Assignment 2

Use the same labor supply model as Assignment 1. We now assume that net log wages follow a process: \[ \log(\tilde{W}_{n,t}) = \gamma_{0} + \gamma_{1} Z_{n,t} + \epsilon_{n,t} \] where \(Z_{n,t}\) is independent of the pair (\(\alpha_{n},\epsilon_{n,t}\)). You can think of \(Z_{n,t}\) as an instrument for labor demand or an indicator for the presence of a wage subsidy. As before, we assume no uncertainty, so the path of \(Z_{n,t}\) is treated as known.

To begin, suppose that you have a single cross-section of data: \[ (H_{n,0},\tilde{W}_{n,0},Z_{n,0},C_{n,0},A_{n,0})_{n=1}^{N} \] where \(Z_{n,0}\in\{0,1\}\) is perceived to be a permanent wage subsidy (so \(Z_{n,0}=Z_{n,t}\) for all \(t>0\)).

Question 1

Suppose we estimate the following system via 2SLS: \[ \log(\tilde{W}_{n,0}) = \gamma_{0} + \gamma_{1} Z_{n,t} + \epsilon_{n,t} \] \[ \log(H_{n,0}) = \alpha_{0} + \alpha_{1}\log(W_{n,0}) + \varepsilon_{n,0} \] What is the population limit of the 2SLS estimator? Does it recover a structural or policy parameter of interest?

Question 2

You are talking to your friend from Assignment 1 (Mr Straw Man) and you point out to him that the correct specification is the structural one: \[ \log(H_{n,t}) = \mu + \psi\log(\tilde{W}_{n,0}) - \psi\sigma\log(C_{n,0}) + \eta_{n,0}.\] Your friend tells you that you have no hope here because you have two endogenous variables and only one instrument. You are going to use the model to prove him wrong! Define \(\tilde{Z}_{n,0} = M_{n,0}\times Z_{n,0}\) where \(M_{n,0}\in\{0,1\}\) is an indicator for whether \(A_{n,0}\) is above or below the median. Note that the conditional expectation of log consumption can be written wlog as: \[ \mathbb{E}[\log(C)|M,Z] = \delta_{0} + \delta_{1}M + \delta_{2}Z + \delta_{3}\underbrace{MZ}_{=\tilde{Z}}\] Use the model to argue that \(\delta_{3}\neq0\).

Question 3

Now show that one can write: \[ \mathbb{E}[\log(H)|M,Z] = \kappa_{0} + \kappa_{1}M + \psi\gamma_{1} Z - \psi\sigma \delta_{2}Z - \psi\sigma \delta_{3}\tilde{Z}.\] And combine these two expressions with the wage equation to argue that \(\psi\) and \(\sigma\) are identified. Why is important that \(\delta_{3}\neq 0\)?

Question 4

Write code to estimate the structural parameters (via GMM or 2SLS) and use a monte-carlo simulation to evaluate the performance of your estimator. You can adapt the code I used in the example here. The example assumes wages are constant, which you can assume also. You will have to adapt the example to include the variable \(Z\in\{0,1\}\) and make your own choices for \((\gamma_{0},\gamma_{1})\). In order to get reasonable precision, you will probably want to evaluate the estimator using a larger sample size (e.g. \(N=10,000\)). If you want you can also try a richer set of instruments by including more interactions with wealth (such as dummies indicating wealth quartile, for example).