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Generalized method of moments estimation

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.

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Generalized method of moments estimation

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  1. Generalized method of moments estimation FINA790C HKUST Spring 2006

  2. 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

  3. Basic motivation • Method of moments: suppose we want to estimate the population mean  and population variance 2of a random variable vt • These two parameters satisfy the population moment conditions E[vt] -  = 0 E[vt2] – (2+2) = 0

  4. Method of moments • So let’s think of the analogous sample moment conditions: T-1vt -  = 0 T-1vt2 – ( 2 + 2 ) = 0 • This implies  = T-1vt 2 = T-1(vt - )2 Basic idea: population moment conditions provide information which can be used for estimation of parameters

  5. Example • Simple supply and demand system: qtD = a pt + utD qtS = 1nt + 2pt + utS qtD = qtS = qt • qtD, qtS are quantity demanded and supplied; pt is price and qt equals quantity produced • Problem: how to estimate a, given qt and pt • OLS estimator runs into problems

  6. Example • One solution: find ztD such that cov(ztD, utD) = 0 • Then cov(ztD,qt) – acov(ztD,pt) = 0 • So if E[utD] = 0 then E[ztDqt] – aE[ztDpt] = 0 • Method of moments leads to a* = (T-1ztDqt)/(T-1ztDpt) which is the instrumental variables estimator of a with instrument ztD

  7. 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-1f(vt,θ*T) = 0 (1) where T is the sample size • Intuitively θ*T→ in probability to the solution θ0 of (1)

  8. Generalized method of moments • Now suppose f is a qx1 vector and q>p • The GMM estimator chooses the value of θwhich comes closest to satisfying (1) as the estimator of θ0, where “closeness” is measured by QT(θ) = T-1f(vt,θ)WTT-1f(vt,θ) = gT(θ)WTgT(θ) and WT is psd and plim(WT) = W pd

  9. GMM example 1 • Power utility based asset pricing model • Et[ (Ct+1/Ct)-aRit+1 – 1] = 0 with unknown parameters b, a • The population unconditional moment conditions are E[((Ct+1/Ct)-aRit+1 – 1)zjt] = 0 for j=1,…,q where zjt is in information set

  10. GMM example 2 • CAPM says E[ Rit+1 - 0(1-bi) – biRmt+1 ] =0 • Market efficiency E[( Rit+1 - 0(1-bi) – biRmt+1 )zjt] =0

  11. 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

  12. GMM estimation • The GMM estimator θ*T = argminθεQT(θ) generates the first order conditions {T-1∂f(vt,θ*T)/∂θ´}´WT{T-1f(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

  13. 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

  14. IV estimation • The GMM estimator θ*T = argminθε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

  15. 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)

  16. 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

  17. 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 or F(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.

  18. 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-1f(vt,θ*T)= {Iq - FT(θ*T)[FT(θ*T)’FT(θ*T)]-1FT(θ*T)’}WT½T-1f(vt,θ*T) where FT(θ)= WT½T-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.

  19. Asymptotic properties of GMM instrumental variables estimator It can be shown that: • θ*T is consistent forθ0 • (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]

  20. Covariance matrix estimation • Assuming ut is serially uncorrelated Var[T-½Z’u] = E{T-½ztut}{T-½ztut} = T-1E[ut2ztzt’} • Therefore, S*T = T-1ut(θ*T)2ztzt’

  21. 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

  22. 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 gives θ*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.

  23. 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

  24. (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 + Etct+j - ½ 2 vart[lnct+j] - ½vart[lnRit,t+j] + covt[ lnct+j,lnRit,t+j ]

  25. Example: power utility + lognormal distributions • First order condition from investor maximizing power utility with log-normal distribution gives: ln(Rit+1) = ui + a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

  26. 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

  27. 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

  28. GMM estimation – general case • As before, let gT(θ) = T-1f(vt,θ). The GMM estimator is θ*T = argminθεQT(θ) • The FOCs are GT()’WTgT() = 0 where GT() is a matrix of partial derivatives with i, j element δgTi/δj

  29. GMM asymptotics It can be shown that: • θ*T is consistent forθ0 • asyvar(θ*T,ii) = MSM’ where • M = (G0’WG0)-1G0’W • G0 = E[∂f(vt,θ0)/∂θ’] • S = limT→∞ Var[T-½gT(θ0)]

  30. 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).

  31. 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)

  32. 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-1ft(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) }

  33. 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

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