Econometrics

We have examined the notion of local uniform convergence  in probability, and provided with (a useful for example in the case of the OLSE and the IVE when the parameter space is compact), result that establishes the aforementioned convergence via pointwise convergence in probability combined with a (with probability one) Lipschitz continuity property  for the empirical criterion (i.e. the function from which the estimator is derived) and a form of boundedness with high probability condition for the associated Lipschitz coefficient.

We then provided our basic consistency theorem that amounts to the case where the parameter space is compact, the empirical criterion converges locally uniformly in probability to a deterministic limiting criterion, and the latter satisfies an asymptotic identification condition. Under those conditions the estimator is weakly consistent.

The condition was discussed (and corrected-please amend you Tuesday lecture notes with the correction that was provided in the Friday lecture if you haven't already done it! See also the notes linked to below that contain the relevant discussion). We have also remarked that it holds for example whenever (i) the limiting criterion is continuous in the parameter, it is uniquely minimized at the true parameter value, and the parameter space is compat, or, (ii) when the criterion is strictly convex, it is minimized at the true parameter value and the parameter space is closed and convex.

We have then proven the theorem and discussed (without proving it) a modification that involves convexity. We have used the results to establish weak consistency for the OLSE (please check at the endnote (1) in the linked to notes below to make sure that you understand the reason for the use of a peculiar version of the OLSE criterion for the establishment-it essentially says that convergence results about empirical second moment are not essentially needed in this model in order to have consistency for the OLSE) when the parameter space is compact, or when the parameter space is closed and convex, and via the use of a geometric argument to extend the validity of the property to more general parameter spaces. We discussed a glimpse of (parameter) misspecification; a situation where the linear specification of the conditional mean is correct, yet the closed and convex parameter space is erroneously chosen so that it does not contain the true parameter value. The convexity of the OLS criterion establishes that the estimator will converge in probability to the unique point of the parameter space that lies closest to the true parameter value. When the parameter space is not convex the limit theory becomes trickier.

Notes for the above can be found here (v. 24/01/23).

Can you use an argument similar to the one about parameter misspecification in order to obtain consistency for the OLSE for an arbitrary parameter space (that remember, it must contain the true parameter value), via the weakly consistent unrestricted OLSE?