Method of Moments
Method of Moments
Both the method of moments and its generalization, namely the generalized method of moments (GMM), have taken a prominent place in statistical inference in the social sciences and have been applied in almost every field of economics, including asset pricing, business cycle, commodity market, education, and labor market. The (standard) method of moments consists of estimating a parameter β by equating sample moments with population moments and solving these equations for β. This method was introduced at the end of the nineteenth century by the British mathematician Karl Pearson and was partly abandoned when the British biologist Ronald Fisher (1890–1962) showed that the maximum likelihood estimator (MLE) is more efficient than the method of moments estimator because its variance is smaller. Since its introduction by Lars Peter Hansen in 1982, however, GMM has been extremely popular in economics for three reasons. The first reason is that economic models are often too complex to be completely specified and an attempt to describe the full model is likely to yield misspecification errors. GMM provides a way to estimate partially specified models. The second reason is that, even if the model is completely specified, MLE may be too cumbersome to implement, whereas GMM provides a practical method to perform inference. Finally, GMM embeds many familiar estimation techniques as special cases, including the method of moments, ordinary least squares, instrumental variables estimation, and even maximum likelihood estimation.
GMM may be illustrated in a time series context where observations x 1, x 2, ..., x T are available. Assume that for a given function g, the moment condition E[g(x,β)] = 0 holds for a unique solution β. To solve this equation, the population mean needs to be replaced by the sample mean Two cases are distinguished, depending on whether the dimension of g is the same as or larger than the dimension of β. In the first case (just identified), the equation gT(β ) = 0 can be solved to obtain the method of moments estimator of β. In the second case (overidentified), the previous equation does not have a solution and the standard method of moments needs to be modified. The GMM estimator is defined as the solution where WT is a positive definite matrix that attaches weights to moments. For any WT’ bT is consistent, that is, it approaches the true value β as the sample size T grows. Additionally, the GMM estimator has minimal variance if WT is an estimator of the inverse of the variance of gT (β )
In the overidentified case, g T (bT ) is not exactly equal to zero but should go to zero as T goes to infinity. This provides the basis for the overidentifying restrictions test. This test consists of rejecting the hypothesis that the moment conditions hold in the population if T g T(bT )' W Tg T(b T) is greater than the critical value given by a chi-square distribution with degrees of freedom equal to the difference in the dimensions of g and β.
GMM provides a framework that encompasses most estimation techniques used in economics. Instrumental variables estimation, although a predecessor to GMM, can be recast as a special case of GMM. Consider the regression
where x t is endogenous, that is, correlated with the residual εt. As a consequence of endogeneity, the ordinary least squares estimator is not consistent. A consistent estimator, however, may be obtained by using a vector of so called instruments z. t To be a valid instrument, z t must be correlated with x t but not with the error ε Then, β can be estimated by GMM using g (yt, xt, zt, β ) = (yt – x t'β) zt. The resulting estimator is called the instrumental variables estimator.
MLE itself can be interpreted as a GMM estimator because the expectation of the derivative of the log-likelihood is equal to zero, giving rise to a moment condition.
To overcome computational difficulties, Daniel McFadden (1989) and Ariel Pakes and David Pollard (1989) have proposed the method of simulated moments (MSM), which consists of replacing population moments with moments computed from simulated data.
Consider a population where each subject is equally likely to be a male or a female. We observe a measure x (for instance, the weight) but not the sex and we wish to estimate the difference between males and females. This is particularly relevant in anthropology where most fossil specimens lack indicators of sex. Assume that observations are normally independently distributed with mean μF and variance σ2 if the subject is a female and with mean μM and the same variance σ2 if the subject is a male. The method of moments that matches the expectations of x,x2, and x 4 with their sample counterparts permits to estimate μF , μM , and σ2
Lars Peter Hansen and Kenneth Singleton (1982) explain how to apply GMM to estimate behavioral parameters of economic agents in a general equilibrium model, without having to describe the full economic environment. This approach may be used to study how agents allocate their spending. Consider an economy where a representative agent chooses consumption and investment plans so as to maximize , where ct is consumption in period t, U is a utility function, δ is a discount factor, and Ώt is the information at t. Maximization is performed under a budget constraint ct + ptqt = rtqt - 1 + w t
Where w t is the income, q t the quantity of asset held at the end of period t, pt the price of this asset, and rt its return. The first order condition is
Assume that U(ct ) = (crt – 1)/γ. The parameter of interest β =(δ,γ) may be estimated by GMM using
where zt is a vector of instruments. Any element of Ώt may be used as instrument, for example the constant and
Consider a model where each individual has the choice among J alternatives (for example, occupations or means of transportation). The individual chooses the alternative with the greatest value. The value uij of occupation j for individual i depends on a set of observed variables x ij (such as sex, race, age, and education) so that where εj = (ε1j,ε2j, … εnj ) is normally distributed with mean zero and covariance matrix Σ. We observe that individual i chooses alternative j, if uij ≥ uil for l= 1,.....J. The probability of choosing alternative j, denoted by Pij, involves a J- 1 dimensional integral. If J is large, this integral is cumbersome to compute and hence maximum likelihood estimation is intractable. By contrast, Pij can be estimated using simulations. Let r = 1, …, Rij be independently drawn from a normal distribution with mean zero and covariance Σ, and let Then, an estimate P̂ij of Pij is given by the proportion of cases where exceeds uilϒ for all l different from j. An MSM estimator of (β1, β2, …, βJ ), is given by the solution to where zij is exogenous (equal to xij, for instance).
SEE ALSO Inference, Statistical; Instrumental Variables Regression; Large Sample Properties; Pearson, Karl
Hansen, Lars Peter. 1982. Large Sample Properties of Generalized Method of Moments Estimators. Econometrica 50 (4): 1029–1054.
Hansen, Lars Peter, and Kenneth J. Singleton. 1982. Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models. Econometrica 50 (5): 1269–1286.
McFadden, Daniel L. 1989. A Method of Simulated Moments for Estimation of Discrete Response Models without Numerical Integration. Econometrica 57(5): 995–1026.
Pakes, Ariel, and David Pollard. 1989. Simulation and the Asymptotics of Optimization Estimators. Econometrica 57 (5): 1027–1057.