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Unformatted text preview: Economics 241B Hypothesis Testing Under Normality Often, the economic theory that gives rise to a model also yields the values of the coe cients. As the probability that the estimate equals the hypothesized value is zero, we cannot conclude the theory is false simply because the estimated coe cient is not equal to the value speci&ed by theory. To determine the adequacy of the theory, we study the sampling error B & & = ( X X ) & 1 X U: To determine if the sampling error is so large that we reject the theory, we need to construct (from the sampling error) a test statistic whose distribution is known given the truth of the hypothesis. It seems that to do so we need to know the joint distribution of ( X;U ) as the sampling error is a function of both random variables. Surprisingly, if the distribution of U conditional on X is Gaussian, then we can derive the distribution of the test statistic without knowledge of the distribution of X . The restriction to be tested, such as H : & 2 = 0 is termed the null hypothesis. It is a restriction on the set of maintained hypothe- ses, a set of assumptions, which together with the null, produce the test statistic with known distribution. For testing hypotheses about regression coe cients, we need add only that the error, conditional on X , is Gaussian Assumption 5 (Normality of the Error): The distribution of U, condi- tional on X, is jointly normal. Together, Assumptions 1-5 form the maintained hypotheses. The model is cor- rectly speci&ed if the maintained hypotheses are true. Too large a value of the test statistic is taken as evidence that the null is false. Such a conclusion is only correct if the model is correctly speci&ed. If one of the maintained hypotheses is false, then it is possible that the test statistic does not have the supposed distribution under the null. A large value of the test statistic can arise from the failure of the other maintained hypothesis, rather than the null. Implications of Normality Assumption & The distribution depends only on the mean and variance. The mean and variance can be functions of X . If , however, the mean and variance do not depend on X , then the marginal distribution of U is identical to the conditional distribution. & If two random variables are jointly Gaussian, then lack of correlation im- plies independence. This carries over to the conditional distribution, if two random variables are jointly Gaussian and uncorrelated conditional on X , then they are independent conditional on X . & Linear combinations preserve normality. If A is a function of X , then AU is Gaussian conditional on X . With the speci&ed mean and variance of U from preceding assumptions, we have U j X N & ;& 2 I n : Because the conditional distribution does not depend on X , U and X are inde- pendent, which implies U N & ;& 2 I n : It also follows that the sampling error is Gaussian, conditional on X ( B ) j X N ;& 2 ( X X ) & 1 : (0.1) Tests for Individual Regression Coe cient Hypotheses...
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This note was uploaded on 12/26/2011 for the course ECON 241b taught by Professor Staff during the Fall '08 term at UCSB.
- Fall '08