ECON 626: Problem Set 8

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

Problem 1

Consider the linear model \[ Y_i = \beta X_i^* + \epsilon_i \] with \(\var(X_i^*)>0\), \(\Er[\epsilon_i X_i^*] = 0\), and \(\Er[\epsilon_i]=0\). Furthermore suppose \(X_i^* \in \{0,1\}\), but \(X_i^*\) is unobserved. Instead, you observe \(X_i \in \{0,1\}\) and know that \[ P(X_i = 1 | X_i^* = 1) = P(X_i=0 | X_i^*=0) = p \] where \(p > 1/2\) is unknown.

OLS is Inconsistent

Let \(\hat{\beta}^{OLS} = \frac{\sum X_i Y_i}{\sum X_i^2}\) be the least squares estimator. Let \(\pi = P(X_i^*=1)\). Compute \(\plim \hat{\beta}^{OLS}\) in terms of \(p\), \(\pi\), and \(\beta\).

Instrument?

Suppose you also observe \(Z_i \in \{0,1\}\) with \[ P(Z_i = 1 | X_i^* = 1) = P(Z_i=0 | X_i^*=0) = q \] where \(q>1/2\), and \(Z_i\) and \(X_i\) are independent conditional on \(X_i^*\). Let \(\hat{\beta}^{IV} = \frac{\sum Z_i Y_i}{\sum Z_i X_i}\) be the instrumental variable estimator. Compute \(\plim \hat{\beta}^{IV}\)

Or Something Else?

Describe how \(X\), \(Y\), and \(Z\) could be used to estimate \(\beta\).

Hint: think of \(\beta\), \(p\), \(q\), and \(\pi = P(X^*_i = 1)\) as four parameters to estimate, and come up with four moment conditions that involve these parameters.

Problem 2

Consider the model: \[ y_{it} = \rho y_{it-1} + x_{it}'\beta + \alpha_i + u_{it} \] for \(i=1,.., N\) and \(t=1,...,T\). Assume observations are independent accross \(i\), \(x\) is strictly exogenous, \(\Er[x_{it} u_{is}] = 0 \forall s,t\), and \(y_{it-1}\) is weakly exogenous, \(\Er[y_{it-1} u_{it+s}] = 0\) for \(s \geq 0\).

First Differences

Let \(\Delta y_{it} = y_{it} - y_{it-1}\). Take differences of the model to eliminate \(\alpha_i\), leaving \[ \Delta y_{it} = \rho \Delta y_{it-1} + \Delta x_{it}'\beta + \Delta u_{it}. \] Will OLS on this equation be consistent?

Fixed Effects Inconsistent

Show that for \(T\) fixed and \(N \to \infty\), the fixed effects estimator, i.e. regressing \(y_{it}\) on \(y_{it-1} - \bar{y}_i\) and \(x_{it} - \bar{x}_i\), is not consistent. Hint: what is \(\Er[\bar{y}_i u_{it}]\)?

GMM

Assume \(T \geq 3\). Argue that \(\Er[\Delta u_{it} y_{it-\ell}]=0\) for \(\ell \geq 2\), and that \(\Er[\Delta y_{it-1} y_{it-\ell}] \neq 0\). Use this fact, along with the assumptions above to derive a GMM estimator for \(\rho\) and \(\beta\).