The VARMAX Procedure
- Overview
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Getting Started
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Syntax
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Details
Missing Values VARMAX Model Dynamic Simultaneous Equations Modeling Impulse Response Function Forecasting Tentative Order Selection VAR and VARX Modeling Bayesian VAR and VARX Modeling VARMA and VARMAX Modeling Model Diagnostic Checks Cointegration Vector Error Correction Modeling I(2) Model Multivariate GARCH Modeling Output Data Sets OUT= Data Set OUTEST= Data Set OUTHT= Data Set OUTSTAT= Data Set Printed Output ODS Table Names ODS Graphics Computational Issues
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Examples
- References
| Cointegration |
This section briefly introduces the concepts of cointegration (Johansen 1995b).
- Definition 1.
(Engle and Granger 1987): If a series
with no deterministic components can be represented by a stationary and invertible ARMA process after differencing 
times, the series is integrated of order , that is, 
. - Definition 2.
(Engle and Granger 1987): If all elements of the vector
are 
and there exists a cointegrating vector 
such that 
for any 
, the vector process is said to be cointegrated 
.
A simple example of a cointegrated process is the following bivariate system:
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with 
and 
being uncorrelated white noise processes. In the second equation, 
is a random walk, 
, 
. Differencing the first equation results in
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Thus, both 
and are 
processes, but the linear combination 
is stationary. Hence 
is cointegrated with a cointegrating vector 
.
In general, if the vector process has 
components, then there can be more than one cointegrating vector 
. It is assumed that there are 
linearly independent cointegrating vectors with 
, which make the 
matrix 
. The rank of matrix is , which is called the cointegration rank of .
Common Trends
This section briefly discusses the implication of cointegration for the moving-average representation. Let be cointegrated 
, then 
has the Wold representation:
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where 
is 
, 
with 
, and 
.
Assume that 
if 
and 
is a nonrandom initial value. Then the difference equation implies that
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where 
and 
is absolutely summable.
Assume that the rank of 
is 
. When the process is cointegrated, there is a cointegrating matrix such that 
is stationary.
Premultiplying by results in
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because 
and 
.
Stock and Watson (1988) showed that the cointegrated process has a common trends representation derived from the moving-average representation. Since the rank of is , there is a matrix 
with rank such that 
. Let 
be a 
matrix with rank 
such that 
; then 
has rank . The 
has rank . By construction of 
,
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where 
. Since and , 
lies in the column space of and can be written
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where 
is a -dimensional vector. The common trends representation is written as
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and
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where 
, 
, 
, and 
.
Stock and Watson showed that the common trends representation expresses as a linear combination of random walks (
) with drift 
plus 
components (
.
Test for the Common Trends
Stock and Watson (1988) proposed statistics for common trends testing. The null hypothesis is that the -dimensional time series 
has common stochastic trends, where 
and the alternative is that it has 
common trends, where 
. The test procedure of versus common stochastic trends is performed based on the first-order serial correlation matrix of . Let 
be a matrix orthogonal to the cointegrating matrix such that 
and 
. Let 
and 
. Then
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Combining the expression of 
and 
,
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The Stock-Watson common trends test is performed based on the component by testing whether 
has rank against rank .
The following statements perform the Stock-Watson test for common trends:
proc iml;
sig = 100*i(2);
phi = {-0.2 0.1, 0.5 0.2, 0.8 0.7, -0.4 0.6};
call varmasim(y,phi) sigma=sig n=100 initial=0
seed=45876;
cn = {'y1' 'y2'};
create simul2 from y[colname=cn];
append from y;
quit;
data simul2;
set simul2;
date = intnx( 'year', '01jan1900'd, _n_-1 );
format date year4. ;
run;
proc varmax data=simul2;
model y1 y2 / p=2 cointtest=(sw);
run;
In Figure 35.51, the first column is the null hypothesis that has common trends; the second column is the alternative hypothesis that has common trends; the third column contains the eigenvalues used for the test statistics; the fourth column contains the test statistics using AR(
) filtering of the data. The table shows the output of the case 
.
| Common Trend Test | |||||
|---|---|---|---|---|---|
| H0: Rank=m |
H1: Rank=s |
Eigenvalue | Filter | 5% Critical Value | Lag |
| 1 | 0 | 1.000906 | 0.09 | -14.10 | 2 |
| 2 | 0 | 0.996763 | -0.32 | -8.80 | |
| 1 | 0.648908 | -35.11 | -23.00 | ||
The test statistic for testing for 2 versus 1 common trends is more negative (–35.1) than the critical value (–23.0). Therefore, the test rejects the null hypothesis, which means that the series has a single common trend.