Consider a random variable f(x|θ

_{1}….θ

_{m}) and a sample x

_{1}…x

_{n}

Define the k

^{th}raw population moment as: μ_{k}’ =E(x^{k}) which is some function of θ.This is referred to as the population moment condition and is often expressed as an ‘orthogonality’ condition:μ

_{k}’ - E(x

^{k}) = 0 or more compactly as E[Ψ (x

_{i},θ)] = 0

The sample analog of μ

_{k}’ =m_{k}’ = (1/n) Σ x_{i}^{k}.
The MM estimator is derived by equating the sample moments
to the population moments and solving for the parameters of interest.

This gives us the sample
analog to the orthogonality condition:

m

_{k}’ - μ_{k}’ = 0 or (1/n)Σ Ψ (x_{i}’,θ) =0**Example: OLS**

Let e = y-xb and Ψ (x,θ) = x(y-xb)

Our moment condition is then: E(xe) = 0 (which is one of the
familiar assumptions of OLS)

The sample analog:

1/n Σ x(y-xb) = 0

X’(Y-Xb) = 0 (matrix
representation)

X’Y-X’Xb = 0

b = (X’X)

^{-1}X’Y**Generalized Method of Moments**

The sample analog to the moment condition (1/n)Σ Ψ (x

_{i}’,θ) often represents a system of ‘q’ equations and ‘p’ parameters. If q =p then we have an identified system and can derive the MM estimator. If q>p then the system is overidentified and the following quadratic form is specified:
min[(1/n)Σ Ψ (x

_{i}’,θ)]’**W**[(1/n)Σ Ψ (x_{i}’,θ)] where the matrix**W**is a weigthing matrix.
GMM estimates are
derived from the above iteratively.

**References:**

Jeff Thurk provides a very useful set of notes related to
GMM on his website: https://webspace.utexas.edu/jmt597/www/teaching.htm

Econometric Analysis,
William H. Greene . 1990.