GMM广义矩估计介绍

Econometrics2—Fall2005
Generalized Method of Moments
(GMM)Estimation
Heino Bohn Nielsen
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Outline
(1)Introduction and motivation
(2)Moment Conditions and Identification
(3)A Model Class:Instrumental Variables(IV)Estimation
(4)Method of Moment(MM)Estimation
Examples:Mean,OLS and Linear IV
(5)Generalized Method of Moment(GMM)Estimation
Properties:Consistency and Asymptotic Distribution
(6)Efficient GMM
Examples:Two-Stage Least Squares
(7)Comparison with Maximum Likelihood
Pseudo-ML Estimation
(8)Empirical Example:C-CAPM Model
Introduction
Generalized method of moments(GMM)is a general estimation principle.
Estimators are derived from so-called moment conditions.
Three main motivations:
(1)Many estimators can be seen as special cases of GMM.
Unifying framework for comparison.
(2)Maximum likelihood estimators have the smallest variance in the class of consistent
and asymptotically normal estimators.
But:We need a full description of the DGP and correct specification.
GMM is an alternative based on minimal assumptions.
(3)GMM estimation is often possible where a likelihood analysis is extremely difficult.
We only need a partial specification of the model.
Models for rational expectations.
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Moment Conditions and Identification
•A moment condition is a statement involving the data and the parameters:
g(θ0)=E[f(w t,z t,θ0)]=0.(∗) whereθis a K×1vector of parameters;f(·)is an R dimensional vector of(non-linear)functions;w t contains model variables;and z t contains instruments.
•If we knew the expectation then we could solve the equations in(∗)tofindθ0.•If there is a unique solution,so that
E[f(w t,z t,θ)]=0if and only ifθ=θ0,
then we say that the system is identified.
•Identification is essential for doing econometrics.Two ideas:
(1)Is the model constructed so thatθ0is unique(identification).
(2)Are the data informative enough to determineθ0(empirical identification).
Example:Under-Identification •Consider again a regression model
y t=x0tβ0+ t=x01tγ0+x02tδ0+ t.
•Assume that the K1variables in x1t are predetermined,while the K2=K−K1 variables in x2t are endogenous.That implies
E[x1t t]=0(K1×1)(†)
E[x2t t]=0(K2×1).(††)
•We have K parameters inβ0=(γ00,δ00)0,but only K1<K moment conditions
(i.e.K1equations to determine K unknowns).
The parameters are not identified and cannot be estimated consistently.
11of32 Example:Simple IV Estimator •Assume K2new variables,z2t,that are correlated with x2t but uncorrelated with t:
E[z2t t]=0.(†††) The K2moment conditions in(†††)can replace(††).To simplify notation,we define
=µx1t x2t¶and z t(K×1)=µx1t z2t¶.
x t
(K×1)
x t are model variables,z2t are new instruments,and z t are instruments.
We say that x1t are instruments for themselves.
•Using(†)and(†††)we have K moment conditions:
g(β0)=µE[x1t t]E[z2t t]¶=E[z t t]=E[z t(y t−x0tβ0)]=0,
which are sufficient to identify the K parameters inβ.
Distances and Weight Matrices •Consider a simple example with2moment conditions
g T(θ)=µg a g b¶,
where the dependence of T andθis suppressed.
•First consider a simple weight matrix,W T=I2:
Q T(θ)=g T(θ)0W T g T(θ)=¡g a g b¢µ1001¶µg a g b¶=g2a+g2b,
which is the square of the simple distance from g T(θ)to zero.
Here the coordinates are equally important.
•Alternatively,look at a different weight matrix:
Q T(θ)=g T(θ)0W T g T(θ)=¡g a g b¢µ2001¶µg a g b¶=2·g2a+g2b,
which attaches more weight to thefirst coordinate in the distance.
15of32 Consistency:Why Does it Work?•Assume that a law of large numbers(LLN)applies to f(w t,z t,θ),i.e.
T−1
T
X t=1f(w t,z t,θ)→E[f(w t,z t,θ)]for T→∞.
That requires IID or stationarity and weak dependence.
•If the moment conditions are correct,g(θ0)=0,then GMM is consistent,
bθGMM(W T)→θ0as T→∞,
for any W T positive definite.
•Intuition:If a LLN applies,then g T(θ)converges to g(θ). Since bθGMM(W T)minimizes the distance from g T(θ)to zero,it will be a consistent estimator of the solution to g(θ0)=0.
•The weight matrix,W T,has to be positive definite,so that we put a positive and。