Monday, December 5, 2016

Monte Carlo Simulation Basics, III: Regression Model Estimators

This post is the third in a series of posts that I'm writing about Monte Carlo (MC) simulation, especially as it applies to econometrics. If you've already seen the first two posts in the series (here and here) then you'll know that my intention is to provide a very elementary introduction to this topic. There are lots of details that I've been avoiding, deliberately.

In this post we're going to pick up from where the previous post about estimator properties based on the sampling distribution left off. Specifically, I'll be applying the ideas that were introduced in that post in the context of regression analysis. We'll take a look at the properties of the Least Squares estimator in three different situations. In doing so, I'll be able to illustrate, through simulation, some "text book" results that will know about already.

If you haven't read the immediately preceding post in this series already, I urge you to do so before continuing. The material and terminology that follow will assume that you have.

Saturday, December 3, 2016

December Reading List

Goodness me! November went by really quickly!
© 2016, David E. Giles

Monday, November 28, 2016

David Hendry on "Economic Forecasting"

Today I was reading a recent discussion paper by Neil Ericcson, titled "Economic Forecasting in Theory and Practice: An Interview With David F. Hendry". The interview is to be published in the International Journal of Forecasting.

Here's the abstract:

"David Hendry has made major contributions to many areas of economic forecasting. He has developed a taxonomy of forecast errors and a theory of unpredictability that have yielded valuable insights into the nature of forecasting. He has also provided new perspectives on many existing forecast techniques, including mean square forecast errors, add factors, leading indicators, pooling of forecasts, and multi-step estimation. In addition, David has developed new forecast tools, such as forecast encompassing; and he has improved existing ones, such as nowcasting and robustification to breaks. This interview for the International Journal of Forecasting explores David Hendry’s research on forecasting."

Near the end of the wde-rangng, and thought-provoking, interview David makes the following point:
"Many top econometricians are now involved in the theory of forecasting, including Frank Diebold, Hashem Pesaran, Peter Phillips, Lucrezia Reichlin, Jim Stock, Timo Teräsvirta, KenWallis, and MarkWatson. Their technical expertise as well as their practical forecasting experience is invaluable in furthering the field. A mathematical treatment can help understand economic forecasts, as the taxonomy illustrated. Recent developments are summarized in the books by Hendry and Ericsson (2001), Clements and Hendry (2002), Elliott, Granger, and Timmermann (2006), and Clements and Hendry (2011b). Forecasting is no longer an orphan of the profession."
(My emphasis added; DG)

Neil's interview makes great reading, and I commend it to you.

© 2016, David E. Giles

Friday, November 18, 2016

The Dead Grandmother/Exam Syndrome

Anyone who's had to deal with students will be familiar with the well-known problem that biologist Mike Adams discussed in his 1999 piece, "The Dead Grandmother/Exam Syndrome", in the Annals of Improbable Research. 😊

As Mike noted,
"The basic problem can be stated very simply:
A student’s grandmother is far more likely to die suddenly just before the student takes an exam, than at any other time of year."
Based on his data, Mike observed that:
"Overall, a student who is failing a class and has a final coming up is more than 50 times more likely to lose a family member than is an A student not facing any exams. Only one conclusion can be drawn from these data. Family members literally worry themselves to death over the outcome of their relative's performance on each exam.
Naturally, the worse the student’s record is, and the more important the exam, the more the family worries; and it is the ensuing tension that presumably causes premature death."
I'll leave you to read the rest, and to find out why grandmothers are more susceptible to this problem than are grandfathers.

Enjoy Mike's research - and then make sure that you put a link to his paper on your course outlines!

© 2016, David E. Giles

Thursday, November 17, 2016

Inside Interesting Integrals

In some of my research - notably that relating to statistical distribution theory, and that in Bayesian econometrics - I spend quite a bit of time dealing with integration problems. As I noted in this recent post, integration is something that we really can't avoid in econometrics - even if it's effectively just "lurking behind the scenes", and not right in our face.

Contrary to what you might think, this can be rather interesting!

We can use software, such as Maple, or Mathematica, to help us to evaluate many complicated integrals. Of course, that wasn't always so, and in any case it's a pity to let your computer have all the fun when you could get in there and get your hands dirty with some hands-on work. Is there anything more thrilling than "cracking" a nasty looking integral?

I rely a lot on the classic book, Table of Integrals, Series and Products, by Gradshteyn Ryzhik. It provides a systematic tabulation of thousands of integrals and other functions. I know that there are zillions of books that discuss various standard methods (and non-standard tricks) to help us evaluate integrals. I'm not qualified to judge which ones are the best, but here's one that caught my attention some time back and which I've enjoyed delving into in recent months.

It's written by an electrical engineer, Paul J. Nahin, and it's called Inside Interesting Integrals.

I just love Paul's style, and I think that you will too. For instance, he describes his book in the following way -
"A Collection of Sneaky Tricks, Sly Substitutions, and Numerous Other Stupendously Clever, Awesomely Wicked, and Devilishly Seductive Maneuvers for Computing Nearly 200 Perplexing Definite Integrals From Physics, Engineering, and Mathematics. (Plus 60 Challenge Problems with Complete, Detailed Solutions.)"
Well, that certainly got my attention!

And then there's the book's "dedication":
"This book is dedicated to all who, when they read the following line from John le Carre´’s 1989 Cold War spy novel The Russia House, immediately know they have encountered a most interesting character:
'Even when he didn’t follow what he was looking at, he could relish a good page of mathematics all day long.'
as well as to all who understand how frustrating is the lament in Anthony Zee’s book Quantum Field Theory in a Nutshell:
'Ah, if we could only do the integral … . But we can’t.' "
What's not to love about that?

Take a look at Inside Interesting Integrals - it's a gem.

© 2016, David E. Giles

Saturday, November 12, 2016

Monte Carlo Simulation Basics, II: Estimator Properties

In the early part of my recent post on this series of posts about Monte Carlo (MC) simulation, I made the following comments regarding its postential usefulness in econometrics:
".....we usually avoid using estimators that are are "inconsistent". This implies that our estimators are (among other things) asymptotically unbiased. ......however, this is no guarantee that they are unbiased, or even have acceptably small bias, if we're working with a relatively small sample of data. If we want to determine the bias (or variance) of an estimator for a particular finite sample size (n), then once again we need to know about the estimator's sampling distribution. Specifically, we need to determine the mean and the variance of that sampling distribution. 
If we can't figure the details of the sampling distribution for an estimator or a test statistic by analytical means - and sometimes that can be very, very, difficult - then one way to go forward is to conduct some sort of MC simulation experiment."
Before proceeding further, let's recall just what we mean by a "sampling distribution". It's a very specific concept, and not all statisticians agree that it's even an interesting one.

Tuesday, November 8, 2016

Monte Carlo Simulation Basics, I: Historical Notes

Monte Carlo (MC) simulation provides us with a very powerful tool for solving all sorts of problems. In classical econometrics, we can use it to explore the properties of the estimators and tests that we use. More specifically, MC methods enable us to mimic (computationally) the sampling distributions of estimators and test statistics in situations that are of interest to us. In Bayesian econometrics we use this tool to actually construct the estimators themselves. I'll put the latter to one side in what follows.

Sunday, November 6, 2016

The BMST Package for Gretl

As a follow-up to this recent post, I heard again from Artur Tarassow.

You'll see from his email message below that he's extended his earlier work and has prepared a new package for Gretl called "Binary Models Specification Tests".

It's really good to see tests of this type being made available for users of different software - especially free software such as Gretl.

Artur writes:

Saturday, November 5, 2016

Snakes in a Room

Teachers frequently use analogies when explaining new concepts. In fact, most people do. A good analogy can be quite eye-opening.

The other day my wife was in the room while I was on the 'phone explaining to someone why we often like to apply BOTH the ADF test and the KPSS test when we're trying to ascertain whether a partcular time-series is stationary or non-stationary. (More specifically, whether it is I(0) or I(1).) The conversation was, not surprisingly, relatively technical in nature.

After the call was over, it occurred to me that my wife (who is an artist, and not an econometrician) might ask me what the heck all that gobbly-gook was all about. As it happened, she didn't - she had more important things on her mind, no doubt. But it forced me to think about a useful analogy that one might use in this particular instance.

I'm not suggesting that what I came up with is the best possible analogy, but for what it's worth I'll share it with you.

Friday, November 4, 2016

November Reading

You'll see that this month's reading list relates, in part, to my two recent posts about Ted Anderson and David Cox.
  • Acharya, A., M. Blackwell, & M. Sen, 2015. Explaining causal findings without bias: Detecting and assessing direct effects. RWP15-194, Harvard Kennedy School.
  • Anderson, T.W., 2005. Origins of the limited information maximum likelihood and two-stage least squares estimators. Journal of Econometrics, 127, 1-16.
  • Anderson, T.W. & H. Rubin, 1949. Estimation of the parameters of a single equation in a complete system of stochastic equations. Annals of Mathematical Statistics, 20, 46-63.
  • Cox, D.R., 1972. Regression models and life-tables (with discussion). Journal of the Royal Statistical Society B, 34, 187–220.
  • Malsiner-Walli, G. & H. Wagner, 2011. Comparing spike and slab priors for Bayesian variable selection. Austrian Journal of Statistics, 40, 241-264.
  • Psaradakis, Z. & M. Vavra, 2016. Portmanteau tests for linearity of stationary time series. Working Paper 1/2016, National Bank of Slovakia.
© 2016, David E. Giles