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## Generalized method of moments estimation

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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 2of 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 - )2

Basic idea: population moment conditions provide information which can be used for estimation of parameters

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

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-1ztDqt)/(T-1ztDpt)

which is the instrumental variables estimator of a with instrument ztD

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 θwhich 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)-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

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

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 = argminθεQT(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 = argminθεQT(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 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.

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)’}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.

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]

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 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.

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) = ui + alnCt+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 = argminθεQT(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 forθ0
- 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

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