SLN F-test Spring 2007

# SLN F-test Spring 2007 - F-test and Joint Hypotheses...

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F-test and Joint Hypotheses Supplementary Lecture Notes: Spring 2007 Joint Hypothesis Test: Statistical procedure for testing more than one coefficient at a time. This involves an F-test. Consider the following regression model: E(Y) = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + β 4 X 4 + β 5 X 5 Suppose we want to test the following joint hypotheses. Ho: β 2 = 0, β 4 = 0, β 5 = 0 H 1 : At least one coefficient in Ho 0 Note: The null states that the parameters, β 2 , β 4 , and β 5 , are simultaneously (i.e., jointly) equal to zero, i.e., β 2 = 0 and β 4 = 0 and β 5 = 0. The alternative hypothesis states that at least one of the parameters in the null is not zero. “At least” means either one of the parameters is not zero or a combination of any of the parameters is not zero or all the parameters stated in the null are not zero. In other words, the alternative hypothesis says that β 2 0 OR β 4 0 OR β 5 0 OR ( β 2 0 and β 4 0) OR ( β 2 0 and β 5 0) OR ( β 4 0 and β 5 0) OR ( β 2 0 and β 4 0 and β 5 0 ) The null hypothesis imposes parameter restrictions on the model. As such, the null implies what is called a Restricted Model. In this case since the parameters β 2 , β 4 , and β 5 , are being restricted to zero values, the restricted model excludes three terms from the original model. In other words we are left with a model with only two explanatory variables. Restricted Model: E(Y) = β 0 + β 1 X 1 + β 2 X 2 The model without restrictions is simply the original model and is called the Unrestricted Model. Unrestricted Model: E(Y) = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 3 + β 4 X 4 + β 5 X 5 Thus the test involves estimating two regressions; the restricted and unrestricted model. The basic idea underlying the test is to see whether these two models are not statistically different.

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SLN F-test Spring 2007 - F-test and Joint Hypotheses...

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