Because the coefficient of determination R2 always increases when a new independent variable is added to the model, it is tempting to include many variables in a model to force R2 to be near 1. However, doing so reduces the degrees of freedom available for estimating σ2, which adversely affects our ability to make reliable inferences. Suppose you want to use 18 economic indicators to predict next year's Gross Domestic Product (GDP). You fit the model
where y = GDP and x1, x2, . . ., X18 are the economic indicators. Only 20 years of data (n = 20) are used to fit the model, and you obtain R2 = .95. Test to see whether this impressive-looking R2 is large enough for you to infer that the model is useful, that is, that at least one term in the model is important for predicting GDP. Use α = .05.