Tentative Order Selection
Previous Page|Next Page

The VARMAX Procedure

Tentative Order Selection

Sample Cross-Covariance and Cross-Correlation Matrices
Partial Autoregressive Matrices
Partial Correlation Matrices
Partial Canonical Correlation Matrices
The Minimum Information Criterion (MINIC) Method

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 36.39 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 36.39: 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 > 2*std error,  - is < -2*std 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 $m$,

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

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 36.40 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 36.40: 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 > 2*std error,  - is < -2*std error,  . is between


Partial Correlation Matrices

Define the forward autoregression

$\displaystyle \mb {y} _{t} = \sum _{i=1}^{m-1} \Phi _{i,m-1} \mb {y} _{t-i} + \mb {u} _{m,t} $

and the backward autoregression

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

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

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

where

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

and

$\displaystyle \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}^{*} $

$P(m)$ 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 36.41 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 36.41: 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 > 2*std error,  - is < -2*std 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 36.42 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 36.42: 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. 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, $\bepsilon _ t$, 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 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

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


Previous Page|Next Page|Top of Page
Copyright © SAS Institute Inc. All Rights Reserved.