Generalized method of moments

"GMM" redirects here. For the Thai entertainment company, see GMM Grammy. For Gaussian mixture models, see mixture model.

In econometrics, generalized method of moments (GMM) is a generic method for estimating parameters in statistical models. Usually it is applied in the context of semiparametric models, where the parameter of interest is finite-dimensional, whereas the full shape of the distribution function of the data may not be known, and therefore the maximum likelihood estimation is not applicable.

The method requires that a certain number of moment conditions were specified for the model. These moment conditions are functions of the model parameters and the data, such that their expectation is zero at the true values of the parameters. The GMM method then minimizes a certain norm of the sample averages of the moment conditions.

The GMM estimators are known to be consistent, asymptotically normal, and efficient in the class of all estimators that don’t use any extra information aside from that contained in the moment conditions.

GMM was developed by Lars Peter Hansen in 1982 as a generalization of the method of moments.

Description

Suppose the available data consists of T iid observations {Y_t }_{t = 1,...,T}, where each observation Y_t is an n-dimensional multivariate random variable. The data comes from a certain statistical model, defined up to an unknown parameter θ ∈ Θ. The goal of the estimation problem is to find the “true” value of this parameter, θ₀, or at least a reasonably close estimate.

In order to apply GMM there should exist a vector-valued function g(Y,θ) such that

$m(\theta_0) \equiv \operatorname{E}[\,g(Y_t,\theta_0)\,]=0,$

where E denotes expectation, and Y_t is a generic observation, which are all assumed to be iid. Moreover, function m(θ) must not be equal to zero for θ ≠ θ₀, or otherwise parameter θ will not be identified.

The basic idea behind GMM is to replace the theoretical expected value E[⋅] with its empirical analog — sample average:

$\hat{m}(\theta) = \hat{\operatorname{E}}\big[\,g(Y_t,\theta)\,\big] \equiv \frac{1}{T}\sum_{t=1}^T g(Y_t,\theta)$

and then to minimize the norm of this expression with respect to θ.

By the law of large numbers, $\scriptstyle\hat{m}(\theta)\,\approx\;\operatorname{E}[g(Y_t,\theta)]\,=\,m(\theta)$ for large values of T, and thus we expect that $\scriptstyle\hat{m}(\theta_0)\;\approx\;m(\theta_0)\;=\;0$. The generalized method of moments looks for a number $\scriptstyle\hat\theta$ which would make $\scriptstyle\hat{m}(\;\!\hat\theta\;\!)$ as close to zero as possible. Mathematically, this is equivalent to minimizing a certain norm of $\scriptstyle\hat{m}(\theta)$ (norm of m, denoted as ||m||, measures the distance between m and zero). The properties of the resulting estimator will depend on the particular choice of the norm function, and therefore the theory of GMM considers an entire family of norms, defined as

$\| \hat{m}(\theta) \|^2_{W} = \hat{m}(\theta)'\,W\hat{m}(\theta),$

where W is a positive-definite weighting matrix, and m′ denotes transposition. In practice, the weighting matrix W is computed based on the available data set, which will be denoted as $\scriptstyle\hat{W}$. Thus, the GMM estimator can be written as

$\hat\theta = \operatorname{arg}\min_{\theta\in\Theta} \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\theta)\bigg)' \hat{W} \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\theta)\bigg)$

Under suitable conditions this estimator is consistent, asymptotically normal, and with right choice of weighting matrix $\scriptstyle\hat{W}$ asymptotically efficient.

Properties

Consistency

Consistency is a statistical property of an estimator stating that, having sufficient number of observations, the estimator will get arbitrarily close to the true value of parameter:

$\hat\theta \xrightarrow{p} \theta_0\ \text{as}\ T\to\infty$

(see Convergence in probability). Necessary and sufficient conditions for a GMM estimator to be consistent are as follows:

1. $\hat{W}_T \xrightarrow{p} W,$ where W is a positive semi-definite matrix,
2. $\,W\operatorname{E}[\,g(Y_t,\theta)\,]=0$  only for $\,\theta=\theta_0,$
3. $\theta_0\in\Theta,$ which is compact,
4. $\,g(Y,\theta)$  is continuous at each θ with probability one,
5. $\operatorname{E}[\,\textstyle\sup_{\theta\in\Theta} \lVert g(Y,\theta)\rVert\,]<\infty.$

The second condition here (so-called Global identification condition) is often particularly hard to verify. There exist simpler necessary but not sufficient conditions, which may be used to detect non-identification problem:

• Order condition. The dimension of moment function m(θ) should be at least as large as the dimension of parameter vector θ.
• Local identification. If g(Y,θ) is continuously differentiable in a neighborhood of θ₀, then matrix $W\operatorname{E}[\nabla_\theta g(Y_t,\theta_0)]$ must have full column rank.

In practice applied econometricians often simply assume that global identification holds, without actually proving it.

Asymptotic normality

Asymptotic normality is a useful property, as it allows us to construct confidence bands for the estimator, and conduct different tests. Before we can make a statement about the asymptotic distribution of the GMM estimator, we need to define two auxiliary matrices:

$G = \operatorname{E}[\,\nabla_{\!\theta}\,g(Y_t,\theta_0)\,], \qquad \Omega = \operatorname{E}[\,g(Y_t,\theta_0)g(Y_t,\theta_0)'\,]$

Then under conditions 1–6 listed below, the GMM estimator will be asymptotically normal with limiting distribution

$\sqrt{T}\big(\hat\theta - \theta_0\big)\ \xrightarrow{d}\ \mathcal{N}\big[0, (G'WG)^{-1}G'W\Omega WG(G'WG)^{-1}\big]$

(see Convergence in distribution). Conditions:

1. $\hat\theta$ is consistent (see previous section),
2. $\,\theta_0$ lies in the interior of set $\,\Theta,$
3. $\,g(Y,\theta)$ is continuously differentiable in some neighborhood N of θ₀ with probability one,
4. $\operatorname{E}[\,\lVert g(Y_t,\theta) \rVert^2\,]<\infty,$
5. $\operatorname{E}[\,\textstyle\sup_{\theta\in N}\lVert \nabla_\theta g(Y_t,\theta) \rVert\,]<\infty,$
6. matrix G'WG is nonsingular.

Efficiency

So far we have said nothing about the choice of matrix W, except that it must be positive semi-definite. In fact any such matrix will produce a consistent and asymptotically normal GMM estimator, the only difference will be in the asymptotic variance of that estimator. It can be shown that taking

$W \propto\ \Omega^{-1}$

will result in the most efficient estimator in the class of all asymptotically normal estimators. Efficiency in this case means that such an estimator will have the smallest possible variance (we say that matrix A is smaller than matrix B if B–A is positive semi-definite).

In this case the formula for the asymptotic distribution of the GMM estimator simplifies to

$\sqrt{T}\big(\hat\theta - \theta_0\big)\ \xrightarrow{d}\ \mathcal{N}\big[0, (G'\,\Omega^{-1}G)^{-1}\big]$

The proof that such a choice of weighting matrix is indeed optimal is quite elegant, and is often adopted with slight modifications when establishing efficiency of other estimators. As a rule of thumb, a weighting matrix is optimal whenever it makes the “sandwich formula” for variance collapse into a simpler expression.

Proof. We will consider the difference between asymptotic variance with arbitrary W and asymptotic variance with W = Ω − 1. If we can factor this difference into a symmetric product of the form CC' for some matrix C, then it will guarantee that this difference is nonnegative-definite, and thus W = Ω − 1 will be optimal by definition.
$\,V(W)-V(\Omega^{-1})$	$\,=(G'WG)^{-1}G'W\Omega WG(G'WG)^{-1} - (G'\Omega^{-1}G)^{-1}$
	$\,=(G'WG)^{-1}\Big(G'W\Omega WG - G'WG(G'\Omega^{-1}G)^{-1}G'WG\Big)(G'WG)^{-1}$
	$\,=(G'WG)^{-1}G'W\Omega^{1/2}\Big(I - \Omega^{-1/2}G(G'\Omega^{-1}G)^{-1}G'\Omega^{-1/2}\Big)\Omega^{1/2}WG(G'WG)^{-1}$
	$\,=A(I-B)A',$
where we introduced matrices A and B in order to slightly simplify notation; I is an identity matrix. We can see that matrix B here is symmetric and idempotent: B2 = B. This means I–B is symmetric and idempotent as well: I − B = (I − B)(I − B)'. Thus we can continue to factor the previous expression as
	$\,=A(I-B)(I-B)'A' = \Big(A(I-B)\Big)\Big(A(I-B)\Big)' \geq 0$

Implementation

One difficulty with implementing the outlined method is that we cannot take W = Ω⁻¹ because, by the definition of matrix Ω, we need to know the value of θ₀ in order to compute this matrix, and θ₀ is precisely the quantity we don’t know and are trying to estimate in the first place.

Several approaches exist to deal with this issue, the first one being the most popular:

• Two-step feasible GMM:
  • Step 1: Take W = I (the identity matrix), and compute preliminary GMM estimate $\scriptstyle\hat\theta_{(1)}$. This estimator is consistent for θ₀, although not efficient.
  • Step 2: Take

    $\hat{W} = \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\hat\theta_{(1)})g(Y_t,\hat\theta_{(1)})'\bigg)^{-1},$

    where we have plugged in our first-step preliminary estimate $\scriptstyle\hat\theta_{(1)}$. This matrix converges in probability to Ω⁻¹ and therefore if we compute $\scriptstyle\hat\theta$ with this weighting matrix, the estimator will be asymptotically efficient.
• Iterated GMM. Essentially the same procedure as 2-step GMM, except that the matrix  $\hat{W}_T$ is recalculated several times. That is, the estimate obtained in step 2 is used to calculate the weighting matrix for step 3, and so on. Such estimator, denoted $\scriptstyle\hat\theta_{(i)}$, is equivalent to solving the following system of equations:

  $\bigg(\frac{1}{T}\sum_{t=1}^T \frac{\partial g}{\partial\theta'}(Y_t,\hat\theta_{(i)})\bigg)' \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\hat\theta_{(i)})g(Y_t,\hat\theta_{(i)})'\bigg)^{\!-1} \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\hat\theta_{(i)})\bigg) = 0$

  Asymptotically no improvement can be achieved through such iterations, although certain Monte-Carlo experiments suggest that finite-sample properties of this estimator are slightly better.^{[citation needed]}
• Continuously Updating GMM (CUGMM, or CUE). Estimates $\scriptstyle\hat\theta$ simultaneously with estimating the weighting matrix W:

  $\hat\theta = \operatorname{arg}\min_{\theta\in\Theta} \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\theta)\bigg)' \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\theta)g(Y_t,\theta)'\bigg)^{\!-1} \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\theta)\bigg)$

  In Monte-Carlo experiments this method demonstrated a better performance than the traditional two-step GMM: the estimator has smaller median bias (although fatter tails), and the J-test for overidentifying restrictions in many cases was more reliable.

Another important issue in implementation of minimization procedure is that the function is supposed to search through (possibly high-dimensional) parameter space Θ and find the value of θ which minimizes the objective function. No generic recommendation for such procedure exists, it is a subject of its own field, numerical optimization.

J-test

When the number of moment conditions is greater than the dimension of the parameter vector θ, the model is said to be over-identified. Over-identification allows us to check whether the model's moment conditions match the data well or not.

Conceptually we can check whether $\hat{m}(\hat\theta)$ is sufficiently close to zero to suggest that the model fits the data well. The GMM method has then replaced the problem of solving the equation $\hat{m}(\theta)=0$, which chooses θ to match the restrictions exactly, by a minimization calculation. The minimization can always be conducted even when no θ₀ exists such that m(θ₀) = 0. This is what J-test does. The J-test is also called a test for over-identifying restrictions.

Formally we consider two hypotheses:

• $H_0:\ m(\theta_0)=0$  (the null hypothesis that the model is “valid”), and
• $H_1:\ m(\theta)\neq 0,\ \forall \theta\in\Theta$  (the alternative hypothesis that model is “invalid”; the data do not come close to meeting the restrictions)

Under hypothesis H₀, the following so-called J-statistic is asymptotically chi-squared with k–l degrees of freedom. Define J to be:

$J \equiv T \cdot \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\hat\theta)\bigg)' \hat{W}_T \bigg(\frac{1}{T}\sum_{t=1}^T g(Y_t,\hat\theta)\bigg)\ \xrightarrow{d}\ \chi^2_{k-\ell}$   under H₀,

where $\hat\theta$ is the GMM estimator of the parameter θ₀, k is the number of moment conditions (dimension of vector g), and l is the number of estimated parameters (dimension of vector θ). Matrix $\hat{W}_T$ must converge in probability to Ω ^{− 1}, the efficient weighting matrix (note that previously we only required that W be proportional to Ω ^{− 1} for estimator to be efficient; however in order to conduct the J-test W must be exactly equal to Ω ^{− 1}, not simply proportional).

Under the alternative hypothesis H₁, the J-statistic is asymptotically unbounded:

$J\ \xrightarrow{p}\ \infty$  under H₁

To conduct the test we compute the value of J from the data. It is a nonnegative number. We compare it with (say) the 0.95 quantile of the $\chi^2_{k-\ell}$ distribution:

• H₀ is rejected at 95% confidence level if $J > q_{0.95}^{\chi^2_{k-\ell}}$
• H₀ cannot be rejected at 95% confidence level if $J < q_{0.95}^{\chi^2_{k-\ell}}$

Scope

Many other popular estimation techniques can be cast in terms of GMM optimization:

• Ordinary Least Squares (OLS) is equivalent to GMM with moment conditions:

  $\operatorname{E}[\,x_t(y_t - x_t'\beta)\,]=0$

• Generalized Least Squares (GLS)

  $\operatorname{E}[\,x_t(y_t - x_t'\beta)/\sigma^2(x_t)\,]=0$

• Instrumental variables regression (IV)

  $\operatorname{E}[\,z_t(y_t - x_t'\beta)\,]=0$

• Non-linear Least Squares (NLLS):

  $\operatorname{E}[\,\nabla_{\!\beta}\, g(x_t,\beta)\cdot(y_t - g(x_t,\beta))\,]=0$

• Maximum likelihood estimation (MLE):

  $\operatorname{E}[\,\nabla_{\!\theta} \ln f(x_t,\theta) \,]=0$

Implementations

• R Programming wikibook, Method of Moments
• R
• Stata
• EViews

See also

• Method of maximum likelihood
• Generalized empirical likelihood

References

• Kirby Faciane (2006): Statistics for Empirical and Quantitative Finance. H.C. Baird: Philadelphia. ISBN 0-9788208-9-4.
• Alastair R. Hall (2005). Generalized Method of Moments (Advanced Texts in Econometrics). Oxford University Press. ISBN 0-19-877520-2.
• Lars Peter Hansen (1982): Large Sample Properties of Generalized Method of Moments Estimators, Econometrica 50, 1029-1054.
• Lars Peter Hansen (2002): Method of Moments in International Encyclopedia of the Social and Behavior Sciences, N. J. Smelser and P. B. Bates (editors), Pergamon: Oxford.
• Hansen, Lars Peter; Heaton, John; Yaron, Yaron (1996). "Finite-sample properties of some alternative GMM estimators". Journal of Business & Economic Statistics 14 (3): 262–280. JSTOR 1392442. 
• Imbens, Guido W.; Spady, Richard H.; Johnson, Phillip (1998). "Information theoretic approaches to inference in moment condition models". Econometrica 66 (2): 333–357. JSTOR 2998561. 
• Newey W., McFadden D. (1994). Large sample estimation and hypothesis testing, in Handbook of Econometrics, Ch.36. Elsevier Science.
• Special issues of Journal of Business and Economic Statistics: vol. 14, no. 3 and vol. 20, no. 4.