.35491 0.10941 3.24 0.0019 y2(t-2) AR2_3_3 -0.02223 0.13612 -0.16 0.8708 y3(t-2)

Output 32.2.4 shows the innovation covariance matrix estimates, the various information criteria results, and the tests for white noise residuals The residuals are uncorrelated except at lag 3 for y2 variable.

Output 32.2.4 Diagnostic Checks

Covariances of Innovations

Information Criteria

AICC -24.4884 HQC -24.2869 AIC -24.5494 SBC -23.8905 FPEC 2.18E-11

Cross Correlations of Residuals

Schematic Representation of Cross Correlations of Residuals Variable/

+ is > 2*std error, - is <

-2*std error, is between

Portmanteau Test for Cross Correlations of Residuals

Up To

2210 F Chapter 32: The VARMAX Procedure

Output 32.2.5 describes how well each univariate equation fits the data The residuals are off from the normality, but have no AR effects The residuals for y1 variable have the ARCH effect.

Output 32.2.5 Diagnostic Checks Continued

Univariate Model ANOVA Diagnostics

Standard Variable R-Square Deviation F Value Pr > F

Univariate Model White Noise Diagnostics

Variable Watson Chi-Square Pr > ChiSq F Value Pr > F

Univariate Model AR Diagnostics

Variable F Value Pr > F F Value Pr > F F Value Pr > F F Value Pr > F

Output 32.2.6 is the output in a matrix format associated with the PRINT=(IMPULSE=) option for the impulse response function and standard errors The y3 variable in the first row is an impulse variable The y1 variable in the first column is a response variable The numbers, 0.96122, 0.41555, –0.40789 at lag 1 to 3 are decreasing.

Output 32.2.6 Impulse Response Function

Simple Impulse Response by Variable

Variable

The proportions of decomposition of the prediction error covariances of three variables are given

in Output 32.2.7 If you see the y3 variable in the first column, then the output explains that about 64.713% of the one-step-ahead prediction error covariances of the variable y3tis accounted for by its own innovations, about 7.995% is accounted for by y1t innovations, and about 27.292% is accounted for by y2tinnovations.