复旦大学：《投资学讲义》（英文版） Chapter 18 GMM in explicit discount factor models

Chapter 18 GMM in explicit discount factor models Fan longzhen

Chapter 18 GMM in explicit discount factor models Fan Longzhen

Our task How to estimate and test discount factor model Ep, =e(m data,, parameter)x, 1. Bring an asset pricing model to data to estimate free parameters. For example, parameter B, r in m=B(c /c Or the b inm=o'f 2. Evaluate the model, is it a good model or not? Is another model better?

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? • β,γ γ β − = + ( / ) t 1 t m c c m = b' f ( ( , ) ) t = t+1 t+1 t+1 Ep E m data parameter x

GMM in explicit discount factor model Asset pricing model predicts that E(p)=E[m(data, l,parameters)x,+1 The most natural way to check this prediction is to examine sample average, 1,. e, to calculate ∑na0.m(bamr

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 ( ) [ ( , ) ] t = t+1 t+1 E p E m data parameters x ∑= T t t p T 1 1 ∑= + + T t t t m data parameters x T 1 1 1 [ ( , ) ] 1

GMM and asset pricing model Any asset pricing model implies E(p)=E[m, (6) t+1 Equivalently E[p, -m, (6)x 1=0 or E(m. (6)R4 1-1]=0 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 u, (6)=m,.(6)*+, The sample mean is 8(6)=2u,(b)=ErJu,(L The first stage estimate of b minimizes a quadratic form of the sample mean of the errors b,=arg min6 87(6)Wg(b) For some arbitrary matrix w(often W=l)

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) ( ) [ ( ) ] t = t+1 t+1 E p E m b x E [ pt − mt+1( b ) xt+1] = 0 t t t t u b = m b x − p +1 +1 ( ) ( ) ∑= = = T t T ut b ET ut b T g b 1 ( ) [ ( )] 1 ( ) { } ) ˆ )' ( ˆ argmin ( ˆ b1 ˆ g T b Wg T b b = E [ mt+1( b ) Rt+1 − 1 ] = 0

GMM and asset pricing model continued USing b,, form an estimate s of S=∑Eu(6)x-( Second-stage estimate b2=arg min gr(b)sg,(b) 6 is a consistent, asymptotically normal, and asymptotically efficient estimate of the parameter vector b The variance-covariance matrix of b. is var(b)=-(dsd) · Where ab

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 ˆ b S ˆ { } ) ˆ ( ˆ )' ˆ argmin ( ˆ 1 b 2 ˆ g T b S g T b b − = ∑ ∞ =−∞ = − j t t j S E[u ( b ) u ( b)'] 2 ˆ b 2 ˆ b 1 1 2 ( ' ) 1 ) ˆ var( − − = d S d T b b g b 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. vla var(b) No,)b, [var(b)]b, x(#ofb, Finally, the test of overidentifying restriction is a test of the overall fit of the model TU=I min[gr(b)'58(6)]-x(of moments-#of parameters {b}

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, ~ ( 0,1) ) ˆ var( ˆ N b b ii i ~ (# ) ˆ ) ] ˆ [var( ˆ 1 2 b j b jj b j χ ofb j − { } [ ( )' ( )] ~ (# # ) 1 2 TJ T min g b S g b of moments of parameters T T b T = − − χ

Interpreting the gMM procedure--pricing errors g(6) =E[m(6), ]-EIP, In the language of expected returns 8,(bS proportional to the difference between actual and predicted returns Jensens alp · Because So we can write E(R )=-coV(m, R /E(m) gb=e(mr ) =e(me(R )-(cov(m, R/E(m) Factual mean return-predicted mean return Rf If we express the model in expected return-beta language E(R2)=a1+B then the GMM object is proportional to the Jensen's alpha measure of mispricing g6=a /R

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)/R f • If we express the model in expected return-beta language then the GMM object is proportional to the Jensen’s alpha measure of mispricing ( ) [ ( ) ] [ ] T T t 1 t 1 T t g b = E m b x − E p + + g (b) T E ( R ) cov( m, R ) / E ( m ) e e = − g ( b ) E (mR ) E ( m)( E ( R ) ( cov( m, R ) / E ( m))) e e e = = − − ( ) αi β i 'λ ei E R = + f g ( b ) = αi / R

Why va(g)=wa(∑un)→n∑Bun)=nS This fact suggests that a good weighting matrix might be inverse of S. Hansen(1982)shows formally that the choice W=S-is statistically optimal weighting matrix, meaning that it produces estimates with lowest asymptotic variance

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 t t j T t T t S T E u u T u T g 1 ( ' ) 1 ) 1 var( ) var( 1 1 −1 W = S

Standard errors The formula for the standard error of the estimate va (2)=(dSa) Where it come from?-Delta method Delta method the asymptotic variance of f(x)is f'(x)'var(x) S/T is the variance matrix of the moment 8T Is [Og /ab] Then the delta method gives ab ab gz T ogr agr t

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 1 1 2 ( ' ) 1 ) ˆ var( − − = d S d T b '( ) var( ) 2 f x x T g −1 d T T g b g b ∂ ∂ ∂ ∂ = −1 [ / ] 1 1 2 1 var( ) 1 ) ˆ var( − − = ∂ ∂ ∂ ∂ = d Sd g T b g g b T b T T T

test T 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 bi The J asks whether they are big by statistical method. If it is big the model is rejected The test is T=minlE(b)s"gT (6)x(# of moments-#of parameters)

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 { } [ ( )' ( )] ~ (# # ) 1 2 TJ T min g b S g b of moments of parameters T T b T = − − χ