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Econometric theory and methods, Russell Davidson - Chapter 4

Chapter 4 Hypothesis Testing in Linear Regression Models 4.1 Introduction ˆ As we saw in Chapter 3, the vector of OLS parameter estimates β is a random ˆ vector. Since it would be an astonishing coincidence if β were equal to the true parameter vector β0 in any finite sample | Chapter 4 Hypothesis Testing in Linear Regression Models 4.1 Introduction As we saw in Chapter 3 the vector of OLS parameter estimates 3 is a random vector. Since it would be an astonishing coincidence if 3 were equal to the true parameter vector 30 in any finite sample we must take the randomness of 3 into account if we are to make inferences about 3- In classical econometrics the two principal ways of doing this are performing hypothesis tests and constructing confidence intervals or more generally confidence regions. We will discuss the first of these topics in this chapter as the title implies and the second in the next chapter. Hypothesis testing is easier to understand than the construction of confidence intervals and it plays a larger role in applied econometrics. In the next section we develop the fundamental ideas of hypothesis testing in the context of a very simple special case. Then in Section 4.3 we review some of the properties of several distributions which are related to the normal distribution and are commonly encountered in the context of hypothesis testing. We will need this material for Section 4.4 in which we develop a number of results about hypothesis tests in the classical normal linear model. In Section 4.5 we relax some of the assumptions of that model and introduce large-sample tests. An alternative approach to testing under relatively weak assumptions is bootstrap testing which we introduce in Section 4.6. Finally in Section 4.7 we discuss what determines the ability of a test to reject a hypothesis that is false. 4.2 Basic Ideas The very simplest sort of hypothesis test concerns the population mean from which a random sample has been drawn. To test such a hypothesis we may assume that the data are generated by the regression model yt 3 ut ut IID 0 a2 4-01 Copyright 1999 Russell Davidson and James G. MacKinnon 123 124 Hypothesis Testing in Linear Regression Models where yt is an observation on the dependent variable 3 is the population .