mmtest - Introduction to Financial Econometrics Hypothesis...

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Introduction to Financial Econometrics Hypothesis Testing in the Market Model Eric Zivot Department of Economics University of Washington February 29, 2000 1 Hypothesis Testing in the Market Model In this chapter, we illustrate how to carry out some simple hypothesis tests concerning the parameters of the excess returns market model regression. 1.1 A Review of Hypothesis Testing Concepts To be completed. 1.2 Testing the Restriction α =0 . Using the market model regression, R t = α + β R Mt + ε t ,t =1 ,...,T ε t iid N (0 , σ 2 ε ) , ε t is independent of R Mt (1) consider testing the null or maintained hypothesis α = 0 against the alternative that α 6 =0 H 0 : α =0 vs. H 1 : α 6 =0 . If H 0 is true then the market model regression becomes R t = β R Mt + ε t and E [ R t | R Mt = r Mt ]= β r Mt . We will reject the null hypothesis, H 0 : α =0 ,i f the estimated value of α is either much larger than zero or much smaller than zero. Assuming H 0 : α = 0 is true, ˆ α N (0 ,SE α ) 2 ) and so is fairly unlikely that ˆ α
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be more than 2 values of SE α ) from zero. To determine how big the estimated value of α needs to be in order to reject the null hypothesis we use the t-statistic t α =0 = b α 0 d SE ( b α ) , where b α is the least squares estimate of α and d SE ( b α ) is its estimated standard error. The value of the t-statistic, t α =0 , gives the number of estimated standard errors that b α is from zero. If the absolute value of t α =0 is much larger than 2 then the data cast considerable doubt on the null hypothesis α =0whereasi fitislessthan2thedata are in support of the null hypothesis 1 . To determine how big | t α =0 | needs to be to reject the null, we use the fact that under the statistical assumptions of the market model and assuming the null hypothesis is true t α =0 Student t with T 2 degrees of freedom If we set the signi f cance level (the probability that we reject the null given that the null is true) of our test at, say, 5% then our decision rule is Reject H 0 : α =0atthe5%leve lif | t α =0 | >t T 2 (0 . 025) where t T 2 is the 2 1 2 % critical value from a Student-t distribution with T 2degrees of freedom. Example 1 Market Model Regression for IBM Consider the estimated MM regression equation for IBM using monthly data from January 1978 through December 1982: b R IBM,t = 0 . 0002 (0 . 0068) +0 . 3390 (0 . 0888) · R Mt ,R 2 =0 . 20 , b σ ε =0 . 0524 where the estimated standard errors are in parentheses. Here b α = 0 . 0002, which is very close to zero, and the estimated standard error, d SE α ) = 0.0068, is much larger than b α . The t-statistic for testing H 0 : α =0vs . H 1 : α 6 =0is t α =0 = 0 . 0002 0 0 . 0068 = 0 . 0363 so that b α is only 0.0363 estimated standard errors from zero. Using a 5% signi f cance level, t 58 (0 . 025) 2and | t α =0 | =0 . 0363 < 2 so we do not reject H 0 :
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mmtest - Introduction to Financial Econometrics Hypothesis...

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