Forecasting with linear regression with ar(1) error

[Gelfan]

Hi,

I'm trying to understand in detail how Eviews calculates the forecast standard error for a linear regression equation that includes an ar(1) error term, I.e.

Y_t = X_t' * B + u_t where u_t = p * u_{t-1} + e_t, e_t is normal(0,sigma^2) (*)

I am using the example House1.wf1 workfile from Chapter 5 "Forecasting from an Equation" of the User II guide. I fit the model listed on page 111 minus the lagged dependent variable term HS(-1), i.e the model HS = C SP AR(1) using nonlinear least squares and the data through 1990m01. Then I created a dynamic forecast for the period 1990m02 to 1996m06 and saved the point forecasts of HS and the standard errors.

I am able to replicate in excel the point forecasts, but I can't replicate the forecast s.e. My questions are:

(1) How does eviews calculate the forecast s.e. in the case of not including the coefficient variance and in the case of including the coefficient variance?
(2) What are the formulas and/or procedure to calculate the forecast s.e. for each point in the forecast? Further does eviews assume that the forecast error is a normal or t-distribution?

I have Economic Models and Economic Forecasts by Pindyck and Rubenstien. Is there another text that would go into more detail about forecasting with ar and sar terms in general?

Thanks,
Gelfan

[EViews Glenn]

I assume you are doing dynamic forecasting...

. Without coefficient variances, you just get the residual standard error squared (from the unbiased estimate), run through the AR filter.

In the first period, we initialize the lagged residual variance to zero so you'll get s0 = sigma^2.
In the second period, you get a variance of s1 = sigma^2*(1 + p^2) = sigma^2 + p^2*s0
In the third period, you'll get s2 = sigma^2 + p^2*(sigma^2*(1 + p^2)) = sigma^2 + p^2*s1
...

These follow from the standard result var(Y + bX) = var(Y) + b^2*var(X) provided Y and X are uncorrelated.

. Add coefficient uncertainty at each step by finding the variance of the linear combination of regressors. For nonlinear models, use the delta method.

. The distribution doesn't matter for computation of variances.

. Without looking, I'm certain that Box and Jenkins will have a discussion.