Forecasting with AR errors

[srey]

In the Users Guide II on page 126 the residuals in a dynamic forecast for a model with AR errors are said to be formed as:

\hat{u}_t= \hat{y}_t - x'_t b

using the forecast value of y_t.

My question is, I am not sure why the forecasted residuals would not be 0 since the forecast value of y_t would be determined from the model (i.e., x'_t b)?
Am I reading the explanation correctly?

[startz]

In the Users Guide II on page 126 the residuals in a dynamic forecast for a model with AR errors are said to be formed as:

\hat{u}_t= \hat{y}_t - x'_t b

using the forecast value of y_t.

My question is, I am not sure why the forecasted residuals would not be 0 since the forecast value of y_t would be determined from the model (i.e., x'_t b)?
Am I reading the explanation correctly?

I'm not sure whether you have the current version of the manual or not, but the current version doesn't have a hat on the y.

[srey]

Thanks for the reply. I do have the 7.1 docs dated April 2, 2010. And, in my copy of the Users Guide II on page 126 after the table that follows equation 22.9 the sentence reads:

"where the residuals \hat{u}_t = \hat{y}_t - x_t' b are formed using the forecasted values of y_t."

I see where the line after equation (22.9) omits the hat on the y_t but this is for e_t and not \hat{u}_t.

Can you let me know if this clarifies my question or if my copy is outdated?

Thanks in advance.

[startz]

There's a very simple answer. You're right. (I was reading in the wrong spot.) It must be a typo.

[srey]

I assumed it was a typo as well. But, I'm still not clear since in an ex ante context where y_t is not yet observed, it would seem eviews is doing something different for \hat{y}_t from x_{t}' b. Otherwise if \hat{y}_t = x_{t}'b then clearly \hat{u}_t will be zero. So perhaps in the ex ante period the ar terms are ignored?

[EViews Glenn]

It looks right me...(but that may not be saying much)...

The e_t are the fitted residuals using actual data.
The u_t are the residuals using the forecasted values of the dependent variable.

In the static forecast, you will note that we *always* use the fitted residuals in forecasting using the AR process. Thus, all of the residuals that appear in the Static column are e_t's.

In the dynamic forecast, we use fitted residuals for the pre-forecast-sample values, but the forecasted residuals for insample values. That's why the first row of the Dynamic has the pre-sample fitted residuals for both lagged terms, the second row uses a mix of the fitted and lagged forecast residuals, and the third row uses only forecast residuals. In the latter case, we both of the lagged observations are in the forecast period.

Note that the reason that dynamically forecasted values don't equal the contemporaneous x'b is the presence of the AR components. The whole idea of the AR forecasting exercise is that lagged residuals provide additional information to aid in refining the forecast. You are correct that the residual values would be 0 in that case, but in that case we wouldn't need values for those residuals for our forecasts.

[srey]

Thanks for the detailed explanation.

By forecasted residuals, I'm assuming this means (for an AR(1) process)

\hat{u}_t = \hat{\rho_1} e_{t-1}

\hat{u}_{t+1} = \hat{\rho_1}^2 e_{t-1}

with \hat{\rho_1} the estimated AR coefficient, e_{t-1} the estimation residual, and
the forecast period beginning in t.

Is this interpretation consistent with what is done in Eviews when solving a model for forecast periods without observations on y_t?