Chapter 14, Generalized Method of Moments and Minimum Distance Estimation Video Solutions, Econometric Analysis of Cross Section and Panel Data

Problem 1

Consider the system in equations (14.34) and (14.35).
a. How would you estimate equation (14.35) using single-equation methods? Give a few possibilities, ranging from simple to more complicated. State any additional assumptions relevant for estimating asymptotic variances or for efficiency of the various estimators.
b. Is equation (14.34) identified if $\gamma_1=0$ ?
c. Now suppose that $\gamma_3=0$, so that the parameters in equation (14.35) can be consistently estimated by OLS. Let $\hat{y}_2$ be the OLS fitted values. Explain why NLS estimation of

$$
y_1=\mathbf{x}_1 \boldsymbol{\delta}_1+\gamma_1 \hat{y}_2^{\gamma_2}+\text { error }
$$

does not consistently estimate $\delta_1, \gamma_1$, and $\gamma_2$ when $\gamma_1 \neq 0$ and $\gamma_2 \neq 1$.

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Problem 2

Consider the following labor supply function nonlinear in parameters:

$$
\text { hours }=\mathbf{z}_1 \delta_1+\gamma_1\left(\text { wage }^{\rho_1}-1\right) / \rho_1+u_1, \quad \mathrm{E}\left(u_1 \mid \mathbf{z}\right)=0,
$$

where $\mathbf{z}_1$ contains unity and $\mathbf{z}$ is the full set of exogenous variables.
a. Show that this model contains the level-level and level-log models as special cases. (Hint: For $w>0,\left(w^\rho-1\right) / \rho \rightarrow \log (w)$ as $\rho \rightarrow 0$.)
b. How would you test $\mathrm{H}_0: \gamma_1=0$ ? (Be careful here; $\rho_1$ cannot be consistently estimated under $\mathrm{H}_0$.)
c. Assuming that $\gamma_1 \neq 0$, how would you estimate this equation if $\operatorname{Var}\left(u_1 \mid \mathbf{z}\right)=\sigma_1^2$ ? What if $\operatorname{Var}\left(u_1 \mid \mathbf{z}\right)$ is not constant?
d. Find the gradient of the residual function with respect to $\boldsymbol{\delta}_1, \gamma_1$, and $\rho_1$. (Hint: Recall that the derivative of $w^\rho$ with respect to $p$ is $w^\rho \log (w)$.)
e. Explain how to obtain the score test of $\mathrm{H}_0: p_1=1$.

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Problem 3

Use Theorem 14.3 to show that the optimal instrumental variables based on the conditional moment restrictions (14.60) are given by equation (14.63).

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Problem 4

a. Show that, under Assumptions WNLS.1-WNLS. 3 in Chapter 12, the weighted NLS estimator has asymptotic variance equal to that of the efficient IV estimator based on the orthogonality condition $\mathrm{E}\left[\left(y_i-m\left(\mathbf{x}_i, \boldsymbol{\beta}_o\right)\right) \mid \mathbf{x}_i\right]=0$.
b. When does the NLS estimator of $\boldsymbol{\beta}_0$ achieve the efficiency bound derived in part a?
c. Suppose that, in addition to $\mathrm{E}(y \mid \mathbf{x})=m\left(\mathbf{x}, \boldsymbol{\beta}_{\mathrm{o}}\right)$, you use the restriction $\operatorname{Var}(y \mid \mathbf{x}) =\sigma_{\mathrm{o}}^2$ for some $\sigma_{\mathrm{o}}^2>0$. Write down the two conditional moment restrictions for estimating $\boldsymbol{\beta}_o$ and $\sigma_o^2$. What are the efficient instrumental variables?

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Problem 5

Write down $\boldsymbol{\theta}, \boldsymbol{\pi}$, and the matrix $\mathbf{H}$ such that $\boldsymbol{\pi}=\mathbf{H} \boldsymbol{\theta}$ in Chamberlain's approach to unobserved effects panel data models when $T=3$.

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Problem 6

Let $\boldsymbol{\pi}$ and $\boldsymbol{\pi}$ be two consistent estimators of $\boldsymbol{\pi}_0$, with Avar $\sqrt{N}\left(\boldsymbol{\pi}-\boldsymbol{\pi}_0\right)=\boldsymbol{\Xi}_0$ and $\mathrm{Avar} \sqrt{N}\left(\tilde{\boldsymbol{\pi}}-\boldsymbol{\pi}_0\right)=\boldsymbol{\Lambda}_0$. Let $\hat{\boldsymbol{\theta}}$ be the CMD estimator based on $\hat{\boldsymbol{\pi}}$, and let $\tilde{\boldsymbol{\theta}}$ be the CMD estimator based on $\tilde{\boldsymbol{\pi}}$, where $\boldsymbol{\pi}_{\mathrm{o}}=\mathbf{h}\left(\boldsymbol{\theta}_{\mathrm{o}}\right)$. Show that, if $\boldsymbol{\Lambda}_{\mathrm{o}}-\boldsymbol{\Xi}_{\mathrm{o}}$ is p.s.d., then so is Avar $\sqrt{N}\left(\tilde{\boldsymbol{\theta}}-\boldsymbol{\theta}_0\right)$ - Avar $\sqrt{N}\left(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_0\right)$. (Hint: Twice use the fact that, for two positive definite matrices $\mathbf{A}$ and $\mathbf{B}, \mathbf{A}-\mathbf{B}$ is p.s.d. if and only if $\mathbf{B}^{-1}-\mathbf{A}^{-1}$ is p.s.d.)

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Problem 7

Show that when the mapping from $\boldsymbol{\theta}_0$ to $\boldsymbol{\pi}_0$ is linear, $\boldsymbol{\pi}_0=\mathbf{H} \boldsymbol{\theta}_0$ for a known $S \times P$ matrix $\mathbf{H}$ with $\operatorname{rank}(\mathbf{H})=P$, the CMD estimator $\hat{\boldsymbol{\theta}}$ is

$$
\hat{\boldsymbol{\theta}}=\left(\mathbf{H}^{\prime} \hat{\boldsymbol{\Xi}}^{-1} \mathbf{H}\right)^{-1} \mathbf{H}^{\prime} \hat{\boldsymbol{\Xi}}^{-1} \hat{\boldsymbol{\pi}}
$$

Equation (14.89) looks like a generalized least squares (GLS) estimator of $\hat{\boldsymbol{\pi}}$ on $\mathbf{H}$ using variance matrix $\overline{\boldsymbol{E}}$, and this apparent similarity has prompted some to call the minimum chi-square estimator a "generalized least squares" (GLS) estimator. Unfortunately, the association between CMD and GLS is misleading because $\hat{\pi}$ and $\mathbf{H}$ are not data vectors whose row dimension, $S$, grows with $N$. The asymptotic properties of the minimum chi-square estimator do not follow from those of GLS.

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Problem 8

In Problem 13.9, suppose you model the unconditional distribution of $y_0$ as $f_0\left(y_0 ; \boldsymbol{\theta}\right)$, which depends on at least some elements of $\boldsymbol{\theta}$ appearing in $f_t\left(y_t \mid y_{t-1} ; \boldsymbol{\theta}\right)$. Discuss the pros and cons of using $f_0\left(y_0 ; \boldsymbol{\theta}\right)$ in a maximum likelihood analysis along with $f_t\left(y_t \mid y_{t-1} ; \boldsymbol{\theta}\right), t=1,2, \ldots, T$.

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Problem 9

Verify that, for the linear unobserved effects model under Assumptions RE.1RE.3, the conditions of Lemma 14.1 hold for the fixed effects $\left(\hat{\boldsymbol{\theta}}_2\right)$ and the random effects $\left(\hat{\boldsymbol{\theta}}_1\right)$ estimators, with $p=\sigma_u^2$. (Hint: For clarity, it helps to introduce a cross section subscript, $i$. Then $\mathbf{A}_1=\mathrm{E}\left(\tilde{\mathbf{X}}_i^{\prime} \tilde{\mathbf{X}}_i\right)$, where $\tilde{\mathbf{X}}_i=\mathbf{X}_i-\lambda_j \overline{\mathbf{x}}_i ; \mathbf{A}_2=\mathrm{E}\left(\tilde{\mathbf{X}}_i^{\prime} \tilde{\mathbf{X}}_i\right)$, where $\ddot{\mathbf{X}}_i=\mathbf{X}_i-\mathbf{j}_T \overline{\mathbf{x}}_i ; \mathbf{s}_{i 1}=\ddot{\mathbf{X}}_i^{\prime} \mathbf{r}_i$, where $\mathbf{r}_i=\mathbf{v}_i-\ddot{\mathbf{j}}_T \bar{v}_i ;$ and $\mathbf{s}_{i 2}=\ddot{\mathbf{X}}_i^{\prime} \mathbf{u}_i ;$ see Chapter 10 for further notation. You should show that $\ddot{\mathbf{X}}_i^{\prime} \mathbf{u}_i=\ddot{\mathbf{X}}_i^{\prime} \mathbf{r}_i$ and then $\ddot{\mathbf{X}}_i^{\prime} \ddot{\mathbf{X}}_i=\ddot{\mathbf{X}}_i^{\prime} \ddot{\mathbf{X}}_i$.)

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Problem 10

Consider the model in (14.81) under the strict exogeneity condition (14.82). In addition, assume that $\mathrm{E}\left(c_i \mid \mathbf{x}_i\right)=0$ (and so $\mathbf{x}_{i t}$ should contain a full set of time dummies, but we do not show them explicitly).
a. If $v_{i t}=\eta_t c_i+u_{i t}$, show that $\mathrm{E}\left(v_{i t} \mid \mathbf{x}_i\right)=0, t=1, \ldots, T$.
b. Assume that $\operatorname{Var}\left(\mathbf{u}_i \mid \mathbf{x}_i, c_i\right)=\sigma_u^2 \mathbf{I}_T$ and $\operatorname{Var}\left(c_i \mid \mathbf{x}_i\right)=\sigma_c^2$. Find $\operatorname{Var}\left(v_{i t}\right)$ and $\operatorname{Cov}\left(v_{i t}, v_{i s}\right), t \neq s$.
c. Under parts a and b , propose an estimator that is symptotically more efficient than the usual RE estimator.

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Problem 11

Consider a multivariate regression model nonlinear in the parameters,

$$
\mathbf{y}_i=\mathbf{X}_i \mathbf{g}\left(\boldsymbol{\theta}_{\mathrm{o}}\right)+\mathbf{u}_i, \quad \mathrm{E}\left(\mathbf{u}_i \mid \mathbf{X}_i\right)=\mathbf{0}
$$

where $\mathbf{y}_i$ is $G \times 1, \mathbf{X}_i$ is $G \times K$, and $\mathbf{g}: \mathbb{R}^P \rightarrow \mathbb{R}^K$ is continuously differentiable. Explain how to estimate $\boldsymbol{\theta}_{\mathrm{o}}$ using CMD.

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