Wednesday, December 30, 2015

The Econometric Game, 2016

I like to think of The Econometric Game as the World Championship of Econometrics.

There have been 17 annual Econometric Games to date, and some of these have been featured previously in this blog. For instance in 2015 there were several posts, such as this one. You'll find links in that post to earlier posts for other years.

I also discussed the cases that formed the basis for the 2015 competition here.

In 2016, the 18th Econometric Game will be held at the University of Amsterdam between 6 and 8 April.

The competing teams will be representing the following universities:

Requests I Ignore

About six months ago I wrote a post titled, "Readers' Forum Page".

Part of my explanation for the creation of the page was as follows:

Tuesday, December 29, 2015

Job Market for Economics Ph.D.'s

In a post in today's Inside Higher Ed, Scott Jaschik discusses the latest annual jobs report from the American Economic Association.

Ne notes:
"A new report by the American Economic Association found that its listings for jobs for economics Ph.D.s increased by 8.5 percent in 2015, to 3,309. Academic jobs increased to 2,458, from 2,290. Non-academic jobs increased to 846 from 761."
(That's an 11.1% increase for non-academic jobs, and a 7.3% increase for academic positions.)

The bounce-back in demand for graduates since 2008 is impressive:
"Economics, like most disciplines, took a hit after 2008. Between then and 2010, the number of listings fell to 2,285 from 2,914. But this year's 3,309 is greater not only than the 2008 level, but of every year from 2001 on. The number of open positions also far exceeds the number of new Ph.D.s awarded in economics."
And here's the really good news for readers of this blog:
"As has been the case in recent years, the top specialization in job listings is mathematical and quantitative methods."

Monday, December 28, 2015

Correlation Isn't Necessarily Transitive

If X is correlated with Y, and Y is correlated with Z, does it follow that X and Z are correlated?

No, not necessarily. That is, the relationship of correlation isn't necessarily transitive.

In a blog post from last year the Fields Medallist, Terrence Tao, discusses the question: "When is Correlation Transitive?", and provides a thorough mathematical answer.

He also provides this simple example of correlation intransitivity:

This is something for students of econometrics to keep in mind!

Sunday, December 27, 2015

Bounds for the Pearson Correlation Coefficient

The correlation measure that students typically first encounter is actually Pearson's product-moment correlation coefficient. This coefficient is simply a standardized version of the covariance between two random variables (say, X and Y):

ρXY = cov.(X,Y) / [s.d.(X) s.d.(Y)] ,                                                  (1)

where "s.d." denotes "standard deviation".

In the case of sample data, this formula will be:

ρXY = Σ[(Xi - X*)(Yi - Y*)] / {[Σ(Xi - X*)2][Σ(Yi - Y*)2]}1/2 ,                 (2)

where the summations run from 1 to n (the sample size); and X* and Y* are the sample averages of the X and Y variables.

Scaling the covariance in this way to create the correlation coefficient ensures that (i) the latter is unitless; and (ii) it takes values in the interval [-1, +1]. The first of these two properties facilitates meaningful comparisons of correlations involving data measured in different units. The second property provides a metric that enables us to think about the "degree" of correlation in a meaningful way. (In contrast, a covariance can take any real value - there are no upper or lower bounds.)

Result (i) above is obvious. Result (ii) can be established in a variety of ways.

(a)  If you're familiar with the Cauchy-Schwarz inequality, the result that -1 ≤ ρ ≤ 1 is immediate.

(b)  If you like working with vectors, then it's easy to show that ρ is the cosine of the angle between two vectors in the X-Y plane. As cos(θ) is bounded below by -1 and above by +1 for any θ, we have our result for the range of ρ right away. See this post by Pat Ballew for access to the proof.

(c)  However, what about a proof that requires even less background knowledge? Suppose that you're a student who knows how to solve for the roots of a quadratic equation, and who knows a couple of basic results relating to variances. Then, proving that  -1 ≤ ρ ≤ 1 is still straightforward:

Let Z = X + tY, for any scalar, t. Note that var.(Z) = t2var.(Y) +2tcov.(X,Y) + var.(X) ≥ 0.

Or, using obvious notation, at2 + bt + c ≥ 0

This implies that the quadratic must have either one real root or no real roots, and this in turn implies that b2 - 4ac ≤ 0.

Recalling that a = var.(Y); b = 2cov.(X,Y); and c = var.(X), some simple re-arrangement of the last inequality yields the result that  -1 ≤ ρ ≤ 1.

A complete version of this proof is provided by David Darmonhere.

Saturday, December 26, 2015

Gretl Update

The Gretl econometrics package is a great resource that I've blogged about from time to time. It's free to all users, but of a very high quality.

Recently, I heard from Riccardo (Jack) Lucchetti - one of the principals of Gretl. He wrote:
"In the past, you had some nice words on Gretl, and we are grateful for that.
Your recent post on HEGY made me realise that you may not be totally aware of the recent developments in the gretl ecosystem: we now have a reasonably rich and growing array of "addons". Of course, being a much smaller project than, say, R, you shouldn't expect anything as rich and diverse as CRAN, but we, the core team, are quite pleased of the way things have been shaping up."
The HEGY post that Jack is referring to is here, and he's quite right - I haven't been keeping up sufficiently with some of the developments at the Gretl project.

There are now around 100 published Gretl "addons", of "function packages". You can find a list of those currently supported here. By way of example, these packages include ones as diverse as Heteroskedastic I.V. Probit; VECM for I(2) Analysis; and the Moving Blocks Bootstrap for Linear Panels.

If you go to this link you'll be able to download the Gretl Function Package Guide. This will tell you everything you want to know about using function packages in Gretl, and it also provides the information that you need if you're thinking of writing and contributing a package yourself.

Congratulations to Jack and to Allin Cottrell for their continuing excellent work in making Grelt available to all of us!

Tuesday, December 22, 2015

Wishing all readers a very special holiday season!

• Agiakloglou, C., and C. Agiropoulos, 2016. The balance between size and power in testing for linear association for two stationary AR(1) processes. Applied Economics Letters, 23, 230-234.
• Allen, D., M. McAleer, S. Peiris, and A. K. Singh, 2015. Nonlinear time series and neural-network models of exchange rates between the US dollar and major currencies. Discussion Paper No. 15-125/III, Tinbergen Institute.
• Basu, D., 2015. Asymptotic bias of OLS in the presence of reverse causality. Working Paper 2015-18, Department of Economics, University of Massachusetts, Amherst.
• Giles, D. E., 2005. Testing for a Santa Claus effect in growth cycles. Economics Letters, 87, 421-426.
• Kim, J., and I Choi, 2015. Unit roots in economic and financial time series: A re-evaluation based on enlightened judgement. MPRA Paper No. 68411.
• Triacca, U., 2015. A pitfall in using the characterization of Granger non-causality in vector autoregressive models. Econometrics, 3, 233-239.

Wednesday, December 9, 2015

Seasonal Unit Root Testing in EViews

When we're dealing with seasonal data - e.g., quarterly data - we need to distinguish between "deterministic seasonality" and "stochastic seasonality". The first type of seasonality is what we try to remove when we "seasonally adjust" the series. It's also what we're trying to account for when we include seasonal dummy variables in a regression model.

On the other hand, "stochastic seasonality" refers to unit roots at the seasonal frequencies. This is a whole different issue, and it's been well researched in the time-series econometrics literature.

This distinction is similar to that between a "deterministic trend" and a "stochastic trend" in annual data. The former can be removed by "de-tending" the series, but the latter refers to a unit root (at the zero frequency).

The most widely used procedure for testing for seasonal unit roots is that proposed by Hylleberg et al. (HEGY) (1990), and extended by Ghysels et al. (1994).

In my graduate-level time-series course we always look at stochastic seasonality. Recently, Nicolas Ronderos has written a new "Add-in" for EViews to make it easy to implement the HEGY testing procedure (see here). This will certainly save some coding for EViews users.

Of course, stochastic seasonality can also arise in the case of monthly data - this is really messy - see Beaulieu and Miron (1993). In the case of half-yearly data, the necessary theoretical framework and critical values are developed and illustrated by Feltham and Giles (2003)

And if you have unit roots at the seasonal frequencies in two or more time-series, you might also have seasonal cointegration. The seminal contribution relating to this is by Engle et al. (1993), and an short empirical application is provided by Reinhardt and Giles (2001)

I plan to illustrate the application of seasonal unit root and cointegration tests in a future blog post.

(Also, note the comment from Jack Lucchetti, below, that draws attention to a HEGY addon for Gretl, written by Ignacio Diaz Emparanza.)

References

Beaulieu, J. J., and J. A. Miron, 1993. Seasonal unit roots in aggregate U.S. data. Journal of Econometrics, 55, 305-328.

Engle, R. F., C. W. J. Granger, S. Hyleberg, H. S. Lee, 1993. Seasonal cointegration: The Japanese consumption function. Journal of Econometrics, 55, 275-298.

Feltham, S. G. and D. E. A. Giles, 2003. Testing for unit roots in semi-annual data. in D.E.A. Giles
(ed.), Computer-Aided Econometrics. Marcel Dekker, New York, 175-208. (Pre-print here.)

Ghysels, E., H. S. Lee, and J. Noh, 1994. Testing for unit roots in seasonal time series: Some theoretical extensions and a Monte Carlo investigation. Journal of Econometrics, 62, 415-442.

Hylleberg, S., R. F. Engle, C. W. J. Granger, and B. S. Yoo, 1990. Seasonal integration and cointegration. Journal of Econometrics, 44, 215-238.

Reinhardt, F. S. and D. E. A. Giles, 2001. Are cigarette bans really good economic policy?. Applied Economics, 33, 1365-1368. (Pre-print here.)