Assignment 3

Estimating a Dynamic Discrete Choice Model

The Model

Consider the following infinite horizon, dynamic labor supply problem.

• There are two choices in the model, work \(j=0\) or don’t work (\(j=0\))
• The wage \(W\) is a state variable with two values, \(w_{L}<w_{H}\).
• The evolution of the wage depends on work behavior as follows:
  • \(P(w_{t+1}=w_{H}|w_{t}=w_{H},j=1) = 1\)
  • \(P(w_{t+1}=w_{L}|w_{t}=w_{H},j=0) = \pi\)
  • \(P(w_{t+1}=w_{L}|w_{t}=w_{L},j=0) = 1\)
  • \(P(w_{t+1}=w_{H}|w_{t}=w_{L},j=1) = \pi\)
• Thus, human capital appreciates and depreciates with symmetric probabilities given by \(\pi\) when working / not working.
• Per-period payoffs are given by: \[ U_{n,t,j}(w) = j \times w - \kappa (1-j_{t-1}) + \epsilon_{ntj} \] where \(\epsilon_{ntj}\) is a type 1 extreme value shock with location parameter 0 and scale parameter \(\sigma\), and \(j_{t-1}\) indicates whether person \(n\) worked in the previous period.

Part 1

Write code to solve this model and calculate the conditional probability of working given human capital (\(w\)) and lagged work behavior \(j_{-1}\).

Part 2

Write code to simulate panel data from the model consisting of \(N\) individuals and \(T\) time periods per individual. You can make whatever assumption you like regarding initial conditions.

Part 3

Has this model specification already imposed a location normalization on utilities? What about a scale normalization? If not, state the normalization you are going to make.

Part 4

Assume you know the values of \(\beta\) and \(\pi\). Write code to estimate – by maximizing the log-likelihood of observed choices – the remaining model parameters and write a monte-carlo simulation to document the performance of your estimator.

Part 5

Now try estimating \(\beta\) in addition to the parameters from Part 4. Does \(\beta\) appear to be identified? Do you have some intuition for why / why not?