\(\newcommand{\Eb}{{\bf E}}\)This post was written jointly with Enrique Pinzon, Senior Econometrician, StataCorp.

The generalized method of moments (**GMM**) is a method for constructing estimators, analogous to maximum likelihood (**ML**). **GMM** uses assumptions about specific moments of the random variables instead of assumptions about the entire distribution, which makes **GMM** more robust than **ML**, at the cost of some efficiency. The assumptions are called moment conditions.

**GMM** generalizes the method of moments (**MM**) by allowing the number of moment conditions to be greater than the number of parameters. Using these extra moment conditions makes **GMM** more efficient than **MM**. When there are more moment conditions than parameters, the estimator is said to be overidentified. **GMM** can efficiently combine the moment conditions when the estimator is overidentified.

We illustrate these points by estimating the mean of a \(\chi^2(1)\) by **MM**, **ML**, a simple **GMM** estimator, and an efficient **GMM** estimator. This example builds on Efficiency comparisons by Monte Carlo simulation and is similar in spirit to the example in Wooldridge (2001). Read more…

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\newcommand{\Eb}{{\bf E}}\) This post was written jointly with Joerg Luedicke, Senior Social Scientist and Statistician, StataCorp.

**Overview**

We provide an introduction to parameter estimation by maximum likelihood and method of moments using **mlexp** and **gmm**, respectively (see **[R] mlexp** and **[R] gmm**). We include some background about these estimation techniques; see Pawitan (2001, Casella and Berger (2002), Cameron and Trivedi (2005), and Wooldridge (2010) for more details.

Maximum likelihood (ML) estimation finds the parameter values that make the observed data most probable. The parameters maximize the log of the likelihood function that specifies the probability of observing a particular set of data given a model.

Method of moments (MM) estimators specify population moment conditions and find the parameters that solve the equivalent sample moment conditions. MM estimators usually place fewer restrictions on the model than ML estimators, which implies that MM estimators are less efficient but more robust than ML estimators. Read more…

**Overview**

In this post, I show how to use Monte Carlo simulations to compare the efficiency of different estimators. I also illustrate what we mean by efficiency when discussing statistical estimators.

I wrote this post to continue a dialog with my friend who doubted the usefulness of the sample average as an estimator for the mean when the data-generating process (DGP) is a \(\chi^2\) distribution with \(1\) degree of freedom, denoted by a \(\chi^2(1)\) distribution. The sample average is a fine estimator, even though it is not the most efficient estimator for the mean. (Some researchers prefer to estimate the median instead of the mean for DGPs that generate outliers. I will address the trade-offs between these parameters in a future post. For now, I want to stick to estimating the mean.)

In this post, I also want to illustrate that Monte Carlo simulations can help explain abstract statistical concepts. I show how to use a Monte Carlo simulation to illustrate the meaning of an abstract statistical concept. (If you are new to Monte Carlo simulations in Stata, you might want to see Monte Carlo simulations using Stata.) Read more…