OUTAR= Data Set

The STATESPACE Procedure

OUTAR= Data Set

The OUTAR= data set contains the estimates of the preliminary autoregressive models. The OUTAR= data set contains the following variables:

ORDER, a numeric variable containing the order p of the autoregressive model that the observation represents
AIC, a numeric variable containing the value of the information criterion AIC_p
SIGFl, numeric variables containing the estimate of the innovation covariance matrices for the forward autoregressive models. The variable SIGFl contains the lth column of ${\hat{{\Sigma}}_{p}}$ in the observations with ORDER=p.
SIGBl, numeric variables containing the estimate of the innovation covariance matrices for the backward autoregressive models. The variable SIGBl contains the lth column of ${\hat{{\Omega}}_{p}}$ in the observations with ORDER=p.
FORk_l, numeric variables containing the estimates of the autoregressive parameter matrices for the forward models. The variable FORk_l contains the lth column of the lag k autoregressive parameter matrix ${ \hat{{\Phi}}^p_{k}}$ in the observations with ORDER=p.
BACk_l, numeric variables containing the estimates of the autoregressive parameter matrices for the backward models. The variable BACk_l contains the lth column of the lag k autoregressive parameter matrix ${ \hat{{\Psi}}^p_{k}}$ in the observations with ORDER=p.

The estimates for the order p autoregressive model can be selected as those observations with ORDER=p. Within these observations, the k,lth element of ${ {\Phi}^p_{i}}$ is given by the value of the FORi_l variable in the kth observation. The k,lth element of ${ {\Psi}^p_{i}}$ is given by the value of BACi_l variable in the kth observation. The k,lth element of ${\Sigma}$ _p is given by SIGFl in the kth observation. The k,lth element of ${\Omega}$ _p is given by SIGBl in the kth observation.

Table 18.1 shows an example of the OUTAR= data set, with ARMAX=3 and x_t of dimension 2. In Table 18.1, (i,j) indicate the i,jth element of the matrix.

Table 18.1: Values in the OUTAR= Data Set

Obs	ORDER	AIC	SIGF1	SIGF2	SIGB1	SIGB2	FOR1_1	FOR1_2	FOR2_1	FOR2_2	FOR3_1
1	0	AIC₀	${\Sigma}$ _0(1,1)	${\Sigma}$ _0(1,2)	${\Omega}$ _0(1,1)	${\Omega}$ _0(1,2)	.	.	.	.	.
2	0	AIC₀	${\Sigma}$ _0(2,1)	${\Sigma}$ _0(2,2)	${\Omega}$ _0(2,1)	${\Omega}$ _0(2,2)	.	.	.	.	.
3	1	AIC₁	${\Sigma}$ _1(1,1)	${\Sigma}$ _1(1,2)	${\Omega}$ _1(1,1)	${\Omega}$ _1(1,2)	${\Phi}^1_{1}$ _(1,1)	${\Phi}^1_{1}$ _(1,2)	.	.	.
4	1	AIC₁	${\Sigma}$ _1(2,1)	${\Sigma}$ _1(1,2)	${\Omega}$ _1(2,1)	${\Omega}$ _1(1,2)	${\Phi}^1_{1}$ _(2,1)	${\Phi}^1_{1}$ _(2,2)	.	.	.
5	2	AIC₂	${\Sigma}$ _2(1,1)	${\Sigma}$ _2(1,2)	${\Omega}$ _2(1,1)	${\Omega}$ _2(1,2)	${\Phi}^2_{1}$ _(1,1)	${\Phi}^2_{1}$ _(1,2)	${\Phi}^2_{2}$ _(1,1)	${\Phi}^2_{2}$ _(1,2)	.
6	2	AIC₂	${\Sigma}$ _2(2,1)	${\Sigma}$ _2(1,2)	${\Omega}$ _2(2,1)	${\Omega}$ _2(1,2)	${\Phi}^2_{1}$ _(2,1)	${\Phi}^2_{1}$ _(2,2)	${\Phi}^2_{2}$ _(2,1)	${\Phi}^2_{2}$ _(2,2)	.
7	3	AIC₃	${\Sigma}$ _3(1,1)	${\Sigma}$ _3(1,2)	${\Omega}$ _3(1,1)	${\Omega}$ _3(1,2)	${\Phi}^3_{1}$ _(1,1)	${\Phi}^3_{1}$ _(1,2)	${\Phi}^3_{2}$ _(1,1)	${\Phi}^3_{2}$ _(1,2)	${\Phi}^3_{3}$ _(1,1)
8	3	AIC₃	${\Sigma}$ _3(2,1)	${\Sigma}$ _3(1,2)	${\Omega}$ _3(2,1)	${\Omega}$ _3(1,2)	${\Phi}^3_{1}$ _(2,1)	${\Phi}^3_{1}$ _(2,2)	${\Phi}^3_{2}$ _(2,1)	${\Phi}^3_{2}$ _(2,2)	${\Phi}^3_{3}$ _(2,1)

Obs FOR3_2 BACK1_1 BACK1_2 BACK2_1 BACK2_2 BACK3_1 BACK3_2

1 . . . . . . .

2 . . . . . . .

3 . ${\Psi}^1_{1}$ _(1,1) ${\Psi}^1_{1}$ _(1,2) . . . .

4 . ${\Psi}^1_{1}$ _(2,1) ${\Psi}^1_{1}$ _(2,2) . . . .

5 . ${\Psi}^2_{1}$ _(1,1) ${\Psi}^2_{1}$ _(1,2) ${\Psi}^2_{2}$ _(1,1) ${\Psi}^2_{2}$ _(1,2) . .

6 . ${\Psi}^2_{1}$ _(2,1) ${\Psi}^2_{1}$ _(2,2) ${\Psi}^2_{2}$ _(2,1) ${\Psi}^2_{2}$ _(2,2) . .

7 ${\Phi}^3_{3}$ _(1,2) ${\Psi}^3_{1}$ _(1,1) ${\Psi}^3_{1}$ _(1,2) ${\Psi}^3_{2}$ _(1,1) ${\Psi}^3_{2}$ _(1,2) ${\Psi}^3_{3}$ _(1,1) ${\Psi}^3_{3}$ _(1,2)

8 ${\Phi}^3_{3}$ _(2,2) ${\Psi}^3_{1}$ _(2,1) ${\Psi}^3_{1}$ _(2,2) ${\Psi}^3_{2}$ _(2,1) ${\Psi}^3_{2}$ _(2,2) ${\Psi}^3_{3}$ _(2,1) ${\Psi}^3_{3}$ _(2,2)

The estimated autoregressive parameters can be used in the IML procedure to obtain autoregressive estimates of the spectral density function or forecasts based on the autoregressive models.

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Obs	FOR3_2	BACK1_1	BACK1_2	BACK2_1	BACK2_2	BACK3_1	BACK3_2
1	.	.	.	.	.	.	.
2	.	.	.	.	.	.	.
3	.	${\Psi}^1_{1}$ _(1,1)	${\Psi}^1_{1}$ _(1,2)	.	.	.	.
4	.	${\Psi}^1_{1}$ _(2,1)	${\Psi}^1_{1}$ _(2,2)	.	.	.	.
5	.	${\Psi}^2_{1}$ _(1,1)	${\Psi}^2_{1}$ _(1,2)	${\Psi}^2_{2}$ _(1,1)	${\Psi}^2_{2}$ _(1,2)	.	.
6	.	${\Psi}^2_{1}$ _(2,1)	${\Psi}^2_{1}$ _(2,2)	${\Psi}^2_{2}$ _(2,1)	${\Psi}^2_{2}$ _(2,2)	.	.
7	${\Phi}^3_{3}$ _(1,2)	${\Psi}^3_{1}$ _(1,1)	${\Psi}^3_{1}$ _(1,2)	${\Psi}^3_{2}$ _(1,1)	${\Psi}^3_{2}$ _(1,2)	${\Psi}^3_{3}$ _(1,1)	${\Psi}^3_{3}$ _(1,2)
8	${\Phi}^3_{3}$ _(2,2)	${\Psi}^3_{1}$ _(2,1)	${\Psi}^3_{1}$ _(2,2)	${\Psi}^3_{2}$ _(2,1)	${\Psi}^3_{2}$ _(2,2)	${\Psi}^3_{3}$ _(2,1)	${\Psi}^3_{3}$ _(2,2)