Course Blog Econometrics

Synopsis: Lectures 5 and 6

Wed, 25 Jan 2023 00:28:07 +0300

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?

Synopsis: Lectures 3 and 4

Tue, 17 Jan 2023 11:49:32 +0300

We continued exploring the class of optimization based estimators via-among others-the examination of the semi-parametric linear model with instrumental variables.

In the general case we commented on issues of existence of the estimator, based on properties of the underlying empirical criterion and the parameter space. We examined a generalization of the estimator enabling the consideration of optimization errors, that among others are usually met in numerical procedures.

We begun the exploration of asymptotic properties of such-like estimators, having in mind the usual linear model. We reminded ourselves of the high level conditions that ensure weak consistency of the OLSE when the parameter space is maximal. We asked whether this remains true in the general case when the parameter space is arbitrary yet the true parameter value lies in it. In such cases we generally do not have at our disposal an analytical form of the estimator as a measurable function of the sample. Thus we work with limiting properties of the criterion that are relevant to optimization.

We begun working with the general case, and examine the mode of locally uniform convergence in probability.

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

Synopsis: Lectures 1-2 (Introductory Stuff)

Sun, 11 Dec 2022 23:46:11 +0300

We are interested in developing a quite general theory of optimization based estimation and testing procedures. To this end we begun the construction of an appropriate framework consisting of the notion of the sample, the notion of the parametric and semi-parametric statistical model and the subsequent notions of the estimator and testing procedure in such-like models.

Given the complexity incurred in models that even slightly deviate from the standard forms of the linear model; this among others implies that several estimators and testing procedures are only approximable via numerical optimization, resampling, and non-analytically derivable, and are in any case quite non-linear functions of the sample, the general properties upon which we will rely in order to evaluate our inferential procedures are asymptotic and minimal; namely (weak) consistency, rates and asymptotic normality for estimators, and asymptotic conservativeness and consistency for tests.

We begun exploring the class of optimization based estimators by noting that in the standard form of the linear model, due to its structure and under the usual identification condition, there exists a function that is uniquely minimized at the unknown parameter value. If this function were hence analytically known, the unknown value of the parameter would be accurately retrieved. However this function is not analytically known since it also depends on the unknown parameter value. It can be however empirically approximated by its sample counterpart, the optimization of which leads to the OLS estimator.

Notes for the above can be found here (v.24/1/23).