Note: Figures in parentheses are the estimated standard errors of forecast values.

Note: Figures in parentheses are the estimated standard errors of forecast values.

47For further discussion, see, Robert S. Pindyck and Daniel L. Rubinfeld, Econometric Models and Economic Forecasts, McGraw-Hill, 4th ed., 1998, pp. 214-217.

As you can see from the preceding exercise, the dynamic forecasts are closer to their actual values than the static forecasts and the standard errors of dynamic forecasts are smaller than their static counterpart. So, it may be profitable to incorporate the AR(1) scheme (or higher-order schemes) for the purpose of forecasting. However, note that for both types of forecasts the standard errors of forecastfor 1998 are greater than that for 1997, which suggests, not surprisingly, that forecasting into the distant future may be hazardous.

In Chapter 9 we considered dummy variable regression models. In particular, recall the U.S. savings-income regression model for 1970-1995 that we presented in (9.5.1), which for convenience is reproduced below:

where Y = savings X = income

D = 1 for observations in period 1982-1995 D = 0 for observations in period 1970-1981

The regression results based on this model are given in (9.5.4). Of course, this model was estimated with the usual OLS assumptions.

But now suppose that ut follows a first-order autoregressive, AR(1), scheme. That is, ut = put—1 + st. Ordinarily, if p is known or can be estimated by one of the methods discussed above, we can use the generalized difference method to estimate the parameters of the model that is free from (first-order) autocorrelation. However, the presence of the dummy variable D poses a special problem: Note that the dummy variable simply classifies an observation as belonging to the first or second period. How do we transform it? One can follow the following procedure.48

1. In (12.13.1), values of D are zero for all observations in the first period; in period 2 the value of D for the first observation is 1/(1 — p) instead of 1, and 1 for all other observations.

2. The variable Xt is transformed as (Xt — pXt—1). Note that we lose one observation in this transformation, unless one resorts to Prais-Winsten transformation for the first observation, as noted earlier.

3. The value of DtXt is zero for all observations in the first period (note: Dt is zero in the first period); in the second period the first observation takes the value of DtXt = Xt and the remaining observations in the second period are set to (DtXt — DtpXt—1) = (Xt — pXt— 1). (Note: the value of Dt in the second period is 1.)

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