Wednesday, November 19, 2014

The Rise of Bayesian Econometrics

A recent discussion paper by Basturk et al. (2014) provides us with (at least) two interesting pieces of material. First, they give a very nice overview of the origins of Bayesian inference in econometrics. This is a topic dear to my heart, given that my own Ph.D. dissertation was in Bayesian Econometrics; and I began that work in early 1973 - just two years after the appearance of Arnold Zellners' path-breaking book (Zellner, 1971).

Second, they provide an analysis of how the associated contributions have been clustered, in terms of the journals in which they have been published. The authors find, among other things, that: 
"Results indicate a cluster of journals with theoretical and applied papers, mainly consisting of Journal of Econometrics, Journal of Business and Economic Statistics, and Journal of Applied Econometrics which contains the large majority of high quality Bayesian econometrics papers."
A couple of the paper coming out of my dissertation certainly fitted into that group - Giles (19??) and Giles and Rayner (19??)

The authors round out their paper as follows:
"...with a list of subjects that are important challenges for twenty-first century Bayesian conometrics: Sampling methods suitable for use with big data and fast, parallelized and GPU, calculations, complex models which account for nonlinearities, analysis of implied model features such as risk and instability, incorporating model incompleteness, and a natural combination of economic modeling, forecasting and policy interventions."
So, there's lots more to be done!


Basturk, N., C. Cacmakli, S. P. Ceyhan, and H. K. van Dijk, 2014. On the rise of Bayesian econometrics after Cowles Foundation monographs 10 and 14. Tinbergen Institute Discussion Paper TI 2014-085/III.

Giles, D.E.A., 1975. Discriminating between autoregressive forms: A Monte Carlo comparison of Bayesian and ad hoc methods”, Journal of Econometrics, 3, 229-248.

Giles, D.E.A.and A.C. Rayner, 1979. The mean squared errors of the maximum likelihood and natural-conjugate Bayes regression estimators”, Journal of Econometrics, 11, 319-334.

Zellner, A., 1971. An Introduction to Bayesian Inference in Econometrics. Wiley, New York.

© 2014, David E. Giles

Sunday, November 16, 2014

Orthogonal Regression: First Steps

When I'm introducing students in my introductory economic statistics course to the simple linear regression model, I like to point out to them that fitting the regression line so as to minimize the sum of squared residuals, in the vertical direction, is just one possibility.

They see, easily enough, that squaring the residuals deals with the positive and negative signs, and that this prevents obtaining a "visually silly" fit through the data. Mentioning that one could achieve this by working with the absolute values of the residuals provides the opportunity to mention robustness to outliers, and to link the discussion back to something they know already - the difference between the behaviours of the sample mean and the sample median, in this respect.

We also discuss the fact that measuring the residuals in the vertical ("y") direction is intuitively sensible, because the model is purporting to "explain" the y variable. Any explanatory failure should presumably be measured in this direction. However, I also note that there are other options - such as measuring the residuals in the horizontal ("x") direction.

Perhaps more importantly, I also mention "orthogonal residuals". I mention them. I don't go into any details. Frankly, there isn't time; and in any case this is usually the students' first exposure to regression analysis and they have enough to be dealing with. However, I've thought that we really should provide students with an introduction to orthogonal regression - just in the simple regression situation - once they've got basic least squares under their belts. 

The reason is that orthogonal regression comes up later on in econometrics in more complex forms, at least for some of these students; but typically they haven't seen the basics. Indeed, orthogonal regression is widely used (and misused - Carroll and Ruppert, 1966) to deal with certain errors-in-variables problems. For example, see Madansky (1959).

That got me thinking. Maybe what follows is a step towards filling this gap.

Friday, November 14, 2014

Cointegration - The Definitive Overview

Recently released, this discussion paper from Søren Johansen, will give you the definitive overview of cointegration that you've been waiting for.

Titiled simply, "Time Series: Cointegration", Johansen's paper has been prepared for inclusion in the 2nd. edition of The International Encyclopedia of the Social and Behavioural Sciences, 2014. In the space of just sixteen pages, you'll find pretty much everything you need or want to know about cointegration.

To get you started, here's the abstract:
"An overview of results for the cointegrated VAR model for nonstationary I(1) variables is given. The emphasis is on the analysis of the model and the tools for asymptotic inference. These include: formulation of criteria on the parameters, for the process to be nonstationary and I(1), formulation of hypotheses of interest on the rank, the cointegrating relations and the adjustment coefficients. A discussion of the asymptotic distribution results that are used for inference. The results are illustrated by a few examples. A number of extensions of the theory are pointed out."

© 2014, David E. Giles

Tuesday, November 11, 2014

Normality Testing & Non-Stationary Data

Bob Jensen emailed me about my recent post about the way in which the Jarque-Bera test can be impacted when temporally aggregated data are used. Apparently he publicized my post on the listserv for Accounting Educators in the U.S.. He also drew my attention to a paper from Two former presidents of the AAA: "Some Methodological Deficiencies in Empirical Research Articles in Accounting", by Thomas R. Dyckman and Stephen A. Zeff, Accounting Horizons, September 2014, 28 (3), 695-712. (Here.) 

Bob commented that an even more important issue might be that our data may be non-stationary. Indeed, this is always something that should concern us, and regular readers of this blog will know that non-stationary data, cointegration, and the like have been the subject of a lot of my posts.

In fact, the impact of unit roots on the Jarque-Bera test was mentioned in this old post about "spurious regressions". There, I mentioned a paper of mine (Giles, 2007) in which I proved that:

Read Before You Cite!

Note to self - file this post in the "Look Before You Leap" category!

Looking at The New Zealand Herald newspaper this morning, this headline caught my eye:

"How Did Sir Owen Glenn's Domestic Violence Inquiry Get $7 Billion Figure Wrong?"

$7 Billion? Even though that's (only) New Zealand dollars, it still sounds like a reasonable question to ask, I thought. And (seriously) this is a really important issue, so, I read on.

Here's part of what I found (I've added the red highlighting):

Monday, November 10, 2014

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":

Sunday, November 9, 2014

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

Saturday, November 8, 2014

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:

            yi = βxi + εi   .                                                              (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(xiyi) / Σ(xi2) ,

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

Now consider the "reverse regression":

           xi = αyi + ui   .                                                             (2)

The OLS estimator of α is

           a = Σ(xiyi) / Σ(yi2).

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.

Friday, November 7, 2014

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.

               Yt = yt + yt - 1 + ......+ yt - 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 oneat this point. However, the J-B test is very widely used by econometricians, and it warrants some further consideration.

Consider the following a small Monte Carlo experiment.