Wednesday, May 27, 2009

Finite sample theory: A discussion

9:59 PM Amy: about Prof. Ullah's paper last Friday, what's the main point? Have he and Yongbao developed some estimation even when the error term is nonnormal?

Yundong Tu to Amy show details 10:18 PM (19 hours ago) Reply

Their papers are not to develope estimation result but to provide finite sample approximation for higher order moments of the estimators ( estimators, say beta hat, are taken as given, which can be MLE, GMM, IV, LS, etc.). The other paper is about expectation of quadradict form. They also provide finite sample approximation for these terms. Finite sample approximation differs when the error terms are nonnormally distributed from normally distributed case. These approximation results, however, could be used to study the properties of some other estimators, for example, the estimator of rho in the spatial autoregressive model, or the estimators of the coefficients in the MA or AR models.

10:29 PM Amy: so when error tems are nonnormal, we can still estimate the coefficients such as in VAR models. What 's the difference between this way and other methods approximating nonnormal errors to a normal distribution?
10:31 PM me: yes, you still can estimate using the same method as if the error term is normal
Amy: in Ullah's way?
me: no the classical way he is sillent about the estimation approach he is only concerned with the moments of the estimators
10:32 PM which is not quite a concern in macro, i think
Amy: but you say we can still estimate the coefficients me: yes
Amy: That's what I am considering
me: but we do not know the higher oder moments the properties of that is provided by Ullah and Bao
10:33 PM Amy: when the error term is nonnormal, could we use some methods of finite sample to estimate> Since they mention the MLE
me: finite sample is not to estimate the coefficients but to approximate higher order moments, say skewness and kurtosis of the estimators
10:34 PM Amy: I see. I am not familiar with finte sample
me: yes, you can still use MLE, GMM, IV and LS, etc. for the estimation purpose but once you get these estimators, you might be interested in its higher order properties the estimator you get would be not normally distributed
10:35 PM especially when the error term is not normally distributed and when the sample is small or even moderate large not even close to normal
Amy: I c.
me: finite sample approach is one method to tell how far is your estimator from a normal random variable
10:36 PM typical way to examine this is to check the property of skewness and kurtosis and see how far they are from those of normal distribution
10:37 PM in large sample, everything (estimator) is normally distributed, so there is no difference but what we have usually is not large sample observations
10:38 PM this leaves a room for finite sample theory to improve upon the large sample theory to get more accurate properties of the estimators we derived
Amy: But what will we do if we find the estimator is not normal distribution?
10:39 PM me: we impose finite sample corrections then
10:41 PM you know that for normally distributed random variable, skewness is zero and kurtosis is 3. if it is not normally distributed Ullah and Bao provide formula for those, which should also be used when sample size is small
10:42 PM Amy: I know that. But how to corret them?
10:43 PM me: use the fomula in Ullah and Bao for Skewness and Kurtosis
Amy: and then?
me: In Ullah's book, there should be fomula for mean and variance
10:45 PM Amy: I see. Maybe I should read the book firstly. But is there any way to use this correction way for estimation?
me: no it is not for estimation of the coefficient in econometric models like y=xb+e
10:46 PM their method is silent about the estimation approach, as i said earlier it can be used for a general class of estimators called extremum estimators
10:47 PM Amy: I c. Thank you. me: their approximation result depends on the following assumption sqrt(n)(bhat-b) converges to a normal distribution it's a pleasure
10:48 PM Amy: see. Thanks a million~~ I am looking at your pics in picasa
10:49 PM me: :)
10:51 PM Amy: ok, night~~ good dream.
me: u2

Tuesday, May 26, 2009

An interesting corelation question

Today, I was confronted by an interesting macroeconomic question, which turns out to be a statistical question. The setup of the question is as follows: say x, and z are procyclical for y (output), that is, cov(x,y)>0 and cov(z,y)>0. Is it possible that cov(x,z)<0? If it is, find out the conditions for this to hold. Note that the covariance operator only captures the linear relationship while the nonlinear relationship is here to paly a role to make x and z negatively related. The solution would simplify once we assume that y is a symmetrically distributed random variable with mean zero. Also, the relationship between x and y and that between z and y are charactorized by x=f(y), z=g(y), respectively. Assuming f() and g() having zero derivatives for order higher than 2 will easily give the following condition for cov(x,z)<0 to hold:
00 and g'>0.
That is, the curvature of f and g must be different, with one being convex and the other being concave. Other than this, we need have abs(f ''(0)g''(0)) very large or y with high kurtosis for the condition to hold.

Thursday, May 21, 2009

Finite-Sample Asymptotics

Small Sample Asymptotics
Author(s): Michael A. FlignerSource: Journal of Educational Statistics, Vol. 13, No. 1 (Spring, 1988), pp. 53-61

Tuesday, May 19, 2009

Vector Generalized Linear and Additive Models

The litteratur that is new to econometricans has already there in statistics, with link
http://www.iq.harvard.edu/blog/sss/archives/2009/04/yee_on_vector_g.shtml

Thomas Yee's abstract of the article is also given at the above link.

Fisher's exact test

Fisher's exact test is described at
http://en.wikipedia.org/wiki/Fisher's_exact_test
with references

Fisher, R. A. (1922). "On the interpretation of χ2 from contingency tables, and the calculation of P". Journal of the Royal Statistical Society 85 (1): 87–94. JSTOR: 2340521.
Fisher, R. A. 1954 Statistical Methods for Research Workers. Oliver and Boyd.
^ Mehta, Cyrus R; Patel, Nitin R; Tsiatis, Anastasios A (1984), "Exact significance testing to establish treatment equivalence with ordered categorical data", Biometrics 40: 819–825, doi:10.2307/2530927, http://www.jstor.org/stable/2530927
^ Mehta, C. R. 1995. SPSS 6.1 Exact test for Windows. Englewood Cliffs, NJ: Prentice Hall.
^ mathworld.wolfram.com Page giving the formula for the general form of Fisher's exact test for m x n contingency tables

Andrew Gelman wrote a blog about it and introduces some of his work,
http://www.stat.columbia.edu/~cook/movabletype/archives/2009/05/i_hate_the_so-c.html

College is also part of consumption

Econjeff wrote a blog which share the same idea I have had since 2006. The article is available at
http://econjeff.blogspot.com/2009/05/college-as-consumption.html

So given the fact that the return on the ivestment in college is less than the cost, Chinese people are still rational to maximize their utility. This argument kills the debate whether it is worthwhile to go to college or not that prevailing in the last few years.

Monday, May 18, 2009

The deterioration continues

James D. Hamilton, Econbrowser
http://www.econbrowser.com/archives/2009/05/the_deteriorati.html

Asymmetric Loss Functions

Optimal prediction under asymmetric loss: Christoffersen, FX Diebold - Econometric Theory, 1997 - jstor.org
Further results on forecasting and model selection under asymmetric loss: Christoffersen, FX Diebold - Journal of Applied Econometrics, 1996 - jstor.org
Financial Asset Returns, Direction-of-Change Forecasting, and ... :Christoffersen, FX Diebold - Management Science, 2006

Friday, May 15, 2009

Normal distribution and dependence

Normally distributed and uncorrelated does not imply independent
http://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent

A counterexample
The fact that two random variables X and Y both have a normal distribution does not imply that the pair (X, Y) has a joint normal distribution. A simple example is one in which Y = X if X > 1 and Y = −X if X < 1.
http://en.wikipedia.org/wiki/Multivariate_normal_distribution

Perturbation methods

Chapter II: Introduction to perturbation methods by Johan Byström, Lars-Erik Persson, and Fredrik Strömberg
Introduction to regular perturbation theory by Eric Vanden-Eijnden (PDF)
Duality in Perturbation Theory
Perturbation Method of Multiple Scales
Retrieved from "http://en.wikipedia.org/wiki/Perturbation_theory"

Wednesday, May 13, 2009

Hotelling, Harold, 1895-1973

Leading mathematical economists, also known as mathematical statistician. Famous in economics for his Hotelling's Lemma and in statistics for his T_squared statistic.

The teaching of statistics and probability, Statistical Science 3 (1) (1988), 63-71 reprinted from Annals of Mathematical Statistics 11 (1940), 457-470.
The place of statistics in the university, Statistical Science 3 (1) (1988), 72-83 reprinted from Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability (ed. J Neyman), Berkeley: California U. Press 1949, pp. 21-40.
Comment: Academic politics and the teaching of statistics, Statistical Science 3 (1) (1988), 92-95
Harold Hotelling 1895-1973 by Adrian C Darnell Statistical Science 3 (1) (1988), 57-62.

Kolmogorov, Andrei Nikolaevich, 1903-1987

(``Andrei Nikolaevich Kolmogorov,'' CWI Quarterly, 1(1988), pp. 3-18.) by Paul M.B. Vitanyi, CWI and University of Amsterdam

Andrei Nikolaevich Kolmogorov, born 25 April 1903 in Tambov, Russia, died 20 October 1987 in Moscow. He was perhaps the foremost contemporary Soviet mathematician and counts as one of the great mathematicians of this century. His many creative and fundamental contributions to a vast variety of mathematical fields are so wide-ranging that I cannot even attempt to treat them either completely or in any detail. For now let me mention a non-exhaustive list of areas he enriched by his fundamental research: The theory of trigonometric series, measure theory, set theory, the theory of integration, constructive logic (intuitionism), topology, approximation theory, probability theory, the theory of random processes, information theory, mathematical statistics, dynamical systems, automata theory, theory of algorithms, mathematical linguistics, turbulence theory, celestial mechanics, differential equations, Hilbert's 13th problem, ballistics, and applications of mathematics to problems of biology, geology, and the crystallization of metals.
Full Article

More about Kolmogorov
Site devoted to the life and work of Kolmogorov

Friday, May 8, 2009

R-np package

The following are from Racine's webpage:

Software
The np package (current version 0.30-1) for R (www.r-project.org) Obtaining: Available directly from the Comprehensive R Archive Network (cran.r-project.org)
Direct link to the np package on CRAN
Announcement on the R-packages mailing list
October 2007 Rnews article (pdf)
Manual (pdf)Vignette (pdf)FAQ (pdf) (html)

Nonparametric Econometrics: A Primer

Racine, J. S. (2008), "Nonparametric Econometrics: A Primer," Foundations and Trends in Econometrics: Vol. 3: No 1, pp 1-88. http://dx.doi.org/10.1561/0800000009

Wednesday, May 6, 2009

Jefferey Racine is Visiting UCR on May 7th and 8th

The world's most well-known computational/Nonparametric Econometrician, Jefferey Racine, a prior student of Aman Ullah, is visiting UCR on May 7th and 8th. Racine is known already for his nonparamtric-package in R, beside his nonparametric book, Nonparamtric Econometrics---Theory and Practice, written together with Qi Li.

This time he will present a paper on economic constraints in nonparametric modelling. His visit to US including UCLA, UCSD and UCR. Although he has visited UCR for several times, his coming this time is highly appreciated and expected.

Behave as an Econometrician

Dream of being an Econometrican? Look at what they do, as follows,
1. Study Economics
2. Develope Econometric tools
3. Reading econometric journal articles
4. Learning mathematics and statistics
5. Improve programming skills
6. Go to seminars and comment on others work
7. Referee journal articles
8. Reading history of Econometrics
9. Teach econometrics in a way that could be understood to child and the old as well
10. Go to conferences and present papers
11. Educate young economics Ph. D. students
12. Sell Econometrics to applied economists, financialists, government agents and other institutions as well
13. Travel for tour sight-seeing
14. Help to develope economic research centers and institutes
15. Provide deeper insight into economic questions
16. Writing and publishing articles on top journals
17. Passing down anecdotes of great econometricans to the next generations
18. Create their own history of contributing to a better/more desirable world to live
19. Talk and write in symbols
20. Enjoy their lives with their families
21. Invest their time and money in a way to minimize SSR
22. Estimate whatever they do not know
23. Live on statistics and produce statistics as well
24. Being smart everyday
25. Ask quantitive questions
26. Writing books for the young and others interested
27. Welcomed everywhere except by those economists in history
28. Great heros

Not enough? Add more to the list!

Saturday, May 2, 2009

Seashells: the Plainness and Beauty of Their Mathematical Description

by Jorge Picado
Departamento de Matemática
Universidade de Coimbra
picado@mat.uc.pt
Fulltext
Abstract:
One might at first tend to think that the growth of plants and animals, because of their elaborate forms, are ruled by highly complex laws. However, this is surprisingly not always true: many aspects of the growth of plants and animals may be described by remarkably simple mathematical laws. An obvious example of this are the seashells and snails, as we show here: with a very simple model it is possible to describe and generate any of the many types of seashells that one may find classified in a good seashell bookguide. The fact that the animal which lives at the open edge of the shell places new shell material always in that edge, and faster on one side than the other, makes the shell to grow in a spiral. The rates at which shell material is secreted at different points of the open edge are presumably determined by the anatomy of the animal. And, surprisingly, even fairly small changes in such rates can have quite tremendous effects on the overall shape of the shell, which is in the origin of the existence of a great diversity of shells.

The ET Interview: Professor H. O. A. Wold: 1908-1992 The ET Interview: Professor H. O. A. Wold: 1908-1992

David F. Hendry, Mary S. Morgan, H. O. A. Wold
Econometric Theory, Vol. 10, No. 2 (Jun., 1994), pp. 419-433

Sadly, Herman Wold died on February 16, 1992, before agreeing to the final version of this interview. We are indebted to his son, Professor Svante Wold, for his kind permission to publish. We hope that this record of our discussions with Herman Wold, who, together with his two great Norwe- gian compatriots Ragnar Frisch and Trygve Haavelmo, helped lay the sta- tistical foundations of modern econometrics, will contribute to his memory. From our personal perspective, Herman Wold was an enthusiastic suppor- ter of our early incursions into the history of econometrics (see The History of Econometric Ideas by Mary Morgan, 1990), and we know that there are many like us who will greatly miss his stimulating contributions. Herman Wold was born on Christmas day, 1908, at Skien, Norway. His family moved to a small town outside Stockholm in 1912, and he lived in Sweden for the remainder of his life. He enrolled at Stockholm University in 1927 to study physics, mathematics, and economics but switched to study- ing statistics with Harald Cramer. After his undergraduate degree, he stud- ied the theory of risk with Cramer, then worked for an insurance company for a period, returning to Stockholm University in 1936. His doctoral the- sis of 1938, A Study in the Analysis of Stationary Time Series, embodies the famous Wold Decomposition theorem. In 1942, he moved to the Chair of Statistics in Uppsala and held that post until 1970, when he went to Goteborg for five years, finally becoming Professor Emeritus at Uppsala in 1975. He became a Fellow and later President of the Econometric Society; was Vice-President of the International Statistical Institute; a Foreign Honor- ary Member of both the American Economic Association and the Ameri- can Academy of Arts and Sciences; an Honorary Fellow of the Royal Statistical Society; a member of the Swedish Royal Academy of Sciences, serving on the Nobel Prize Committee in Economics from 1968 until 1980; and was the recipient of several honorary doctorates. In retirement, he was Professeur Invite at the University of Geneva until 1980.