Reverse Regression Follow-up

At the end of my recent post on Reverse Regression, I posed three simple questions - homework for the students among you, if you will. 

Here they are again, with brief "solutions":

A Source of Irritation

I very much liked one of ECONJEFF's posts last week, titled "Epistemological Irritation of the Day".

The bulk of it reads:

" "A direct test of the hypothesis is looking for significance in the relationship between [one variable] and {another variable]."

No, no, no, no, no. Theory makes predictions about signs of coefficients, not about significance levels, which also depend on minor details such as the sample size and the amount of variation in the independent variable of interest present in the data."

He was right to be upset - and see his post for the punchline!

© 2014, David E. Giles

Econometric Society World Congress

Every five years, the Econometric Society holds a World Congress. In those years, the usual annual European, North American, Latin American, and Australasian meetings are held over.

The first World Congress was held in Rome, in 1960. I've been to a few of these gatherings over the years, and they're always great events.

The next World Congress is going to be held in Montréal, Canada, in August of 2015. You can find all of the details right here.

Something to look forward to!

© 2014, David E. Giles

A Reverse Regression Inequality

Suppose that we fit the following simple regression model, using OLS:

            y_i = βx_i + ε_i   .                                                              (1)

To simplify matters, suppose that all of the data are calculated as deviations from their respective sample means. That's why I haven't explicitly included an intercept in (1). This doesn't affect any of the following results.

The OLS estimator of β is, of course,

            b = Σ(x_iy_i) / Σ(x_i²) ,

where the summations are for i = 1 to n (the sample size).

Now consider the "reverse regression":

           x_i = αy_i + u_i   .                                                             (2)

The OLS estimator of α is

           a = Σ(x_iy_i) / Σ(y_i²).

Clearly, a ≠ (1 / b), in general. However, can you tell if a ≥ (1 / b), or if a ≤ (1 / b)?

The answer is, "yes", and here's how you do it.

The Econometrics of Temporal Aggregation V - Testing for Normality

This post is one of a sequence of posts, the earlier members of which can be found here, here, here, and here. These posts are based on Giles (2014).

Some of the standard tests that we perform in econometrics can be affected by the level of aggregation of the data. Here, I'm concerned only with time-series data, and with temporal aggregation. I'm going to show you some preliminary results from work that I have in progress with Ryan Godwin. Although these results relate to just one test, our work covers a range of testing problems.

I'm not supplying the EViews program code that was used to obtain the results below - at least, not for now. That's because what I'm reporting is based on work in progress. Sorry!

As in the earlier posts, let's suppose that the aggregation is over "m" high-frequency periods. A lower case symbol will represent a high-frequency observation on a variable of interest; and an upper-case symbol will denote the aggregated series.

So,

               Y_t = y_t + y_{t - 1} + ......+ y_{t - m + 1} .

If we're aggregating monthly (flow) data to quarterly data, then m = 3. In the case of aggregation from quarterly to annual data, m = 4, etc.

Now, let's investigate how such aggregation affects the performance of the well-known Jarque-Bera (1987) (J-B) test for the normality of the errors in a regression model. I've discussed some of the limitations of this test in an earlier post, and you might find it helpful to look at that post (and this one) at this point. However, the J-B test is very widely used by econometricians, and it warrants some further consideration.

Consider the following small Monte Carlo experiment.

The Village Idiot Hypothesis

Yesterday, I received an email from Michael Belongia (Economics, U. Mississippi). With it, he kindly sent a copy of the Presidential Address to the American Agricultural Economics Association in 1979. The talk, given by Richard A. King, was titled "Choices and Consequences". It makes interesting reading, and many of the points that King makes are just as valid today as they were in 1979.

He has a lot to say about empirical consumer demand studies, especially as they relate to agricultural economics. In particular, he's rightly critical of the very restrictive characteristics of the Linear Expenditure System (Stone, 1954), and the Rotterdam Model (Theil, 1975). However, many of the objections that King raised were overcome just a year later with the "Almost Ideal Demand System" introduced by Deaton and Muellbauer (1980). 

However, it was my recent post on hypothesis testing that prompted Michael to email me, and King makes some telling observations on this topic in his address.

I liked this remark about the need to be explicit about the hypotheses that we have in mind when undertaking empirical work:

King also talks about "The Village Idiot Hypothesis", in relation to the preoccupation with testing hypotheses such as β = 0. 

As Michael said to me in his email, "When, as in one example, decades of research have indicated that some elasticity is -0.2, why do new papers test whether β = 0 rather than β = -0.2?"

If you have access to the American Journal of Agricultural Economics, I recommend that you take a look at Richard King's address, as he makes several other important points that practitioners should take to heart.

References

King, R. A., 1979. Choices and consequences. American Journal of Agricultural Economics, 61, 839-848.

Deaton, A. and J. Muellbauer, 1980. An almost ideal demand system. American Economic Review, 70, 312-326.

Stone, R., 1954. Linear expenditure systems and demand analysis: An application to the pattern of British demand".Economic Journal, 64, 511-527.

Theil, H., 1975. Theory and Measurement of Consumer Demand, Vol. 1. North-Holland, Amsterdam.

© 2014, David E. Giles

Update to ARDL Add-In for EViews

In a post back in January, I drew attention to an Add-In for EViews that allows you to estimate ARDL models. The Add-In was written by Yashar Tarverdi. At that time, one limitation was that the Add-In handles only two variables, X and Y.

Judging by the questions and feedback I get about ARDL models, I know you'll be delighted to know that this limitation has been eased considerably. News out of @IHSEViews on Twitter this morning announces that the Add-In will now handle up to ten variables.

Good job! And thanks!

© 2014, David E. Giles

Computing Power Curves

In a recent post I discussed some aspects of the distributions of some common test statistics when the null hypothesis that's being tested is actually false. One of the things that we saw there was that in many cases these distributions are "non-central", with a non-centrality parameter that increases as we move further and further away from the null hypothesis being true.

In such cases, it's the value of the non-centrality parameter that determines the power of tests. For a particular sample size and choice of significance level, this parameter usually depends on the all of the other features of the testing problem in question.

To illustrate this in more detail, let's consider a linear multiple regression model:

Central and Non-Central Distributions

Let's imagine that you're teaching an econometrics class that features hypothesis testing. It may be an elementary introduction to the topic itself; or it may be a more detailed discussion of a particular testing problem. We're not talking here about a course on Bayesian econometrics, so in all likelihood you'll be following the "classical" Neyman-Pearson paradigm.

You set up the null and alternative hypotheses. You introduce the idea of a test statistic, and hopefully, you explain why we try to find one that's "pivotal". You talk about Type I and Type II errors; and the trade-off between the probabilities of these errors occurring. 

You might talk about the idea of assigning a significance level for the test in advance of implementing it; or you might talk about p-values. In either case, you have to emphasize to the classt that in order to apply the test itself, you have to know the sampling distribution of your test statistic for the situation where the null hypothesis is true.

Why is this?

Confusing Charts

Today's on-line edition of The New Zealand Herald includes an article titled "Junior rugby putting little kids in harm's way". The article included two charts, presented one after the other, and explicitly intended to be viewed as a a pair. Here they are:

Why on earth didn't they use the same colour-coding for the four age groups in both charts?



© 2014, David E. Giles