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Moment-based estimation

Related pages: Numerical methods

§1. GMM

GMM estimator

Let \(Y_i \in \mathbb{R}^D\) be the data for a typical individual \(i\), and let \(\theta^* \in \Theta \subseteq \mathbb{R}^K\) be the true parameter value. Let \(g\colon \mathbb{R}^D \times \mathbb{R}^K \to \mathbb{R}^M\) be a vector-valued function. \(g\) typically captures the distance between observed quantities and their model-predicted counterparts, which we expect to equal in expectation. Formally, the \(M\) moment conditions are given by

\[\begin{equation*} \text{E} [ g(Y_i, \theta^*) ] = 0 \in \mathbb{R}^M. \end{equation*}\]

As empirical researchers, we do not know the true parameter value \(\theta^*\). Instead, we observe an i.i.d. sample \((Y_i)_{i=1}^N\). With a given parameter value \(\theta\), the sample analogue of these \(M\) moments is:

\[\begin{equation*} G(\theta | (Y_i)_{i=1}^N) = \frac{1}{N} \sum_{i=1}^{N} g(Y_i, \theta) \in \mathbb{R}^M. \end{equation*}\]

For simplicity, we write \(G(\theta | (Y_i)_{i=1}^N)\) as \(G(\theta)\). Ideally, we want to find a parameter value \(\theta\) that satisfies \(G(\theta) = 0\). However, when there are more moment conditions than parameters (\(M > K\); over-identification), there is usually no parameter value \(\theta\) that exactly satisfies \(G(\theta) = 0\). Instead, we resort to minimizing the distance between \(G(\theta)\) and \(0\). The GMM estimator is given by

\[\begin{equation} \hat\theta_{GMM} = \arg \min_{\theta \in \Theta} G(\theta)^\top W G(\theta), \end{equation}\]

where \(W\in \mathbb{R}^{M\times M}\) is a positive semidefinite weighting matrix.

Choice of moments

TBD

Weighting matrix

TBD

2-Step GMM

TBD

§2. SMM

§3. Quantile regression