Forecasting

The VARMAX Procedure

Forecasting

The optimal (MMSE) l-step-ahead forecast of y_t+l is

$y_{t+l| t} &=& \sum_{j=1}^p \Phi_j y_{t+l-j| t} + \sum_{j=0}^s \Theta_j^* x_{t... ...{j=1}^p \Phi_j y_{t+l-j| t} + \sum_{j=0}^s \Theta_j^* x_{t+l-j| t}, l \gt q$

with $y_{t+l-j| t}=y_{t+l-j}$ and $x_{t+l-j| t}=x_{t+l-j}$ for $l\leq j$ .For the forecasts $x_{t+l-j| t}$ , see the previous section.

Impulse Response Function

The VARMAX(p,q,s) model has a convergent representation

$y_{t}=\Psi^{*}(B)x_{t} + \Psi(B) {\epsilon}_{t}$

where $\Psi^{*}(B)=\Phi(B)^{-1}\Theta^{*}(B)=\sum_{j=0}^{\infty}\Psi_j^{*} B^j$ and $\Psi(B)=\Phi(B)^{-1}\Theta(B)=\sum_{j=0}^{\infty}\Psi_j B^j$ .

The elements of the matrices $\Psi_{j}$ from the operator $\Psi(B)$ , called the impulse response, can be interpreted as the impact that a shock in one variable has on another variable. Let $\psi_{j,in}$ be the element of the $\Psi_{j}$ .The notation i is the index for the impulse variable, and n is the index for the response variable (impulse response). For instance, $\psi_{j,11}$ is an impulse response to $y_{1t} arrow y_{1t}$ ,and $\psi_{j,12}$ is an impulse response to $y_{1t} arrow y_{2t}$ .

The accumulated impulse response function is the cumulative sum of the impulse response function, $\Psi^a_l=\sum_{j=0}^l\Psi_j$ .

The MA representation with a standardized white noise innovation process offers a further possibility to interpret a VARMA(p,q) model. Since $\Sigma$ is positive-definite, there is a lower triangular matrix P such that $\Sigma=PP'$ . The alternate MA representation is written as

$y_{t}=\Psi^o(B) u_{t}$

where $\Psi^o(B)=\sum_{j=0}^{\infty}\Psi^o_j B^j$ , $\Psi^o_j=\Psi_j P$ , and $u_{t}=P^{-1}{\epsilon}_t$ .

The elements of the matrices $\Psi^o_{j}$ , called the orthogonal impulse response, can be interpreted as the effects of the components of the standardized shock process u_t on the process y_t at the lag j.

The coefficient matrix $\Psi_j^{*}$ from the transfer function operator $\Psi^{*}(B)$ can be interpreted as the effects that changes in the exogenous variables x_t have on the output variable y_t at the lag j, and is called an impulse response matrix in the transfer function.

The accumulated impulse response in the transfer function is the cumulative sum of the impulse response in the transfer function, $\Psi^{*a}_l=\sum_{j=0}^l\Psi_j^{*}$ .

The asymptotic distributions of the impulse functions can be seen in the "VAR Modeling" section.

The following statements provide the impulse response and the accumulated impulse response in the transfer function for a VARX(1,0) model. Parts of the VARMAX procedure output are shown in Figure 4.24 and Figure 4.25.

   proc varmax data=grunfeld;
      model y1-y3 = x1 x2 / p=1 print=(impulsx=(all)) lagmax=15
                            printform=univariate;
   run;

The VARMAX Procedure

Impulse Response Matrices in Transfer Function by Variable
Variable	Lead	x1	x2
y1	0	1.69281	-0.00859
	1	0.35399	0.01727
	2	0.09090	0.00714
	3	0.05136	0.00214
	4	0.04717	0.00071540
	5	0.04620	0.00039601
	6	0.04487	0.00033224
	7	0.04331	0.00031467
	8	0.04171	0.00030319
	9	0.04016	0.00029220
	10	0.03866	0.00028139
	11	0.03721	0.00027090
	12	0.03582	0.00026077
	13	0.03448	0.00025102
	14	0.03319	0.00024164
	15	0.03195	0.00023260
y2	0	-6.09850	2.57980
	1	-5.15484	0.45445
	2	-3.04168	0.04391
	3	-2.23797	-0.01376
	4	-1.98183	-0.01647
	5	-1.87415	-0.01453
	6	-1.79926	-0.01334
	7	-1.73170	-0.01266
	8	-1.66707	-0.01214
	9	-1.60479	-0.01168
	10	-1.54481	-0.01125
	11	-1.48705	-0.01083
	12	-1.43145	-0.01042
	13	-1.37793	-0.01003
	14	-1.32641	-0.00966
	15	-1.27682	-0.00930
y3	0	-0.02317	-0.01274
	1	1.57476	-0.01435
	2	1.80231	0.00398
	3	1.77024	0.01062
	4	1.70435	0.01197
	5	1.63913	0.01187
	6	1.57727	0.01148
	7	1.51815	0.01105
	8	1.46136	0.01064
	9	1.40672	0.01024
	10	1.35412	0.00986
	11	1.30349	0.00949
	12	1.25476	0.00914
	13	1.20784	0.00879
	14	1.16268	0.00846
	15	1.11921	0.00815

Figure 4.24: Impulse Response in Transfer Function (IMPULSX= option)

The VARMAX Procedure

Accumulated Impulse Response Matrices in Transfer Function by Variable
Variable	Lead	x1	x2
y1	0	1.69281	-0.00859
	1	2.04680	0.00868
	2	2.13770	0.01582
	3	2.18906	0.01796
	4	2.23623	0.01867
	5	2.28243	0.01907
	6	2.32730	0.01940
	7	2.37061	0.01972
	8	2.41232	0.02002
	9	2.45247	0.02031
	10	2.49113	0.02059
	11	2.52834	0.02087
	12	2.56416	0.02113
	13	2.59864	0.02138
	14	2.63183	0.02162
	15	2.66378	0.02185
y2	0	-6.09850	2.57980
	1	-11.25334	3.03425
	2	-14.29502	3.07816
	3	-16.53299	3.06440
	4	-18.51482	3.04793
	5	-20.38897	3.03340
	6	-22.18823	3.02006
	7	-23.91994	3.00741
	8	-25.58701	2.99526
	9	-27.19180	2.98358
	10	-28.73661	2.97233
	11	-30.22366	2.96150
	12	-31.65511	2.95108
	13	-33.03304	2.94105
	14	-34.35946	2.93139
	15	-35.63628	2.92210
y3	0	-0.02317	-0.01274
	1	1.55159	-0.02709
	2	3.35390	-0.02311
	3	5.12414	-0.01249
	4	6.82848	-0.00051706
	5	8.46762	0.01135
	6	10.04489	0.02283
	7	11.56304	0.03389
	8	13.02440	0.04453
	9	14.43112	0.05477
	10	15.78524	0.06463
	11	17.08874	0.07412
	12	18.34349	0.08325
	13	19.55133	0.09205
	14	20.71402	0.10051
	15	21.83323	0.10866

Figure 4.25: Accumulated Impulse Response in Transfer Function (IMPULSX= option)

The following statements provide the impulse response function, the accumulated impulse response function, and the orthogonalized impulse response function with their standard errors for a VAR(1) model. Parts of the VARMAX procedure output are shown in Figure 4.26 through Figure 4.28.

   proc varmax data=simul1;
      model y1 y2 / p=1 noint lagmax=15 print=(impulse=(all)) 
                    printform=univariate;
   run;

The VARMAX Procedure

Impulse Response by Variable
Variable	Lead	y1	y2
y1	1	1.15977	-0.51058
	STD	0.05508	0.05898
	2	1.06612	-0.78872
	STD	0.10450	0.10702
	3	0.80555	-0.84798
	STD	0.14522	0.14121
	4	0.47097	-0.73776
	STD	0.17191	0.15864
	5	0.14315	-0.52450
	STD	0.18214	0.16115
	6	-0.12053	-0.27501
	STD	0.17757	0.15498
	7	-0.29004	-0.04434
	STD	0.16333	0.14731
	8	-0.36060	0.13102
	STD	0.14655	0.14203
	9	-0.34663	0.23455
	STD	0.13382	0.13812
	10	-0.27387	0.26728
	STD	0.12773	0.13267
	11	-0.17160	0.24273
	STD	0.12566	0.12422
	12	-0.06640	0.18106
	STD	0.12293	0.11376
	13	0.02191	0.10361
	STD	0.11643	0.10345
	14	0.08202	0.02870
	STD	0.10579	0.09483
	15	0.11080	-0.03083
	STD	0.09277	0.08778
y2	1	0.54634	0.38499
	STD	0.05779	0.06188
	2	0.84396	-0.13073
	STD	0.08481	0.08556
	3	0.90738	-0.48124
	STD	0.10307	0.09865
	4	0.78943	-0.64856
	STD	0.12318	0.11661
	5	0.56123	-0.65275
	STD	0.14236	0.13482
	6	0.29428	-0.53785
	STD	0.15455	0.14475
	7	0.04744	-0.35732
	STD	0.15690	0.14428
	8	-0.14019	-0.16179
	STD	0.15039	0.13701
	9	-0.25098	0.00929
	STD	0.13856	0.12843
	10	-0.28601	0.13172
	STD	0.12594	0.12184
	11	-0.25974	0.19674
	STD	0.11606	0.11677
	12	-0.19375	0.20836
	STD	0.10987	0.11095
	13	-0.11087	0.17914
	STD	0.10565	0.10314
	14	-0.03071	0.12557
	STD	0.10081	0.09391
	15	0.03299	0.06403
	STD	0.09375	0.08487

Figure 4.26: Impulse Response Function (IMPULSE= option)

Figure 4.26 is the part of output in a univariate format associated with the IMPULSE= option for the impulse response function. The keyword STD stands for the standard errors of the elements.

The VARMAX Procedure

Accumulated Impulse Response by Variable
Variable	Lead	y1	y2
y1	1	2.15977	-0.51058
	STD	0.05508	0.05898
	2	3.22589	-1.29929
	STD	0.21684	0.22776
	3	4.03144	-2.14728
	STD	0.52217	0.53649
	4	4.50241	-2.88504
	STD	0.96922	0.97088
	5	4.64556	-3.40953
	STD	1.51137	1.47122
	6	4.52503	-3.68455
	STD	2.06983	1.95299
	7	4.23500	-3.72889
	STD	2.55669	2.33660
	8	3.87440	-3.59787
	STD	2.89878	2.57280
	9	3.52776	-3.36331
	STD	3.05526	2.65464
	10	3.25389	-3.09603
	STD	3.02480	2.61158
	11	3.08229	-2.85330
	STD	2.84129	2.48797
	12	3.01589	-2.67223
	STD	2.55954	2.31657
	13	3.03780	-2.56862
	STD	2.23480	2.10442
	14	3.11982	-2.53992
	STD	1.90364	1.84193
	15	3.23062	-2.57074
	STD	1.57743	1.52719
y2	1	0.54634	1.38499
	STD	0.05779	0.06188
	2	1.39030	1.25426
	STD	0.17614	0.18392
	3	2.29768	0.77302
	STD	0.36166	0.36874
	4	3.08711	0.12447
	STD	0.65129	0.65333
	5	3.64834	-0.52829
	STD	1.07510	1.06309
	6	3.94262	-1.06614
	STD	1.61541	1.56798
	7	3.99006	-1.42346
	STD	2.20790	2.09305
	8	3.84987	-1.58525
	STD	2.76481	2.55123
	9	3.59889	-1.57595
	STD	3.20063	2.87328
	10	3.31288	-1.44423
	STD	3.45276	3.02575
	11	3.05315	-1.24749
	STD	3.49344	3.01360
	12	2.85940	-1.03913
	STD	3.33163	2.86852
	13	2.74853	-0.86000
	STD	3.00578	2.62918
	14	2.71782	-0.73442
	STD	2.57030	2.32369
	15	2.75080	-0.67040
	STD	2.08022	1.96462

Figure 4.27: Accumulated Impulse Response Function (IMPULSE= option)

Figure 4.27 is the part of output in a univariate format associated with the IMPULSE= option for the accumulated impulse response function.

The VARMAX Procedure

Orthogonalized Impulse Response by Variable
Variable	Lead	y1	y2
y1	0	1.13523	0
	STD	0.08068	0
	1	1.13783	-0.58120
	STD	0.10666	0.14110
	2	0.93412	-0.89782
	STD	0.13113	0.16776
	3	0.61756	-0.96528
	STD	0.15348	0.18595
	4	0.27633	-0.83981
	STD	0.16940	0.19230
	5	-0.02115	-0.59705
	STD	0.17432	0.18830
	6	-0.23313	-0.31306
	STD	0.16731	0.17942
	7	-0.34478	-0.05047
	STD	0.15193	0.17144
	8	-0.36349	0.14914
	STD	0.13501	0.16628
	9	-0.31138	0.26700
	STD	0.12345	0.16158
	10	-0.21732	0.30426
	STD	0.11949	0.15419
	11	-0.10981	0.27631
	STD	0.11916	0.14321
	12	-0.01198	0.20611
	STD	0.11688	0.13033
	13	0.06115	0.11794
	STD	0.10969	0.11822
	14	0.10316	0.03267
	STD	0.09791	0.10849
	15	0.11499	-0.03509
	STD	0.08426	0.10065
y2	0	0.35016	1.13832
	STD	0.11676	0.08855
	1	0.75503	0.43824
	STD	0.06949	0.10937
	2	0.91231	-0.14881
	STD	0.10553	0.13565
	3	0.86158	-0.54780
	STD	0.12266	0.14825
	4	0.66909	-0.73827
	STD	0.13305	0.15846
	5	0.40856	-0.74304
	STD	0.14189	0.16765
	6	0.14574	-0.61225
	STD	0.14785	0.17108
	7	-0.07126	-0.40674
	STD	0.14787	0.16692
	8	-0.21580	-0.18417
	STD	0.14099	0.15808
	9	-0.28167	0.01058
	STD	0.12927	0.14904
	10	-0.27856	0.14994
	STD	0.11684	0.14202
	11	-0.22597	0.22395
	STD	0.10757	0.13587
	12	-0.14699	0.23718
	STD	0.10251	0.12832
	13	-0.06314	0.20392
	STD	0.09939	0.11847
	14	0.00910	0.14294
	STD	0.09504	0.10739
	15	0.05987	0.07288
	STD	0.08779	0.09695

Figure 4.28: Orthogonalized Impulse Response Function (IMPULSE= option)

Figure 4.28 is the part of output in a univariate format associated with the IMPULSE= option for the orthogonalized impulse response function.

Covariance Matrices of Prediction Errors without Exogenous (Independent) Variables

Under the stationarity assumption, the optimal (MMSE) l-step-ahead forecast of y_t+l has an infinite moving-average form, $y_{t+l| t}=\sum_{j=l}^{\infty}\Psi_{j}{\epsilon}_{t+l-j}$ .The prediction error of the optimal l-step-ahead forecast is $e_{t+l| t}=y_{t+l}-y_{t+l| t}=\sum_{j=0}^{l-1}\Psi_{j}{\epsilon}_{t+l-j}$ , with zero mean and covariance matrix

$\Sigma(l)={\rm Cov}(e_{t+l| t} )=\sum_{j=0}^{l-1}\Psi_{j}\Sigma\Psi_{j}'=\sum_{j=0}^{l-1}\Psi^o_{j}\Psi_{j}^{o'}$

where $\Psi_j^o=\Psi_j P$ with a lower triangular matrix P such that $\Sigma=PP'$ .Under the assumption of normality of the ${\epsilon}_t$ , the l-step-ahead prediction error $e_{t+l| t}$ is also normally distributed as multivariate $N(0, \Sigma(l))$ . Hence, it follows that the diagonal elements $\sigma^2_{ii}(l)$ of $\Sigma(l)$ can be used, together with the point forecasts y_i,t+l|t, to construct l-step-ahead prediction interval forecasts of the future values of the component series, y_i,t+l.

The following statements use the COVPE option to compute the covariance matrices of the prediction errors for a VAR(1) model. The parts of the VARMAX procedure output are shown in Figure 4.29 and Figure 4.30.

   proc varmax data=simul1;
      model y1 y2 / p=1 noint print=(covpe(15))  
                    printform=both;
   run;

The VARMAX Procedure

Prediction Error Covarinace Matrices
Lead	Variable	y1	y2
1	y1	1.28875	0.39751
	y2	0.39751	1.41839
2	y1	2.92119	1.00189
	y2	1.00189	2.18051
3	y1	4.59984	1.98771
	y2	1.98771	3.03498
4	y1	5.91299	3.04856
	y2	3.04856	4.07738
5	y1	6.69463	3.85346
	y2	3.85346	5.07010
6	y1	7.05154	4.28845
	y2	4.28845	5.78914
7	y1	7.20389	4.44614
	y2	4.44614	6.18523
8	y1	7.32531	4.49124
	y2	4.49124	6.35575
9	y1	7.47968	4.54222
	y2	4.54222	6.43624
10	y1	7.64792	4.63275
	y2	4.63275	6.51569
11	y1	7.78772	4.73890
	y2	4.73890	6.61576
12	y1	7.87613	4.82560
	y2	4.82560	6.71698
13	y1	7.91875	4.87624
	y2	4.87624	6.79484
14	y1	7.93640	4.89643
	y2	4.89643	6.84041
15	y1	7.94811	4.90204
	y2	4.90204	6.86092

Figure 4.29: Covariances of Prediction Errors (COVPE option)

Figure 4.29 is the output in a matrix format associated with the COVPE option for the prediction error covariance matrices.

The VARMAX Procedure

Prediction Error Covariances by Variable
Variable	Lead	y1	y2
y1	1	1.28875	0.39751
	2	2.92119	1.00189
	3	4.59984	1.98771
	4	5.91299	3.04856
	5	6.69463	3.85346
	6	7.05154	4.28845
	7	7.20389	4.44614
	8	7.32531	4.49124
	9	7.47968	4.54222
	10	7.64792	4.63275
	11	7.78772	4.73890
	12	7.87613	4.82560
	13	7.91875	4.87624
	14	7.93640	4.89643
	15	7.94811	4.90204
y2	1	0.39751	1.41839
	2	1.00189	2.18051
	3	1.98771	3.03498
	4	3.04856	4.07738
	5	3.85346	5.07010
	6	4.28845	5.78914
	7	4.44614	6.18523
	8	4.49124	6.35575
	9	4.54222	6.43624
	10	4.63275	6.51569
	11	4.73890	6.61576
	12	4.82560	6.71698
	13	4.87624	6.79484
	14	4.89643	6.84041
	15	4.90204	6.86092

Figure 4.30: Covariances of Prediction Errors Continued

Figure 4.30 is the output in a univariate format associated with the COVPE option for the prediction error covariances. This printing format more easily explains the forecast limit of each variable.

Covariance Matrices of Prediction Errors in Presence of Exogenous (Independent) Variables

Exogenous variables can be both stochastic and nonstochastic (deterministic) variables. Considering the forecasts in the VARMAX(p,q,s) model, there are two cases.

When exogenous (independent) variables are stochastic (future values not specified)

As defined in the "State-space Modeling" section, $y_{t+l| t}$ has the representation

$y_{t+l| t}=\sum_{j=l}^{\infty}V_{j}a_{t+l-j} + \sum_{j=l}^{\infty}\Psi_{j}{\epsilon}_{t+l-j}$

and hence

$e_{t+l| t}=\sum_{j=0}^{l-1}V_{j}a_{t+l-j} + \sum_{j=0}^{l-1}\Psi_{j}{\epsilon}_{t+l-j}.$

Therefore, the covariance matrix of the l-step-ahead prediction error is given as

$\Sigma(l)={\rm Cov}(e_{t+l| t} )=\sum_{j=0}^{l-1}V_{j}\Sigma_{a} V_{j}' + \sum_{j=0}^{l-1}\Psi_{j}\Sigma_{\epsilon}\Psi_{j}'$

where $\Sigma_{a}$ is the covariance of the white noise series a_t, where a_t is the white noise series for the VARMA(p,q) model of exogenous (independent) variables, which is assumed not to be correlated with ${\epsilon}_t$ or its lags. See the "Forecasting" section for details.

When future exogenous (independent) variables are specified

The optimal forecast $y_{t+l| t}$ of y_t conditioned on the past information and also on known future values x_t+1, ... , x_t+l can be represented as

$y_{t+l| t}=\sum_{j=0}^{\infty}\Psi_{j}^{*}x_{t+l-j} + \sum_{j=l}^{\infty}\Psi_{j}{\epsilon}_{t+l-j}$

and the forecast error is

$e_{t+l| t}=\sum_{j=0}^{l-1}\Psi_{j}{\epsilon}_{t+l-j}.$

Thus, the covariance matrix of the l-step-ahead prediction error is given as

$\Sigma(l)={\rm Cov}(e_{t+l| t} )=\sum_{j=0}^{l-1}\Psi_{j}\Sigma_{\epsilon}\Psi_{j}'.$

Decomposition of Prediction Error Covariances

In the relation $\Sigma(l)=\sum_{j=0}^{l-1}\Psi^o_{j}\Psi_{j}^{o'}$ , the diagonal elements can be interpreted as providing a decomposition of the l-step-ahead prediction error covariance $\sigma_{ii}^2(l)$ for each component series y_it into contributions from the components of the standardized innovations ${\epsilon}_t$ .

If you denote the (i,n)th element of $\Psi^o_{j}$ by $\psi_{j,in}$ ,the MSE of y_i,t+h|t is

${\rm MSE}(y_{i,t+h| t})={\rm E}(y_{i,t+h} - y_{i,t+h| t})^2=\sum_{j=0}^{l-1} \sum_{n=1}^k \psi_{j,in}^2.$

Note that $\sum_{j=0}^{l-1} \psi_{j,in}^2$ is interpreted as the contribution of innovations in variable n to the prediction error covariance of the l-step-ahead forecast of variable i.

The proportion, $\omega_{l,in}$ , of the l-step-ahead forecast error covariance of variable i accounting for the innovations in variable n is

$\omega_{l,in}=\sum_{j=0}^{l-1} \psi_{j,in}^2/{\rm MSE}(y_{i,t+h| t}).$

The following statements use the DECOMPOSE option to compute the decomposition of prediction error covariances and their proportions for a VAR(1) model:

   proc varmax data=simul1;
      model y1 y2 / p=1 noint print=(decompose(15))
                    printform=univariate;
   run;

The VARMAX Procedure

Proportions of Prediction Error Covariances by Variable
Variable	Lead	y1	y2
y1	1	1.00000	0
	2	0.88436	0.11564
	3	0.75132	0.24868
	4	0.64897	0.35103
	5	0.58460	0.41540
	6	0.55508	0.44492
	7	0.55088	0.44912
	8	0.55798	0.44202
	9	0.56413	0.43587
	10	0.56440	0.43560
	11	0.56033	0.43967
	12	0.55557	0.44443
	13	0.55260	0.44740
	14	0.55184	0.44816
	15	0.55237	0.44763
y2	1	0.08644	0.91356
	2	0.31767	0.68233
	3	0.50247	0.49753
	4	0.55607	0.44393
	5	0.53549	0.46451
	6	0.49781	0.50219
	7	0.46937	0.53063
	8	0.45757	0.54243
	9	0.45909	0.54091
	10	0.46567	0.53433
	11	0.47035	0.52965
	12	0.47087	0.52913
	13	0.46865	0.53135
	14	0.46611	0.53389
	15	0.46473	0.53527

Figure 4.31: Decomposition of Prediction Error Covariances (DECOMPOSE option)

The proportions of decomposition of prediction error covariances of two variables are given in Figure 4.31. The output explains that about 91% of the one-step-ahead prediction error covariances of the variable y_2t is accounted for by its own innovations and about 9% is accounted for by y_1t innovations. For the long-term forecasts, 53.5% and 46.5% of the error variance is accounted for by y_2t and y_1t innovations.

Forecasting of the Centered Series

If the CENTER option is specified, the sample mean vector is added to the forecast.

Forecasting of the Differenced Series

If endogenous (dependent) variables are differenced, the final forecasts and their prediction error covariances are produced by integrating those of the differenced series. However, if the PRIOR option is specified, the forecasts and their prediction error variances of the differenced series are produced.

Let z_t be the original series with some zero values appended corresponding to the unobserved past observations. Let $\Delta(B)$ be the k ×k matrix polynomial in the backshift operator corresponding to the differencing specified by the MODEL statement. The off-diagonal elements of $\Delta_{i}$ are zero and the diagonal elements can be different. Then $y_{t}=\Delta(B)z_{t}$ .

This gives the relationship

$z_{t}=\Delta^{-1}(B) y_{t}=\sum_{j=0}^{\infty} \Lambda_{j}y_{t-j}$

where $\Delta^{-1}(B)=\sum_{j=0}^{\infty}\Lambda_{j} B^j$ and $\Lambda_{0}=I_{k}$ .

The l-step-ahead prediction of z_t+l is

$z_{t+l| t}=\sum_{j=0}^{l-1}\Lambda_{j} y_{t+l-j| t} + \sum_{j=l}^{\infty}\Lambda_{j} y_{t+l-j}.$

The l-step-ahead prediction error of z_t+l is

$\sum_{j=0}^{l-1} \Lambda_{j} (y_{t+l-j} - y_{t+l-j| t})=\sum_{j=0}^{l-1} (\sum_{u=0}^j \Lambda_{u} \Psi_{j-u}) {\epsilon}_{t+l-j}.$

Letting $\Sigma_{z}(0)=0$ ,the covariance matrix of the l-step-ahead prediction error of z_t+l, $\Sigma_{z}(l)$ , is

$\Sigma_{z}(l) &=&\sum_{j=0}^{l-1}{(\sum_{u=0}^j \Lambda_{u}\Psi_{j-u}) \Sigma_... ... \Psi_{l-1-j}) \Sigma_{\epsilon} (\sum_{j=0}^{l-1} \Lambda_{j} \Psi_{l-1-j})'.$

If there are stochastic exogenous (independent) variables, the covariance matrix of the l-step-ahead prediction error of z_t+l, $\Sigma_{z}(l)$ , is

$\Sigma_{z}(l) &=& \Sigma_{z}(l-1) +(\sum_{j=0}^{l-1}\Lambda_{j} \Psi_{l-1-j}) ... ...-1}\Lambda_{j} V_{l-1-j}) \Sigma_{a} (\sum_{j=0}^{l-1}\Lambda_{j} V_{l-1-j})'.$

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