get reliable estimates of the betas -but if too long leads to a problem of non-constant betas (see structural stability tests below). STAGE 2: Estimate a second pass cross-section regression using the estimated betasfrom stage 1 (using data for a later period than used in stage 1)where x is a vector of other risk factors. This vector could include e.g., the own variance of the security obtained from stage 1 (measuring diversifiablerisk). ()itftMtiiftitrrrrεβα+−+=−iiiifixrrεγβγβγγ++++=−32210ˆˆWarwick Business School 7
Basic procedure for testing CAPM
Key predictions: If CAPM is true then we’d expect at stage 2:Also important to…Test the stage 1 disturbances for linear dependence (joint test of market efficiency and CAPM) + stability of beta estimates.Before carrying out inferences on stage 2 coefficients test disturbances for heteroscedasticityand non-normality (use an alternative estimator if there is heteroscedasticity…)risk) blediversifia e.g.,risk beta-non of influence systematic (no 0off/SML)-dereturn tra-risk of (linearity 0off)-dereturn tra-risk (positive 003210==>=γγγγSecurity Market LineWarwick Business School 8
Measurement error in the betas
The fundamental problem with this procedure is that the true betas are estimated with errorThis will lead to biased (and inconsistent) estimates of using these betas as regressorsat stage 2 ⇒iiiv+=ββˆ 1γMeasurement errorTrue betaConsequences of measurement error in the explanatory variables (see Gujarati Chp 13.5) Consider the model iiiXYεγγ++=*10 Suppose we only observe the explanatory variable with measurement error: ()()()()()00,,0 where,**22*=====+=iiiiiiviiiiiXEvXEvEvEvEvXXεεσ Therefore the model estimated is: ()iiiiiiivXvXY11010γεγγεγγ−++=+−+= The OLS estimator of 1γis given by ()⎟⎠⎞⎜⎝⎛−+=⎥⎦⎤⎢⎣⎡−+=−=−==ΣΣΣΣΣΣΣiiiiiiiiiiiiiiiiiiiiiiiivxxxvxxxYYyXXxxyx1211122111, ,ˆγεγγεγγ Now ()()()0*=+=iiiiivxExEεεand ()()()()22*viiiiiivEvvxEvxEσ==+=. Consequently the OLS estimator is biased downwards: ()12212121111ˆγσσγσγγγ<⎟⎟⎠⎞⎜⎜⎝⎛−=⎟⎠⎞⎜⎝⎛−=Σ−xviivxTEEWarwick Business School 9
Measurement error in the betas
One solution is to sort the securities into portfolios and estimate betas for the portfolios.For example, the beta for an equally weighted portfolio of msecurities isAssuming the v are thenThe bias in the stage 2 cross-sectional OLS estimator is therefore reduced (see previous slide) Σ==miipm1ˆ1ˆββ()2,0viidσ()221ˆvarvvpmσσβ<=Warwick Business School 10
Testing CAPM on portfolios of securities
Portfolios are often formed by ranking securities into portfolios sorted by:
Size (market cap)
Betas
Book-to-Market (B-M) value
Size andbeta/B-M value
The procedure involves sorting the data by the ranking variable(s).
Then dividing the sorted data into portfolios.
Example:You have 20 stocks sorted in ascending order of size.
A portfolio consisting of the first4 stocks corresponds to the lowest quintile of the size distribution.
A portfolio consisting of the last4 stocks corresponds to the highest quintile of the size distribution.Warwick Business School 11
Problems with using portfolios
Loss of information/variation in betas. Ideally want portfolioswhich average out measure
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