LinearRegression3

9252012 p kolm 63 the test we test whether the

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Unformatted text preview: est! In order to compute the F statistic, we first run the regression (1) without restriction and (2) rerun it with restriction. The output is shown below VER. 9/25/2012. © P. KOLM 64 Regression without restriction Ordinary Least-squares Estimates Dependent Variable = return R-squared = 0.0395 Rbar-squared = 0.0114 sigma^2 = 1536.1089 Durbin-Watson = 1.6854 Nobs, Nvars = 142, 5 *************************************************************** Variable Coefficient t-statistic t-probability intercept -14.370212 -2.084568 0.038965 dkr 0.320544 1.595457 0.112914 eps 0.042699 0.546448 0.585647 netinc -0.005109 -1.092805 0.276397 salary 0.003499 1.595289 0.112952 ------------------------------------------- VER. 9/25/2012. © P. KOLM 65 Regression with restriction Ordinary Least-squares Estimates Dependent Variable = return R-squared = 0.0000 Rbar-squared = 0.0000 sigma^2 = 1553.8736 Durbin-Watson = 1.6957 Nobs, Nvars = 142, 1 *************************************************************** Variable Coefficient t-statistic t-probability intercept -4.042686 -1.222099 0.223709 ------------------------------------------- F statistic p value VER. 9/25/2012. © P. KOLM = = 1.4077 0.2347 66 The F statistic for the overall significance of the regression with n = 142 and k = 4: (SSRr - SSRur ) / k SSRur / (n - k - 1) (219,096-219,097)/4 = = 1.41 219,097/(142-4-1) F= Critical value at the 5% level with 4 numerator df and 137 denominator df, is F4,137 = 2.44 , which is well above the value of F Reject if F > F4,137 = 2.44 . Therefore, we fail to reject H0 at the 5% level What do we conclude about the EMH from this example? VER. 9/25/2012. © P. KOLM 67 Hypothesis (“individually”): H0: bi = 0 at the 5% level H1: bi ¹ 0 is not true Critical value at the 5% level with 137 df, is t137 = 1.98 Reject if t > t137 = 1.98 . From the outputs we see that no explanatory variable is individually significant at the 5% level The largest absolute t statistic is on dkr, tdkr » 1.60, which is not significant at the 5% level against a two-sided alternative VER. 9/25/2012. © P. KOLM 68 The R 2 form of the F statistic Because the SSR’s may be large and unwieldy, an alternative form of the formula is useful Using the fact that SSR = SST (1 - R2 ) for any regression, we can substitute in for SSRu and SSRur Fº (R 2 ur ) - Rr2 q (1 - R ) (n - k - 1) 2 ur , where again r is restricted and ur is restricted VER. 9/25/2012. © P. KOLM 69 Overall Significance A special case of exclusion restrictions is to test H 0 : b1 = b2 = ... = bk = 0 In this case, our F-statistic takes the form R2 k (SST - SSR) / k F= = 2 SSR / (n - k - 1) 1 - R (n - k - 1) ( ) In these formulae everything refers to the unrestricted model VER. 9/25/2012. © P. KOLM 70 General Linear Restrictions The basic form of the F statistic will work for any set of linear restrictions First estimate the unrestricted model and then estimate the restricted model In each case, make note of the SSR Imposing the restrictions...
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