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By Hall E. H.

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1. 22) - p - 1) is the lOO(1 - a / 2 ) percentage point of a t distri- THE REGRESSION MODEL AND ITS APPLICATION IN FORECASTlNG 26 bution with n - p - 1 degrees of freedom. If in repeated samples such confidence intervals were constructed, these random intervals would cover the unknown, but fixed, parameter Piin lOO(1 - a) percent of the cases. 2. 23) If a. It) > t,/,(n - p - l), we reject H , in favor of H I at significance level If 1‘1 f t , / , ( n - p - l), there is not enough evidence for rejecting the null hypothesis H , in favor of HI.

If one or more are insignificant (for a chosen significance level a), only the least significant gets dropped from the model. Then the simplified model is reestimated and the procedure is repeated until no variable can be dropped. Most computer packages include programs for backward elimination; usually only the significance level has to be specified. Another strategy for constructing empirical models is to start with the simplest model and then add variables as necessary. Such an approach is called forward selection.

24). Since it is well known [see Hogg and Craig (1978, Chap. 4)] that FJl, n - p - 1) = t;,,(n - p - l), it follows that in this case the ordinary t test and the extra sum of squares test are equivalent. This result gives us additional insight into the meaning of f tests. 'The individual t statistic ti, = b,/s& actually tests whether the variable X, leads to a significant reduction in the error sum of squares (or equivalently, a significant increase in the regression sum of squares) after all other variables XI,...

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