Given a stationary multivariate time series , cross-covariance matrices are
where , and cross-correlation matrices are
where D is a diagonal matrix with the standard deviations of the components of on the diagonal.
The sample cross-covariance matrix at lag l, denoted as , is computed as
where is the centered data and is the number of nonmissing observations. Thus, has th element . The sample cross-correlation matrix at lag l is computed as
The following statements use the CORRY option to compute the sample cross-correlation matrices and their summary indicator plots in terms of and , where indicates significant positive cross-correlations, indicates significant negative cross-correlations, and indicates insignificant cross-correlations.
proc varmax data=simul1; model y1 y2 / p=1 noint lagmax=3 print=(corry) printform=univariate; run;
Figure 35.39 shows the sample cross-correlation matrices of and . As shown, the sample autocorrelation functions for each variable decay quickly, but are significant with respect to two standard errors.
Figure 35.39: Cross-Correlations (CORRY Option)
Cross Correlations of Dependent Series by Variable |
|||
---|---|---|---|
Variable | Lag | y1 | y2 |
y1 | 0 | 1.00000 | 0.67041 |
1 | 0.83143 | 0.84330 | |
2 | 0.56094 | 0.81972 | |
3 | 0.26629 | 0.66154 | |
y2 | 0 | 0.67041 | 1.00000 |
1 | 0.29707 | 0.77132 | |
2 | -0.00936 | 0.48658 | |
3 | -0.22058 | 0.22014 |
Schematic Representation of Cross Correlations |
||||
---|---|---|---|---|
Variable/Lag | 0 | 1 | 2 | 3 |
y1 | ++ | ++ | ++ | ++ |
y2 | ++ | ++ | .+ | -+ |
+ is > 2*std error, - is < -2*std error, . is between |
For each you can define a sequence of matrices , which is called the partial autoregression matrices of lag m, as the solution for to the Yule-Walker equations of order m,
The sequence of the partial autoregression matrices of order m has the characteristic property that if the process follows the AR(p), then and for . Hence, the matrices have the cutoff property for a VAR(p) model, and so they can be useful in the identification of the order of a pure VAR model.
The following statements use the PARCOEF option to compute the partial autoregression matrices:
proc varmax data=simul1; model y1 y2 / p=1 noint lagmax=3 printform=univariate print=(corry parcoef pcorr pcancorr roots); run;
Figure 35.40 shows that the model can be obtained by an AR order since partial autoregression matrices are insignificant after lag 1 with respect to two standard errors. The matrix for lag 1 is the same as the Yule-Walker autoregressive matrix.
Figure 35.40: Partial Autoregression Matrices (PARCOEF Option)
Partial Autoregression | |||
---|---|---|---|
Lag | Variable | y1 | y2 |
1 | y1 | 1.14844 | -0.50954 |
y2 | 0.54985 | 0.37409 | |
2 | y1 | -0.00724 | 0.05138 |
y2 | 0.02409 | 0.05909 | |
3 | y1 | -0.02578 | 0.03885 |
y2 | -0.03720 | 0.10149 |
Schematic Representation of Partial Autoregression |
|||
---|---|---|---|
Variable/Lag | 1 | 2 | 3 |
y1 | +- | .. | .. |
y2 | ++ | .. | .. |
+ is > 2*std error, - is < -2*std error, . is between |
Define the forward autoregression
and the backward autoregression
The matrices defined by Ansley and Newbold (1979) are given by
where
and
are the partial cross-correlation matrices at lag m between the elements of and , given . The matrices have the cutoff property for a VAR(p) model, and so they can be useful in the identification of the order of a pure VAR structure.
The following statements use the PCORR option to compute the partial cross-correlation matrices:
proc varmax data=simul1; model y1 y2 / p=1 noint lagmax=3 print=(pcorr) printform=univariate; run;
The partial cross-correlation matrices in Figure 35.41 are insignificant after lag 1 with respect to two standard errors. This indicates that an AR order of can be an appropriate choice.
Figure 35.41: Partial Correlations (PCORR Option)
Partial Cross Correlations by Variable | |||
---|---|---|---|
Variable | Lag | y1 | y2 |
y1 | 1 | 0.80348 | 0.42672 |
2 | 0.00276 | 0.03978 | |
3 | -0.01091 | 0.00032 | |
y2 | 1 | -0.30946 | 0.71906 |
2 | 0.04676 | 0.07045 | |
3 | 0.01993 | 0.10676 |
Schematic Representation of Partial Cross Correlations |
|||
---|---|---|---|
Variable/Lag | 1 | 2 | 3 |
y1 | ++ | .. | .. |
y2 | -+ | .. | .. |
+ is > 2*std error, - is < -2*std error, . is between |
The partial canonical correlations at lag m between the vectors and , given , are . The partial canonical correlations are the canonical correlations between the residual series and , where and are defined in the previous section. Thus, the squared partial canonical correlations are the eigenvalues of the matrix
It follows that the test statistic to test for in the VAR model of order is approximately
and has an asymptotic chi-square distribution with degrees of freedom for .
The following statements use the PCANCORR option to compute the partial canonical correlations:
proc varmax data=simul1; model y1 y2 / p=1 noint lagmax=3 print=(pcancorr); run;
Figure 35.42 shows that the partial canonical correlations between and are {0.918, 0.773}, {0.092, 0.018}, and {0.109, 0.011} for lags 1 to 3. After lag 1, the partial canonical correlations are insignificant with respect to the 0.05 significance level, indicating that an AR order of can be an appropriate choice.
The minimum information criterion (MINIC) method can tentatively identify the orders of a VARMA(p,q) process (Spliid, 1983; Koreisha and Pukkila, 1989; Quinn, 1980). The first step of this method is to obtain estimates of the innovations series, , from the VAR(), where is chosen sufficiently large. The choice of the autoregressive order, , is determined by use of a selection criterion. From the selected VAR() model, you obtain estimates of residual series
In the second step, you select the order () of the VARMA model for p in and q in
which minimizes a selection criterion like SBC or HQ.
The following statements use the MINIC= option to compute a table that contains the information criterion associated with various AR and MA orders:
proc varmax data=simul1; model y1 y2 / p=1 noint minic=(p=3 q=3); run;
Figure 35.43 shows the output associated with the MINIC= option. The criterion takes the smallest value at AR order 1.