generalized method of moments estimation fina790c hkust spring 2006
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Generalized method of moments estimation
FINA790C
HKUST Spring 2006
Outline
• Basic motivation and examples• First case: linear model and instrumental
variables– Estimation– Asymptotic distribution– Test of model specification
• General case of GMM– Estimation– Testing
Basic motivation
• Method of moments: suppose we want to estimate the population mean and population variance 2 of a random variable vt
• These two parameters satisfy the population moment conditions
E[vt] - = 0
E[vt2] – (2+2) = 0
Method of moments
• So let’s think of the analogous sample moment conditions:
T-1vt - = 0
T-1vt2 – ( 2 + 2 ) = 0
• This implies
= T-1vt
2 = T-1(vt - )2Basic idea: population moment conditions provide information which can be used for estimation of parameters
Example
• Simple supply and demand system:
qtD = pt + ut
D
qtS = 1nt + 2pt + ut
S
qtD = qt
S = qt
• qtD, qt
S are quantity demanded and supplied; pt is price and qt equals quantity produced
• Problem: how to estimate , given qt and pt
• OLS estimator runs into problems
Example
• One solution: find ztD such that cov(zt
D, utD) = 0
• Then cov(ztD,qt) – cov(zt
D,pt) = 0
• So if E[utD] = 0 then E[zt
Dqt] – E[ztDpt] = 0
• Method of moments leads to * = (T-1zt
Dqt)/(T-1ztDpt)
which is the instrumental variables estimator of with instrument zt
D
Method of moments
• Population moment condition: vector of observed variables, vt, and vector of p parameters θ, satisfy a px1 element vector of conditions E[f(vt,θ)] = 0 for all t
• The method of moments estimator θ*T solves the analogous sample moment conditions
gT(θ*) = T-1f(vt,θ*T) = 0 (1)where T is the sample size
• Intuitively θ*T → in probability to the solution θ0 of (1)
Generalized method of moments
• Now suppose f is a qx1 vector and q>p
• The GMM estimator chooses the value of θ which comes closest to satisfying (1) as the estimator of θ0, where “closeness” is measured by
QT(θ) = T-1f(vt,θ)WTT-1f(vt,θ) = gT(θ)WTgT(θ)
and WT is psd and plim(WT) = W pd
GMM example 1
• Power utility based asset pricing model
• Et[ (Ct+1/Ct)-Rit+1 – 1] = 0 with unknown parameters
• The population unconditional moment conditions are
E[((Ct+1/Ct)-Rit+1 – 1)zjt] = 0 for j=1,…,q where zjt is in information set
GMM example 2
• CAPM says
E[ Rit+1 - 0(1-i) – iRmt+1 ] =0
• Market efficiency
E[( Rit+1 - 0(1-i) – iRmt+1 )zjt] =0
GMM example 3
• Suppose the conditional probability density function of the continuous stationary random vector vt, given Vt-1={vt-1,vt-2,…} is p(vt;θ0,Vt-1)
• The MLE of θ0 based on the conditional log likelihood function is the value of which maximizes LT(θ) = ln{p(vt;θ,Vt-1)}, i.e. which solves ∂LT(θ)/∂θ = 0
• This implies that the MLE is just the GMM estimator based on the population moment condition
E[ ∂ln{p(vt;θ,Vt-1)}/∂θ} ] =0
GMM estimation
• The GMM estimator θ*T = argminθ ε QT(θ) generates the first order conditions
{T-1∂f(vt,θ*T)/∂θ´}´WT{T-1f(vt,θ*T)} = 0 (2)
where ∂f(vt,θ*T)/∂θ´ is the q x p matrix with i,j element ∂fi(vt,θ*T)/∂θj´
• There is typically no closed form solution for θ*T so it must be obtained through numerical optimization methods
Example: Instrumental variable estimation of linear model
• Suppose we have yt = xt´θ0 + ut, t=1,…,T• Where xt is a (px1) vector of observed
explanatory variables, yt is an observed scalar, ut is an unobserved error term with E[ut] = 0
• Let zt be a qx1 vector of instruments such that E[zt´ut]=0 (contemporaneously uncorrelated)
• Problem is to estimate θ0
IV estimation
• The GMM estimator θ*T = argminθ ε QT(θ)
where QT(θ) = {T-1u(θ)’Z}WT{T-1Z’u(θ)}
• The FOCs are
(T-1X’Z)WT(T-1Z’y) = (T-1X’Z)WT(T-1Z’X)θ*T
• When # of instruments q = # of parameters p (“just-identified”) and (T-1Z’X) is nonsingular then
θ*T = (T-1Z’X)-1(T-1Z’y)
independently of the weighting matrix WT
IV estimation
• When # instruments q > # parameters p (“over-identified”) then
θ*T = {(T-1X’Z)WT(T-1Z’X)}-1(T-1X’Z)WT(T-1Z’y)
Identifying and overidentifying restrictions
• Go back to the first-order conditions• (T-1X’Z)WT(T-1Z’u(θ*T)) = 0• These imply that GMM = method of
moments based on population moment conditions E[xtzt’]WE[ztut(θ0)] = 0
• When q = p GMM = method of moments based on E[ztut(θ0)] = 0
• When q > p GMM sets p linear combinations of E[ztut(θ0)] equal to 0
Identifying and over-identifying restrictions: details
• From (2), GMM is method of moments estimator based on population moments
{E[∂f(vt,θ0)/∂θ’]}’W{E[f(vt,θ0)]} = 0 (3)
• Let F(θ0)=W½E[∂f(vt,θ0)/∂θ´] and rank(F(θ0))=p. (The rank condition is necessary for identification of θ0). Rewrite (3) as
F(θ0)’W½E[f(vt,θ0)] = 0 orF(θ0)[F(θ0)’F(θ0)]-1F(θ0)’W½E[f(vt,θ0)]=0 (4)
• This says that the least squares projection of W½E[f(vt,θ0)] on to the column space of F(θ0) is zero. In other words the GMM estimator is based on rank{F(θ0)[F(θ0)’F(θ0)]-1F(θ0)’} = p restrictions on the (transformed) population moment condition W½E[f(vt,θ0)]. These are the identifying restrictions; GMM chooses θ*T to satisfy them.
Identifying and over-identifying restrictions: details
• The restrictions that are left over are
{Iq - F(θ0)[F(θ0)’F(θ0)]-1F(θ0)’}W½E[f(vt,θ0)]=0
• This says that the projection of W½E[f(vt,θ0)] on to the orthogonal complement of F(θ0) is zero, generating q-p restrictions on the transformed population moment condition. These over-identifying restrictions are ignored by the GMM estimator, so they need not be satisfied in the sample
• From (2),
WT½T-1f(vt,θ*T)= {Iq - FT(θ*T)[FT(θ*T)’FT(θ*T)]-1FT(θ*T)’}WT
½T-1f(vt,θ*T) where FT(θ)= WT
½T-1∂f(vt,θ)/ ∂θ´. Therefore QT(θ*T) is like a sum of squared residuals, and can be interpreted as a measure of how far the sample is from satisfying the over-identifying restrictions.
Asymptotic properties of GMM instrumental variables estimator
It can be shown that:
• θ*T is consistent for θ0
• (θ*T,i – θ0,i)/√V*T,ii/T ~ N(0,1) asymptotically where– V*T = (X’ZWTZ’X)-1X’ZWTS*TWTZ’X (X’ZWTZ’X)-1
– S*T = limT→∞ Var[T-½Z’u]
Covariance matrix estimation
• Assuming ut is serially uncorrelated
Var[T-½Z’u] = E{T-½ztut}{T-½ztut}
= T-1E[ut2ztzt’}
• Therefore,
S*T = T-1ut(θ*T)2ztzt’
Two step estimator
• Notice that the asymptotic variance depends on the weighting matrix WT
• The optimal choice is WT = S*T-1 to give V*T = (X’Z
S*T-1Z’X)-1
• But we need θ*T to construct S*T This suggests a two-step (iterative) procedure– (a) estimate with sub-optimal WT, get θ*T(1),get S*T(1)
– (b) estimate with WT = S*T(1)-1
– (c) iterated GMM: from (b) get θ*T(2), get S*T(2), set WT = S*T(2)-1, repeat
GMM and instrumental variables
• If ut is homoskedastic: var[u]=2IT then S*T = *2ZtZt’ where Zt is a (Txq) matrix with t-th row = zt’ and *2 is a consistent estimate of 2
• Choosing this as the weighting matrix WT = S*T-1
givesθ*T = {X’Z(Z’Z)-1Z’X)}-1 X’Z(Z’Z)-1Z’y = {X^’X^}-1X^’y
where X^=Z(Z’Z)-1Z’X is the predicted value of X from a regression of X on Z. This is two-stage least squares (2SLS): first regress X on Z, then regress y on the fitted value of X from the first stage.
Model specification test
• Identifying restrictions are satisfied in the sample regardless of whether the model is correct
• Over-identifying restrictions not imposed in the sample
• This gives rise to the overidentifying restrictions test
JT = TQT(θ*T) = T-½u(θ*T)’Z S*T-1 T-½ Z’u(θ*T)
• Under H0: E[ztut(θ0)] = 0, JT asymptotically follows a 2(q-p) distribution
(From last time)
• In this case the MRS is
mt,t+j = j{ Ct+j /Ct } -
• Substituting in above gives:
Et[ln(Rit,t+j)] =
-jln + Etct+j - ½ 2 vart[lnct+j] - ½vart[lnRit,t+j]
+ covt[ lnct+j,lnRit,t+j ]
Example: power utility + lognormal distributions
• First order condition from investor maximizing power utility with log-normal distribution gives:
ln(Rit+1) = i + lnCt+1 + uit+1
• the error term uit+1 could be correlated with lnCt+1 so we can’t use OLS
• However Et[uit+1] = 0 means uit+1 is uncorrelated with any zt known at time t
Instrumental variables regressions for returns and consumption growth
Return Stage 1 Stage 2 JT test r Δlnc γ r Δlnc
CP 0.297 0.102 -0.953 -0.118 0.221 0.091 (0.00) (0.15) (0.57) (0.11) (0.00) (0.10)
Stock 0.110 0.102 -0.235 -0.008 0.105 0.097 (0.11) (0.15) (1.65) (0.06) (0.06) (0.08)
Annual US data 1889-1994 on growth in log real consumption of nondurables & services, log real return on S&P500, real return on 6-month commercial paper. Instruments are 2 lags each of: real commercial paper rate, real consumption growth rate and log of
dividend-price ratio
GMM estimation – general case
• Go back to GMM estimation but let f be a vector of continuous nonlinear functions of the data and unknown parameters
• In our case, we have N assets and the moment condition is that E[(m(xt,0)Rit-1)zjt-1]=0 using instruments zjt-1 for each asset i=1,…,N and each instrument j=1,…,q
• Collect these as f(vt,)= zt-1’ut(Xt,) where zt is a 1xq vector of instruments and ut is a Nx1 vector. f is a column vector with qN elements – it contains the cross-product of each instrument with each element of u.
• The population moment condition is
E[f(vt,0)] = 0
GMM estimation – general case
• As before, let gT(θ) = T-1f(vt,θ). The GMM estimator is θ*T = argminθ ε QT(θ)
• The FOCs are
GT()’WTgT() = 0
where GT() is a matrix of partial derivatives with i, j element δgTi/δj
GMM asymptotics
It can be shown that:
• θ*T is consistent for θ0
• asyvar(θ*T,ii) = MSM’ where
– M = (G0’WG0)-1G0’W
– G0 = E[∂f(vt,θ0)/∂θ’]
– S = limT→∞ Var[T-½gT(θ0)]
Covariance matrix estimation for GMM
In practice V*T = M*TS*TM*T’ is a consistent estimator of V, where
• M*T = (GT(θ*T)’WTGT(θ*T))-1 GT(θ*T)’WT
• S*T is a consistent estimator of S• Estimator of S depends on time series
properties of f(vt,0). In general it is
S = Γ0 + ( Γi + Γi’)
where Γi = E{ft-E(ft)}{ft-i-E(ft-i)}’=E[ftft-I’] is the i-th autocovariance matrix of ft = f(vt,0).
GMM asymptotics
• So (θ*T,i – θ0,i)/√V*T,ii/T ~ N(0,1) asymptotically
• As before, we can choose WT = S*T-1 and
– V*T is then (GT(θ*T)’ST-1GT(θ*T))-1
– a test of the model’s over-identifying restrictions is given by JT = TQT(θ*T) which is asymptotically 2(qN-p)
Covariance matrix estimation
• We can consistently estimate S with
S*T = Γ*0(θ*T) + ( Γ*i(θ*T) + Γ*i(θ*T)’)
where Γ*i(θ*T) = T-1ft(vt,θ*T)ft-i(vt-i,θ*T)’• If theory implies that the autocovariances
of f(vt,θ0) = 0 for some lag j then we can exclude these from S*T
– e.g. ut are serially uncorrelated implies S*T = T-1{ ut(vt,θ*T)ut(vt,θ*T)’ (zt’zt) }
GMM adjustments
• Iterated GMM is recommended in small samples
• More powerful tests by subtracting sample means of ft(vt,θ*T) in calculating Γ*i(θ*T)
• Asymptotic standard errors may be understated in small samples: multiply asymptotic variances by “degrees of freedom adjustment” T/(T-p) or (N+q)T/{(N+q)T – k} where k = p+((Nq)2+Nq)/2