Chapter 19 GMM and regression- based tests of linear factor model Fan Longzhen General GMM formula ? Let be an h-vector of variables that are observed at date t, let θ denote an unknown vector of coefficients, ? Be an r-vector real function. Let denote true value of θ, and suppose this true value is characterized by the property that ? A sample of size T is ? Denote sample average of is ? The GMM estimator is the value of that minimizes the scalar t y ),( t yh θ 0 θ { } 0),( 0 = t yhE θ )',...,','( 11 yyy TTT ? =Υ ),( t yh θ )];([)]';([);( TTTT gWgQ ΥΥ=Υ θθθ ),( 1 ),( 1 ∑ = =Υ T t tT yh T g θθ T θ ? θ Where is a sequence of positive definite matrices which may be a function of sample T W example ? Sample is from a standard t-distributions with v degrees of freedom, so that its density is ? If ,its mean is zero, and its variance is ? For large sample T, the sample moment should be close to the population moment ? So we have ? This is classical moment estimator. t y 2/)1(2 2/1 )]/(1[ )2/()( ]2/)1[( );( +? + Γ +Γ = v ttY vy vv v vyf t π 2>v )2/()( 2 2 ?== vvyE t μ 2 1 2 ,2 1 ? μμ ?→?= ∑ = p T t tT y T 1 ? ? 2 ? ,2 ,2 ? = T T T v μ μ Example:Generalized Method of Moments ? If , the population fourth moment of the t-distribution is ? If we want to choose v to match both moment, we have following minimization problem ?where ? W is positive define symmetric matrix reflecting the importance given to matching each of the moments 4>ν )4)(2( 3 )( 2 4 4 ?? == vv v yE t μ {} WggyyvQ T v '),...,,( 1 min = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? = )4)(2( 3 ? 2 ? 2 ,4 ,2 vv v v v g T T μ μ 22× Number of parameter and Number of equation ? If the number of parameter a is the same as the number of the equations r, we simply estimate θ by solving ? Usually r>a, so we have to set estimates to minimize the 0); ? ( =Υ TT g θ )];([)]';([);( TTTT gWgQ ΥΥ=Υ θθθ Optimal weighting matrix ? The optimal weighting matrix is the inverse of asymptotic variance of ? It turn out to be ? Where ? If were serially uncorrelated, then the matrix S could be consistently estimated by ),( 0 T g Υθ {} ∑ ∞ ?∞= ∞→ Γ=ΥΥ= v vTT t ggTES )]';()][;([ 00 lim θθ { })]',()][,([ 00 vttv yhyhE ? =Γ θθ { } ∞ ?∞=t t yh ),( 0 θ (){ })]',()][,([/1 00 1 * tt T t T yhyhTS θθ ∑ = = Optimal weighting matrix--- continued ? If the vector process is serially correlated, the New-West (1987) estimate of S could be ?Where ? Why? ? { } ∞ ?∞=t t yh ),( 0 θ {}) ?? ()]1/([1 ?? ' ,, 1 ,0 TvTv q v TT qvS Γ+Γ+?+Γ= ∑ = { })]', ? ()][, ? ([/1 ? 1 , vtt T vt Tv yhyhT ? += ∑ =Γ θθ )'( )]'()'([...)]()()[1()(]var[ ' 1 ' 1 ' 1 ktt q qv tqtqtttttttt q v v uuE q vq q uuEuuEuuEuuEquuqEu ? ?= ???? = ∑ ∑ ? = ++++?+= Asymptotic distribution of the GMM estimates ? Let be the value that minimizes ? With regarded fixed with respect θ and ? The GMM estimates is typically a solution to the following system of nonlinear equations : ? In many situations( stationary of y, continuity of h(), and restriction on higher moments) it should be the case T θ ? )];([ ? )]';([ 1 TTT gSg ΥΥ ? θθ 1 ? ? T S SS p T ?→? ? T θ ? [ ] 0); ? ( ? ' ' );( 1 ? =Υ×× ? ? ? ? ? ? ? Υ? ? = TTT T gS g T θ θ θ θθ ),0();( 0 SNgT L T ?→?Υθ proposition ? With suitable conditions ?(1) ?(2) ? (3) For any sequence satisfying ? It is case that ?Then ? where 0 ? θθ ?→? P T ),0();( 0 SNgT L T ?→?Υθ { } ∞ =1 * T T θ 0 * θθ ?→? P T ' 0 * ' );( lim ' );( lim ar TT D g p g p T × == = ? ? ? ? ? ? ? Υ? = ? ? ? ? ? ? ? Υ? θθθθ θ θ θ θ ),0() ? ( 0 VNT L T ?→??θθ { } 1 1 ' ? ? = DDSV Delta method ? We want to estimate a quantity that is a nonlinear function of sample means ? The estimates is ? The sample variance is ? For example, a correlation coefficient can be written as a function of sample means as ? Just take )()]([ μφφ == t xEb )] 1 ([ ? 1 ∑ = = T t t x T Eb φ ')',cov(' 1 ) ? var( ∑ ∞ ?∞= ? ? ? ? ? ? ? ? ? ? ? ? ? = j jttT d d xx d d T b μ φ μ φ 2222 ))(()()( )( ),( tttt tttt tt yEyEExEx EyExyxE yxcorr ?? ? = ][ 22 tttttt yExEyEyExEx=μ Using GMM for regressions ? OLS pick parameters to minimize the variance of the residual: ? is derived from the first-order condition,which states that the residual is orthogonal to the right-hand variable ? This condition is exactly identified-the number of moments equals the number of parameters,thus we set the sample moments exactly to zero, and solve the estimate analytically, ? The rest of the ingredients are ? β {} ])'[( 2 min ttT xyE β β ? β )]'([) ? ( ββ tttTT xyxEg ?= [])()'( ? 1 ttTttT yxExxE ? =β tttttttt exxyxxfxxEd =?== )'(),(),'( ββ continued ? The variance of the parameter is ? β 11 )'()()'( 1 )var( ? ∞ ?∞= ?? ? ? ? ? ? ? ? = ∑ tt j jtjttttt xxEexxExxE T εβ Serially uncorrelated, homoskedastic errors ? Formally, the OLS assumptions are ? With these assumptions ? We obtain the familiar formula ? The last notation is typical of econometrics texts, in which ? represents the data matrix. 0,...),,...,,( 211 = ??? ttttt eexxeE 2 211 2 ,...),,...,,( ettttt eexxeE σ= ??? )'()''( 2 ttt j jtjttt xxeEexxeE = ∑ ∞ ?∞= ?? )'()'()()'( 222 ttetttttt xxExxEeExxeE σ== 1212 )'()'( 1 ) ? var( ?? == XXxxE T ette σσβ ]',...,,[ 21 T xxxX = Serially uncorrelated, heteroskedastic errors ? Formally, the assumption is ? With this assumption ? We obtain the formula ? This is known as “ heteroskedastic consistent standard errors” or “white standard errors” (White 1980) 0,...),,...,,( 211 = ??? ttttt eexxeE )'()''( 2 ttt j jtjttt xxeEexxeE = ∑ ∞ ?∞= ?? 121 )'()'()'( 1 ) ? var( ?? = ttttttt xxExxeExxE T β Hansen-Hodrick errors ? Hansen and Hodrick (1982) run forecasting regression of six-month returns, using monthly data, we write it in regression notation as ? Fama-French(1988) also use regressions of overlapping long-horizon returns on variables such as dividend/price ratio: ? Such regressions are are an important part of the evidence for predictability in asset returns, ? Under the nulls that one-period returns are unforecastable, we will still see correlation in the due to overlapping data. Unforecastable returns imply ? So the standard errors are Ttexy kttkkt ,...,2,1' =+= ++ β t e kjforeeE jtt >= ? 0)( ∑ ?= ? ?? ? = k kj ttjtjtttttk xxEexxeExxE T 11 )'()]'([)'( 1 )var(β Regression-based tests of linear factor models-time series regression ? For simple, consider a model with a single factor, we all use excess returns. The beta are defined by regression coefficients ?(1) ? The model states that expected returns are linear in the betas: ? Since the factor is also excess return, the model applies to the factor as well, so ? The model has one and only one implication for the data: all the regression intercepts should be zero; ? Black, Jensen, and Scholes (1972) suggested a natural strategy for estimation and evaluation: run time-series regressions for each asset, the estimate of the factor risk premium is just the sample mean of the factor: ? Then use standard t-test to test if the intercept is zero . i ttii ei t efR ++= βα )()( fERE i ei β= λ×=1)( fE )( ? fE T =λ For a group of assets ? We want to know whether all the pricing errors are jointly zero. ? We can use the following test ? Where is a vector of the estimated intercepts, ? is the residual covariance matrix, I.e. the sample estimate of ?Where () 21 1 ~? ? '? ? )( 1 N T f fE T χαα σ ? ? Σ ? ? ? ? ? ? ? ? ? ? ? ? ? ? + α ? ]' ? ,..., ? , ? [ ? 21 N αααα = Σ ? Σ=)'( tt eeE ]'...,, ? [ ? 21 N ttt eeee= continued ? In regression test, we can derive ? This is the Gibbons, Ross, and Shanken (1989) or “GRS” test statistic. ? If there are many factors, the same ideas work. The regression equation is ? The asset pricing model again predicts that the intercepts should be zero. We can estimate with OLS time- series regression. Assuming normal I.I.d. errors, the quadratic form has the distribution 1, 11 1 2 ~? ? ? )( )( 1 1 ?? ?? ? Σ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ?? NTN T T F f fE N NT αα σ i ttii ei efR ++= 'βα βα, KNTNTT FfEfE N KNT ?? ??? Σ?+ ?? , 111 ~? ? '?))( ? )'(1( αα Where N=number of assets, K=number of factors [][]')()( 1 ? 1 fEffEf T Tt T t Tt ??=? ∑ = Derivation of the Chi-square statistic, and distributions with general errors ? It derived it with GMM. This approach allows us to generate straightforwardly the required corrections for auto correlated and heteroskedastic disturbances. It also serves to remind us that one can do a GMM　estimate of an expected return-beta model. ? Write the equations for all N assets together in vector form ? We use the usual OLS moments to estimate the coefficients tt e t fR εβα ++= 0 ])[( )( )( = ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ?? ?? = tt t T tt e tT t e tT T f E ffRE fRE bg ε ε βα βα continued ? These moments exactly identify the parameters , so weighting matrix is identity matrix. Solving the equation, we obtain the OLS estimates ? The d matrix in the general GMM formula is ? The S matrix is βα, )(var ),(cov ]))([( ]))([( ? )( ? )( ? tT t e tt ttTtT t e tT e tT tT e tT f fR ffEfE fRERE fERE = ? ? = ?= β βα N tt t tNtN tNN T I EfEf Ef fEIfEI fEII b bg d ? ? ? ? ? ? ? = ? ? ? ? ? ? ?= ? ? = 22 1 )()( )( ' )( ∑ ∞ ?∞= ??? ??? ? ? ? ? ? ? ? ? = j jtjtttjttt jtjttjtt ffEfE fEE S )''()'( '' εεεε εεεε continued ? Using GMM variance formula with I=a, we have ? If we assume that errors e and factor f are independent, and the errors are independent over time ? Then the S matrix simplifies to ? Using ' 1 )var( 11 ?? = ? ? ? ? ? ? Sdd Tβ α )'()()'( εεεε EfEfE = )'()()'( 22 εεεε EfEfE = Σ? ? ? ? ? ? ? = ? ? ? ? ? ? = 22 1 )'()'( )'()'( tt t tttttt ttttt EfEf Ef EfEEEf EfEE S εεεε εεεε 111 )( ??? ?=? BABA BDACDCBA ?=?? ))(( continued ? We obtain ? Evaluating the inverse ? We are interested in the top left corner. Using ? we have ? This is the traditional formula, but there is no reason to assume that the errors are I.I.d. or independent of the factors. We can easily construct standard errors and test statistics that do not require these assumptions. Σ? ? ? ? ? ? ? = ? ? ? ? ? ? ?1 2 1 1 ) ? ? var( tt t EfEf Ef T β α Σ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ?1 2 1)var( 11 ) ? ? var( t tt Ef EfEf fT β α va)( 22 EfEf += Σ+= ) )var( 1( 1 ) ? var( 2 f Ef T α Cross-sectional regressions ? Start with the K-factor model, written as ? The central economic question is why average returns vary across assets; expected returns of an asset should be high if that asset has high betas or a large risk exposure to factors that carry high risk premia; ? For the case of a single factor CAPM, the model says the avergae returns should be proportional to betas, so plot the sample average returns against the betas. Even if the model is true, this plot will not work out perfectly in each sample, so there will be some spread as shown. ? Given this fact, a natural idea is to run a cross-sectional regression to fit a line through the scatterplot of the figure. NiRE i ei ,...,3,2,1)( ' == λβ continued ? First find estimates of the betas from time-series regressions ? Then estimate the factor risk premia from a regression across assets of average returns on the betas , ? As in the figure, are right-hand variables, are the regression coefficients, and the cross-sectional regression residuals are the pricing errors. This is known as a two- pass regression estimate. ieachforTtfR i ttii ei t ,...,2,1' =++= εβα λ ii ei T RE αλβ += ')( β λ i α Derivations and formulas for the variance of estimated parameters ? The easy and elegant way to account for the effects of the generated regressors such as beta is to map the whole thing into GMM. ? To keep the algebra manageable, I treat the case of a single factor. The moments are ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ?? ?? = 0 0 0 )( ])[ )( )( βλ β β e tt e t t e t T RE ffaRE faRE bg