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 that contains the order p of the autoregressive model that the observation represents

  • AIC, a numeric variable that contains the value of the information criterion ${AIC_{p}}$

  • SIGFl, numeric variables that contain the estimate of the innovation covariance matrices for the forward autoregressive models. The variable SIGFl contains the lth column of ${\widehat{\bSigma }_{p}}$ in the observations with ORDER=p.

  • SIGBl, numeric variables that contain the estimate of the innovation covariance matrices for the backward autoregressive models. The variable SIGBl contains the lth column of ${\widehat{\bOmega }_{p}}$ in the observations with ORDER=p.

  • FORk _l, numeric variables that contain 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 ${ \widehat{\bPhi }^{p}_{k}}$ in the observations with ORDER=p.

  • BACk _l, numeric variables that contain 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 ${ \widehat{\bPsi }^{p}_{k}}$ 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 ${ \bPhi ^{p}_{i}}$ is given by the value of the FORi _l variable in the kth observation. The k,lth element of ${ \bPsi ^{p}_{i}}$ is given by the value of BACi _l variable in the kth observation. The k,lth element of $\bSigma $$_{\mi {p} }$ is given by SIGFl in the kth observation. The k,lth element of $\bOmega $$_{\mi {p}}$ is given by SIGBl in the kth observation.

Table 28.2 shows an example of the OUTAR= data set, with ARMAX=3 and ${\mb {x}_{t}}$ of dimension 2. In Table 28.2, ${(i,j)}$ indicate the i,jth element of the matrix.

Table 28.2: 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

AIC$_{0}$

$\bSigma $$_{0(1,1)}$

$\bSigma $$_{0(1,2)}$

$\bOmega $$_{0(1,1)}$

$\bOmega $$_{0(1,2)}$

.

.

.

.

.

2

0

AIC$_{0}$

$\bSigma $$_{0(2,1)}$

$\bSigma $$_{0(2,2)}$

$\bOmega $$_{0(2,1)}$

$\bOmega $$_{0(2,2)}$

.

.

.

.

.

3

1

AIC$_{1}$

$\bSigma $$_{1(1,1)}$

$\bSigma $$_{1(1,2)}$

$\bOmega $$_{1(1,1)}$

$\bOmega $$_{1(1,2)}$

$\bPhi ^{1}_{1}$$_{(1,1)}$

$\bPhi ^{1}_{1}$$_{(1,2)}$

.

.

.

4

1

AIC$_{1}$

$\bSigma $$_{1(2,1)}$

$\bSigma $$_{1(1,2)}$

$\bOmega $$_{1(2,1)}$

$\bOmega $$_{1(1,2)}$

$\bPhi ^{1}_{1}$$_{(2,1)}$

$\bPhi ^{1}_{1}$$_{(2,2)}$

.

.

.

5

2

AIC$_{2}$

$\bSigma $$_{2(1,1)}$

$\bSigma $$_{2(1,2)}$

$\bOmega $$_{2(1,1)}$

$\bOmega $$_{2(1,2)}$

$\bPhi ^{2}_{1}$$_{(1,1)}$

$\bPhi ^{2}_{1}$$_{(1,2)}$

$\bPhi ^{2}_{2}$$_{(1,1)}$

$\bPhi ^{2}_{2}$$_{(1,2)}$

.

6

2

AIC$_{2}$

$\bSigma $$_{2(2,1)}$

$\bSigma $$_{2(1,2)}$

$\bOmega $$_{2(2,1)}$

$\bOmega $$_{2(1,2)}$

$\bPhi ^{2}_{1}$$_{(2,1)}$

$\bPhi ^{2}_{1}$$_{(2,2)}$

$\bPhi ^{2}_{2}$$_{(2,1)}$

$\bPhi ^{2}_{2}$$_{(2,2)}$

.

7

3

AIC$_{3}$

$\bSigma $$_{3(1,1)}$

$\bSigma $$_{3(1,2)}$

$\bOmega $$_{3(1,1)}$

$\bOmega $$_{3(1,2)}$

$\bPhi ^{3}_{1}$$_{(1,1)}$

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

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

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

$\bPhi ^{3}_{3}$$_{(1,1)}$

8

3

AIC$_{3}$

$\bSigma $$_{3(2,1)}$

$\bSigma $$_{3(1,2)}$

$\bOmega $$_{3(2,1)}$

$\bOmega $$_{3(1,2)}$

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

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

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

$\bPhi ^{3}_{2}$$_{(2,2)}$

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


Obs

FOR3_2

BACK1_1

BACK1_2

BACK2_1

BACK2_2

BACK3_1

BACK3_2

1

.

.

.

.

.

.

.

2

.

.

.

.

.

.

.

3

.

$\bPsi ^{1}_{1}$$_{(1,1)}$

$\bPsi ^{1}_{1}$$_{(1,2)}$

.

.

.

.

4

.

$\bPsi ^{1}_{1}$$_{(2,1)}$

$\bPsi ^{1}_{1}$$_{(2,2)}$

.

.

.

.

5

.

$\bPsi ^{2}_{1}$$_{(1,1)}$

$\bPsi ^{2}_{1}$$_{(1,2)}$

$\bPsi ^{2}_{2}$$_{(1,1)}$

$\bPsi ^{2}_{2}$$_{(1,2)}$

.

.

6

.

$\bPsi ^{2}_{1}$$_{(2,1)}$

$\bPsi ^{2}_{1}$$_{(2,2)}$

$\bPsi ^{2}_{2}$$_{(2,1)}$

$\bPsi ^{2}_{2}$$_{(2,2)}$

.

.

7

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

$\bPsi ^{3}_{1}$$_{(1,1)}$

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

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

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

$\bPsi ^{3}_{3}$$_{(1,1)}$

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

8

$\bPhi ^{3}_{3}$$_{(2,2)}$

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

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

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

$\bPsi ^{3}_{2}$$_{(2,2)}$

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

$\bPsi ^{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.