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 The STATESPACE Procedure

## Canonical Correlation Analysis

Given the order p, let pt be the vector of current and past values relevant to prediction of xt+1:

pt = ( x't, x't-1, ... , x't-p)'

Let ft be the vector of current and future values:

ft = ( x't, x't+1, ... , x't+p)'

In the canonical correlation analysis, consider submatrices of the sample covariance matrix of pt and ft. This covariance matrix, V, has a block Hankel form:

### State Vector Selection Process

The canonical correlation analysis forms a sequence of potential state vectors, zjt. Examine a sequence, fjt, of subvectors of ft, and form the submatrix, Vj, consisting of the rows and columns of V corresponding to the components of fjt, and compute its canonical correlations.

The smallest canonical correlation of Vj is then used in the selection of the components of the state vector. The selection process is described in the following. For more details about this process, refer to Akaike (1976).

In the following discussion, the notation denotes the wide sense conditional expectation (best linear predictor) of xt+k, given all xs with s less than or equal to t. In the notation xi,t+1, the first subscript denotes the ith component of xt+1.

The initial state vector z1t is set to xt. The sequence fjt is initialized by setting

That is, start by considering whether to add x1,t+1|t to the initial state vector z1t.

The procedure forms the submatrix V1 corresponding to f1t and computes its canonical correlations. Denote the smallest canonical correlation of V1 as .If is significantly greater than 0, x1,t+1|t is added to the state vector.

If the smallest canonical correlation of V1 is not significantly greater than 0, then a linear combination of f1t is uncorrelated with the past, pt. Assuming that the determinant of C0 is not 0, (that is, no input series is a constant), you can take the coefficient of x1,t+1|t in this linear combination to be 1. Denote the coefficients of z1t in this linear combination as l. This gives the relationship:

Therefore, the current state vector already contains all the past information useful for predicting x1,t+1 and any greater leads of x1,t. The variable x1,t+1|t is not added to the state vector, nor are any terms x1,t+k|t considered as possible components of the state vector. The variable x1 is no longer active for state vector selection.

The process described for x1,t+1|t is repeated for the remaining elements of ft. The next candidate for inclusion in the state vector is the next component of ft corresponding to an active variable. Components of ft corresponding to inactive variables that produced a zero in a previous step are skipped.

Denote the next candidate as xl,t+k|t. The vector fjt is formed from the current state vector and xl,t+k|t as follows:

The matrix Vj is formed from fjt and its canonical correlations are computed. The smallest canonical correlation of Vj is judged to be either greater than or equal to 0. If it is judged to be greater than 0, xl,t+k|t is added to the state vector. If it is judged to be 0, then a linear combination of fjt is uncorrelated with the pt, and the variable xl is now inactive.

The state vector selection process continues until no active variables remain.

### Testing Significance of Canonical Correlations

For each step in the canonical correlation sequence, the significance of the smallest canonical correlation, , is judged by an information criterion from Akaike (1976). This information criterion is

where q is the dimension of fjt at the current step, r is the order of the state vector, p is the order of the vector autoregressive process, and is the value of the SIGCORR= option. The default is SIGCORR=2. If this information criterion is less than or equal to 0, is taken to be 0; otherwise, it is taken to be significantly greater than 0. (Do not confuse this information criterion with the AIC.)

Variables in are not added in the model, even with positive information criterion, because of the singularity of V. You can force the consideration of more candidate state variables by increasing the size of the V matrix by specifying a PASTMIN= option value larger than p.

### Printing the Canonical Correlations

To print the details of the canonical correlation analysis process, specify the CANCORR option in the PROC STATESPACE statement. The CANCORR option prints the candidate state vectors, the canonical correlations, and the information criteria for testing the significance of the smallest canonical correlation.

Bartlett's and its degrees of freedom are also printed when the CANCORR option is specified. The formula used for Bartlett's is

with r (p+1)-q+1 degrees of freedom.

Figure 18.11 shows the output of the CANCORR option for the introductory example shown in the "Getting Started" section of this chapter.

 The STATESPACE Procedure Canonical Correlations Analysis

 x(T;T) y(T;T) x(T+1;T) Information Criterion Chi Square DF 1 1 0.237045 3.566167 11.4505 4

 x(T;T) y(T;T) x(T+1;T) y(T+1;T) Information Criterion Chi Square DF 1 1 0.238244 0.056565 -5.35906 0.636134 3

 x(T;T) y(T;T) x(T+1;T) x(T+2;T) Information Criterion Chi Square DF 1 1 0.237602 0.087493 -4.46312 1.525353 3
Figure 18.11: Canonical Correlations Analysis

New variables are added to the state vector if the information criteria are positive. In this example, and are not added to the state space vector because the information criteria for these models are negative.

If the information criterion is nearly 0, then you may want to investigate models that arise if the opposite decision is made regarding .This investigation can be accomplished by using a FORM statement to specify part or all of the state vector.

### Preliminary Estimates of F

When a candidate variable xl,t+k|t yields a zero and is not added to the state vector, a linear combination of fjt is uncorrelated with the pt. Because of the method used to construct the fjt sequence, the coefficient of xl,t+k|t in l can be taken as 1. Denote the coefficients of zjt in this linear combination as l.

This gives the relationship:

The vector l is used as a preliminary estimate of the first r columns of the row of the transition matrix F corresponding to xl,t+k-1|t.

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