Given a stationary multivariate time series , cross-covariance matrices are
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where , and cross-correlation matrices are
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where is a diagonal matrix with the standard deviations of the components of on the diagonal.
The sample cross-covariance matrix at lag , denoted as , is computed as
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where is the centered data and is the number of nonmissing observations. Thus, has th element . The sample cross-correlation matrix at lag is computed as
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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 36.39 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 36.39: Cross-Correlations (CORRY Option)
Cross Correlations of Dependent Series by Variable |
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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 |
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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 , as the solution for to the Yule-Walker equations of order ,
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The sequence of the partial autoregression matrices of order has the characteristic property that if the process follows the AR(), then and for . Hence, the matrices have the cutoff property for a VAR() 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 36.40 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 36.40: Partial Autoregression Matrices (PARCOEF Option)
Partial Autoregression | |||
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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 |
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Variable/Lag | 1 | 2 | 3 |
y1 | +- | .. | .. |
y2 | ++ | .. | .. |
+ is > 2*std error, - is < -2*std error, . is between |
Define the forward autoregression
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and the backward autoregression
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The matrices defined by Ansley and Newbold (1979) are given by
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where
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and
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are the partial cross-correlation matrices at lag between the elements of and , given . The matrices have the cutoff property for a VAR() 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 36.41 are insignificant after lag 1 with respect to two standard errors. This indicates that an AR order of can be an appropriate choice.
Figure 36.41: Partial Correlations (PCORR Option)
Partial Cross Correlations by Variable | |||
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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 |
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---|---|---|---|
Variable/Lag | 1 | 2 | 3 |
y1 | ++ | .. | .. |
y2 | -+ | .. | .. |
+ is > 2*std error, - is < -2*std error, . is between |
The partial canonical correlations at lag 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
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It follows that the test statistic to test for in the VAR model of order is approximately
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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 36.42 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.
Figure 36.42: Partial Canonical Correlations (PCANCORR Option)
Partial Canonical Correlations | |||||
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Lag | Correlation1 | Correlation2 | DF | Chi-Square | Pr > ChiSq |
1 | 0.91783 | 0.77335 | 4 | 142.61 | <.0001 |
2 | 0.09171 | 0.01816 | 4 | 0.86 | 0.9307 |
3 | 0.10861 | 0.01078 | 4 | 1.16 | 0.8854 |
The minimum information criterion (MINIC) method can tentatively identify the orders of a VARMA(,) process. Note that Spliid (1983), Koreisha and Pukkila (1989), and Quinn (1980) proposed this method. 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
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In the second step, you select the order () of the VARMA model for in and in
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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 36.43 shows the output associated with the MINIC= option. The criterion takes the smallest value at AR order 1.
Figure 36.43: MINIC= Option
Minimum Information Criterion Based on AICC | ||||
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Lag | MA 0 | MA 1 | MA 2 | MA 3 |
AR 0 | 3.3574947 | 3.0331352 | 2.7080996 | 2.3049869 |
AR 1 | 0.5544431 | 0.6146887 | 0.6771732 | 0.7517968 |
AR 2 | 0.6369334 | 0.6729736 | 0.7610413 | 0.8481559 |
AR 3 | 0.7235629 | 0.7551756 | 0.8053765 | 0.8654079 |