Generalized Method of Moments

Paul Schrimpf

2024-10-21

Reading

  • Required: Song (2021) chapter 13
  • Suggested:
    • Originated in Hansen (1982), building on method of moments which has a longer history
    • Important early application Hansen and Singleton (1982)
    • Review from Hansen (2010) ungated version

\[ \def\Er{{\mathrm{E}}} \def\En{{\mathbb{E}_n}} \def\cov{{\mathrm{Cov}}} \def\var{{\mathrm{Var}}} \def\R{{\mathbb{R}}} \def\arg{{\mathrm{arg}}} \newcommand\norm[1]{\left\lVert#1\right\rVert} \def\rank{{\mathrm{rank}}} \newcommand{\inpr}{ \overset{p^*_{\scriptscriptstyle n}}{\longrightarrow}} \def\inprob{{\,{\buildrel p \over \rightarrow}\,}} \def\indist{\,{\buildrel d \over \rightarrow}\,} \DeclareMathOperator*{\plim}{plim} \]

Generalized Method of Moments

Moment Conditions

\[ \Er\left[g(Z_i,\theta_0) \right] = 0 \]

  • Parameter \(\theta_0 \in \R^d\)

  • Data \(\tilde{Z}_i \in \R^m\)

  • Moment function \(g: \R^m \times \R^d \to \R^k\)

  • Identification: \(\Er\left[g(Z_i,\theta) \right] = 0\) iff \(\theta=\theta_0\)

Example: IV

\[ Y_i = X_i' \beta_0 + u_i \] \[ \Er\left[Z_i(Y_i - X_i'\beta_0) \right] = 0 \]

Example: IV for Nonlinear Regression

\[ \Er\left[Z_i(Y_i - h(X_i,\beta_0)) \right] = 0 \]

Example: Binary Choice

  • \(D_i = 1\{X_i'\beta_0 > u_i\}\)
  • \(u_i \sim N(0,1)\) \[ \Er\left[ \left(D_i - \Phi(X_i'\beta_0) \right) h(X_i) \right] = 0 \]

Example: Consumption and Assets

  • Hansen and Singleton (1982)
  • Model \[ \begin{align*} \max_{c_t, q_t} & \Er\left[ \sum_{t=0}^\infty \beta^t u(c_t) | \mathcal{I}_0 \right] \\ \text{s.t. } & \;\; p_t q_t + c_t \leq (p_t + d_t)q_{t-1} + y_t \end{align*} \]

Example: Consumption and Assets

  • Cleverly rearrange first order conditions: \[ \Er\left[\beta \frac{u'(c_{t+1})}{u'(c_t)} \underbrace{\frac{p_{t+1} + d_{t+1}}{p_t}}_{R_t} | \mathcal{I}_s \right] = 1 \text{ for } s \leq t \]

Example: Consumption and Assets

  • Assume \(u(c) = \frac{c^{1-\gamma}}{1-\gamma}\) \[ \Er\left[\beta \frac{c_{t+1}^{-\gamma}}{c_t^{-\gamma}} R_t | \mathcal{I}_s \right] = 1 \text{ for } s \leq t \]
  • Model implies \[ \Er\left[\left(\beta \frac{c_{t+1}^{-\gamma}}{c_t^{-\gamma}} R_t -1 \right)Z_t \right] = 0 \] for any \(Z_t \in \mathcal{I}_t\)

Econometric Theory

Identification

  • Assume:

\(\exists \theta_0 \in \Theta\) s.t. \(\forall \epsilon>0\), \[ \inf_{\theta: \norm{\theta-\theta_0} > \epsilon} \norm{\Er[g(Z_i,\theta)]} > \norm{\Er[g(Z_i,\theta_0)]} \]

  • Slightly easier assumption to verify is (loosely) \(g(z,\theta)\) continuous and \(\theta_0\) is unique minimizer of \(\norm{\Er[g(Z_i,\theta)]}\), see lemma 1 in Song (2021)

Estimation

  • Population objective function \[ Q^{GMM}(\theta) = \frac{1}{2} \norm{\Er[g(Z_i,\theta)]}^2_s = \frac{1}{2} \Er[g(Z_i,\theta)]'S'S\Er[g(Z_i,\theta)] \]
  • Sample objective function \[ \hat{Q}^{GMM}(\theta) = \frac{1}{2} \En[g(Z_i,\theta)]'S_n'S_n\En[g(Z_i,\theta)] \]
  • Estimator \[ \hat{\theta} = \arg\min_{\theta \in \Theta}\hat{Q}^{GMM}(\theta) \]

Consistency

Suppose:

  1. \(\exists \theta_0 \in \Theta\) s.t. \(\forall \epsilon>0\), \(\inf_{\theta: \norm{\theta-\theta_0} > \epsilon} \norm{\Er[g(Z_i,\theta)]} > \norm{\Er[g(Z_i,\theta_0)]}\)

  2. \(\sup_{\theta \in \Theta} \norm{\En[g(Z_i,\theta)] - \Er[g(Z_i,\theta)]} \inprob 0\)

  3. \(S_n \inprob S\)

Then \(\hat{\theta} \inprob \theta_0\)

Asymptotic Normality

Suppose:

  1. \(\theta_0 \in int(\Theta)\), & \(g(z,\theta)\) is twice continuously differentiable

  2. \(\sqrt{n} \frac{\partial}{\partial \theta} \hat{Q}^{GMM}(\theta_0) \indist N(0,\Omega)\)

  3. \(\sup_{\theta \in int(\Theta)} \norm{\frac{\partial^2}{\partial \theta \partial \theta'} \hat{Q}^{GMM}(\theta) - B(\theta)} \inprob 0\) with \(B(\cdot)\) continuous at \(\theta_0\) and \(B(\theta_0) > 0\)

Then, \[ \sqrt{n}(\hat{\theta} - \theta_0) \indist N(0, B_0^{-1} \Omega_0 B_0^{-1}) \]

Optimal Weighting Matrix

Let \[ \begin{align*} M & = (\Gamma'C\Gamma)^{-1} \Gamma' C \Sigma C \Gamma (\Gamma'C\Gamma)^{-1} \\ M^* & = (\Gamma' \Sigma^{-1} \Gamma)^{-1} \end{align*} \] then \(M \geq M^*\)

References

Hansen, Lars Peter. 1982. “Large Sample Properties of Generalized Method of Moments Estimators.” Econometrica 50 (4): 1029–54. http://www.jstor.org/stable/1912775.
———. 2010. “Generalized Method of Moments Estimation.” In Macroeconometrics and Time Series Analysis, edited by Steven N. Durlauf and Lawrence E. Blume, 105–18. London: Palgrave Macmillan UK. https://doi.org/10.1057/9780230280830_13.
Hansen, Lars Peter, and Kenneth J. Singleton. 1982. “Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models.” Econometrica 50 (5): 1269–86. http://www.jstor.org/stable/1911873.
Song, Kyunchul. 2021. “Introduction to Econometrics.”