Guy's Econometrics blog

Econometrics job

2011-11-15T03:35:00.001-08:00

Econmetricians looking for a job might like to check out http://www.technopolis-group.com/site/working/index.htm

2nd Winter School on Bayesian Methods for Empirical Macroeconomics

2011-09-22T05:35:00.000-07:00

The 2nd Winter School on Bayesian Methods for Empirical Macroeconomics will take place from the 14th-16th December 2011 at Queen Mary University of London. The course will be given by Professor Gary Koop of the University of Strathclyde.

The course will describe techniques of Bayesian Time Series Econometrics, starting from basic Bayesian Econometrics and dealing also with the estimation of VARs, linearised DSGE models, stochastic volatility and Time-Varying Parameter-VARs. It will provide insight into the methods used, and will be an opportunity for learning how to estimate these models using Matlab.

Similar versions of this course where recently given by Prof. Koop at the Bundesbank, the Bank of England, the Czech National Bank and the Polish Ministry of Finance and Queen Mary University of London. During this last event there were many requests for a future session from colleagues who were unable to attend due to space restrictions; as a result it was decided to organise this Winter School.

Gary Koop is a Professor of Economics at the University of Strathclyde and a world leader in Bayesian Econometrics. With this approach, he has published numerous articles in journals such as the Journal of Econometrics, the Journal of Applied Econometrics and the Journal of Business and Economic Statistics. He is an associate editor for several journals and is currently co-editing (with John Geweke and Herman van Dijk) the soon-to-be-released Handbook of Bayesian Econometrics.

The course will describe techniques on Bayesian Time Series Econometrics, starting from basic Bayesian Econometrics and dealing also with the estimation of VARs, linearised DSGE models, stochastic volatility and Time-Varying Parameter-VARs. It will provide insight into the methods used, and will be an opportunity for learning how to estimate these models using Matlab.

Similar versions of this course where recently given by Prof. Koop at the Bundesbank, the Bank of England, the Czech National Bank and the Polish Ministry of Finance and Queen Mary University of London. During this last event we received many requests for a futuresession from colleagues who were unable to attend due to space restrictions, and have therefore decided to host this Winter School.

Gary Koop is a Professor of Economics at the University of Strathclyde and a world leader in Bayesian Econometrics. With this approach, he has published numerous articles in journals such as the Journal of Econometrics, the Journal of Applied Econometrics and the Journal of Business and Economic Statistics. He is an associate editor for several journals and is currently co-editing (with John Geweke and Herman van Dijk) the soon-to-be-released Handbook of Bayesian Econometrics.

Application forms and further information about the course are available at
http://hosted.busman.qmul.ac.uk/cgr/Summer%20Schools/44157.html

Or you can e-mail gr@qmul.ac.uk

Confidence Intervals

2011-08-22T05:01:00.000-07:00

In his Bad Science column in the Guardian Newspaper last Saturday Ben Goldacre wrote
"Those figures are called 95% confidence intervals, and these are one of the most useful inventions of modern life."

The column both explains what a confidence interval is and why it is so important. One must be very careful to check that increases (or decreases) over time in a statistic based on a sample are actually significant. In this case the statistic relates to unemployment and the changes observed are not big enough to conclude that the underlying population value has changed.

NOTE: The article is also available via the Bad Science web page. By the way I just love his phrase "...the gentle static fuzz of random variation"

Why you shouldn't judge econometric results just in terms of P-values and R squared

2011-08-17T04:29:00.000-07:00

Both The Undercover Economist (Tim Harford) and Freakonmics (Stephen J Dubner) have drawn attention to a Working Paper by Tatu Westling of the University of Helsinki which has the eye-catching title Male Organ and Economic Growth: Does Size Matter?.

I think this paper should help us to remember that P values and the size of R squared are not the only things to focus on when evaluating the results of an applied econometrics study.

How bizarre

2011-03-22T01:11:00.000-07:00

Oh dear, another econometrics poem or song. I was talking to a student yesterday who has just started working on his econometrics project where he has to formulate, estimate and test suitable applied econometric models of his own and then write up a report. He was complaining that it was so much harder than doing the weekly computer practicals set by me where all the answers seemed to come out right straight away and each week there was only one issue to worry about, whether it be multicollinearity, autocorrelation or heteroskedastcity. Now in this project he seemed to be facing all these problems at once and none of his results seemed to make sense. Some of the results, he said, were quite bizarre. Then on the way home, listening to the radio, I heard the old OMC hit How bizzare.

So here are some alternative econometrics lyrics (with apologies to OMC):

In all the computer labs
Everything just came out well
Coefficient signs and sizes
Looked OK from what I could tell
Now I’m working on my project
Everything’s just gone awry
The estimates are all crazy
Not in the ranges they should lie

How bizarre, how bizarre, how bizarre
Ooh baby,
It’s making me crazy
Every time I look around
It’s hard to face

I removed a few outliers
Tried different functional forms
Checked for multicollinearity
All the tests I could perform
Econometrics isn’t easy
Nothing seems to turn out right
Toss the output in the waste bin
I’m in a right old plight

How bizarre, how bizarre, how bizarre
Ooh baby,
It’s making me crazy
Every time I look around
It’s hard to face

Stay calm and think about it
Read the text book once again
Check the data file’s not scrambled
Breathe in deep and count to ten
Don’t rush to run regressions
Use the theory to help you out
And a bit of imagination
Don’t get mad and start to shout

How bizarre, how bizarre, how bizarre
Ooh baby,
It’s making me crazy
Every time I look around
It’s hard to face

Buying and selling second-hand books?

2011-02-03T00:34:00.000-08:00

Econometrics textbooks can be quite expensive so it is often worth looking out for a second-hand (used) copy instead of buying a new one. In many universities students will do this through their local bookshop, or via messages pinned to noticeboards. For a wider reach you might try eBay. Another place that you might try is iSwapBooks which was set up last year by a couple of Durham University graduates. This might be a good place to go too if you want to sell some of your textbooks for courses you have now completed.

Talk like an econometrician

2010-08-02T04:50:00.001-07:00

Here is the latest effort, loosely based on the Bangles hit "Walk like an Egyptian". With apologies to the Bangles and Songwriters Sternberg and Liam.

All the graphics on the screen
They show the patterns don't you know
If they spread too much (oh whey oh)
They're telling you there is hetero

Run Shazam with your data files
Get ready to do the tests
Stata/MP, oh whey oh
Turn your back on SPSS

Reading data from a pipe
Ay oh whey oh, ay oh whey oh
Talk like an econometrician

GARCH processes they have their place
For clustered points you can't ignore
They match the moves (oh whey oh)
You click the mouse to see some more

The ones at school who read their books
They like the forecast CI band
And the hazard rate (oh whey oh)
Talk like an econometrician

Iterate til the program stops
Then go down to the donut shop
You can sing and dance (oh whey oh)
See which regressors you can drop

When you get to the marketplace say
Ay oh whey oh, ay oh whey oh
Talk like an econometrician

Slide your feet and bootstrap jack
Shift to ARMA then pull it back
Modelling is hard you know (oh whey oh)
We've moved on since the UNIVAC

And the Japanese with their yen
And the Indian friends of Amartya Sen
And the Chinese too (oh whey oh)
They all talk like an econometrician

Post docs up in Berkeley say
Ay oh whey oh
Ay oh whey oh
Talk like an econometrician

The Sir Clive Granger Memorial Conference

2010-06-21T09:25:00.000-07:00

On Monday 24th and Tuesday 25th May this year the Granger Centre for Time Series Econometrics at the University of Nottingham and the Department of Economics at the University of California San Diego jointly hosted the Sir Clive Granger Memorial Conference at the East Midlands Conference Centre at the University of Nottingham. The event came almost exactly one year after the demise of Sir Clive Granger, joint Nobel prize winner in Economics (with Robert Engle) in 2003. I was fortunate enough to be amongst the several dozen people invited to the conference, where I was in the exalted company of Sir David Hendry, Halbert White, Peter Philllips, James Stock, Mark Watson and Hashem Pesaran (to mention just a few of the big names who were there).

The conference began with laudations in honour of Sir Clive Granger provided by David Hendry, Ken Wallis, Peter Phillips, Jim Stock, Cheng Hsiao, Hal White and John Bates. Each speaker revealed when they had first met Sir Clive and how he had influenced both their own work and the wider subject of econometrics. It is, of course, virtually impossible to undertake any time series econometrics without making use of Granger's work. David Hendry remarked upon Granger's ability not only to innovate but also to communicate clearly his ideas (possibly one of the reasons that he had such an impact). Peter Phillips described Granger as a thinker and a leader, emphasizing in particular his leadership qualities. We can learn from him that ideas are paramount and that one should display intellectual courage in promoting new ideas. Phillips also admired the simplicity and elegance in much of Granger's work. He spoke for everyone at the conference in affirming that all who knew Sir Clive loved and revered him. Jim Stock spoke of Sir Clive's willingness to challenge conventional wisdom and of the help and support that he had been given by Sir Clive when Stock was an Associate Professor early in his career. He praised his scientific integrity and the importance that he attached to forecasting. Cheng Hsiao said that Sir Clive was a source of inspiration to students and colleagues alike, and Hal White also noted the encouragement that he provided to colleagues and visitors in his time at UCSD. He reminded us that Granger's work had both reach and depth. He told us that he had recently attended a conference on Neuroscience and reported that the concept of Granger causaility is providing new insights for those attempting to understand the functioning of the brain. John Bates told the conference how he had first met Granger back in 1956 during Granger's early years as a lecturer at Nottingham when Bates was student. He noted that Granger was always generous with his time for students. He also liked the way that Granger mixed theory and applications and recalled how inspiring he was with the ideas that he tossed out. At the end of the session Dame Patricia (Pat) Granger, Sir Clive's widow, told of how she had first met Sir Clive when she was a research assistant working for an economic historian on the subject of smallpox deaths in 18th century England. She had been recommended to go and see him as he would be someone who would be willing to spend time giving advice about statistics in language that could be understood by a non-specialist. The rest, we were told, was history!

We were then treated to the first of ten keynote lectures that had been prepared for the conference, Sir David Hendry speaking on the subject of Empirical Model Discovery. Hendry began by noting that many features of models are not derivable from theory. Instead, a more data-based approach could help us to discover them. Hendry recalled the dialogue between Granger and himself reported in the special 2005 issue of Econometric Theory on Automated Inference and the Future of Econometrics, in which Hendry responded to a series of questions prepared by Clive Granger on the use of PcGets in data modelling and as a new research tool (see also Hendry and Krolzig, 2001). He went on to explain how the Autometrics software can be used for model formulation, estimation, selection and evaluation. Using the PcGive data set on consumption, income and inflation he showed how, after allowing for up to 20 lags and with impulse indicator saturation, the software could arrive at a congruent, parsimonious and encompassing model, finding a local DGP from the initial General Unrestricted Model (GUM). He emphasized that the approach was not based on repeated testing but on selecting the best model on the basis of squared t values and the notion of "sequential factorization". This approach does not impose theory on the data (other than the initial choice of variables to include) and can deal with structural breaks in a "constantly evolving world" . He mentioned that he had used this approach in practical applications with a big US insurance company, and in work with the UK Office for National Statistics to improve the quality of the latest available data. No printed papers were distributed at the conference but interested readers might like to view the video lecture at the University of Aarhus which covers similar ground.

The second keynote session had papers by Peter Phillips on "Implicit maps and new unit root limit theory" and Marcus Chambers on "Testing for seasonal unit roots by frequency domain regression". Phillips talked about embedded simulation techniques in estimation, bias correction and the failure of the delta method. In response to a question from James Davidson, Phillips said that these ideas were not yet ready to be unleashed upon the general public via EViews, it was more a matter of raising awareness at this stage. I must admit that at times during this talk I began to appreciate the way that some of my students must feel when they tell me that they can understand all the individual words that I am saying but still can't get what it is all about. As someone who is at the more applied end of the econometrics spectrum I shall have to wait for a future edition of Davidson's textbook before I can comment further on this topic.

In the first afternoon session Hal White gave a paper with the title "Granger causality, exogeneity, cointegration and policy analysis" - a bit of a "Granger kitchen sink" title as he commented himself. Relating back to the 1998 paper by Ericsson, Hendry and Mizon in the Journal of Business and Economic Statistics, White proposed a slightly different framework in which he emphasized the concept of "structural causality", distinguishing it from Granger causality and EHM's conditional super exogeneity. This would mean that Granger Causality is not an essential property but a consequence of conditional exogeneity. The second paper was by Norman Swanson with the title "Diffusion index based data reduction with shrinkage: new empirical evidence". My notes on this paper seem to have shrunk to almost nothing (!) but I do recall Swanson mentioning that Granger was an expert on ex ante prediction of car parking spaces when he was at UCSD. Swanson also noted that he was relieved on one occasion when coming out of grocery store into the car park and forgetting exactly where he had left his car to discover that Clive Granger admitted to also suffering this same fate on a different occasion.

During the various coffee, lunch and tea breaks at the conference the younger and less established attendees were able to display their work in poster sessions. It seemed to me that most of these were devoted to unit root tests that were able to deal with various structural breaks, outliers or other abnormalities, but all were very professionally put together and I'm sure that the experience of talking through their work with colleagues as they walked round will have been very valuable.

On the Monday night we were treated to a very nice dinner at the Hart's restaurant in Nottingham, during and after which there were some more informal reminiscences of Sir Clive Granger. The following day there were further keynote lectures from James Stock, Mark Watson, Cheng Hsaio, Graham Elliott and Jesus Gonzalo, as well as the 4th Annual Granger lecture which was given by Hashem Pesaran. Pesaran talked about "Aggregation in Large Dynamic Panels", referring back to a conjecture that Granger had made on page 237 of his seminal 1980 Journal of Econometrics paper concerning fractional integrated processes. Pesaran showed that it was true.

Stock talked about "Forecast for time series with smooth spectral densities". The motivation for his work was the observation that the Akaike Information Criterion often seems to point to longer lags being needed in models than the Bayesian Information Criterion. Another puzzle was that the average of AR(1), AR(2) and AR(3) models (with equal weights) appeared to be working better in forecasting than any of the individual models. From simulations it seemed to be that the BIC was capturing big effects with the AIC picking up more local effects. Watson's paper "Estimating turning points using large data sets" started with a review of the various attempts to date the business cycle, going back to the early work of Thorp, and then Burns and Mitchell, through Moore and Zarnovitch and on to the Business Cycle dating committee approach. Early approaches were based on eye-balling the graphs looking for clusters of peaks (or troughs) while more recent attempts have been based on more analytical and computer-based methods. We were shown a nice "temperature" colour plot which illustrated very clearly the clustering of series with peaks and troughs.

Hsiao looked again at the question of whether there is an optimal forecast combination while Elliott looked at a similar issue relating to the averaging and optimal combination of forecasts. He showed that forecast averages beat OLS combinations.

In the last paper of the conference Gonzalo introduced the concepts of "summability" and "co-summability" which, he said, are the extensions of the concepts of integration and co-integration to non-linear systems. After two days of concentraion on heavy-duty econometrics (not to mention the several glasses of red wine consumed at the dinner on the Monday night) I was struggling to keep up with the detail of this talk, but I was able to appreciate the general idea. If we are dealing with non-linear processes of any kind - maybe x^2(t) where x(t) is I(d) - then we want to ensure that any estimated model is balanced on both sides of the = sign.

The conference was organised by Rob Taylor and Dave Harvey of the Granger Centre for time series econometrics. I am sure that all attendees will join with me in thanking them for all their hard work in putting together the programme and ensuring that everything ran smoothly over the two days. I appreciate too the efforts of all the presenters. Although I didn't instantly understand everything that I heard at the conference, I am hopeful that with more thought and more reading (when some of these papers come out in printed form) in my head there will be convergence to a point not too far away from the authors' equilibrium position!

References
[1] Ericsson, N R, Hendry, D F and Mizon, G E (1998) Exogeneity, Cointegration, and Economic Policy Analysis. Journal of Business and Economic Statistics, 16, 370-387.
[2] Granger, C W J (1980) Long memory relationships and the aggregation of dynamic models. Journal of Econometrics, Volume 14, Issue 2, pages 227-238
[3] Granger, C W J and Hendry, D F (2005) A dialogue concerning a new instrument for econometric modeling, Econometric Theory Volume 21 pp 278-297
[4] Hendry, D.F. & H.-M. Krolzig (2001) Automatic Econometric Model Selection. Timberlake Consultants Press.

Cricket and econometrics

2010-04-18T01:15:00.000-07:00

There are many applied econometrics papers that have a sports theme. Here is a link to a recent paper about cricket.

Aiyar, Shekhar and Ramcharan, Rodney (2009) What Can International Cricket Teach Us About the Role of Luck in Labor Markets?. IMF Working Paper, September 25, 2009

Wonderful econometrics (video) podcast archive

2010-04-15T02:48:00.000-07:00

The Center for Research in Econometric Analysis of Time Series at the University of Aarhus has a wonderful collection of (video) podcasts by distinguished econometricians such as Sir David Hendry and Soren Johansen.

Go to http://www.econ.au.dk/da/research/research-centres/creates/podcast-archive/

Econometrics Hall of Fame

2010-04-13T04:24:00.000-07:00

For the very first verse we could do a lot worse
Than to remember Jevons and Moore
Were sunspots what made the cycles in trade?
Or was their approach too flawed?

And then came the turn of Mitchell and Burns
And Persons played a big part in it too
But Slutsky and Yule knew correlations could fool
With spurious findings thought to be true.

With a Nobel prize for the specially wise
Jan Tinbergen was one of the first
And if Ragnar Frisch could have just one wish
Confluence analysis would still be rehearsed.

And Koopmans and Klein would think it just fine
To be remembered as part of the story
Haavelmo and Stone should also be known
And given their share of the glory.

So remember these names for the big hall of fame
Econometrics Hall of Fame
They were all very smart and played a big part
In the Econometrics Hall of Fame, oh yeah
Econometrics Hall of Fame

Now Engle and Granger could never be strangers
To students with time series data
You can feel the elation of cointegration
We all get it sooner or later.

But the list'd be in vain if we forgot the Great Dane
Soren Johansen, both modest and brilliant a man
who, with great intuition, proposed an ingenious solution
to determine dimension of a vector-spanned space

And Sir David Hendry is one of the best
Remember his mantra to test, test and test
With Durbin and Watson and Sargan before
The LSE school should not be ignored.

And the authors of textbooks deserve our respect
For keeping us all in the loop
Judge, Goldberger and Johnston I would never neglect
Now its Wooldridge, Kennedy and Koop.

Now I may have missed someone off of this list
Who should be here or more widely known
If that is your view then what you should do
Is to add in a verse of your own.

So remember these names for the big hall of fame
Econometrics Hall of Fame
They were all very smart and played a big part
In the Econometrics Hall of Fame, oh yeah
Econometrics Hall of Fame

Guy Judge, April 2010 (with thanks to Joachim Grammig for the Johansen verse)

Econometric poetry

2010-03-28T00:58:00.000-07:00

Following my recent postings of a poetic nature I have been contacted by Professor Joachim Grammig of the University of Tuebingen in Germany telling me about the wonderful website that he and his team have devoted to Econometric Poetry. Lots of good ones there. Please take a look.

Bootstrapping

2010-03-22T01:37:00.000-07:00

As a follow up to Carry on Regressing, We are the disturbances, Heteroskedasticity Blues and Facebook econometricians I give you "Bootstrapping"

Ten thousand iterations, that should be enough
Can't rely on asymptotics or any of that stuff
Let's work our simulation, set our random seed
And run the batch program, because of the need

For bootstrapping
bootstrapping
bootstrapping.

Resample the data many thousands of times
Get a sampling distribution along the right lines
Its computer intensive but that's what we love
No need for large sample mathematical proofs

Wer'e bootstrapping
bootstrapping
Just bootstrapping

Professor Bradley Efron, he showed us the way
With Robert Tibshirani from the US of A
Statisticians and doctors, economists too
Medical scientists, they all want to do

Bootstrapping
Bootstrapping
Lots of bootstrapping

Guy Judge
March 2010

Facebook econometricians

2010-02-19T01:32:00.000-08:00

In one of my ECONMET computer workshops this week I discovered that some students were looking at Facebook rather than working on their econometrics practical exercises. I told them that they could glance at Facebook every now and then but only AFTER they had obtained their regression results, and for no more than 5% of the class time.

This morning I saw that one of them had started to use some econometrics words on his Facebook wall (but, he said, it would be no more than 5% of them !!). This "inspired" the following reply (say it like a rap!).

"Don't be bombastic
It's way too drastic
To write stochastic...
Or heteroskedastic
Iconoclastic
It's way too classic
For Facebook traffic
Just stick to plastic
Or strong elastic
Don't try to boot-lick
Or be hedonic
Trust me to nitpick
It's just the metric"

Why so few comments?

2010-01-10T04:03:00.000-08:00

I sometimes wonder whether anyone reads my blog as I get so few comments. Maybe I will adopt the strategy suggested by this cartoon!

The One Thing Internet Cant Ignoere

99 problems

2009-10-22T02:15:00.000-07:00

If you want a distraction from actual econometric modelling then check out this YouTube video from the Metrics Gang - Skit Party 2009.

Instrumental Variables and "stylometrics"

2009-09-17T00:25:00.000-07:00

A short piece in yesterday's Freakonomics column in the New York Times drew attention to some work by James Stock and Francesco Trebbi, who compared the writing styles of Philip Wright and his son Sewall in an attempt to settle once and for all who was responsible for an appendix to Philip Wright's book The Tariff on Animal and Vegetable Oils published in 1928.

The appendix (B) is the first known articulation of the method of estimation that we now know as Instrumental Variables. Some authors have previously suggested that Sewall Wright must have written the appendix (there is evidence that he used the method in some work that he had undertaken on corn and hog cycles during World War I). However the statistical analysis of the writing styles of the two men suggest that Philip rather than Sewall wrote the appendix. You can read the Stock and Trebbi paper online .

But of course the paper doesn't fully resolve the problem of who thought of the method first.

Guinnessometrics

2009-03-25T02:06:00.000-07:00

Back in February 2007 I blogged about the history of Student's t-test under the heading Guinness is good for you!.

If you want to read a fuller (and more authoratative) piece on it you can do so in the Fall 2008 issue of the Journal of Economic Perspectives, in which there is an article by Stephen T Ziliak with the title Guinessometrics: The Economic Foundation of "Student's " t

Heteroskedasticity blues

2009-03-24T09:05:00.000-07:00

Well I woke up this morning
Econometric model on my mind
So I got myself some data
To see what I could find

Then I booted up my computer
And loaded up EViews
I ran a few regressions
But here comes the bad news

I got those old
heteroskedasticity blues
You can see from my diagnostics
I got those old heteroskedasticity blues

I tried logarithmic transforms
Expressed everything per head
Even interactive dummies
Everything the textbook said
And the pattern wasn't clear enough
For weighted least squares instead
I was tearing all the hair out
From my poor old achin' head

I got those old
heteroskedasticity blues
Yes you can see from my diagnostics
I got those old heteroskedasticity blues

But then I got to thinking
Everything's gonna be all right
I'll just follow the procedure
Of Professor Halbert White
Get robust standard errors
It's an option you can choose
'Cos it takes away the terror
Of t stats you can't use

Now they're gone
Those old heteroskedasticity blues
There's no need to be disheartened
By those old heteroskedasticity blues

Quantitative easing

2009-03-11T05:05:00.000-07:00

The following is not about econometrics at all, but due to the fact that a poem of mine about econometrics posted here has been picked up by the Freakonomics column on the New York Times website, I was encouraged by my one of my colleagues to write another one. The challenge was to write a poem about "quantitative easing" - here it is!

Quantitative easing
Is not very pleasing
But I guess it's just what has to be done.

'Cos when your tummy's bunged up
With no movement at all
A little syrup of figs
Can unblock the wall.

So Mr Darling buys gilts
And pumps in some money
But don't you agree
That his eyebrows are funny.

Are you taking the P? P-values versus t-values: the “sting in the tail”

2009-03-03T01:02:00.000-08:00

A maths joke with an econometrics twist

2009-02-05T02:26:00.000-08:00

This joke is my own version of a joke to be found in Professor Ian Stewart's excellent book Professor Stewart's Cabinet of Mathematical Curiosities. (I follow the "folk tradition" of adapting songs and stories to fit local conditions and points of reference).

Two mathematicians were having lunch together in a local restaurant. One of them was lamenting the fact that most people in the population are afraid of mathematics, and was wondering what could be done to address the problem. The other mathematician took the view that people actually know more mathematics than they might think, but perhaps don't recognise how much they know. For example, take our waitress. She would immediately know if the tip left for her was generous or not even if she denied having a good understanding of percentages. "Perhaps you are right about that" said the first mathematician "but what about her knowledge of the maths that they teach in school? How much of that do you think she understands? Look, I just need to answer a call of nature, but when I get back maybe we can put it to the test."

While the first mathematician was out of the room the second mathematician called over the waitress. "Look", he said, "when my friend returns I am going to ask you a question. All you have to do is say 'A third of x cubed'. There will be an extra tip for you if you can do this for me."

"Sure. No problem" replied the waitress. "A third ice cube".

"No, no" said the mathematician "A third of x cubed". "OK" , said the waitress "a third of eggs scooped".

"Oh, this isn't going to work" thought the mathematician as he saw his friend returning.

"Ah, good, here's the waitress", said the first mathematician. "Let's test your theory. Just ask her any typical school maths question and we shall see how she can do".

"Right" said the second mathematician, addressing the waitress. "Can you tell me what is the integral of x squared please?".

"Sure" said the waitress." It is one third of x cubed" adding, after a short pause, "plus the constant of integration!".

"Fantastic" said the first mathematician. "Maybe I have been underestimating the general public's maths skills".

"Probably not" said the waitress. "I'm doing a PhD in financial econometrics at the university. I only work here part-time!".

Make sure that you press ALL the right buttons!

2008-09-10T03:15:00.000-07:00

Last Monday I was in Bristol at a meeting of all the Economics Network key contacts (representatives from each university economics department in the UK). The programme included a very good talk from Andy Ross, the Deputy Director of the UK Government Economic Service (GES) on the theme of "Employability of Economics Graduates".

One of the things that he mentioned was the rather disappointing number of graduate economists who apply to the GES but fail the entrance test and interview process - around 50% of the just over 300 applicants that they get each year. Quite often this isn't because they are not sufficiently proficient in the technical skills that they need, but because they can't get the basics right. Of course the GES does want people who have technical skills in econometrics, but that in itself is not enough. Quantitative economics and econometrics is about more than just pressing buttons on a computer keyboard to generate unit root tests or run regressions. You have to be able to explain, interpret and communicate to people what you are doing. Although of course each year the GES employs about 160 excellent intelligent and energetic young economists, they could take more. But they find that too many applicants can't get the basics right. So don't neglect basic data skills, make sure that you are comfortable with index numbers, rates of growth and general spreadsheet skills. Make sure that you really understand basic economic concepts and principles like opportunity cost, the difference between real and nominal interest rates etc. A lot of what they do in the GES involves cost-benefit analysis. So if working in the GES sounds interesting and you apply there, make sure that you know about CBA and how it can be used in practical decision-making (especially how you would deal with conditions of risk and uncertainty).

To test your communication skills why not seek out a friend on a different degree course (maybe a medical student, a psychology student or a geographer) and see if you can explain in simple terms some basic economic ideas. Read one of the "poponomics" books like Freakonomics or The Undercover Economist (or maybe just browse through the blog that goes with the book).

Look at the stars

2008-03-26T02:18:00.000-07:00

Modern econometrics software, such as PcGive (part of the Oxmetrics suite) helpfully adds asterisks (or stars) to test statistics to indicate statistical significance - one star for significance at the 5% level and two stars for significance at the 1% level. This can be helpful for quickly showing you whether or not a model fails a diagnostic test, for example whether or not the residuals appear to exhibit heteroskedasticity.

Using the data on house prices in Baton Rouge, Louisiana in 1985, supplied by Hill, Griffiths and Judge (2001) and running a simple regression of house prices (in dollars) on plot size (in square feet) at first sight we appear to have a satisfactory model with a postive coefficient for the the independent variable and a t-value showing statistical significance even at the 1% level. The F statistic for the regression (here equivalent to the square of the t-value for the coefficient of the explanatory variable) is large enough for us to reject the null hypothesis of no relationship even at the 1% level. The P value shown here after the F statistic as 0.000 to three decimal places is clearly less than 0.01 (1%). PcGive highlights this by attaching two stars.

EQ( 1) Modelling price by OLS-CS (using b_rouge.xls)
       The estimation sample is: 1 to 213



  Coefficient Std.Error t-value   t-prob Part.R^2
Constant  -426.708  5061 -0.0843 0.933 0.0000
sqft 46.0050 2.803 16.4 0.000 0.5608

sigma                 8163.25  RSS           1.40607587e+010
R^2                  0.560768  F(1,211) =    269.4 [0.000]**

However a post regression test for heteroskedastcity tells a different story. Using the test based on White (1980) PcGive runs an auxiliary regression of the squared residuals on the original regressor (in this case plot size in square feet) and the squared value of this variable. If the null hypothesis of homoskedasticity is correct we should get a low value for the test statistic (Chi squared and an F form version of the test are available) and associated high probability values. However in this case because the spread of house prices is larger at higher plot sizes we find that we must reject the null hypothesis and accept the alternative, i.e. we must conclude that there is a problem of heteroskedasticty. PcGive helpfully makes sure that we can see this instantly by attaching the stars to the computer output.

Testing for heteroscedasticity using squares
 Chi^2(2) =   11.048 [0.0040]** and F-form F(2,208) =   5.6892 [0.0039]**

Clearly we must reconsider our model. A simple linear regression model is inadequate and we should consider an alternative functional form (perhaps a log-linear model). We should also add in and test for the relevance of many other possible variables to explain variations in house prices, some of them in dummy variable form (such as the presence or not of central heating).

The asterisks (or stars) that come in the PcGive output are a great help, but they should just be used to draw your attention to certain features of the results and must not be used mechanically.

As this example illustrates, the presence or absence of stars can be something we welcome (as with the overall F statistic in the basic regression) or something that causes us to pause and consider what to do next (as in the case of the heteroskedastcity test result). In all cases it is NOT appropriate to report your results just by talking about the presence or absence of the stars. A full consideration of the null and alternative hypotheses, test statistic and its meaning should be provided. Unfortunately I have seen assessed work handed in by some students where this doesn't happen and all I get is star gazing. I call this approach to econometrics "Yellow Econometrics"*. It is a kind of technological upgrade on the "cowboy econometrics" that I discussed in an earlier blog (where people would just quickly look to see if t-values were bigger than 2 or not rather than looking up the exact t statistic from the tables).

* Based on the opening line of Coldplay's 2000 hit "Yellow"
"Look at the stars,
Look how they shine for you,
And everything you do,
Yeah they were all yellow"

References

Doornik, J A and Hendry, D F (1994-2008)PcGive Professional. Timberlake Consultants Limited.
Hill, R C, Griffiths, W E and Judge, G G (2001) Undergraduate Econometrics. Second Edition John Wiley and Sons Ltd.
White, H. (1980). "A heteroskedastic-consistent covariance matrix estimator and a direct test for heteroskedasticity" Econometrica, 48, 817--838.

Why so many formulae?

2008-02-21T07:55:00.000-08:00

Something that puzzles many students of statistics and econometrics is why there might be several different formulae available to calculate the same result.

To take a simple example Dougherty (2007) says on p63 that R squared is ESS/TSS (where ESS stands for the Explained Sum of Squares and TSS stands for the Total Sum of Squares). Then on the next page he calculates this by subtracting the ratio of the residual sum of squares (RSS) to TSS from 1. We can see that this is correct using a simple bit of algebra because TSS = ESS + RSS.

Then on page 115 he gives the formula for the F statistic (for testing the overall significance of a multiple regression model) as [(ESS/k-1)/(RSS/n-k)] where n is the number of sample observations and k is the number of parameters to be estimated in the model.

A few lines later he gives another formula for F as [(Rsquared/k-1)/(1 – R squared/n-k)]

Of course in both cases one formula is the definition and the other is a simple algebraic transformation that might, in certain circumstances, be more convenient to work with.

In each of these cases it is not difficult to see that the second version follows from the first. If we can break the Total Sum of Squares into two additive parts ESS+RSS, then R squared = ESS/TSS = (TSS-RSS)/TSS = 1 –(RSS/TSS). For the F statistic if you divide top and bottom of the formula by TSS you leave it unchanged overall. So it follows that you can replace ESS on the top by R squared and RSS on the bottom by 1-R squared.

It might even be argued that working with these different formulae can actually give you a greater understanding of the meaning and interpretation of the various concepts.

Reference

Dougherty, C. (2007) Introduction to Econometrics. Third Edition. Oxford University Press, Oxford

Carry on regressing (econometrician's poem)

2007-06-26T00:28:00.000-07:00

Econometricians are not a Total Sum of Squares
They can be dynamic and follow the trend.
If things are non-stationary they can make a difference.
And they can get their Dickey-Fuller augmented at the end.

They test, test and test again
Adopting Hendry's main refrain
From general to specific, t, F and chi-squared too
They must look for significance in everything they do.

They can transform things
With a bit of Box and Cox
They can take random walks
And they sometimes work with an Ox.

They use dummies for sex
And like a bit of variance and deviation (from the mean)
They prefer to have their parameters stable
But sometimes have a break-point in between.

Although they sometimes have an identification problem
They know the conditions they must inspect
And with the proper instruments find
What they want in two stages or indirect.

Sometimes they can be found in Monte Carlo
Where they play God in their own domain
Creating many thousand replications
Power with small samples hoping to obtain.

And here is another short one.

We are the disturbances

We are the disturbances
You can't see us but you know we are there
Stochastic, homoskedastic, independent and normal
We must be specified with great care.

We must be added to your expectation
To make the model complete
And although we are mostly small
We are spread about with two tails but no feet.

Get Real!

2007-03-29T01:32:00.000-07:00

Many applied econometric models based on aggregate time series data make use of expenditure, income or GDP series. Consumption functions, demand functions or even production functions, for example. It is important to recognise the need to use series that have been expressed at constant prices (relating to a suitable base year) rather than at the current prices ruling in each individual year. That is the series should be in real rather than nominal terms.

For example if you are modelling the total use of energy in the UK you may want to include real GDP as one of your explanatory variables, to indicate the overall level of economic activity in the economy. Using current price GDP figures would overstate the growth of real economic activity as it would include increases due to inflation as well as those due to actual economic activity. You will need a measure of GDP that has been deflated by dividing through by an appropriate measure of inflation – in this case the GDP deflator.

Often suitably deflated series are readily available in published form. For example the series with the code ABMI - UK Gross Domestic Product in £ million at constant 2003 prices - can now be downloaded directly from the UK National Statistics website. Go to http://www.statistics.gov.uk/statbase/tsdtimezone.asp?vlnk=pn2, select Blue Book, select UK national and domestic product, and then pick out the ABMI series.

Sometimes you will have to find an appropriate price deflator for yourself to adjust a current price series for inflation. Here you will have to make sure that you are using the appropriate price deflator as there are many series that track inflation. The CPI (consumer price index) would be appropriate if you were looking at overall consumer expenditure but there will be occasions when some other price series such as the GDP deflator would be appropriate.

Implicit price series
The availability of expenditure series in both current and constant price form means that you will have the possibility of recovering an implicit price series for the expenditure category. For example if you have the two series FOODEXP and FOODEXP2005, where FOODEXP measures total expenditure on food each year at current prices while FOODEXP2005 is total expenditure on food at constant (2005) prices, then the index of food prices would be say PFOOD = FOODEXP/FOODEXP2005 (or you might like to multiply this by 100 so that the base year value of the series is 100 rather than 1). This of course would give you a measure of the nominal price of food rather than an index of the price of food in real terms. To get that you will have to divide your nominal price series by an overall price index for the economy.

Real interest rates
Another area where you might need to think about real rather than nominal values is with interest rates. Real interest rates are nominal (or market) interest rates minus the rate of inflation. If the rate of inflation in a country is quite high, then nominal interest rates would also have to be high in order to provide a real return for lenders.

You will also have to be careful to compute other series in real terms. For example you might need real wages, the real money supply or even a measure of real exchange rates. Make sure that you know how to do this!

How do you spell "heteroskedasticity"?

2007-03-26T07:05:00.000-07:00

Econometrics is full of long and difficult words: stochastic, kurtosis, multicollinearity, autocorrelation and - perhaps worst of all - heteroskedastcity.

You may like to know that the correct spelling of heteroskedasticity was actually the subject of a one page article in the journal Econometrica. In the March 1985 issue J Huston McCulluch argued that there should be a k in the middle of the word and not a c. His argument was that the word had been constructed in English directly from Greek roots rather than coming into the English language indirectly via the French.

The earliest use that McCulloch could find for either heteroskedastcity or heteroscedastcity was in a 1923 statistics text by Truman L Kelley. However John Aldrich in contributing to a website on the earliest known uses of some words in mathematics states that 'The terms heteroscedasticity and homoscedasticity were introduced in 1905 by Karl Pearson in "On the general theory of skew correlation and non-linear regression," Drapers' Company Res. Mem. (Biometric Ser.) II. Pearson wrote, "If ... all arrays are equally scattered about their means, I shall speak of the system as a homoscedastic system, otherwise it is a heteroscedastic system." The words derive from the Greek skedastos (capable of being scattered).'

One might ask the question as to why is it that econometricians, statisticians and other scientists use Greek words such as this as labels for these concepts when ordinary English words might do just as well. In part the answer might be to avoid confusion with the ordinary English usage of a word. For example the word "investment" when used by an economist carries a very specific meaning that might not be apparent even to a well-educated non-economist. Perhaps at least having a special word for the concept makes a reader check exactly what it means rather than making an assumption that it must mean what he thinks it does.

Of course what can happen is that these special words can then make it into ordinary language. In case you think that would be impossible for the word "heteroskedasticty" I invite you to read the paragraph about the distribution of rabbits around the UK in the Smallweed column of the Guardian for 23rd July 2005. "Have these brainboxes never heard of the concept of heteroscedasticity?" Yes, we have but you spell it with a k and not a c!

References
[1] J Huston McCulloch. On Heteros*edastcity. Econometrica 1985, Vol 53 No 2 (March) p483.
[2] John Aldrich. Earliest Known Uses of Some of the Words of Mathematics (H). Accessed March 2007.
[3] David McKie. Armageddon isn't upon us. The Guardian 31st August 2006.
[4] Smallweed column 23rd July 2005

The Power of Logs

2007-03-26T01:53:00.000-07:00

When I was at school (in the 1960s!) we were always using logarithms (or “logs” for short). We used our “log tables” to help us multiply together unpleasant looking numbers that we couldn’t very easily work with using just pen, paper and brain. If we had two nasty looking numbers to multiply together we would look up their logarithms (to the base 10), add together these logarithms and then finally take the anti-logarithm of this sum of logs to give us our answer. Once you know what logarithms are you will quickly see how this method works. You can use a similar method to divide one number by another only in this case you subtract the logs instead of adding them.

Today’s generation of school children have no need to use log tables for this kind of calculation. They can use a calculator (perhaps even one built into a mobile phone) or maybe a computer for unpleasant calculations. But students of economics would still be advised to find out something about logarithms as they might come across them in a variety of situations: when using a logarithmic transformation of a power function equation for example, or you might even meet a logarithmic equation itself. In the first situation you are basically making use of the properties of logarithms to turn a multiplicative function in the original variables into a linear (additive) function of the logarithms. Back to that later when we have had a look at the basic concept of a logarithm.

Put bluntly, the logarithm of a number is the power that a base must be raised to in order to get the number. You can see that we first have to think about what we mean by the base. In theory you could use any positive number as the base but in practice we usually work with one of two numbers: either the number 10 or the exponential constant known as e (it has a value approximately = 2.71828). Let’s come back to e later and stick with the base 10 for now.

From the definition we can see that the logarithm of the number 100, to the base 10, is 2. Why? Because if we raise the base 10 to the power 2, we get the number 100. Think of another simple example. What is the logarithm of the number 0.1 to the base 10? Answer? -1, because 10 raised to the power minus 1 = 1/10 = 0.1

Notice that the log of 1 to the base 10 is 0. Because 10 to the power 0 = 1.
In fact the log of 1 is zero whatever base we work with. Any positive number raised to the power zero is 1. But we are getting ahead of ourselves. Let’s stay with the base 10 for now.

Let’s see how logs to the base 10 were used in log tables. First someone has to produce a set of tables solving the equation 10^(log) = number, for lots of different numbers. For example, the logarithm of 2 to the base 10 is (to four significant figures) 0.3010. [I looked up this logarithm so many times in my youth that it is imprinted in my brain – I don’t even need to refer to a set of log tables to get the result!]. Similarly the log of 3 to the base 10 is 0.4771. So if I add these two logarithms together I should get the logarithm of the number 6 – because the log of a product is the sum of the logarithms. Now 0.3010 + 0.4771 = 0.7781. Actually when I look up the log of 6 to the base 10 I get 0.7782. (Using only four significant figure approximations has caused us to get an approximation error.) Rather than searching through the log tables to find the number that has a log = 0.7781 we were able to make use of the anti-logarithm table in which the results were set out in the other direction. That is the tables were constructed in a way to give you the number corresponding to any particular logarithm that you had calculated. Doing it this way round I would find that the anti-log of 0.7781 comes out as 5.999 – another approximation error due to rounding. Before you start smiling at this too much remember that even using a calculator or a computer there may be some rounding involved – although you will get much more than four significant figures.

The Scottish mathematician John Napier (1550-1617) is generally credited with the invention or discovery of logarithms – although apparently they were known about in eighth century India (see Smoller (2001) or Alfeld (1997) for more details).

Let’s think for a minute about how this all works. Take any base b (>0) .
If we know that b^u = x and b^v = y then simple algebra tells us that xy = b^(u+v). When we multiply two separate powers of b together we just add these powers. The insight in developing logarithms was to see that we could turn hard calculations (multiplication and division) into easier calculations (addition and subtraction) by providing a set of u and v values to go with the set of x and y values that could then be reused time and time again.

In analytical (as opposed to computational) work there may be advantages in working with logs to the base e. This is because the exponential function y = e^x has the special property that its derivative at any point is equal to the function itself – that is dy/x = y for all values of x. If you plot the graph of the function the slope of the curve is always the same as the function itself. This means that the derivative of the inverse function or logarithmic function y = lnx will be 1/x [Logs to the base e are written as lnx – the ln is short for “natural” logarithm]. This is very convenient. Logarithmic functions can be useful themselves in economics as they have the property that as x goes up y goes up but at a declining rate – something that we expect to get in a whole range of economic relationships such as production functions and utility functions.

But for us today it is the logarithmic transformation that is of most interest.

Suppose that we think that two variables are related by a power function equation – say Q = AP^b (maybe here Q is quantity demanded, P is price, A is just a constant of proportionality whose value will depend on the units of measurement for P and Q, and b is negative so as to ensure an inverse relation between the variables). If this is true then the graph showing the relationship between P and Q is a downward sloping (non-linear) curve. In the special case here b = -1 we have a rectangular hyperbola with the graph totally symmetrical around the 45 degree line, but with other values of b the graph will approach one of the axes more steeply than the other.

But if we plot the logarithm of Q against the logarithm of P we will see a downward sloping straight line with gradient b (remember b is negative). This is because the first rule of logarithms is that log(AB) = logA + logB (sticking to the same base. So log Q = logA + log(P^b). Now from the second rule of logarithms log(P^b)= b logP. Now logA is just another constant if A is a constant, so we have found that logQ = a constant plus b times log P. The power function is linear in the logarithms – or as we sometimes say for short – it is log-linear.

In regression analysis of course we don’t expect all our observations to fit exactly on a straight line (or a curve) but if, when we plot the logs of one variable against the logs of another we get points clustered around a straight line then it suggests that the underlying variables are linked by a power function equation – and the power in that equation can be estimated from the slope of the line linking the logs.

References
[1] John Napier and logarithms Laura Smoller UALR March 2001
[2} What on earth is a logarithm? Peter Alfeld, University of Utah 1997

Dummies for dummies

2007-03-07T07:51:00.000-08:00

“Let us remember the unfortunate econometrician who, in one of the major functions of his system, had to use a proxy for risk and a dummy for sex.” Fritz Machlup (1974)

As a student I was really fascinated when I first came across dummy variables. Here was a way of incorporating qualitative effects into regression equations. For example, in examining the factors affecting the hourly earnings of individuals as well as specifying potential influences that can be quantified (e.g. age, number of years of education, number of years experience etc.) you could include dummy variables to distinguish between dichotomous categories (that is situations that fall into two groups). So for example you could have a gender dummy variable to look for differences between the earnings of male and female workers, all other factors having been accounted for. All you must do is assign the value zero to the dummy for female workers and the value 1 for male workers and then include the dummy variable in the regression along with all the other potential influences on earnings. The estimated coefficient of the dummy variable will measure the differential in earnings purely due to the worker being a male. Of course as well as measuring the difference you can also use a standard t-test on the estimated coefficient to see if the measured differential is actually significant.

We can see how this works algebraically and graphically if we focus on just one other regressor – years of experience.

If the relationship is assumed to be a linear one then we can write the model algebraically as Y_i = b₁ + b₂X_i+ b₃D_i + u_i

The dummy variable D_i is assigned the value 0 for all women in the sample and 1 for all men. The affect of this is to make the equation showing how women’s earnings are believed to be generated just Y_i = b₁ + b₂X_i+ u_iwhile for men the equation becomes Yi = b₁ + b₃+b₂X_i+ + u_i.

When you put in the value 0 for D the b₃ D_i term disappears but when you put in the value 1 for D the term is just b₃, which can be grouped together with the b₁ term. Effectively the intercept becomes b+b₃ for men – or to put it another way there is a parallel upward shift in the regression line of b₃ for men as compared with the base line that defines the relationship between earnings and experience for women.

Figure 1. The effect of a dummy variable shown graphically

As well as incorporating dummy variables to look for shifts in a regression line we can also use interactive dummies to test for differences in the slope parameters.

Figure 2 Testing for differences in intercepts and slopes with interactive dummies

The expanded model includes a regressor that is the product of the dummy variable and the X variable. So its value will be zero when D = 0 and X when D = 1. Effectively its parameter b₄ will pick up the differential in slope between the regression lines for men and women. Figure 2 corresponds to a situation where initially men earn less per hour than women (the graph implies b₃ is negative) but the additional payments received by men for every extra year of experience exceed those given to women.

Dummies in models based on time series data
Dummy variables can be very useful to pick up the effects of circumstances that only apply to some time periods and not others when working with time series data. For example suppose we are looking at the relationship between the sales of ice cream in a particular supermarket and the average temperature over a succession of days. Here we have a t subscript for our observations Yt = b₁ + b₂ X_t + u_t and we would be want to estimate the value of b₂ so that, based on weather forecasts for the days ahead, we could predict extra demand that might be expected as temperatures rise.

However it may be worth testing to see if demand is higher at weekends or on public holidays. If for the moment we just treat all such days as the same we could define just one dummy variable that takes the value 0 on ordinary week days but is assigned the value 1 for days that are at the weekend or correspond to public holidays. So the extended model becomes
Y_t = b₁ + b₂X_t+ b₃ D_t + u_t and b₃ would be used to measure the additional demand to be expected on high days and holidays.

If there is enough data points, separate dummy variables could be defined for Saturdays, Sundays and holidays. Within this less restrictive model each dummy would have a separately estimated parameter. But it would also be possible to test whether the different day effects are actually needed by seeing if the restriction implicit in the model given above could be accepted against the alternative of the more general model.

Seasonal dummies
One common use of dummy variables with time series data is to allow for seasonal shifts in a relationship.

Suppose for example that you have quarterly data on energy consumption, the price of energy and consumers’ income for a number of years. You might specify a simple model, perhaps in log-linear rather than linear form, relating energy consumption to the price of energy and consumers’ income. (Perhaps both the price and income variables would be better expressed in real terms i.e. after adjustment for inflation rather than in current nominal prices – but we shall ignore that point here).

For a variety of reasons energy consumption may be higher in some quarters than others even after we have taken note of the values of price and consumers’ income (for example it is colder and darker in the winter leading to higher energy use). We could allow for these differences by incorporating three quarterly dummy variables, say for quarters 1,2 and 3, leaving quarter 4 as the base period.

Notice that we don’t have a dummy for all four quarters. This is sometimes referred to as the dummy variable trap. One quarter has to be kept as the base, just as with the gender dummy we didn’t have two dummy variables but chose one gender category – female – to be the base group. Incidentally it doesn’t really matter which category is assigned to be the base group although it may be convenient to specify the dummy as we did so that its coefficient takes a positive value.

So going back to our energy demand equation, with the dummies the model becomes

Log(Energy)_t= b₁+ b₂Log(Price)_t+ b₃ Log(Income)_t+ d₁D_1t+ d₂D_2t+ d₃D_3t+ u_t

Here D₁ takes the value 1 for all first quarter observations, zero otherwise; D₂ takes the value 1 for all second quarter observations, zero otherwise; D₃ takes the value 1 for all third quarter observations, zero otherwise.

I have used the symbol d for the coefficients of the dummy variables to distinguish them from those of the measured variables.

Interpreting the coefficients of dummy variables in log-linear models
Some care is required in interpreting dummy variable coefficients in log-linear regression models. In the example we have just been looking at suppose that the estimated value of d₁comes out as 0.01. This means that the intercept in the log-linear equation will be increased by 0.01 for all first quarter observations. But what does that mean for energy consumption itself. If Log(Energy) is up by 0.01 then Energy will be up by a factor of exp (0.01) = 1.01005 (assuming we have used logs to the base e – natural logarithms. Because log-linear models imply that the underlying variables interact with each other in multiplicative way the shift in the log-linear equation implies a multiplicative effect in the un-logged version of the equation.

Impulse and step dummies
Dummy variables can be included in regression models based on time series data to account for special circumstances or events that affect an individual observation. An example could be a model looking at quarterly sales of poultry products. In the light of the avian flue scare at the Bernard Matthews factory in Norfolk last month we might want to include an impulse dummy for the first quarter of 2007. The use of an impulse dummy would be based on the assumption that the (negative) effect would disappear and things return to normal in the following month. If that isn’t the case a step dummy might be more suitable. A step dummy takes the value 0 for all periods before a particular event and then the value 1 for periods after that time. Effectively the intercept steps up (or down) after the event. An example might be a dummy variable to measure the affect of banning of smoking in pubs on their revenue.

Dummy dependent variables (i.e. dummies on the left-hand side of regression equations)
Dummy variables can also appear on the left-hand side of regression equations. Limited dependent variable models of this sort can help to explain things like why some households have Internet access at home while others don’t or why some students succeed in passing a course while others don’t. This is interesting stuff but a whole new topic that we will look at another time.

Why don’t you look for interesting examples of the use of dummy variables in published work. Here are a few to be going on with: (1)the effect of computer ownership on college grade point average (Wooldridge, Introductory Econometrics p 235; (2) the effect of physical attractiveness on wages (Hamermesh and Biddle, 1994) reported in Wooldridge p 242; the effect of satellite TV on football attendances (Allan, Applied Economics Letters, 2004)

Online material on dummy variables
[1] Introductory Econometrics, a textbook written by Humberto Barreto and Frank Howland, was published by Cambridge University Press in 2005. The authors have put the introductory section of each chapter online, together with Excel spreadsheets to illustrate their material. Take a look at their material for chapter 8 on dummy variables at http://caleb.wabash.edu/econometrics/EconometricsBook/chap8.htm
[2] Kelly Rathje and Christopher Bruce have a short contribution to the Expert Witness newsletter (Winter 2000) that illustrates the use of dummy variables. Go to http://www.economica.ca/ew54p3.htm.

Notation, notation, notation

2007-03-06T01:05:00.001-08:00

If you are new to econometrics you may well get confused or even irritated by the variations in the notation used by different text book authors. Most (but not all) authors use the Greek letter beta to represent the unknown parameter that is the coefficient of an independent variable in a regression equation, with a subscript to indicate the variable that it is associated with. But different authors accommodate the constant intercept in such equations in different ways. Some give a subscript zero to this first beta, continuing with subscripts 1 to k for the betas linked to the X variables (which have matching subscripts 1 to k). Kennedy, Stock and Watson, and Wooldridge all adopt this form of notation. But an alternative approach is to give a subscript 1 to the constant intercept with the other betas then following on with subscripts 2 to k. In this approach the variable X1 just consists of a column of constant values (=1). This is the convention followed by Dougherty, Greene, Gujarati (Basic Econometrics), Hill, Griffiths and Judge, and Pindyck & Rubinfeld. Yet another variant is for the constant intercept to be labelled alpha, with the beta coefficients numbered from 1 to k (Maddala).

These differences are relatively minor, but when we move on to choice of symbols for the least squares estimates to go with these parameters there is a further lack of consistency. Kennedy, Maddala, Pindyck & Rubinfeld, Stock and Watson, Wooldridge and Gujarati (Basic Econometrics) put a "hat" over the Greek letter to show that we have an estimate (or estimator) of the parameter rather than its unknown value. But Dougherty, Greene, Hill, Griffiths and Judge and Gujarati in his other book (Essential Econometrics) instead use the equivalent Roman letter b for each of the betas.

And then we have the disturbances and their estimated equivalents, the residuals. Of the ten textbooks that I examined six use the letter u to denote the (unobservable) disturbance in the regression equation but three of the others use the Greek letter epsilon. One book (Hill, Griffiths and Judge) uses an ordinary e for the disturbance (or error) term. Now that can be confusing because two of the authors that use u as the disturbance have e to stand for the associated residual, as does one of the authors (Greene) who has epsilon for the disturbance. Hill, Griffiths and Judge put a hat over the e to denote the residual while Gujarati (Basic Econometrics), Maddala, Stock & Watson, and Wooldridge put a hat on the u to denote the residual. Pindyck & Rubinfeld put a hat on the epsilon that they use for the disturbance when they want to indicate the residual that goes with it. You can find a table showing the different symbols used by the various authors on my Introduction to Econometrics website at Portsmouth.

What is to be done about this? Why can't all these authors agree on a common system of notation? Taking the second question first I guess that each would argue that there are advantages of working with the particular convention that they adopt. There is certainly a logic to each of the choices made by the different authors but it does make it difficult for a student who consults more than one text book as he tries to get to grips with the subject. You might advise him to stick just to one textbook until he is confident enough about the meaning of the various symbols to recognise a slightly different label being used elsewhere. But I have never wanted just to recommend a single textbook for the courses that I teach. Different types of exposition suit different students. Some want a formal presentation and can handle the proofs and derivations that go with it. Others need a more intuitive approach with lots of examples and illustrations. And I find that some authors give a better exposition on one topic (perhaps autocorrelation) but maybe not such a good one as elsewhere on another (multicollinearity perhaps). So students can benefit by reading more than one account. In any case at some point they will have to get to grips with different notational systems used in the journal articles that they must read so it might be better to face up to this sooner rather than later.

So my point is...? Let's face up to the fact that there are different notational conventions, look at each of them and compare them explicitly and thereby enable students to become flexible enough to switch form one to another as the situation requires. It may also help students gain a better understanding of the underlying concepts if they have to think more carefully about what they are reading or writing.

Degrees of freedom and cowboy econometrics

2007-03-02T06:49:00.000-08:00

In the glossary at the end of his “A Guide to Econometrics” text (Fifth Edition, 2003, p 545) Peter Kennedy defines degrees of freedom as “..the number of free or linearly independent sample observations used in the calculation of a statistic”. In regression models we often see the number of degrees of freedom defined as “the number of observations minus the number of parameters to be estimated” – so for simple bivariate regression models that means n-2 (there are two parameters here: one fixing the slope of the line relating the variables and one fixing the intercept on the vertical axis).

Take an extreme case where you only have two observations on (X,Y). You have no freedom in fitting the line at all as there is only one line that can be chosen to connect the two points (see Figure 1).

Anybody undertaking any serious applied work would be advised to ensure that they have a great many more observations than 2. Not only do these extra observations provide some freedom in estimating the equation of the line, they also ensure that the 95% confidence intervals for the parameters are not too wide. The confidence interval will be the point estimate plus or minus the product of the standard error of the parameter estimate and the t-value leaving two and a half percent of the distribution in each tail (where we look up t with the appropriate number of degrees of freedom). A glance at the t-tables shows that the t-values fall as the number of degrees of freedom increase. For example t(0.025;10) = 2.228 while t(0.025;60) = 2.

When assessing the statistical significance of a variable in a regression (or more correctly of its accompanying parameter) the calculated t value must be compared with the critical value from the tables. So for example if the calculated t-value was say 3.5 then we would be able to reject the null hypothesis that the parameter is zero and accept the alternative hypothesis. [If we have a strong a priori view about the sign of the parameter, as predicted by theory, we might use a one-tailed test which would put all the 5% significance level area into one tail and thus pick out a smaller critical value that has to be exceeded for the decision to be taken to reject the null.] Your computer software might also produce a figure for the P-value or probability value linked to the calculated statistic. This measures the area beyond the calculated value, in the tail(s). This provides an alternative way for you to decide whether to reject the null or not. You simply compare the P-value with 0.05 (i.e. 5%). If the P-value is < 0.05 then you can reject the null.

When I was student the computer software was not that sophisticated and you definitely had to use the t-tables. A friend of mine on the same course never had his tables with him when he was doing the practical exercises set by the lecturer and was
famous for saying “Just check if its bigger than 2”. His reasoning was that whatever degrees of freedom might be appropriate, the t-value you got from the table was always approximately 2. See the examples I mentioned above and also Figure 2 which shows the tabulated t-values at 5% and 2½% for various degrees of freedom.

Of course if you do this you will be slightly misrepresenting the actual significance level of the test. I called my friend’s approach to the subject “cowboy econometrics” (an analogy with “cowboy builders” like the ones who worked on my house and didn’t properly measure the doors they were fitting. They shut OK but they don’t fit snugly so I get a draft under the gap at the bottom).

These days there really is no excuse not to do the tests properly. And there are also some very nice online Java applets that will calculate either the probability value to go with any t-value (for a given number of degrees of freedom) or the t-value to go with a specified P-value. See for example the one produced by R Webster West of the Department of Statistics at the Texas A&M University, from which I have taken the following screen grab – Figure 3. Screen grab of the t-distribution applet graphic.

So it turns out that they are not all cowboys in Texas!

Guinness is good for you!

2007-02-26T08:52:00.000-08:00

Students taking my course on Introductory Econometrics probably won’t recognise this slogan but it was to be found on all bottles of Guinness when I was a student first studying econometrics at the end of the nineteen-sixties. Actually there is a connection between Guinness and econometrics and an interesting story to go with it.

The link is Student’s t-distribution and its originator Mr W S Gosset who worked as a chemist and mathematician for the Guinness brewery a century ago.

In econometrics we regularly use the t-distribution to assess the significance of individual regression coefficients or to compute confidence intervals based on our sample estimates. We divide the coefficient estimate by its standard error and then check whether this calculated value exceeds (in absolute value) the relevant critical t value from the tables (based on the available degrees of freedom and the agreed significance level for the test – usually 5%). To get a 95% confidence interval for a parameter we take the point estimate plus or minus the estimated standard error multiplied by the appropriate t-value from the table with 0.025 in each tail.

As you may know the t-distribution is rather like the normal distribution in that it is symmetrical around its mean and with most of the values falling quite close to the mean with only a small amount of the area out in the extreme tails. Actually the tails are a bit “fatter” than those of the normal distribution, so if you were to use the normal distribution critical values of + and – 1.96 to separate extreme values in the tails from those in the middle of the distribution you would be slightly out in your assessment of the significance level of your hypothesis test or in setting 95% confidence limits for parameter estimates. However, as you perhaps also know the exact t-values depend also on the number of degrees of freedom available to you in estimating the parameter(s) and the t-distribution does approach the normal distribution as the number of degrees of freedom increases – so with a big enough sample size it perhaps wouldn’t make much difference.

Where does this t-distribution come from and why is it important? Let’s suppose first of all that a factory production line is filling bottles with an amount of liquid (Guinness maybe!) supposed to be 1 pint. Now it is not really possible for the technology to guarantee an exact amount of 1 pint each time – in reality the quantity dispensed will be a random variable which sometimes puts a bit more than a pint into a bottle and sometimes a bit less. It may be OK to assume that this random variable has a constant mean and variance, and even that the distribution is normal. In that case, if these two parameters were known in advance it would be possible to set up the machinery in such a way that we could ensure that say 95% of the time there would be at least 1 pint in each bottle. The mean amount dispensed would have to exceed 1 pint, the gap between this figure and 1 (pint) obviously being smaller the lower is the variance of the distribution of liquid dispensed. The problem is that in most cases we won’t be able to know either the mean value of the distribution or its variance in advance. We will have to take a sample of values and use the sample mean to estimate the population mean, and then base our estimate of the population standard deviation on the standard deviation of our sample. (Actually the estimate of the standard deviation of the sampling distribution of the mean – called the standard error – will be the sample standard deviation divided by the square root of the sample size; see any basic statistics text). In a situation like this it is quite likely that the sample size that we work with will be small. After all if we have to remove a number of bottles of liquid from the production line in order to measure how much liquid is in it, that bottle and its contents cannot be sold. Taking a large sample would just be too costly. Fortunately Gosset discovered how the sampling distribution would be affected by having to use an estimate of its variance from a small sample rather than working with the known population value or an estimate based on a very large sample. This distribution has now come to be known as the t-distribution, or more formally, Student’s t-distribution. It is worth noting that Gosset had no computers to help him undertake the necessary simulations to arrive at his result. All his calculations had to be done by hand.

Having completed this work Gosset naturally wanted to share his findings with other members of the statistics community through the usual method of a published journal article. However his employers at Guinness were not at all keen on this and he had to resort to the ruse of publishing under the pseudonym A. Student. Hence “Student’s” t –distribution. (Actually he didn’t originally call the distribution the t-distribution. It was his fellow statistician Fisher who gave it this name.)

You can read the original paper online if you wish at
Student [W S Gosset] (1908)The probable error of a mean. Biometrika, (1): 1–25.

And you can read more on William Sealy Gosset and his work for Guinness (as well as other leading statisticians of the early days such as Fisher, Pearson and others) in a fascinating book called The Lady Tasting Tea by David Salsburg.

Why least squares?

2007-02-16T04:21:00.000-08:00

Introductory courses in econometrics quickly tell students about the use of the least squares criterion in estimating regression equation parameters. The difference between an actual value of the dependent variable Y and its fitted value Yhat is called the residual. Least squares estimators are produced in such a way as to minimise the sum of the squares of these residuals (RSS = Residual Sum of Squares).

Most students will accept that the slope and intercept of a fitted regression line need to be found by some kind of objective method. It is not very reliable just to put down a ruler and draw in the line that seems to give a good fit balancing in some way positive and negative errors. But why minimise the sum of the squares? There are other possible objective criteria that could be used. For example why not just minimise the sum of the absolute deviations of the actual points from the fitted line? (It is easy enough to show that you couldn’t choose values to minimise the simple sum of the deviations because the positive and negative errors would just cancel each other out so you wouldn’t be able to get a solution this way).

Minimising the squared deviations will apply a greater penalty to points that lie further away from the fitted regression line than if we worked only with the absolute distances of the points from the line. It is sometimes argued that this is a desirable feature of the estimation technique. In some sense the residual that you get is an estimate of the disturbance for that observation so you might alternatively ask why a-typical observations with apparent big disturbances should be given extra importance. As Peter Kennedy points out in his book A Guide to Econometrics ( page 12), even if you use the squared deviations rather than the absolute deviations as a way of getting over the problem of positive and negative errors cancelling out you don’t have to give each of these squared deviations equal weight. Indeed as you will find out later in your studies there may be occasions where it is better to use Weighted Least Squares rather than Ordinary Least Squares as the criterion for determining your estimator.

Of course one advantage of least squares estimators is that they are computationally straightforward. The result of applying the criterion using basic methods of calculus is a simple formula for each regression coefficient in terms of the X and Y data (or more accurately their sums, sums of squares and sums of cross-products). Other estimation techniques might require iterative procedures to arrive at an estimate. Although that would be less of a worry in modern times given the advances in computer power, conceptually it seems attractive to have a formula that can always be used.

Least squares estimation might also seem to have a “natural” justification as W W Sawyer suggested in his wonderful little book The Search for Pattern (published by Penguin Books in 1970 as part of a series called Introducing Mathematics but now unfortunately out of print). In part of the chapter on algebra and statistics (pp312-313) he described a mechanical device that could provide a visual confirmation of the least squares criterion. The device consists of a solid piece of wood with nails or screws inserted at places supposed to correspond with the XY values on a graph. Sawyer suggested that a steel rod could be used for the line and the nails could be connected to the steel rod by elastic bands. To quote what he said “Things must be arranged in such a way that, if the rod actually passed through one of the points, that point’s band would ‘feel satisfied’ – there would be no tension in it. But the further away the rod is the greater the tension in the band must be; in fact the tension must be proportional to the amount by which the rod misses the point. Things must be so arranged that the bands are compelled to remain upright, as shown in the figure. Each band then is doing its best for the point to which it belongs, and under all these conflicting pulls the rod would eventually come to rest in a position which represented a fair compromise.”

I well remember constructing a device of this kind when with great excitement I first began teaching econometrics back in the 1970s. I even painted on the fitted least squares line on the wood so that students could see the rod settling exactly where the line was. Everything worked perfectly with the first group that I used it with but later in the week I guess the elastic bands must have weakened and, just after I had held the board aloft with the metal rod sitting perfectly in place, one of the elastic bands snapped and the rod was fired across the room narrowly missing one of the students. So my experiment with a physical representation of the least squares line was short-lived. Looking back at Sawyer’s book now I see that he does say that “the device may not be too easy to set up in actual practice”, a phrase that I must have missed in my enthusiasm at the time.

Another justification for using OLS that I remember from my own student days was that, despite its alleged limitations it was quite robust to departures from its underlying assumptions. Thus, while it might be better to make use of a more complex estimation technique to overcome problems of heteroskedasticity or autocorrelation, those techniques actually require you to have a good idea of the form of autocorrelation or heteroskedasticty that you were going to allow for – something that you might not have. I remember my tutor at the time, Farouk El Sheikh (sadly now no longer with us) setting us an essay question “’In the country of the blind the one eyed-man is king’. Discuss in relation to the use of OLS and other estimation techniques.” However he was somewhat taken aback by my answer, which pointed out that in H G Wells' short story, The Country of the Blind, the fully blind inhabitants of the remote South American country eventually decided that the one-eyed man was insane because of the visions that he kept talking about, and so decided that he must be operated on to make him ‘normal’.

Perhaps sometimes visual and literary allusions in econometrics can be taken too far!

Nothing to prove *

2007-02-12T08:25:00.000-08:00

I don’t really go in much for proofs in my Introduction to Econometrics (INEMET) course.

That doesn’t mean that proofs aren’t important in econometrics. Without a proof how could we be sure that the least squares estimators are unbiased, given the classical assumptions, or that omitting a relevant explanatory variable from a model will not only cause bias in the estimation of the coefficients of the other variables but will also affect their standard errors and t-values.

Certainly anyone who wishes to pursue the study of econometrics beyond an introductory course will need to become familiar with proofs of these and other important propositions found in the textbooks. But beginners can be overwhelmed by all the technical stuff (as I can still remember from my own initial exposure to the subject back in the late 1960s!). It is more important for students who are just beginning their study of econometrics to get a good intuitive feel for the subject, its scope and methodology, than to grapple with formal proofs. So in this respect I go along completely with Christopher Dougherty, who says in the preface to his book Introduction to Econometrics (Third Edition) p vi “For nearly everyone, there is a limit to the rate at which formal mathematical analysis can be digested. If this limit is exceeded, the student spends much mental energy grappling with the technicalities rather than the substance, impeding the development of a unified understanding of the subject.”

That doesn’t mean that students have to just accept a whole set of results without any attempt being made to justify them. In a number of cases a convincing intuitive argument can be provided for the propositions in question, without the need to resort to a proof. Or alternatively a simple quantitative example can be used to support the argument.

Take the case of the formula for the standard error of the X coefficient in the simple linear regression model. As Dougherty shows mathematically (!) on page 83 of his book the theoretical variance of the X coefficient is the variance of the disturbance term divide by the sum of squares of the deviations of the X variable from its sample mean. We can calculate the latter but we have to estimate the former as we don’t observed the actual disturbances. The square root of this estimate of the variance is then the standard error that we use for the t-test and for computing confidence intervals for the parameter.

In my lectures I have tried to convince students of the result by using a simple spreadsheet (Excel) demonstration which amounts to a simple Monte Carlo experiment. I begin by setting up an assumed model with known intercept and slope parameters – say Y = 2 + 0.8X + u.

Next I create a sample of fixed X values in the spreadsheet. I usually centre the values on a mean of 100 and have maybe 12 values either side of that (so X runs from 88 to 112). Then I use the random number generator to create a large number of sets (say 500) of 25 values of u, initially making u ~ N(0,1). From this I can create 500 sets of Y values to go with the Xs. Now I can run regressions based on these 500 data sets and collect the 500 estimates of the slope coefficient. (You might prefer to set this up as a batch job in EViews or some other specialist econometric software packages if you wish). After that, plot a histogram of the beta hat values, as well as calculating the mean and standard deviation of the 500 values. (An interesting discussion point is whether you should use 0 or the mean of the 500 beta hat values in this calculation). Compare these values with those predicted by the theory for the sampling distribution of beta hat.

Now you can repeat the process, varying first the variance of u (maybe make it smaller than 1 – say 0.5). You should see immediately that the variance of the beta hat estimates falls proportionately.

Then you can illustrate the effect of more spread out values of the Xs. Multiply each of them by 10 and recalculate all of the Y values (go back to the original standard normal distribution for u). Rerun the regressions and compare the distribution of the beta hat values. The standard deviation should be one tenth of what it was initially.

If all this takes too much time to do this interactively you could prepare everything in advance.

Another advantage of this exercise is that it introduces students to the idea of Monte Carlo studies and computer simulation at an early stage.

References
[1] Dougherty, C (2006)Introduction to Econometrics. (Third Edition), Oxford University Press.
[2] Judge, G (1999) Simple Monte Carlo studies on a spreadsheet CHEER Volume 13, 2.

* The phrase "Nothing to prove" is one that I always associate with the Sunderland striker David Connolly. He began his career at Watford, averaging a goal in every two games, before he left for a spell at the Dutch side Feyenoord. A bit of a flop in Holland he returned to England to play for Wimbledon declaring that he had "nothing to prove". This caused some amusement among Watford supporters who think of him as rather arrogant and used to label him W4BS (Watford's 4th Best Striker).

	Coefficient	Std.Error	t-value	t-prob	Part.R^2
Constant	-426.708	5061	-0.0843	0.933	0.0000
sqft	46.0050	2.803	16.4	0.000	0.5608