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 ∂ ∂ = ( )