## F

9The following F test is a special case of the more general F test given in (8.7.9) or (8.7.10) in Sec. 8.7.

264 PART ONE: SINGLE-EQUATION REGRESSION MODELS

which is about the same as that obtained from (8.5.17), except for the rounding errors. This F is highly significant, reinforcing our earlier finding that the variable FLR belongs in the model.

A cautionary note: If you use the R2 version of the F test given in (8.5.11), make sure that the dependent variable in the new and the old models is the same. If they are different, use the F test given in (8.5.16).

When to Add a New Variable. The F-test procedure just outlined provides a formal method of deciding whether a variable should be added to a regression model. Often researchers are faced with the task of choosing from several competing models involving the same dependent variable but with different explanatory variables. As a matter of ad hoc choice (because very often the theoretical foundation of the analysis is weak), these researchers frequently choose the model that gives the highest adjusted R2. Therefore, if the inclusion of a variable increases R2, it is retained in the model although it does not reduce RSS significantly in the statistical sense. The question then becomes: When does the adjusted R2 increase? It can be shown that R2 will increase if the t value of the coefficient of the newly added variable is larger than 1 in absolute value, where the t value is computed under the hypothesis that the population value of the said coefficient is zero [i.e., the t value computed from (5.3.2) under the hypothesis that the true p value is zero].10 The preceding criterion can also be stated differently: R2 will increase with the addition of an extra explanatory variable only if the F( = t2) value of that variable exceeds 1.

Applying either criterion, the FLR variable in our child mortality example with a t value of —10.6293 or an F value of 112.9814 should increase R2, which indeed it does—when FLR is added to the model, R2 increases from 0.1528 to 0.6981.

When to Add a Group of Variables. Can we develop a similar rule for deciding whether it is worth adding (or dropping) a group of variables from a model? The answer should be apparent from (8.5.18): If adding (dropping) a group of variables to the model gives an F value greater (less) than 1, R2 will increase (decrease). Of course, from (8.5.18) one can easily find out whether the addition (subtraction) of a group of variables significantly increases (decreases) the explanatory power of a regression model. 