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04 Hypothesis Testing Under Normality

04 Hypothesis Testing Under Normality - Economics 241B...

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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 0 X ) ° 1 X 0 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 0 : ° 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
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± 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 ² N ° 0 ; ± 2 I n ± : Because the conditional distribution does not depend on X , U and X are inde- pendent, which implies U ² N ° 0 ; ± 2 I n ± : It also follows that the sampling error is Gaussian, conditional on X ( B ° ° ) j X ² N ² 0 ; ± 2 ( X 0 X ) ° 1 ³ : (0.1) Tests for Individual Regression Coe¢ cient Hypotheses We consider H 0 : ° 2 = c; where c is a known value speci°ed by the null hypothesis.
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04 Hypothesis Testing Under Normality - Economics 241B...

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