Two Approaches To Testing Hypotheses

Hypothesis testing of the sort suggested above can be approached from two viewpoints. First, having computed a set of parameter estimates, we can ask whether the estimates come reasonably close to satisfying the restrictions implied by the hypothesis. More formally, we can ascertain whether the failure of the estimates to satisfy the restrictions is simply the result of sampling error or is instead systematic. An alternative approach might proceed as follows. Suppose that we impose the restrictions implied by the theory. Since unrestricted least squares is, by definition, "least squares," this imposition must lead to a loss of fit. We can then ascertain whether this loss of fit results merely from sampling error or whether it is so large as to cast doubt on the validity of the restrictions. We will consider these two approaches in turn, then show that (as one might hope) within the framework of the linear regression model, the two approaches are equivalent.

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