Linear beta pricing models: cross-sectional regression tests

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Linear beta pricing modelscross-sectional
regression tests

• FINA790C
• Spring 2006 HKUST

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Motivation

• The F test and ML likelihood ratio test are not
  without drawbacks
• We need T gt N To solve this we could form
  portfolios (but this is not without problems)
• When the model is rejected we dont know why
  (e.g. do expected returns depend on factor
  loadings or on characteristics?)

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Cross-sectional regression

• Can we use the information from the whole
  cross-section of stock returns to test linear
  beta pricing models?
• Fama-MacBeth two-pass cross-sectional regression
  methodology
• Estimate each assets beta from time-series
  regression
• Cross-sectional regression of asset returns on
  constant, betas (and possibly other
  characteristics)
• Run cross-sectional regressions each period,
  average coefficients over time

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Linear beta pricing model

• At time t the returns on the N securities are Rt
  R1t R2t RNt with variance matrix ?R
• Let ft f1t fKt be the vector of time-t
  values taken by the K factors with variance
  matrix ?f
• The linear beta pricing model is ERit ?0
  ?ßi for i1, ,N or
• ERt ?01 B?
• where B E(Rt-E(Rt))(ft-E(ft))?f

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Return generating process

• From the definition of B the time series for Rt
  is
• Rt ERt B( ft-E(ft) ) ut
• with Eut 0 and Eutft 0NxK
• Imposing linear beta pricing model gives
• Rt ?01 B(ft-E(ft)?) ut

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CSR methoddescription

• Define ? ?0 ? ( (K1)x1 vector ) and
• X 1 B ( N x (K1) matrix )
• Assume N gt K and rank(X) K1
• Then ERt 1 B ?0 ? X ?

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CSR first pass

• In first step we estimate ?f and B through usual
  estimators
• ?f (1/T)?(ftµf)(ftµf)
• µf (1/T) ?ft
• B (1/T)?(RtµR)(ftµf)?f-1
• µR (1/T) ?Rt
• In practice we can use rolling estimation period
  prior to testing period

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CSR - second pass

• In second step, for each t 1, , T we use the
  estimate B of the beta matrix and do
  cross-sectional regression of returns on
  estimated B
• ? (XQX)1XQRt (for feasible GLS with
• weighting matrix Q)
• where X 1 B
• The time-series average is
• ? (1/T)?(XQX)1XQRt
• (XQX)1XQ µR

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Fama-MacBeth OLS

• Fama-MacBeth set Q IN and
• ?OLS (XX)1XRt
• The time-series average is
• ?OLS (XX)1XµR
• And the variance of ?OLS is given by
• (1/T) ? (?OLS - ?OLS )(?OLS - ?OLS )

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Issues in CSR methodology

• Dont observe true beta B, but measured B with
  error what is effect on sampling distribution of
  estimates?
• How is CSR methodology related to maximum
  likelihood methodology?

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Sampling distribution of ?

• Let D (XQX)-1XQ, X 1 B
• Basic Result If (Rt, ft) is stationary and
  serially independent then under standard
  assumptions, as T?8, vT(? - ?) converges in
  distribution to a multivariate normal with mean
  zero and covariance
• V D?RD D?D - D(G G)D

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Where does V come from?

• Write µR X? (µR - E(Rt)) (B-B)?
• So vT( ? - ?)
• (XQX)-1XQ vT(µR - E(Rt))
• - (XQX)-1XQ vT(B - B) ?
• Error in estimating ? comes from
• Using average rather than expected returns
• Using estimated rather than true betas

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Comparing V to Fama-MacBeth variance estimator

• From the definition of ?OLS its asymptotic
  variance is
• (XX) -1X?RX(XX)-1 D?RD
• So in general the Fama-MacBeth standard errors
  are incorrect because of the errors-in-variables
  problem

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Special case conditional homoscedasticity of
residuals given factors

• Suppose we also assume that conditional on values
  of the factors ft, the time-series regression
  residuals ut have zero expectation, constant
  covariance ?U and are serially uncorrelated
• This will hold if (Rt, ft) is iid and jointly
  multivariate normal

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Asymptotic variance for special case

• Recall ? ?0 ? ( (K1)x1 vector ) and define
  the (K1)x(K1) bordered matrix
• ?f 0 0K
• 0K ?f
• Then Basic Result holds with
• V ?f (1 ??f-1?)D?UD
• Asymptotically valid standard errors are
  obtained by substituting consistent extimates for
  the various parameters

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Example Sharpe-Lintner-Black CAPM

• For k1, this simplifies to
• The usual Fama-MacBeth variance estimator
  (ignoring estimation error in betas) understates
  the correct variance except under the null
  hypothesis that ?1 (market risk premium) 0

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Maximum likelihood and two-pass CSR

• MLE estimates B and ? simultaneously and thereby
  solves the errors-in-variables problem.
• Asymptotic covariance matrix of the two-pass
  cross-sectional regression GLS estimator ? is
  the same as that for MLE
• I.e. two-pass GLS is consistent and
  asymptotically efficient as T?8

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Two-pass GLS

• For given T, as N ?8 however, the two-pass GLS
  estimator still suffers from an
  errors-in-variables problem from using B (i.e.
  two-pass GLS is not N-consistent)
• We can make the two-pass GLS estimator
  N-consistent as well through a simple
  modification (see Litzenberger and Ramaswamy
  (1979), Shanken (1992))

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Modified two-pass CSR

• For example Sharpe-Lintner-Black CAPM estimated
  with two-pass OLS
• The errors-in-variable problem applies to betas,
  or the lower right-hand block of XX. Note
  that
• E(ßß) ßß tr(?U)/(TsM2)
• So deduct the last term from the lower-right hand
  block this adjustment corrects for the EIV
  problem as N ?8.