Assignment 3: Estimating a Search Model

In this homework you are going to estimate the parameters of the search model for each demographic group individually. That is, you will not impose the parametric restrictions that mapped demographics \(X\) to deeper parameters using the \(\gamma\) vectors from last week.

Part 1

Fix \(\sigma_\zeta\) (the standard deviation of measurement error in log wages) to 0.05. Following your work from last week (and recitation this week)write a function that calculates the log-likelihood of a single month of data from the CPS given \((h,\delta,\mu,\sigma,w^*)\) where \(w^*\) is the reservation wage and \(h = \lambda\times(1-F_{W}(w^*;\mu,\sigma))\).

Part 2

Use the log-likelihood function to get maximum likelihood estimates of \((\hat{h},\hat{\delta},\hat{\mu},\hat{\sigma},\hat{w^*})\) for white men with a college degree.

What is the advantage of estimating \(h\) and \(w^*\) directly instead of \(\lambda\) and \(b\)?

Part 3

Back out the implied maximum likelihood estimates of \(\hat{\lambda}\) and \(\hat{b}\) as a function of the estimated parameters from part (1).

Part 4

Provide an estimate of the asymptotic variance of \((\hat{h},\hat{\delta},\hat{\mu},\hat{\sigma},\hat{w^*})\) using the standard MLE formula.

Part 5

Recall that the delta method implies that if \(\hat{\theta}\) is asymptotically normal with asymptotic variance \(V\) then the vector-valued function \(F(\hat{\theta})\) is also asymptotically normal with:

\[ \sqrt{N}(F(\hat{\theta}) - F(\theta)) \rightarrow_{d} \mathcal{N}(0,\nabla_{\theta'}FV\nabla_{\theta}F') \]

Use this fact to estimate the asymptotic variance of \((\hat{h},\hat{\delta},\hat{\mu},\hat{\sigma},\hat{w^*},\hat{\lambda},\hat{b})\).

Part 6

Now report all of your estimates and standard errors for this group. Repeat this exercise for each group.

If we thought that the parametric relationships using \(\gamma\) from Homework 2 described the true values of the parameters for each group, how might we use these group-specific estimates to derive estimates of each \(\gamma\)?