Introduction to method of moments, generalized method of moments(GMM) and method of simulated moments (MSM)
Basic
Maximum likelihood is one way to get a point estimate, another way to get point estimate is to use the method of moments.
- Moment
- $E(X)$: the first moment
- $E(X^2)$: the second moment
- $E(X^k)$: the $k^{th}$ moment of a random variable
- Estimate some mean using the method of moment:
- Estimate for the first moment $E(X)$: $\frac{\sum X_i}{n}$
- Estimate of $k^{th}$ moment $E(X^k)$: take a random sample of size $n$, then $\frac{\sum_{i=1}^n X_i^k}{n}$
Example
- A random sample $X_1 \cdots, X_n$
- $\mu=E(X)$
- $\sigma^2=E(X-E[x])^2$ or $E(X^2)-\mu^2$
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Method of moments estimator of $\mu$: $\hat{\mu}=\frac{\sum_{i=1}^n X_i}{n}$
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Method of moments estimator of $E(X^2)$: $\frac{\sum_{i=1}^n X_i^2}{n}$
- Method of moments estimator of $\sigma^2$: $\hat{\sigma^2}=\frac{\sum_{i=1}^n X_i^2}{n}- \left[\frac{\sum_{i=1}^n X_i}{n} \right]^2$