Statistical Analysis of the CAPM I. Sharpe–Linter CAPM

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• As the F –test based on (28) is exact, it is, for realistic sample sizes, clearly preferable compared to the likelihood ratio test relying on asymptotic arguments. • However, for the zero–beta version, an exact test is much more difficult to obtain, and it may be useful to consider what is lost when relying on asymptotic arguments. 32
Finite–sample size of likelihood ratio test for nominal size 5% N = number of assets, T = sample size For example, for N = 10, and T = 60, the critical value for a LRT with asymptotic size 5% is 18.307, which corresponds to a critical value of the exact F – test of cF = ( { } ) T − N − 1 LRT exp − 1 N T = 49 ( { } ) 18.307 exp − 1 = 1.748. 10 60 The actual size of the LRT in this situation is therefore 1 − F cdf (1.748; 49, 10) = 0.096. 33
Page 1 and 2: Statistical Analysis of the CAPM I.
Page 3 and 4: Framework for Estimation and Testin
Page 5 and 6: - Note, however, that we still assu
Page 7 and 8: • The density of excess returns,
Page 9 and 10: • From (11), it is clear that the
Page 11 and 12: • To find the MLE of Σ, we make
Page 13 and 14: The log-likelihood function can be
Page 15 and 16: COV ( β) = = = 1 T σ 4 COV m 1 T
Page 17 and 18: Testing for mean-variance efficienc
Page 19 and 20: • The asymptotic distribution of
Page 21 and 22: • By the same line of arguments,
Page 23 and 24: • Using, in (25), and X = √ T [
Page 25 and 26: • It can be shown that 4 J = ( )
Page 27 and 28: • Next, it is easily checked that
Page 29 and 30: shows that β 0 = β 1 + ¯rm σ 2
Page 31: When this formula is applied to (36
Page 35 and 36: Roll’s Critique • Roll (1977) 5
Page 37 and 38: • On the other hand, often our in

Statistical Analysis of the CAPM I. Sharpe–Linter CAPM

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