Generalized Method of Moments

metrics EC484_Econometrics_Analysis

Or, so called GMM.

The GMM Set Up

Some important assumptions:

• Why we need Continuity? Because once we have assumed there exists continuity, we could then apply the Central Limit Theorem to the sample moment.

Continuous function would have the minimum value in a compact set.

Thus we could estimate the ${\hat{\theta }}_W$.

See a counter example:

$g(z,\theta )=\{\begin{array}{ll}z-\theta & \text{if }\theta \ne 0\\ z& \text{if }\theta =0\end{array}$

The function is not continuous at $\theta =0$.

Another assumption is the Dominance, which is written as:

$$\mathbb{E}[\underset{\theta in\Theta }{\sup }|g(Z,\theta )]<\infty$$

• Why we need Dominance?

Because we need to apply the Uniform Law of Large Numbers, which requires the Dominance condition.

GMM in the linear setting

Suppose we have a linear system:

$$Y_i=X_i^{\prime }\beta +e_i,\mathbb{E}[X_i{e}_i]=0$$

It’s moment conditionis based on that the error term is uncorrelated with the regressors. Thus we have:

$$g(Z_i,\beta )=X_i(Y_i-X_i^{\prime }\beta )$$

When we set the $\mathbf{W}=I$, we would have the GMM estimator:

$${\hat{\beta }}_{GMM}={\left(\sum X_i{X}_i^{\prime }\right)}^{-1}\left(\sum X_i{Y}_i\right)$$

Yes and As you can see, since everything is linear, we could just solve it directly. But in the non-linear setting, we could not solve it directly. Thus we need to use numerical optimization methods to find the estimator.

GMM in the non linear setting

Suppose we have a linear system:

$$Y_i=e^{X_i^{\prime }\theta }+e_i,\mathbb{E}[X_i{e}_i]=0$$

Similarly, we could get the moment condition:

$$g(Z_i,\theta )=X_i(Y_i-e^{X_i^{\prime }\theta })$$

GMM estimator:

$${\hat{\theta }}_{GMM}=\arg \underset{\theta \in \Theta }{\min }{\bar{g}}_n(\theta {)}^{\prime }W{\bar{g}}_n(\theta )$$

The Asymptotic Normality would satisfy, which could be represented by:

$$\sqrt{n}(\hat{\theta }-{\theta }_0)\to N(0,(G^{\prime }WG{)}^{-1}{G}^{\prime }W\Omega WG(G^{\prime }WG{)}^{-1})$$

We would describe it one by one:

• What is $G$? $G=\mathbb{E}[\frac{\partial g(Z,{\theta }_0)}{\partial {\theta }^{\prime }}]$

We denote a Jabobian matrix $\bar{G}(\theta )=\frac{\partial \bar{g}(\theta )}{\partial {\theta }^{\prime }}={\left(\begin{array}{ccc}\frac{\partial g_1}{\partial {\theta }_1}& \cdots & \frac{\partial g_1}{\partial {\theta }_p}\\ ⋮& \ddots & ⋮\\ \frac{\partial g_q}{\partial {\theta }_1}& \cdots & \frac{\partial g_q}{\partial {\theta }_p}\end{array}\right)}_{q\times p}$

Recall the asymptotic variance is $(G^{\prime }WG{)}^{-1}{G}^{\prime }W\Omega WG(G^{\prime }WG{)}^{-1}$, for simplicity, we denote it as $V_W$.

Note that $V_W$ is dependent on the weighting matrix $W$.

How to understand the $\Omega$ ? $\Omega =\mathbb{E}[g(Z,\theta )g(Z,\theta {)}^{\prime }]$, which is the variance covariance matrix of the moment conditions. The larger the $\Omega$, the larger the variance of the estimator, which we don’t want to see. Thus we need to give them a smaller weight. In practice, the optimal Weighting Matrix would be

$${\mathbf{W}}^{\ast }={\Omega }^{-1}$$

Plus, under the optimal weighting matrix, the asymptotic variance could be simplified as:

$$V_{{\mathbf{W}}^{\ast }}=(G^{\prime }{\Omega }^{-1}G{)}^{-1}$$

The optimal weighting strategy: Weight each moment condition inversely proportional to its variance. This is like inverse-variance weighting you might have seen in statistics.