## Hypothesis Testing with Artificial Regressions

Artificial regressions like the GNR are probably employed most frequently for hypothesis testing. Suppose we wish to test a set of r equality restrictions on 9. Without loss of generality, we can assume that these are zero restrictions. This allows us to partition 9 into two subvectors, 91 of length k – r, and 92 of length r, the restrictions being that 92 = 0. If the estimator 0 is not only root-n consistent but also asymptotically normal, an appropriate statistic for testing these restrictions is

0ї(^г (02))-102, (1.19)

which will be asymptotically distributed as x2(r) under the null if ^«(02) is a suitable estimate of the covariance matrix of 02.

Suppose that r(9) and R(9) define an artificial regression for the estimator 0...

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