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The STATESPACE Procedure |
Given the order p, let p_{t} be the vector of current and past values relevant to prediction of x_{t+1}:
Let f_{t} be the vector of current and future values:
In the canonical correlation analysis, consider submatrices of the sample covariance matrix of p_{t} and f_{t}. This covariance matrix, V, has a block Hankel form:
The smallest canonical correlation of V^{j} 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 x_{t+k}, given all x_{s} with s less than or equal to t. In the notation x_{i,t+1}, the first subscript denotes the ith component of x_{t+1}.
The initial state vector z^{1}_{t} is set to x_{t}. The sequence f^{j}_{t} is initialized by setting
That is, start by considering whether to add x_{1,t+1|t} to the initial state vector z^{1}_{t}.
The procedure forms the submatrix V^{1} corresponding to f^{1}_{t} and computes its canonical correlations. Denote the smallest canonical correlation of V^{1} as .If is significantly greater than 0, x_{1,t+1|t} is added to the state vector.
If the smallest canonical correlation of V^{1} is not significantly greater than 0, then a linear combination of f^{1}_{t} is uncorrelated with the past, p_{t}. Assuming that the determinant of C_{0} is not 0, (that is, no input series is a constant), you can take the coefficient of x_{1,t+1|t} in this linear combination to be 1. Denote the coefficients of z^{1}_{t} in this linear combination as l. This gives the relationship:
Therefore, the current state vector already contains all the past information useful for predicting x_{1,t+1} and any greater leads of x_{1,t}. The variable x_{1,t+1|t} is not added to the state vector, nor are any terms x_{1,t+k|t} considered as possible components of the state vector. The variable x_{1} is no longer active for state vector selection.
The process described for x_{1,t+1|t} is repeated for the remaining elements of f_{t}. The next candidate for inclusion in the state vector is the next component of f_{t} corresponding to an active variable. Components of f_{t} corresponding to inactive variables that produced a zero in a previous step are skipped.
Denote the next candidate as x_{l,t+k|t}. The vector f^{j}_{t} is formed from the current state vector and x_{l,t+k|t} as follows:
The matrix V^{j} is formed from f^{j}_{t} and its canonical correlations are computed. The smallest canonical correlation of V^{j} is judged to be either greater than or equal to 0. If it is judged to be greater than 0, x_{l,t+k|t} is added to the state vector. If it is judged to be 0, then a linear combination of f^{j}_{t} is uncorrelated with the p_{t}, and the variable x_{l} is now inactive.
The state vector selection process continues until no active variables remain.
where q is the dimension of f^{j}_{t} 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.
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.
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.
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 x_{l,t+k-1|t}.
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