signal predictions in multivariate state space estimation

[s.wright@bbk.ac.uk]

Hi - I'm using Eviews 14 to estimate a state space model with 3 observables. Each signal series is driven only by its own hidden state variable, but the shocks to the states have a covariance structure, which I'm estimating by default (using evar statements). If I generate the signal predictions (generating using makesignals(t=pred,n)) I get the odd feature that a freely estimated regression of the signal on its predictions results in a coefficient sometimes a long way from one, which is what I, at least would have expected. As a direct result the signal errors (which impose a coefficient of one) are not minimum variance.

Note that this only happens when I assume nonzero covariances: if I set these to zero (so the the predictive regressions are in effect univariate) I get an estimated coefficient of almost precisely one.

For reference I append two examples below,

Now I'm aware that the system is being estimate by maximum likelihood, which does not necessarily result in the same estimates as least squares; but this still seems an odd feature. Are you aware of this? Does it arise from some inherent feature of the Kalman Filter or Maximum Likelihood estimation, that I have missed? Or is it possible there is a bug in the way predictions are generated in eviews?

I can provide the code if that helps.

I'd be very grateful for your thoughts.

Thanks

Stephen


With estimated covariances

Dependent Variable: DL
Method: Least Squares
Date: 03/25/25 Time: 18:13
Sample (adjusted): 2 199
Included observations: 198 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

C 0.01950826880736265 0.03851191845641564 0.5065514674227506 0.6130389163930581
DL_F_SS_M_COV_D 1.499149985339002 0.1481819753881737 10.11695235815197 1.261894928912547e-19

With zero covariances

Dependent Variable: DL
Method: Least Squares
Date: 03/25/25 Time: 18:15
Sample (adjusted): 2 199
Included observations: 198 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

C -0.001162282628598479 0.0414056309920197 -0.02807064161931238 0.977634342377867
DL_F_SS_M_COV_D 1.002957127382335 0.1296628138318804 7.735117708325848 5.313325235671231e-13