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# note04 - 1 Chapter 4 Multiple Regression Analysis Inference...

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1 Chapter 4. Multiple Regression Analysis: Inference We can summarize what we have done so far as follows. Specification of multiple linear regression model OLS (Ordinary Least Squares) estimator minimizes the SSR Once the OLS estimators are computed, we can compute Predicted values of : Regression residuals: Estimator of the variance of error term We have seen that parameter estimators are random and vary from sample to sample. The variances and covariances can be expressed as where depends on the data of explanatory variables. The estimated variance and covariance are computed by replacing the unknown with its estimator In this chapter we will consider the test of hypothesis. Test of Hypotheses Hypothesis, Test Statistic, and Rejection Region Specification of Hypotheses A hypothesis about the parameters is an assertion (or a theory) about the true value of the parameter. For example, which says that the explanatory variable has no effect on the dependent variable, or ( )which says that the marginal effect of and are the same. The assertion we wish to test is called a null hypothesis or a maintained hypothesis , and it is typically denoted by H 0 . The null hypothesis is tested against an alternative hypothesis which is typically denoted by and can take different forms depending on what we wish to test. Examples are

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2 1 Some authors use the term one-tailed and two-tailed test. We will use these terms interchangeably. (a) (b) (c) When a hypothesis specifies a specific value such as , it is called a simple hypothesis . When a hypothesis specifies multiple values such as , it is called a composite hypothesis . Thus, (a) tests a simple null against a simple alternative hypothesis, and (b) and (c) test a simple null against composite alternative hypothesis. When a hypothesis contains more than one statement, it is called a joint hypothesis . An example of a joint null hypothesis . Test Statistic Once the null and alternative hypotheses are specified, we need to choose a test statistic. The choice of the test statistic determines how we use the sample information for the test. This choice can be arbitrary, but a function of the estimator of parameter under consideration is commonly used as a test statistic. For example, we may use the OLS estimator as a test statistic in testing the null hypotheses specified in the examples above. Decision Rule - Choice of Rejection Region Given the test statistic, the decision rule specifies when to reject and when to accept . For example, we may decide to reject in example (a) if the test statistic is closer to 0 than to 1. More specifically, we reject if and accept if for some constant c in an interval [0, 1]. For example (b), we may reject if or if for some constant value c . The set of values for which will be rejected is called a rejection region , or a critical region , of the test. The rejection region usually takes the form of a one- sided or a two-sided rejection region: One-sided rejection region : Two-sided rejection region
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note04 - 1 Chapter 4 Multiple Regression Analysis Inference...

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