Raising the bar

Chapter 13. Bridge to econometrics

Unit 13.1. Problems with LSE econometrics

In the exposition below we refer to Dougherty, C. Introduction to Econometrics. Oxford: Oxford University Press, 4th ed. 2011.

Problem with coverage. Dougherty's book is packed with information (it is 530 pages without appendices). There are many small and large models, math is mixed with economic considerations, and literally any piece of that information may appear on the exam.

Example 13.1. In the UoL exam 2012, zone B there is a question: briefly explain what the error correction model is.

The answer from Dougherty is one paragraph of text on pp. 405$latex -$406. By the time the students study that paragraph, they are fed up with different models. They will not remember that paragraph because it is a small detail in a rich landscape. Professionals encounter many such details in their work, easily understand them and don't try to remember. I would reduce the number of small details and concentrate on a deep study of a small number of most important theories, as it is done in my Chapter 10.

Problem with level.

Example 13.2 (UoL exam 2012, zone B). The model is $latex y_t = \beta x_t + u_t$, $latex E u_t = 0$, $latex E u_t^2 = \sigma^2$, $latex t = 1, \ldots, n$. An econometrician suggests an estimator for $latex \beta$ given by

$latex \tilde \beta = \frac{1}{n} \sum_{t=1}^n y_t / x_t.$

Prove that $latex \tilde \beta$ is an unbiased estimator for $latex \beta$ and that its variance is

$latex V(\tilde \beta) = \frac{\sigma^2}{n^2} \left( \sum_{t=1}^n 1/x_t^2 \right)$

under certain assumptions. What extra assumptions have you used?

Formally, to answer this question, all you need is properties of variances and covariances and a notion of simple regression. Informally, this is a truly research problem. You need to understand the proofs in Boxes 2.2 and 2.3 of the book (this is two pages of dense math; among other things, there is double summation which many students don't understand). Moreover, you have to extract the ideas and apply them to the exam problem. Your understanding of the ideas should be deep enough as to be able to impose appropriate conditions on the errors. If you think it's hard to come up with more difficult exam problems, you are wrong. Look at the next example.

Example 13.3 (UoL exam 2013, zone A). Consider the model $latex y_t = \alpha x_t + u_t$, $latex t = 1, \ldots, T$, where $latex E u_t = 0$, $latex E u_t^2 = \sigma^2 x_t^2$, $latex E u_s u_t = 0$ if $latex s \neq t$, for all $latex s$ and $latex t$. $latex x_t$ is an observed non-random variable. The density function of $latex u_t$ is

$latex f(u_t) = \left( 2 \pi \sigma^2 x_t^2 \right)^{-1/2} \exp \left[ -\frac{1}{2} \left( \frac{u_t}{\sigma x_t} \right)^2 \right].$

$latex \quad$(1) Derive the maximum likelihood (ML) estimators of $latex \alpha$ and $latex \sigma^2$. Show that $latex \hat \alpha$, the ML estimator of $latex \alpha$, is unbiased. (8 marks)

$latex \quad$(2) Compare the ML estimator of $latex \alpha$ with the weighted least squares estimator of $latex \alpha$. (7 marks)

The model in this example is the same as in Example 13.2. However, in Example 13.2 the estimator is given, while in this one it has to be derived. This already represents a major hazard for some students. But they also have to repeat the derivations required for Example 13.2 (unbiasedness and variance) and compare the resulting properties of the OLS estimator with those of the ML estimator (which is to be derived too)!

I love research problems but giving them on the exam is too much. If the LSE wants to use its exams as a screening device for future Nobel prize winners, that's fine. But then the LSE Stats courses should provide an adequate background. It is not adequate, because in affiliate centers the first year Stats is based on AP Stats textbooks, and thus is a waste of time, and the second year Stats does not prepare the students to handle two-page proofs.

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