Tentative Order Selection :: SAS/ETS(R) 13.1 User's Guide

Sample Cross-Covariance and Cross-Correlation Matrices

Given a stationary multivariate time series $\mb {y} _ t$ , cross-covariance matrices are

$\Gamma (l)=\mr {E} [(\mb {y} _ t - \bmu )(\mb {y} _{t+l} -\bmu )’]$

where $\bmu =\mr {E} (\mb {y} _ t)$ , and cross-correlation matrices are

$\rho (l) = D^{-1}\Gamma (l)D^{-1}$

where $D$ is a diagonal matrix with the standard deviations of the components of $\mb {y} _ t$ on the diagonal.

The sample cross-covariance matrix at lag $l$ , denoted as $C(l)$ , is computed as

$\hat\Gamma (l) = C(l) =\frac{1}{T} \sum _{t=1}^{T-l}{{\tilde{\mb {y}} }_{t}{\tilde{\mb {y}} }’_{t+l}}$

where ${\tilde{\mb {y}} }_{t}$ is the centered data and ${T}$ is the number of nonmissing observations. Thus, $\hat\Gamma (l)$ has $(i,j)$ th element $\hat\gamma _{ij}(l)=c_{ij}(l)$ . The sample cross-correlation matrix at lag $l$ is computed as

$\hat\rho _{ij}(l) = c_{ij}(l) /[c_{ii}(0)c_{jj}(0)]^{1/2}, ~ ~ i,j=1,\ldots , k$

The following statements use the CORRY option to compute the sample cross-correlation matrices and their summary indicator plots in terms of $+, -,$ and $\cdot$ , where $+$ indicates significant positive cross-correlations, $-$ indicates significant negative cross-correlations, and $\cdot$ 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 $y_{1t}$ and $y_{2t}$ . 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)

The VARMAX Procedure

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 > 2std error, - is < -2std error, . is between

Partial Autoregressive Matrices

For each $m=1,2,\ldots , p$ you can define a sequence of matrices $\Phi _{mm}$ , which is called the partial autoregression matrices of lag $m$ , as the solution for $\Phi _{mm}$ to the Yule-Walker equations of order $m$ ,

$\begin{eqnarray*} \Gamma (l) = \sum _{i=1}^ m \Gamma (l-i) \Phi _{im}’,~ ~ l=1,2,\ldots ,m \end{eqnarray*}$

The sequence of the partial autoregression matrices $\Phi _{mm}$ of order $m$ has the characteristic property that if the process follows the AR( $p$ ), then $\Phi _{pp}=\Phi _ p$ and $\Phi _{mm}=0$ for $m>p$ . Hence, the matrices $\Phi _{mm}$ 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 $m=1$ 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)

The VARMAX Procedure

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 > 2std error, - is < -2std error, . is between

Partial Correlation Matrices

Define the forward autoregression

$\begin{eqnarray*} \mb {y} _{t} = \sum _{i=1}^{m-1} \Phi _{i,m-1} \mb {y} _{t-i} + \mb {u} _{m,t} \end{eqnarray*}$

and the backward autoregression

$\begin{eqnarray*} \mb {y} _{t-m} = \sum _{i=1}^{m-1} \Phi _{i,m-1}^* \mb {y} _{t-m+i} + \mb {u} _{m,t-m}^{*} \end{eqnarray*}$

The matrices $P(m)$ defined by Ansley and Newbold (1979) are given by

$\begin{eqnarray*} P(m)= \Sigma _{m-1}^{*1/2}\Phi _{mm}’ \Sigma _{m-1}^{-1/2} \end{eqnarray*}$

where

$\begin{eqnarray*} \Sigma _{m-1} = \mr {Cov} (\mb {u} _{m,t}) = \Gamma (0) -\sum _{i=1}^{m-1} \Gamma (-i) \Phi _{i,m-1}’ \end{eqnarray*}$

and

$\begin{eqnarray*} \Sigma _{m-1}^* = \mr {Cov} (\mb {u} _{m,t-m}^*) = \Gamma (0)-\sum _{i=1}^{m-1}\Gamma (m-i)\Phi _{m-i,m-1}^{*} \end{eqnarray*}$

$P(m)$ are the partial cross-correlation matrices at lag $m$ between the elements of $\mb {y} _{t}$ and $\mb {y} _{t-m}$ , given $\mb {y} _{t-1}, \ldots , \mb {y} _{t-m+1}$ . The matrices $P(m)$ 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 $m=1$ can be an appropriate choice.

Figure 35.41: Partial Correlations (PCORR Option)

The VARMAX Procedure

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 > 2std error, - is < -2std error, . is between

Partial Canonical Correlation Matrices

The partial canonical correlations at lag $m$ between the vectors $\mb {y} _{t}$ and $\mb {y} _{t-m}$ , given $\mb {y} _{t-1}, \ldots , \mb {y} _{t-m+1}$ , are $1\geq \rho _1(m) \geq \rho _2(m) \cdots \geq \rho _ k(m)$ . The partial canonical correlations are the canonical correlations between the residual series $\mb {u} _{m,t}$ and $\mb {u} _{m,t-m}^{*}$ , where $\mb {u} _{m,t}$ and $\mb {u} _{m,t-m}^{*}$ are defined in the previous section. Thus, the squared partial canonical correlations $\rho _ i^2(m)$ are the eigenvalues of the matrix

$\{ \mr {Cov} (\mb {u} _{m,t})\} ^{-1} \mr {E} (\mb {u} _{m,t} \mb {u} _{m,t-m}^{*}) \{ \mr {Cov} (\mb {u} _{m,t-m}^{*})\} ^{-1} \mr {E} (\mb {u} _{m,t-m}^{*} \mb {u} _{m,t}^{}) = \Phi _{mm}^{*} \Phi _{mm}^{}$

It follows that the test statistic to test for $\Phi _ m = 0$ in the VAR model of order $m>p$ is approximately

$(T-m)\mr {~ tr~ } \{ \Phi _{mm}^{*} \Phi _{mm}^{} \} \approx (T-m)\sum _{i=1}^ k \rho _ i^2(m)$

and has an asymptotic chi-square distribution with $k^2$ degrees of freedom for $m>p$ .

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 $\rho _ i(m)$ between $\mb {y} _ t$ and $\mb {y} _{t-m}$ are {0.918, 0.773}, {0.092, 0.018}, and {0.109, 0.011} for lags $m=$ 1 to 3. After lag $m=$ 1, the partial canonical correlations are insignificant with respect to the 0.05 significance level, indicating that an AR order of $m=1$ can be an appropriate choice.

Figure 35.42: Partial Canonical Correlations (PCANCORR Option)

The VARMAX Procedure

Partial Canonical Correlations
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

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, $\bepsilon _ t$ , from the VAR( $p_{\epsilon }$ ), where $p_{\epsilon }$ is chosen sufficiently large. The choice of the autoregressive order, $p_{\epsilon }$ , is determined by use of a selection criterion. From the selected VAR( $p_{\epsilon }$ ) model, you obtain estimates of residual series

$\tilde{ \bepsilon _ t} = \mb {y} _ t - \sum _{i=1}^{p_{\epsilon }} {\hat\Phi }_{i}^{p_{\epsilon }} \mb {y} _{t-i} - {\hat{\bdelta }}^{p_{\epsilon }}, ~ ~ t=p_{\epsilon }+1, \ldots , T$

In the second step, you select the order ( $p,q$ ) of the VARMA model for $p$ in $(p_{min}:p_{max})$ and $q$ in $(q_{min}:q_{max})$

$\mb {y} _ t = \bdelta + \sum _{i=1}^{p}{\Phi _{i}}\mb {y} _{t-i} - \sum _{i=1}^{q}{\Theta _{i}} \tilde{\bepsilon }_{t-i} + {\bepsilon _ t}$

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.

Figure 35.43: MINIC= Option

The VARMAX Procedure

Minimum Information Criterion Based on AICC
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