Wednesday, April 16, 2014

An Exercise With the SURE Model

Here's an exercise that I sometimes set for students if we're studying the Seemingly Unrelated Regression equations (SURE) model. In fact, I used it as part of a question in the final examination that my grad. students sat last week.

Suppose that we have a 2-equation SURE model:

                        y1 = X1β1 + ε1

                        y2 = X2β2 + ε2  ,

where the sample is "balanced" (i.e,. we have n observations on all of the variables in both equations), and the errors satisfy the usual assumptions for a SURE model:

                     E[ε] = 0  ;  V(ε) = (Σ ⊗ In
where                ε' = [ε1' , ε2']' .

Exercise:  Prove that the SURE estimators of β1 and β2 are identical to the OLS estimators of β1 and β2 if the condition, X(X1'X1)-1 X1' = X(X2'X2)-1 X2' , is satisfied.

Viren Srivastava and I gave this as Exercise 2.14 in our 1987 book on the SURE model. However, we didn't give the solution there - so don't think you can cheat in that way!

You can see that the above condition is satisfied if X1 = X2, and the latter condition is one that is mentioned in most econometrics textbooks. However, it's much more stringent than is needed to get the result.

Also, the above condition is necessary, as well as sufficient, for the OLS and SURE estimators to coincide. However, that's another matter.

I'll post the "solution" to the exercise in a few days' time.


Reference

Srivastava, V. K. and D. E. A. Giles, 1987. Seemingly Unrelated Regression Equations Models:Estimation and Inference. Marcel Dekker, New York. 



© 2014, David E. Giles

Tuesday, April 15, 2014

Econometric Game, 2014

I've blogged about The Econometric Game previously - see here, here,  and here.

It's April, so the Game is on again - today and the next two days, to be specific. You can check out he details, as they become available, at this site.

Good luck to all of the participating teams!

© 2014, David E. Giles

Sunday, April 13, 2014

Edmond Malinvaud on the Contributions of the Cowles Commission

"A father cannot expect more than to see his son take up his business and find new ways of making it flourish. Cowles econometricians of the forties are truly the fathers of present day econometricians and, like successful fathers, have good reason to be proud."
These are the closing remarks in an insightful historical perspective by Edmond Malinvaud, on the occasion of the fiftieth anniversary of the Cowles Commission. Malinvaud's piece, "Econometric Methodology at the Cowles Commission: Rise and Maturity", appeared in the Cowles Fiftieth Anniversary Volume

I urge all students of econometrics to read this enlightening account of the crucial role played by the affiliates of the Cowles Commission, first in Chicago, and subsequently (and still) at Yale University.

Malinvaud sums up this role succinctly as follows:
"The Cowles Commission contributed to the rise of econometric methodology in two determinate ways. On the one hand, it imposed "the probability approach": each application should begin with the definition of a precise stochastic model representing the phenomenon under study and the generation of the data; the method to be used for inference should then be rigorously determined within the framework of this model. On the other hand, it showed why most models to be built by economists should appear as systems of equations disturbed by additive random terms and often containing the values taken by the same variables in a few successive observations; it then fully determined methods to be recommended for estimation or testing within such models."
If you've been following my posts on "vintage years in econometrics", then you'll know that in the 1940's and 1950's, research at Cowles reigned supreme. Malinvaud, again:
"The major work for the building of simultaneous-equation econometrics at Cowles took place during the five and one-half years that roughly coincided with Jacob Marschak's term as director of research (January 1943 to June 1948). In the report for 1943, it appears that a good half of the research done during the year belonged to the simultaneous-equation econometrics field, although still with a definite orientation toward applied questions (demand functions, production functions, multiplier models). The presentation of this work already stresses the general methodological issues that were becoming the object of major concern. Not much later, the Cowles Commission called what turned out to be the most influential conference on statistical inference in economics ever held. It took place in Chicago from January 27 to February 1, 1945, and was attended by R.L. Anderson, T. Haavelmo, H. Hotelling, L. Hurwicz, L.R. Klein, T.C. Koopmans, R. Leipnik, H.B. Mann, J. Marschak, H. Rubin, G. Tintner, and A. Wald. The Commission research staff had prepared a number of papers dealing with various subjects, concerning in particular time-series analysis. The most novel contribution was certainly the one presented by Tjalling Koopmans and his research assistant Herman Rubin; it discussed both identification and maximum likelihood estimation in simultaneous-equation systems, proving the main theoretical results on both questions.
The report of the conference was prepared, with contributions made by other participants and with additions of subsequent work done at Cowles. These additions concerned mainly some theoretical developments; computational questions, which at this precomputer time might have been a real obstacle; and the "limited information" method of estimation, which was conceived in early 1946 by T.W. Anderson and H. Rubin and subsequently fully worked out by them. This report was to become Cowles Commission Monograph No. 10, Statistical Inference in Dynamic Economic Models, edited by T.C. Koopmans. We know that the manuscript was completed in early 1947, but publication was delayed until 1950 by typographical and other printing difficulties."
And of course, the over-identification test that was proposed in the Anderson-Rubin paper, has received significant, renewed, attention in recent research.

To sum up:
"When we look back and try to give a broad evaluation of the achievement of the simultaneous-equation work of the 1940s, we of course know that the theory was not complete by the end of this period. Alternative estimators had to be discovered, small-sample properties to be investigated, nonlinear simultaneous-equation models to be considered, efficient computational softwares to be built, even pedagogical presentations of the theory and of its algebra to be found. Nevertheless, after thirty more years of theoretical research in this field, the Cowles Commission construction essentially stands untouched; new wings and pinions have been added, good maps have been drawn, but the central building needs no repair. This was a perfectly sound and impressive piece of methodological work. No doubt or questioning can be expressed in this respect."


© 2014, David E. Giles

Law-Breaking Econometricians

I don't follow Lars P. Syll's blog, but the other day I was led there by a Twitter tweet. Lars begins his recent post, "Forecasting Alchemy", with the following statement:
'In New York State, Section 899 of the Code of Criminal Procedure provides that persons “Pretending to Forecast the Future” shall be considered disorderly under subdivision 3, Section 901 of the Code and liable to a fine of $250 and/or six months in prison."
Although the law does not apply to “ecclesiastical bodies acting in good faith and without fees,” I’m not sure where that leaves econometricians and other forecasters …'
I'm not sure either, but I'm not going to lose sleep over this unless I happen to re-locate to NY State.

Other economists have also drawn attention to this rather alarming piece of legislation, including Joan Robinson on page 8 of her 1981 book, What are the Questions?: And Other Essays: Further Contributions to Modern Economics.

Walter A. Friedman's recent book, Fortune Tellers: The Story of America's Economic Forecasters provides some insights into the origins of economic forecasting, and perhaps into that of Section 899. (See here for an excerpt.)



© 2014, David E. Giles

Open Science Through R

There's so much being written about R these days, and justifiably so. If you use R for your econometrics, you should also keep in mind that its applicability is far wider than statistical analysis. 

A big HT to the folks at Quandl for leading me to a nice overview of the way in which R is enabling some big changes in the way in which scientific research is being conducted more generally. The article in question is by Tina Amirtha, "How the Rise of the "R" Language is Bringing Open Source to Science", which you'll find here.

If you think that R is just about statistics, and you can't see the point of investing some time (not money) in getting on board, then read Tina's piece. 

You'll change your mind if you consider yourself a survivor.



© 2014, David E. Giles

Thursday, April 10, 2014

Proof of a Result About the "Adjusted" Coefficient of Determination

In a post last year I discussed the conditions under which the "adjusted" coefficient of determination (RA2) will increase or decrease, when regressors are deleted from (added to) a regression model. Without going over the full discussion again, here is one of the key results:

Adding a group of regressors to the model will increase (decrease) RA2 depending on whether the F-statistic for testing that their coefficients are all zero is greater (less) than one in value. RA2 is unchanged if that  F-statistic is exactly equal to one.

A few days ago, "Zeba" reminded me that I had promised to post a simple proof of this result, but I still hadn't done so. Shame on me! A proof is given below. As a bonus, I've given the proof for a more general result - we don't have to be imposing "zero" restrictions on some of the coefficients - any exact linear restrictions will suffice.

Let's take a look at the proof.

Friday, April 4, 2014

There's an App for That

I was looking for econometrics-related "apps" for my Android tablet. Very little of interest came up for "Econometrics", but there certainly are some nice data-related apps.

Here are a few examples:



© 2014, David E. Giles

Playing With the FRB/US Model

Via both Mark Thoma and Gareth Thomas, I learned about a new initiative at the Federal Reserve Board. The details are available here.

In summary:
"The FRB/US model of the U.S. economy is one of several that Federal Reserve Board staff consults for forecasting and the analysis of macroeconomic issues, including both monetary and fiscal policy. FRB/US has long been available to members of the public upon request. To reduce the costs of providing updates of the model specification and databases, and to make the public more broadly aware of the model's availability, a new page has been introduced on the Federal Reserve Board's website from which interested users can download expanded FRB/US documentation; model equations, coefficients, and data; and sample simulation programs. These simulation programs can be run by anyone with access to the EViews software package, a widely available commercial product. This note provides a brief summary of the main features of the model, illustrates some applications of the model using sample programs provided on the web page, and concludes with an overview of the contents of the web page. Because the model continues to undergo changes as both economic theory and empirical evidence evolve, any given model release reflects only the state of thinking at the time of the release."
This a fabulous resource, and I'll certainly be making use of it in my teaching!


© 2014, David E. Giles

Thursday, April 3, 2014

New Paper

 Another of my papers on analytic bias-correction has now been published. This one is with a former M.A. student, Xiao Ling.

The details are: Xiao Ling and David E. Giles, "Bias reduction for the maximum likelihood estimator of the parameters of the generalized Rayleigh family of distributions. Communications in Statistics - Theory and Methods, 2014, 43, 1778-1792.

You can see the paper here.


© 2014, David E. Giles

Tuesday, April 1, 2014

April Reading List

Here are some of the paper that I've been reading lately:


© 2014, David E. Giles