Generalized Method of Moments

GMM is a method for estimating unknown parameters. GMM is often compared with Maximum likelihood estimation, another method for estimating parameters. In stark contrast to MLE, GMM does not require taking a stance on the probability distributions governing the parameters in question or the DGP. This makes it a much more flexible and widely applicable method, at the cost of lower precision.

GMM estimation sets out to identify parameters by leveraging various “moments,” as in the statistical sense. These moments may not be merely statistical; they could come from credible economic theory or any other source. These moments are functions of the desired parameters and assumed to equal zero in population, suggesting that the correct parameters values are those that most closely get the chosen moments to zero or near zero. $E[g(x)]=0$ The constraints setting these moment conditions to particular parameter values are called moment conditions. The sum of squared deviations from zero across the moment conditions is a common choice for minimization. $\underset{x}{\min }{\sum }_{m}g(x{)}^2$ As a general rule, one needs at least as many moments as there are parameters to estimate (“X equations, $\le$ X unknowns”). If the number of moments and parameters are exactly equal, this is referred to as Method of Moments. If there are more moments than parameters, then we are in the realm of Generalized Method of Moments.

With more moments than parameters, some scheme must be devised to weigh the various moment conditions. In such cases, rather than minimizing the simple sum of squares (implying a identity matrix as the weighting matrix), we can choose a weighting matrix, the optimal of which would assign high weights to low variance moment conditions and low weights to high variance conditions, so as to balance their influence on the minimization objective: $\underset{x}{\min }{\sum }_{m}g(x{)}^{T}Wg(x)$ Since the optimal weighting matrix is unknown / infeasible (since it requires knowing the population variances), we instead estimate it via various methods:

• Weight matrix = Identity matrix (suboptimal, but easy)
• Two-step estimator: Assume weight matrix is the identity matrix, minimize objective function, use sample variances of the moment conditions to calculate new weight matrix as $W=(g(x)g(x{)}^T{)}^{-1}$
• Iterated estimator: Same two-step, but keep iterating until parameters stop moving

Many alternative methods of estimation are effectively special cases of GMM, including OLS and MLE.

Standard errors for GMM can be calculated as follows: $\text{var}(x)=\frac{1}{T}(d^{\prime }Wd{)}^{-1}$ where d is the partial derivative of the moment condition with respect to a particular parameter: $d=\frac{\delta g(x)}{\delta x}$ This implies that parameters that have a large impact on the moment conditions have lower standard errors / more precision.

Asset pricing

In asset pricing, GMM can be interpreted as:

• The moment conditions are pricing errors, which naturally should be as close to zero as possible for as many assets as possible. These errors are proportional to alpha
• GMM picks parameters to minimize the weighted sum of squared pricing errors
• The second stage picks the combination of pricing errors (asset prices) that are the best measured / have the least variance

Note that per Asset Pricing, variables in GMM must all be stationary.

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References

Identification Properties of Recent Production Function Estimators The Dynamics of Productivity in the Telecommunications Equipment Industry Introductory Econometrics: A Modern Approach GMM Notes from John Cochrane Asset Pricing @jordaLocalProjections Delta method

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