Estimation of point-identified Dynamic Discrete Choice models with moment inequalities

Question

I'm trying to understand and implement in code the method in Bajari, Benkard, Levin (2007). The first stage is clear to me, as well as how to forward-simulate to obtain an estimate of the value function. What I don't get is:

• What are the moment conditions? In the paper, the moments are expressed as the difference of the simulated value employing the policy obtained from the first stage, and some other policy:

For example, I tried choosing N randomly selected states and obtaining the simulated value for simulated paths of length T, by using the first-stage policy and some perturbed version of that policy. Is this one way of generating moment inequalities?

• Then, the paper refers to the true parameters as those that minimize the following target function:

I understand that the expected difference in simulated values is strictly positive according to Revealed Preferences, but I don't understand why the true parameters are the ones that minimize the function above. For example, if the value function becomes zero at any state and choice for a given set of parameters, the simulated values will be zero as well under any policy, even if the true parameters are not such.

Any help or reference of a concrete example will be greatly appreciated.

Answer 1

Function $Q$ has to be minimized wrt $(\theta,\alpha)$. The parameters $(\theta,\alpha)$ compatible with Nash equilibrium and rationalizing (or generating) the data have to satisfy $$g(x;\theta,\alpha)\leq0.$$ In the case where, for given $x$, $$g(x;\theta,\alpha)>0,$$ the parameter values have to be changed in order to possibly reverse the strict inequality and minimize $Q$. If $$g(x;\theta,\alpha)\leq0$$ for some $(\theta,\alpha)$, then this should not contribute to increase $Q$, because it is fully compatible with the data and model. Hence the term $\min\{g(x;\theta,\alpha),0\}$ in the expression of $Q$.