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| The VARMAX Procedure |
| VAR and VARX Modeling |
The 
th-order VAR process is written as
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with 
.
Equivalently, it can be written as
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with 
.
Stationarity
For stationarity, the VAR process must be expressible in the convergent causal infinite MA form as
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where 
with 
, where 
denotes a norm for the matrix 
such as 
. The matrix 
can be recursively obtained from the relation 
; it is
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where 
and 
for 
.
The stationarity condition is satisfied if all roots of 
are outside of the unit circle. The stationarity condition is equivalent to the condition in the corresponding VAR(1) representation, 
, that all eigenvalues of the 
companion matrix 
be less than one in absolute value, where 
, 
, and
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If the stationarity condition is not satisfied, a nonstationary model (a differenced model or an error correction model) might be more appropriate.
The following statements estimate a VAR(1) model and use the ROOTS option to compute the characteristic polynomial roots:
proc varmax data=simul1; model y1 y2 / p=1 noint print=(roots); run;
Figure 32.44 shows the output associated with the ROOTS option, which indicates that the series is stationary since the modulus of the eigenvalue is less than one.
| Roots of AR Characteristic Polynomial | |||||
|---|---|---|---|---|---|
| Index | Real | Imaginary | Modulus | Radian | Degree |
| 1 | 0.77238 | 0.35899 | 0.8517 | 0.4351 | 24.9284 |
| 2 | 0.77238 | -0.35899 | 0.8517 | -0.4351 | -24.9284 |
Parameter Estimation
Consider the stationary VAR() model
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where 
are assumed to be available (for convenience of notation). This can be represented by the general form of the multivariate linear model,
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where
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with vec denoting the column stacking operator.
The conditional least squares estimator of 
is
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and the estimate of 
is
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where 
is the residual vectors. Consistency and asymptotic normality of the LS estimator are that
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where 
converges in probability to 
and 
denotes convergence in distribution.
The (conditional) maximum likelihood estimator in the VAR() model is equal to the (conditional) least squares estimator on the assumption of normality of the error vectors.
Asymptotic Distributions of Impulse Response Functions
As before, vec denotes the column stacking operator and vech is the corresponding operator that stacks the elements on and below the diagonal. For any 
matrix , the commutation matrix 
is defined as 
; the duplication matrix 
is defined as 
; the elimination matrix 
is defined as 
.
The asymptotic distribution of the impulse response function (Lütkepohl 1993) is
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where 
and
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where 
is a 
matrix and 
is a 
companion matrix.
The asymptotic distribution of the accumulated impulse response function is
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where 
.
The asymptotic distribution of the orthogonalized impulse response function is
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where 
, 
, 
,
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and 
with 
and 
.
Granger Causality Test
Let 
be arranged and partitioned in subgroups 
and 
with dimensions 
and 
, respectively (
); that is, 
with the corresponding white noise process 
. Consider the VAR() model with partitioned coefficients 
for 
as follows:
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The variables are said to cause , but do not cause if 
. The implication of this model structure is that future values of the process are influenced only by its own past and not by the past of , where future values of are influenced by the past of both and . If the future are not influenced by the past values of , then it can be better to model separately from .
Consider testing 
, where 
is a 
matrix of rank 
and 
is an -dimensional vector where 
. Assuming that
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you get the Wald statistic
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For the Granger causality test, the matrix consists of zeros or ones and is the zero vector. See Lütkepohl(1993) for more details of the Granger causality test.
VARX Modeling
The vector autoregressive model with exogenous variables is called the VARX(,) model. The form of the VARX(,) model can be written as
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The parameter estimates can be obtained by representing the general form of the multivariate linear model,
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where
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The conditional least squares estimator of 
can be obtained by using the same method in a VAR() modeling. If the multivariate linear model has different independent variables that correspond to dependent variables, the SUR (seemingly unrelated regression) method is used to improve the regression estimates.
The following example fits the ordinary regression model:
proc varmax data=one;
model y1-y3 = x1-x5;
run;
This is equivalent to the REG procedure in the SAS/STAT software:
proc reg data=one;
model y1 = x1-x5;
model y2 = x1-x5;
model y3 = x1-x5;
run;
The following example fits the second-order lagged regression model:
proc varmax data=two;
model y1 y2 = x / xlag=2;
run;
This is equivalent to the REG procedure in the SAS/STAT software:
data three;
set two;
xlag1 = lag1(x);
xlag2 = lag2(x);
run;
proc reg data=three;
model y1 = x xlag1 xlag2;
model y2 = x xlag1 xlag2;
run;
The following example fits the ordinary regression model with different regressors:
proc varmax data=one;
model y1 = x1-x3, y2 = x2 x3;
run;
This is equivalent to the following SYSLIN procedure statements:
proc syslin data=one vardef=df sur;
endogenous y1 y2;
model y1 = x1-x3;
model y2 = x2 x3;
run;
From the output in Figure 32.20 in the section Getting Started: VARMAX Procedure, you can see that the parameters, XL0_1_2, XL0_2_1, XL0_3_1, and XL0_3_2 associated with the exogenous variables, are not significant. The following example fits the VARX(1,0) model with different regressors:
proc varmax data=grunfeld; model y1 = x1, y2 = x2, y3 / p=1 print=(estimates); run;
| XLag | |||
|---|---|---|---|
| Lag | Variable | x1 | x2 |
| 0 | y1 | 1.83231 | _ |
| y2 | _ | 2.42110 | |
| y3 | _ | _ | |
As you can see in Figure 32.45, the symbol ‘_’ in the elements of matrix corresponds to endogenous variables that do not take the denoted exogenous variables.
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