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Forecasting with ARMA Errors—557

Any missing values for the explanatory variables will generate an NA for that observation and in all subsequent observations, via the dynamic forecasts of the lagged dependent variable.

Static Forecasting

Static forecasting performs a series of one-step ahead forecasts of the dependent variable:

• For each observation in the forecast sample, EViews computes:

ˆ	=	ˆ	ˆ	ˆ	ˆ	(18.8)
yS + k	=	c(1 ) + c(2 )xS + k		+ c(3)zS + k	+ c(4 )yS + k − 1	(18.8)

always using the actual value of the lagged endogenous variable.

Static forecasting requires that data for both the exogenous and any lagged endogenous variables be observed for every observation in the forecast sample. As above, EViews will, if necessary, adjust the forecast sample to account for pre-sample lagged variables. If the data are not available for any period, the forecasted value for that observation will be an NA. The presence of a forecasted value of NA does not have any impact on forecasts for subsequent observations.

A Comparison of Dynamic and Static Forecasting

Both methods will always yield identical results in the first period of a multi-period forecast. Thus, two forecast series, one dynamic and the other static, should be identical for the first observation in the forecast sample.

The two methods will differ for subsequent periods only if there are lagged dependent variables or ARMA terms.

Forecasting with ARMA Errors

Forecasting from equations with ARMA components involves some additional complexities. When you use the AR or MA specifications, you will need to be aware of how EViews handles the forecasts of the lagged residuals which are used in forecasting.

Structural Forecasts

By default, EViews will forecast values for the residuals using the estimated ARMA structure, as described below.

For some types of work, you may wish to assume that the ARMA errors are always zero. If you select the structural forecast option by checking Structural (ignore ARMA), EViews computes the forecasts assuming that the errors are always zero. If the equation is estimated without ARMA terms, this option has no effect on the forecasts.

558—Chapter 18. Forecasting from an Equation

Forecasting with AR Errors

For equations with AR errors, EViews adds forecasts of the residuals from the equation to the forecast of the structural model that is based on the right-hand side variables.

In order to compute an estimate of the residual, EViews requires estimates or actual values of the lagged residuals. For the first observation in the forecast sample, EViews will use pre-sample data to compute the lagged residuals. If the pre-sample data needed to compute the lagged residuals are not available, EViews will adjust the forecast sample, and backfill the forecast series with actual values (see the discussion of “Adjustment for Missing Values” on page 550).

If you choose the Dynamic option, both the lagged dependent variable and the lagged residuals will be forecasted dynamically. If you select Static, both will be set to the actual lagged values. For example, consider the following AR(2) model:

yt = xt′β + ut

(18.9)

ut = ρ1ut − 1 + ρ2ut − 2 + t

Denote the fitted residuals as et = yt − xt′b , and suppose the model was estimated using data up to t = S − 1 . Then, provided that the xt values are available, the static and dynamic forecasts for t = S, S + 1, … , are given by:

			Static		Dynamic

	ˆ		ˆ	ˆ	ˆ	ˆ
	yS	xS′b + ρ1eS		− 1 + ρ2eS − 2	xS′b + ρ1eS − 1 + ρ2eS − 2
	ˆ		ˆ	ˆ	ˆ	ˆ	ˆ
	yS + 1	xS + 1′b + ρ1eS + ρ2eS − 1			xS + 1′b + ρ	1uS +	ρ2eS − 1
	ˆ		ˆ	ˆ	ˆ	ˆ	ˆ ˆ
	yS + 2	xS + 2′b + ρ1eS + 1 + ρ2eS			xS + 2′b + ρ	1uS + 1	+ ρ2uS
		ˆ	ˆ	− xt′b are formed using the forecasted values of yt . For sub-
where the residuals ut			= yt

sequent observations, the dynamic forecast will always use the residuals based upon the multi-step forecasts, while the static forecast will use the one-step ahead forecast residuals.

Forecasting with MA Errors

In general, you need not concern yourselves with the details of MA forecasting, since EViews will do all of the work for you. For those of you who are interested in the details of dynamic forecasting, however, the following discussion should aid you in relating EViews results with those obtained from other sources.

The first step in computing forecasts using MA terms is to obtain fitted values for the innovations in the pre-forecast sample period. For example, if you are forecasting the values of y , beginning in period S , with a simple MA(q ):

ˆ	ˆ	ˆ	(18.10)
yS = S + φ1 S − 1		+ … + φq S − q ,

Forecasting with ARMA Errors—559

you will need values for the lagged innovations, S − 1, S − 2, …, S − q .

To compute these pre-forecast innovations, EViews will first assign values for the q innovations prior to the start of the estimation sample, 0, −1, −2, …, −q . If your equation is estimated with backcasting turned on, EViews will perform backcasting to obtain these values. If your equation is estimated with backcasting turned off, or if the forecast sample precedes the estimation sample, the initial values will be set to zero.

Given the initial values, EViews will fit the values of subsequent innovations,

1, 2, …, q, …, S − 1 , using forward recursion. The backcasting and recursion procedures are described in detail in the discussion of backcasting in ARMA models in “Backcasting MA terms” on page 510.

Note the difference between this procedure and the approach for AR errors outlined above, in which the forecast sample is adjusted forward and the pre-forecast values are set to actual values.

The choice between dynamic and static forecasting has two primary implications:

•Once the q pre-sample values for the innovations are computed, dynamic forecasting sets subsequent innovations to zero. Static forecasting extends the forward recursion through the end of the estimation sample, allowing for a series of one-step ahead forecasts of both the structural model and the innovations.

•When computing static forecasts, EViews uses the entire estimation sample to backcast the innovations. For dynamic MA forecasting, the backcasting procedure uses observations from the beginning of the estimation sample to either the beginning of the forecast period, or the end of the estimation sample, whichever comes first.

Example

As an example of forecasting from ARMA models, consider forecasting the monthly new housing starts (HS) series. The estimation period is 1959M01–1984M12 and we forecast for the period 1985M01–1991M12. We estimated the following simple multiplicative seasonal autoregressive model,

hs c ar(1) sar(12)

yielding:

560—Chapter 18. Forecasting from an Equation

Dependent Variable: HS

Method: Least Squares

Date: 01/15/04 Time: 16:34

Sample (adjusted): 1960M02 1984M12

Included observations: 299 after adjusting endpoints

Convergence achieved after 5 iterations

Variable	Coefficient	Std. Error	t-Statistic	Prob.


C	7.317283	0.071371	102.5243	0.0000
AR(1)	0.935392	0.021028	44.48403	0.0000
SAR(12)	-0.113868	0.060510	-1.881798	0.0608

R-squared	0.862967	Mean dependent var		7.313496
Adjusted R-squared	0.862041	S.D. dependent var		0.239053
S.E. of regression	0.088791	Akaike info criterion		-1.995080
Sum squared resid	2.333617	Schwarz criterion		-1.957952
Log likelihood	301.2645	F-statistic		932.0312
Durbin-Watson stat	2.452568	Prob(F-statistic)		0.000000


Inverted AR Roots	.94	.81+.22i	.81-.22i	.59+.59i
	.59-.59i	.22-.81i	.22+.81i	-.22+.81i
	-.22-.81i	-.59+.59i	-.59-.59i	-.81-.22i
	-.81+.22i

To perform a dynamic forecast from this estimated model, click Forecast on the equation toolbar, select Forecast evaluation and unselect Forecast graph. The forecast evaluation statistics for the model are shown below:

The large variance proportion indicates that the forecasts are not tracking the variation in the actual HS series. To plot the actual and forecasted series together with the two standard error bands, you can type:

smpl 1985m01 1991m12

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