Chapter 18 GMM in explicit discount factor models Fan Longzhen Our task How to estimate and test discount factor model. 1. Bring an asset pricing model to data to estimate free parameters. For example, parameter in ? Or the b in 2. Evaluate the model, is it a good model or not? Is another model better? ? γβ, γ β ? + = )/( 1 tt ccm fbm '= )),(( 111 +++ = tttt xparameterdatamEEp GMM in explicit discount factor model ? Asset pricing model predicts that ? The most natural way to check this prediction is to examine sample average, i,.e., to calculate ? and ]),([)( 11 ++ = ttt xparametersdatamEpE ∑ = T t t p T 1 1 ∑ = ++ T t tt xparametersdatam T 1 11 ]),([ 1 GMM and asset pricing model ? Any asset pricing model implies ? Equivalently or ? Where x and p are typically vectors; we typically check whether a model for m can price a number of assets simultaneously. So the equation is often called moment conditions. ? Define errors as ? The sample mean is ? The first stage estimate of b minimizes a quadratic form of the sample mean of the errors, ? ? For some arbitrary matrix W (often W=I) ])([)( 11 ++ = ttt xbmEpE 0])([ 11 =? ++ ttt xbmpE tttt pxbmbu ?= ++ 11 )()( ∑ = == T t tTtT buEbu T bg 1 )]([)( 1 )( {} ) ? ()' ? (minarg ? ?1 bWgbgb TT b = 0]1)([ 11 =? ++ tt RbmE GMM and asset pricing model--- continued ? Using , form an estimate of ? ? Second-stage estimate ? is a consistent, asymptotically normal, and asymptotically efficient estimate of the parameter vector b. ? The variance-covariance matrix of is ? ?Where 1 ? b S ? {} ) ? ( ? )' ? (minarg ? 1 ?2 bgSbgb TT b ? = ∑ ∞ ?∞= ? = j jtt bubuES ])'()([ 2 ? b 2 ? b 11 2 )'( 1 ) ? var( ?? = dSd T b b bg d T ? ? = )( Test of parameters ? This variance-covariance matrix can be used to test whether a parameter or a group of parameters is equal to zero, via ? Finally, the test of overidentifying restriction is a test of the overall fit of the model, )1,0(~ ) ? var( ? N b b ii i )(#~ ? ]) ? [var( ? 21 jjjjj ofbbbb χ ? {} )#(#~)]()'([ 21 min parametersofmomentsofbgSbgTTJ TT b T ?= ? χ Interpreting the GMM procedure—pricing errors ? In the language of expected returns, is proportional to the difference between actual and predicted returns: Jensen’s alphas. ? Because ? So we can write ? =(actual mean return-predicted mean return)/Rf ? If we express the model in expected return-beta language then the GMM object is proportional to the Jensen’s alpha measure of mispricing ][])([)( 11 tTttTT pExbmEbg ?= ++ )(bg T )(/),cov()( mERmRE ee ?= )))(/),cov(()()(()()( mERmREmEmREbg eee ??== λβα ')( ii ei RE += f i Rbg /)( α= Why ? This fact suggests that a good weighting matrix might be inverse of S. Hansen (1982) shows formally that the choice is statistically optimal weighting matrix, meaning that it produces estimates with lowest asymptotic variance. 1? S ∑∑ ∞ ?∞= ? = + =→= j jtt T t tT S T uuE T u T g 1 )'( 1 ) 1 var()var( 1 1 1? = SW Standard errors ? The formula for the standard error of the estimate, ? Where it come from? –”Delta method” ? Delta method: the asymptotic variance of f(x) is ? S/T is the variance matrix of the moment . ?is ? Then the delta method gives 11 2 )'( 1 ) ? var( ?? = dSd T b )var()(' 2 xxf T g 1? d T T g b bg ? ? =?? ?1 ]/[ 11 2 1 )var( 1 ) ? var( ?? = ? ? ? ? = Sdd Tg b g g b T b T T T test ? You have estimated parameters that make a model “fit best”. The natural question is how does it fit? ? It is natural to look at pricing errors and see if they are “big”. ? The asks whether they are big by statistical method. If it is big, the model is “rejected”. ? The test is T J T J {} )#(#~)]()'([ 21 min parametersofmomentsofbgSbgTTJ TT b T ?= ? χ Applying GMM ? Simply mapping a given problem into the very general notation: ? Using price ? Using returns (1) ? It is common to add instruments as well, you multiply both sides of ? By any variable observed at time t, and take unconditional expectation, resulting in ?Or 0])([ 11 =? ++ ttt pxbmE 0]1)([ 11 =? ++ tt RbmE 1])([ 11 = ++ ttt RbmE t z )(])([ 11 ttttt zEzRbmE = ++ {})2(0]1)([ 11 =? ++ tttt zRbmE (1) And (2) are both important for asset pricing test For example ? Using returns ? Start with two returns and one instrument z,the model that we will test is ? ? Denote and using the Kronecker product ],[ ba RR ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ++ ++ ++ ++ 0 0 0 0 1 1 )( )( )( )( 11 11 11 11 t t t b tt t a tt b tt a tt z z zRbm zRbm Rbm Rbm E ? 0)])'1,1(())(([ 11 =??? ++ tttt zzRbmE 0]1)([ 11 =? ++ tt RbmE )',1( tt zz =