I am reading the book "An Introduction to Econometric Theory" by A. Ronald Gallant. In the section of the book on the Method of Moments, I get a little confused about the method as I know it. I would like to understand how these two forms are equivalent!
The method of moments estimation that I know:
Given some population $X\sim f(x|\theta)$ and a random sample $(X_j)_{j=1}^n$. Firts, defines the sample moments: $$m_i = \frac{1}{n}\sum_{i = 1}^nX_j^{i}$$ Tipically, the population moments - $E[X^i],\,\,\, i = 1,2,...$- are some funtion of the the true parameter $\theta$. Thus, the moments estimator, $\hat{\theta}_n$ , is the solution of the following equation system:
$$m_i = E[X^i], \quad i = 1,2,...$$ (where $\theta$ is the unknown)
The book's approach
First, one defines a statistic $\bar{W}_n = W(x_1,...,x_n)$, computes its expectation $$\mathcal{M}(\theta)= \int ...\int W(x_1,...,x_n)\Pi_{i=1}^n f(x_i | \theta) dx_1...dx_n$$ and solves the equation: $$\bar{W}_n = \mathcal{M}(\theta).$$ The solutions $\hat{\theta}_n$ is the method of moments estimator.
I think the book's approach is intended for later, in the next section, to introduce the Generalized Method of Moments. Okay, but for me it's a little weird. Note that Gallant starts by setting a statistic and its expectation.
In a real case, I wouldn't know which statistic to start with. It's a difficult approach to understand as this is equivalent to the approach I know.
Some help?
