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The STATESPACE Procedure |
After computing the sample autocovariance matrices, PROC STATESPACE fits a sequence of vector autoregressive models. These preliminary autoregressive models are used to estimate the autoregressive order of the process and limit the order of the autocovariances considered in the state vector selection process.
Let x_{t} be the r-component stationary time series given by the VAR statement after differencing and subtracting the vector of sample means. (If the NOCENTER option is specified, the mean is not subtracted.) Let n be the number of observations of x_{t} from the input data set.
Let e_{t} be a vector white noise sequence with mean vector 0 and variance matrix , and let n_{t} be a vector white noise sequence with mean vector 0 and variance matrix .Let p be the order of the vector autoregressive model for x_{t}.
The forward autoregressive form based on the past observations is written as follows:
The backward autoregressive form based on the future observations is written as follows:
Letting E denote the expected value operator, the autocovariance sequence for the x_{t} series, , is
The Yule-Walker equations for the autoregressive model that matches the first p elements of the autocovariance sequence are
and
Here are the coefficient matrices for the past observation form of the vector autoregressive model, and are the coefficient matrices for the future observation form. More information on the Yule-Walker equations in the multivariate setting can be found in Whittle (1963) and Ansley and Newbold (1979).
The innovation variance matrices for the two forms can be written as follows:
The autoregressive models are fit to the data using the preceding Yule-Walker equations with replaced by the sample covariance sequence C_{i}. The covariance matrices are calculated as
Let , ,, and represent the Yule-Walker estimates of ,, , and respectively. These matrices are written to an output data set when the OUTAR= option is specified.
When the PRINTOUT=LONG option is specified, the sequence of matrices and the corresponding correlation matrices are printed. The sequence of matrices is used to compute Akaike information criteria for selection of the autoregressive order of the process.
Thus, the AIC for the order p model is computed as
You can use the printed AIC array to compute a likelihood ratio test of the autoregressive order. The log-likelihood ratio test statistic for testing the order p model against the order p-1 model is
This quantity is asymptotically distributed as a with r^{2} degrees of freedom if the series is autoregressive of order p-1. It can be computed from the AIC array as
You can evaluate the significance of these test statistics with the PROBCHI function in a SAS DATA step, or with a table.
By default, PROC STATESPACE selects the order, p, producing the autoregressive model with the smallest AIC_{p}. If the value p for the minimum AIC_{p} is less than the value of the PASTMIN= option, then p is set to the PASTMIN= value. Alternatively, you can use the ARMAX= and PASTMIN= options to force PROC STATESPACE to use an order you select.
The partial autocorrelations are from the sample partial autoregressive matrices .The standard errors used for the significance limits of the partial autocorrelations are computed from the sequence of matrices and .
Under the assumption that the observed series arises from an autoregressive process of order p-1, the pth sample partial autoregressive matrix has an asymptotic variance matrix .
The significance limits for used in the schematic plot of the sample partial autoregressive sequence are derived by replacing and with their sample estimators to produce the variance estimate, as follows:
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