Econometrics Beat: Dave Giles' Blog

You had to be there!

2012-03-03T09:38:00.000-08:00

It was the Fall of 1983. I was on Study Leave from Monash University in Australia, and I was spending about 6 months in the Economics Department at the University of Western Ontario. My good friend, Aman Ullah, had arranged the visit and I ha a wonderful time.

There were several seminar programs running on a weekly basis, including one in econometrics. Some time that term there was an econometrics seminar presented by two young Assistant Professors in the department. Here is how it went down.

The research that was presented suggested that the usual Wald test has an unfortunate property when the parametric restrictions that are being tested are non-linear. Specifically, the authors claimed that the value taken by the Wald test statistic could change, according to how the restrictions are written. In other words, the value of the test statistic, and possibly the outcome of the test itself, are not invariant to the way in which the restrictions are expressed.

For example, consider the following single non-linear restriction on the coefficients of a linear regression model: (β₁ / β₂) = β₃. An alternative (and absolutely equivalent) way of expressing this restriction is: (β₁ - β₂*β₃) = 0. You'll get different values for the Wald test statistic depending on which of these equivalent restrictions you test. Moreover, the actual size and the power of the Wald test in finite samples can be very different depending on the choice you make.

Now, remember that this was 1983. What was being put forward in that seminar was something of a shock! Indeed, those of us who had "been around a while" were - to put it mildly - somewhat skeptical.

How could this be? The Wald test had been in widespread use by statisticians and econometricians since its introduction in 1943. Here we were, forty years later, finding out that it had a major flaw - some might say a fatal flaw - if applied to a really important class of problems.

After all, this "problem" doesn't arise with competing testing principles, such as the Likelihood Ratio test or the Lagrange Multiplier (Score) test. Surely these two guys couldn't be right?

There was the usual lively discussion, and then we retired to the faculty club for a jug or two of Labatt's IPA. The discussion continued, and after a few more beers the claim about the flawed Wald test looked more and more convincing!

Of course, as most of you will know already, Alan Gregory and Mike Veall were perfectly correct in their claim, and their paper subsequently appeared in Econometrica. The 1983 seminar was acknowledged in the opening footnote of the paper. Other, related, contributions followed. These included papers by Phillips and Park (), Bresusch and Schmidt (1985) and Lafontaine and White (1986). Alan and Mike have gone on to extremely successful academic careers at Queen's University and McMaster University, respectively.

Today, thanks to their insight and persistence, we're justifiably cautious about our use of the Wald test. Some of us who were at that 1983 seminar should have been equally cautious in our initial reactions to the initial airing of their early draft paper!

It was a privilege to be there!

References

Breusch, T. S., and P. Schmidt, 1985. Alternative forms of the Wald test: How long is a piece of string? Mimeo., Department of Economics, University of Southampton.

Gregory, A. and M. Veall, 1985. Formulating Wald tests of nonlinear restrictions. Econometrica, 53, 1465-1468.

Lafontaine, F. and K. J. White. 1986. Obtaining any Wald statistic you want. Economics Letters, 21, 35-40.

Phillips, P. C. B.. and J. Y. Park, 1988. On the formulation of Wald tests of nonlinear restrictions. Econometrica, 56, 1065-1083.

Wald, A., 1943. Tests of statistical hypotheses concerning several parameters when the number of observations is large. Transactions of the American Mathematical Society, 54, 426-482.

Leap Year Econometrics

2012-02-29T08:56:00.000-08:00

I guess there must be a few econometricians who were "leap year babies" - that is, born on 29 February. I'm not one of them, but a former co-author of mine was. This is the first February 29th since he passed away, and it's a good day to remember how much fun we had working together.

Peter Hampton was my very first co-author (in 1976), and we published seven paper together. Peter's primary interests were in trade, regional economics, and urban economics. He loved working with data, and our joint work covered some diverse topics (as is illustrated by the references below).

Peter had a wicked sense of humour and could always be relied upon for an endless supply of good jokes. His sense of humour was matched only by his absolute modesty, and his care for his students.

Not only was Peter a leap year baby, but he was a twin. I never met his mother, but I like to think that it was from her that he inherited his sense of fun. After all, she had the wit to name her twins Peter and Wendy, knowing that they would "never grow old".

References

Giles, D.E.A. and P. Hampton, 1981. Interval Estimation in the Calibration of Certain Trip Distribution Models. Transportation Research, B, 15, 203-219.

Giles, D.E.A. and P. Hampton, 1984. Regional Production Relationships During the Industrialization of New Zealand, 1935-1948. Journal of Regional Science, 24, 519-533.

Giles, D.E.A. and P. Hampton, 1985. An Engel Curve Analysis of Household Expenditure in New Zealand”. Economic Record, 61, 450-462.

Hampton, P. and D.E.A. Giles, 1976. Growth Centres, City Size and Urban Migration in New Zealand. Annals of Regional Science, 10, 41-44.

A Trick With Regression Residuals

2012-02-22T12:34:00.000-08:00

Suppose that you've estimated an econometric model and you want to test the residuals for serial independence, or perhaps for homoskedasticity. The trouble is that for the model and estimator that you;ve used, your favourite computer package doesn't provide such tests. Is there a quick way of "tricking" the package into giving you the information that you want?

Yes, there is. I'll show you how, by looking at a situation that arises when using EViews. Similar examples occur with pretty much any econometrics package that you happen to be using, so the idea here has wide applicability.

If you use EViews to estimate a "Seemingly Unrelated Regression Equations" (SURE) model, then you'll find that you have very few options when it comes to conducting diagnostic tests on the residuals of the equations in the system. For example, no tests for the homoskedasticity of the errors are immediately available. When you run into this sort of problem, here's one way that you can "fool" the package into giving you access to some of the tests you want, with minimal effort.

The trick is to be aware of the following result:

If we fit a regression equation in which the only "regressor" is a column of ones, then the residuals from this curious regression will just be the data for the dependent variable, adjusted so that they are in deviations about the sample mean.

To see this, consider the following "artificial" regression:

y_i = γ + ε_{i ;}i = 1, 2. ...., n.

The OLS estimator of gamma is: c = Σ (y_i 1) / Σ (1) = Σ (y_i) / n.

Then, the residuals are e_i = y_i - Σ (y_i) / n. In particular, if the "y" data had a mean of zero to begin with, then e_i = y_i ; for all i.

So, if we have a residuals series from some estimated model, and we then use this series as the y variable in the above artificial regression, any tests that we then perform on the latter regression's residuals will actually be tests on the y series, or on the residuals from the original model that we estimated. In this way we can get access to any tests available through the OLS command and apply them to the residuals from the other model.

Of course, if we did this, we'd need to make sure that the tests in question were valid when applied to these other residuals.

I have an example using monthly data for the market shares of various web browsers between October 2004 and February 2007. The data are available on the Data page that accompanies this blog. The example involves 6-equation SURE model for these market shares. If you look at the data files, you'll see the definitions of the variables. The EViews workfile that I've used or the following analysis is in the Code page for this blog.

Here is the specification of my model:

The estimated coefficients are as follows:

and the rest of the output is:

The residuals for each equation sum to zero, as can been in the following table:

Now, to illustrate the idea outlined above, I've tested the residuals of each equation for homoskedasticity. Let's consider the first equation. First, I estimate the artificial regression by OLS:

The residuals from the regression are actually just RESID01 - the residuals series for the first equation in the SURE system. I can now select "VIEW / RESIDUAL DIAGNOSTICS / HETEROSKEDASTICITY TESTS", and I get:

Not all of these tests will be appropriate. For example, I can't use White's test because it will simply use the "regressors" from this artificial regression (namely a column of ones), and their squared values. However, the B-P-G, Harvey, and Glesjer tests can all be used. If, for example, I use the B-P-G test, and specify that the heteroskedasticity may be a function of WINXP, I get the following results:

The p-value associated with the nR² version of the test suggests that we should reject the null hypothesis of homoskedasticity at the 3.7% significance level (or higher).

If we apply the same test to the residuals of the fifth equation, postulating that any heteroskedasticity may be associated with the WIN2000 variable, we get:

The small p-value again suggests that we should reject the null hypothesis of homoskedasticity for the errors of this equation of the original SURE model.

Two further comments are in order:

It's worth re-emphasizing that any tests that we perform in this way must be applicable to the residuals being analyzed. Here, the residuals were for the equations of a SURE model. The homoskedasticity tests being applied are asymptotically valid tests, so they will be relevant in this case. Our sample size is small, however, so the result should be treated with caution, just as they should be if we had applied them to an original OLS regression with this sample size.
Having detected the presence of heteroskedasticity, we could now transform the data for the variables in the equations in question, and then re-estimate the SURE model. In other words, we could modify the SURE estimation in a Weighted Least Squares manner.

Computer Update

2012-02-20T20:09:00.000-08:00

Now, this is really scary!!! (Even for nostalgia buffs.)

Tables or Graphs?

2012-02-19T10:42:00.000-08:00

Should I present my results in a table or in a graph? Both have their place, of course.

A recent post, titled "Some Notes on Making Effective Tables", on the Cross Validated Community Blog, makes some interesting points and provides some good advice.

The CVCB is, by the way, an overflow blog for Cross Validated Stack Exchange, ".... a collaboratively edited question and answer site for statisticians, data analysts, data miners and data visualization experts."

Early Contributions to Training in Econometrics at the USDA

2012-02-18T08:10:00.000-08:00

My colleague, Malcolm Rutherford, has recently published a really interesting paper on the history of statistical and economic education at the U.S. Department of Agriculture Graduate School. The school was founded in 1921, and exists to this day. As Malcolm explains, the USDA Graduate School played a seminal role in instruction in statistics in the 1930's, at a time when Econometrics was in its infancy.

I mentioned Malcolm's work in a post last year, titled "An Overly Confident (Future) Nobel Laureate". That post mentioned Milton Friedman's gaffe at a seminar being presented at the USDA Graduate School by Jerzy Neyman in 1937.

Malcolm's paper, titled "The USDA Graduate School: Government Training in Statistics and Economics, 1921-1945", published in the December 2011 issue of the Journal of the History of Economic Thought (vol. 33, no. 4, pp. 419-447). If you don't have online access to this journal, the Working Paper version of the paper is available here.

To give you the flavour of the paper, here's the abstract:

"The USDA Graduate School was founded in 1921 to provide statistical and economic training to the employees of the Department of Agriculture. The school did not grant degrees, but its graduate courses were accepted for credit by a significant number of universities

In subsequent years, the activities of the school grew rapidly to provide training in many different subject areas for employees from almost all federal departments. The training in statistics provided by the school was often highly advanced (instructors included Howard Tolley and, later, Edwards Deming), while the economics taught displayed an eclectic mix of standard and institutional economics. Mordecai Ezekiel taught both economics and statistics at the school, and had himself received his statistical training there. Statistics instruction in 1936 and 1937 included seminar series from R.A. Fisher and J. Neyman, and courses on the probability approach to sampling involving Lester Frankel and William Hurwitz became important after 1939. The instruction in economics was noticeably institutionalist in the period of the New Deal. Towards the end of the period considered here, the instruction in economics became narrower and more focused on agricultural economics.

The activities of the school provide a basis for understanding some of the sources of the relative statistical sophistication of agricultural economists and of the statistical work done in government in the interwar period. It is noteworthy than within the USDA Graduate School, and in contrast to the Cowles Commission, statistical sophistication coexisted with an approach to economics that was not predominantly neoclassical. It also provides a light on the place of institutional economics in the training of government economists through the same time span."

I've had the opportunity to read various versions of Malcolm's paper, and to have access to some of his archival material. I can assure you that this paper is a great read!

The Neyman-Pearson Lemma: An Economic Perspective

2012-02-17T09:54:00.000-08:00

In my graduate-level "Themes in Econometrics" course we've been talking recently about the Neyman-Pearson Lemma. In 1933 Jerzy Neyman and Egon Pearson published one of the most important papers of modern statistics, referenced below. Specifically, they showed that we can use the likelihood ratio to construct the Most Powerful test (for a given significance level), when we are testing a point null hypothesis against a point alternative hypothesis. This set the scene for classical hypothesis testing as we practice it today.

Those of us with Bayesian inclinations find the two opening sentences of this masterpiece in frequentist statistics somewhat interesting! There, Neyman and Pearson begin:

"The problem of testing statistical hypotheses is an old one. Its origin is usually connected with the name of THOMAS BAYES, who gave the well-known theorem on the probabilities a posteriori of the possible "causes" of a given event."

In econometric applications we're usually concerned with testing a point null against a composite alternative. For example, we may wish to test H₀: β₁ = 0 against H₁: β₁ > 0 in a regression model setting. Once we move away from the "point null - point alternative" situation, there's no guarantee that the Likelihood Ratio Test (LRT) will be Uniformly Most Powerful (UMP). However, it often is, and it's certainly a good starting point for our testing in many cases.

To quote Neyman and Pearson (1993, 336-337) again:

"We give the solution of this problem for the case of testing a simple hypothesis.

To solve the same problem in the case where the hypothesis tested is composite, the solution of a further problem is required; this consists in determining what has been called a region similar to the sample space with regard to a parameter.

We have been able to solve this problem only under certain limiting conditions;.......

It has, however been shown that the critical region based on the principle of likelihood satisfies our intuitive requirements of "a good critical region"."

In the example given above, with Normal regression errors, the LRT test can be shown to be be equivalent to the usual t-test (by the monotone likelihood principle), and it is indeed UMP against one-sided alternatives.

About 3 years ago, Cosma Shalizi (Department of Statistics, Carnegie Mellon U.) had an interesting post on his Three-Toed Sloth. That post was titled "The Shadow Price of Power", and it gives an excellent "economic" interpretation and motivation for the Neyman-Person Lemma.

I liked it, and I think you will too!

Note: The links to the following references will be helpful only if your computer's IP address gives you access to the electronic versions of the publications in question. That's why a written References section is provided.

Reference

Neyman, J. and E. S. Pearson (1933). On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London, Series A, 231, 289-337.

ANZecmet

2012-02-16T09:17:00.000-08:00

The following was circulated yesterday by Rob Hyndman, at Monash University:

"ANZecmet is a mailing list intended for econometricians in Australia and New Zealand, but may be of interest to a wider audience. It provides a forum for exchanging views, posting technical questions and responses, job advertisements, conference announcements, new publications, and so on.

The ANZecmet mailing list used to be hosted by Monash University, but is now in the process of moving to a Google group. If anyone on this list would like to join, please head over to

http://groups.google.com/group/anzecmet/about and sign up."

Econometrics has a long-standing and very strong presence in Australia and New Zealand. I've signed up, and I hope some of you will too.

"Asymptotic" Properties of Estimators and Tests

2012-02-14T11:10:00.000-08:00

We're so familiar with "large-sample" asymptotics as a way of characterizing the behaviour of our estimators and tests in econometrics, that we tend to forget that there are other, very interesting ways of evaluating their behaviour, and approximating small-sample behaviour.

I touched on this in an earlier earlier post when I discussed "small-sigma" (or "small error") asymptotics. However, that's by no means the end of the story.

There are at least two other types of "asymptotics" that have proven themselves to be very useful in econometric analysis. The first of these dates back at least as far as the work of Kunitomo (1980), Morimune (1983), and others, during the hey-day of work on simultaneous equations models.

It's what is nowadays called "many instruments" asymptotics. In this set-up, the number of instruments and the sample size are allowed to grow at the same rate. (Of course, the number of regressors, and hence the number of parameters to be estimated, is constant). This type of asymptotic behaviour has been shown to be especially useful in the construction of asymptotic covariance matrices, and hence asymptotic standard errors, for a number of problems. For example, see Hansen et al. (2008).

The other type of "asymptotics" is the so-called "small bandwidth" asymptotics, as exemplified by the work of Cattaneo et al. (2010, 2011). In this case, a non-parametric approach is taken, and the asymptotics are based on a sequence of bandwidths, rather than a sequence of sample sizes (as with the usual large-n asymptotics), or a sequence of values for the disturbance variance (as with "small-sigma" asymptotics).

Very recently, Cattaneo (2012) showed that actually there is a very close connection between "small bandwidth" asymptotics and "many instruments" asymptotics.

One of the many things that is interesting about these other types of asymptotic behaviour is that they enable us to construct asymptotically valid covariance matrix estimators in situations where estimators such as Hal White's "heteroskedasticity-consistent" estimator fails.

A great example of this is in the context of a linear panel-data model with fixed effects. If you thought that White's covariance matrix estimator is consistent in this context, then think again. Better yet, take a look at Stock and Watson (2007). It is inconsistent if T (> 2) is fixed, and the number of cross-section units increases without limit. Stock and Watson supply an alternative covariance matrix estimator that is (nT)^1/2consistent.

In short, the "asymptotic behaviour" of our estimators and tests can depend on what we mean by "asymptotics". There's more than one way to think about this type behaviour.

Note: The links to the following references will be helpful only if your computer's IP address gives you access to the electronic versions of the publications in question. That's why a written References section is provided.

References

Cattaneo, M. D., R. K. Crump, and M. Jansson 2010. Robust data-driven inference for density-weighted average derivatives. Journal of the American Statistical Association, 105, 1070-1083.

Cattaneo, M. D., R. K. Crump, and M. Jansson 2011. Small bandwidth asymptotics for density-weighted average derivatives. Forthcoming in Econometric Theory.

Cattaneo, M. D., M. Jansson and W. K. Newey, 2012. Alternative asymptotics and the partially linear model with many regressors. CREATES Research Paper 2012-02, Department of Economics and Business, Aarhus University.

Hansen, C., J. Hausman, and W. K. Newey, 2008. Estimation with many instrumental variables. Journal of

Business and Economic Statistics, 26, 398-422.

Kunitomo, N., 1980. Asymptotic expansions of the distributions of estimators in a linear functional relationship and simultaneous equations. Journal of the American Statistical Association, 75, 693-700.

Morimune, K., 1983. Approximate distributions of k-class estimators when the degree of overidentifiability is large compared with the sample size. Econometrica, 51, 821-841.

Stock, J. H., and M. W. Watson, 2007. Heteroskedasticity-robust standard errors for fixed effects panel data regression. Econometrica, 76, 155-174.

More on Shortest Confidence Intervals

2012-02-12T17:40:00.000-08:00

I was (pleasantly) surprised by the number of "hits" my recent post on "Minimizing the Length of a Confidence Interval" attracted. As has often been the case, a lot of visitors came by way of Mark Thoma's excellent blog, Economist's View. (Thanks, Mark!)

In that post one of the things I discussed was the issue of constructing a "shortest length" confidence interval in the case where the distribution of the pivotal statistic that's used to start off the interval is asymmetric. In such cases, we have a more difficult task on our hands than when the distribution is symmetric, and uni-modal. In response to this, we usually construct "equal tails" confidence intervals in the asymmetric case.

I'm not going to repeat the previous post! Instead, I'm going to share a few lines of R code that I've put together to deal with this issue in the case of an asymmetric distribution that's of great practical importance to econometricians.

Now, in that earlier post mentioned what we need to do in order to construct a "shortest length" confidence interval using a pivotal statistic whose sampling distribution is asymmetric. Let's use one that follows a Chi-square distribution, for illustrative purposes.

We generally construct an "equal tails" confidence interval. That is, if the confidence level is to be 100(1 - α)%, we choose the "cut-off" quantiles for the distribution so as to get 100(α/2)% in each of the left and right ails of the distribution. Given the asymmetry of the Chi-square density, the "equal tails" interval is not the shortest one that we could possibly construct, while retaining the same coverage probability of 100(1 - α)%. It works very well, of course for large degrees of freedom, as the asymmetry of the Chi-square distribution decreases with an increase in the degrees of freedom.

I explained in the earlier post how to get the shortest possible interval in cases like this:

"We choose the upper and lower quantiles from the (uni-modal) distribution so as to ensure that the height to the density function is the same in each case, while still ensuring that our choice gives us the desired confidence level. As long as the chosen quantiles "straddle" the median of the distribution, we'll have the shortest confidence interval.

In practice, this is going to take quite a lot of effort! The mathematical problem that we face is one of solving two complicated equations for two unknowns - the latter being the two quantiles. One of the equations says that the value of the density function has to be the same when evaluated at the two unknowns. The other equation says that the sum of two areas under the density (two integrals) has to equal one minus the desired confidence level."

Incidentally, the median of the Chi-square distribution with (say) v degrees of freedom is approximately the value, v [1 - 2/(9v)]³. For any v, we can compute the median exactly, of course, as the quantile that cuts off 50% of the area under the density to the left (and hence to the right, too).

So, using the above information, here's a picture that represents our problem, and its solution, for a particular case:

For v = 10, the lower and upper cut-off points (c_L and c_U) that we should use when constructing a 90% confidence interval are 3.107327 and 16.710795 respectively. At each of these quantiles, the height of the density is the same, and equal to 0.02387. For the record, the median of this particular Chi-square distribution is 9.341818, and this value is "straddled" by the two cut-off points of 3.1 and 16.7.

These calculations were done with just a few lines of R code that you can find on the Code page associated with this blog. You can use that code to perform the same calculations for any desired confidence level, and any degrees of freedom for the Chi-square distribution. (If you're not an R user, you can still open the R file with any test editor, to take a look.)

By way of comparison, for v = 10, the "equal tails" cut-off points for a 90% coverage probability are 3.940299 and 18.30704. These give 5% in each of the left and right tails of the distribution.

In addition, you can see a table of illustrative cut-off points for the Chi-square distribution for constructing "shortest length" confidence intervals on this blog's Data page.

Now, let's look at a couple of examples where this information can be used in the construction of a "shortest-length" confidence interval.

Students usually encounter the Chi-square distribution for the first time when they learn how to construct a confidence interval for the variance of a Normal population, under simple random sampling. You'll recall that the pivotal statistic that we use in this case is χ² = (n - 1)s²/σ². This statistic has a sampling distribution that is Chi-square with (n - 1) degrees of freedom.

I'm actually going to look at the related problem of constructing a "shortest length" confidence interval for the "precision" of the population - that is, the parameter τ = (1 / σ²).

To construct a traditional "equal tails" 2-sided confidence interval for τ, with a confidence level (or "coverage probability" of 100(1 - α)% we'd form the interval [c_L/ ((n - 1)s²) , c_U / ((n - 1)s²]. Here, c_L is the quantile that "cuts off" 100(α/2)% of the area under the Chi-square density in the left tail; and c_U is the quantile that cuts off 100(α/2)% of the area in the right tail.

(If this interval has limits that appear to be "inverted" relative to what you're used to, it's because the interval is for τ, not for σ² itself.)

So, using the picture we had earlier, a 90% "shortest length" confidence interval for τ, when v = 20 (say), would be the interval [9.78589 / ((n - 1)s²) , 29.87586 / ((n - 1)s²)]. The length of this interval is approximately 20.09 / [(n - 1)s²]. In contrast, if we constructed the "equal tails" interval in this case we'd use "cut-off" values of 10.85081 and 31.41043, and the length of this interval would be a little longer - namely, approximately 20.56 / [(n - 1)s² ].

The second example I have is one that relates to the "coefficient of variation" of a Normal population. That is, C = [σ / | μ |]. We'd usually estimate this parameter by using the sample coefficient of determination, c = [s / | m |], where m is the sample mean, and s is the sample standard deviation.

You'll no doubt recall that the coefficient of variation is unit-less and adjusted for location, so it provides a useful measure when we want to compare variability across different populations. Sometimes, we use C² and c², instead of C and c. either way, the coefficient of variation finds application in economics - for example to compare income inequality across countries, as in Sala-i-Martin (2002), and elsewhere.

I'm again going to focus on "precision", rather than "variablilty", so I'm going to work with C ^-2, rather than C² itself

Suppose that we want to construct a confidence interval for C ^-2. To do this we need to construct a pivotal statistic, based on c^-2, and (most importantly) we need to know the sampling distribution of that statistic. There's a long and very interesting literature dealing with this sampling distribution. More recent contributions include those of Bai (2009).

The exact sampling distributions of c and c^-2 are very complicated (see Hendricks and Robey, 1936), even if we are sampling from a Normal population. However, McKay (1932) established that a Chi-square approximation can be used in certain circumstances. Specifically, if the population is Normal, and if C is less than (about) one-third, then the statistic, [nc²(C ^-2 + 1) / (1 + c²)], is essentially Chi-square distributed with (n - 1) degrees of freedom. The quality of this approximation was verified numerically by Fieller (1932) and Pearson (1932).

Yes - numerically, in 1932! That would have been lots of fun!

So, in this case you can see right away that a 100(1 - α)% confidence interval for C ^-2 is of the form [c_L {(1 + c²) / nc²} - 1 , c_U {(1 + c²) / nc²} - 1]. Again, we can choose c_L and c_U so as to construct an "equal tails" interval; or we can choose these cut-offs so as to construct a "shortest tails" interval. Let's consider a 95% confidence interval this time. If n = 16, so that the degrees of freedom are (n - 1) = 15, then the cut-off points used for constructing the "shortest length" interval will be 5.31713 and 25.90030, whereas those for the "equal tails" interval will be 6.26214 and 27.48839.

In this case, the length of the two confidence intervals for C ^-2 are 20.58317[(1 + c²) / nc²], and 21.22625[(1 + c²) / nc²], respectively.

With a little thought, you'll be able to guess why I worked with confidence intervals for "precision", rather than "variability", in these two examples. Similar results apply for confidence interval based on pivotal statistics with other asymmetric sampling distributions that econometricians also use a lot, such as the F distribution.

Some more on this in a later post, perhaps.

References

Bao, Y., 2009. Finite-sample moments of the coefficient of variation. Econometric Theory, 25, 291-297.

Fieller, E. C., 1932. A numerical test of the adequacy of A. T. McKay's approximation. Journal of the Royal Statistical Society, 95, 699-702.

Hendricks, W. A. and K. W. Robey, 1936. The sampling distribution of the coefficient of variation. Annals of Mathematical Statistics, 7, 129–132.

McKay, A. T. , 1932. Distribution of the coefficient of variation and the extended "t" distribution. Journal of the Royal Statistical Society, 95, 695-698.

Pearson, E. S., 1932. Comparison of A. T. McKay's approximation with experimental sampling results. Journal of the Royal Statistical Society, 95, 703-704

Sala-i-Martin, X., 2002. The disturbing "rise" of global income inequality. Working Paper 8904, NBER.

More on the Equivalence of GLS & Other Estimators

2012-02-08T08:55:00.000-08:00

In a very recent article, titled "Conditions for the Equality of the OLS, GLS and Amemiya-Cragg Estimators" (currently "in press" at Economics Letters), Cuicui Lu and Peter Schmidt present various conditions under which various regression estimators will be numerically equivalent.

Their results are a nice generalization of the Rao-Kruskal-Zyskind conditions that I discussed in an earlier post, "When is the OLS Estimator BLU?"

Here's the abstract of the Lu-Schmidt paper:

"This paper extends results on the equality of OLS and GLS. We give conditions under which GLS based on two different variance matrices gives the same estimate, and also conditions under which GLS equals a GMM estimator."

The paper is definitely worth reading, and a "pre-print" version can be downloaded from Cuicui Lu's website, here.

On the Asymptotic Properties of Sample Means

2012-02-07T15:35:00.000-08:00

Last month, in a post titled "Extracting the Correct Mean(ing) From the Data" (here), I discussed some aspects of the arithmetic, geometric, and harmonic sample means.

In a subsequent comment, I was asked if the geometric mean (GM) and harmonic mean (HM) are consistent estimators of E[X], the (arithmetic) mean of the population. My first reaction was that they are, but a little further reflection shows otherwise.

We know, from the weak law of large numbers (Khintchine's Theorem) that the AM is a weakly consistent estimator of E[X], provided that the sample values are uncorrelated and E[X] is itself finite. If we strengthen the "uncorrelated" requirement to "independent", then the AM converges almost surely to (is strongly consistent for E[X], by the strong law of large numbers. These results hold for any parent population whose mean is finite.

Now what about the GM and the HM?

The quickest way to show that they are not necessarily consistent for E[X] is to conduct a simulation experiment, and generate a counter-example. Remember that the GM is not defined for negative sample values, so let's make sure that we take this into account when choosing the population from which we sample.

I set up a simple Monte Carlo experiment, using EViews. The EViews workfile and program file can be found in the Code page that goes with this blog.

In the experiment, I used a parent population that was Chi-Square distributed with v = 5 degrees of freedom, so E[X] = 5. Simple random sampling was used, with 5,000 Monte Carlo replications, and with sample sizes of n = 50; 500; and 2,000. In each case, the simulated sampling distributions for GM and HM were constructed. By the time that we have n = 2,000 we should be getting close to the (large-n) asymptotic case.

There is a "READ ME" text-object in the EViews workfile that provides more details, but here are the simulated sampling distributions for the AM, GM, and HM when n = 2,000:

As expected, the mean of the sampling distribution for the AM is 5. The AM is unbiased, consistent, and asymptotically unbiased for E[X]. As far as the GM and HM are concerned, we have:

We see that these two sample statistics are each asymptotically biased (and hence inconsistent) estimators of E[X]. This is just for one situation, but all we needed was one counter-example!

Interestingly, at least for this example, the asymptotic means of the HM (= 3), GM (= 4) and AM (= 5) happen to satisfy the usual inequality for the sample averages themselves: HM < GM < AM.

Influential People in the "Big Data" Field

2012-02-04T18:32:00.000-08:00

Yesterday, Haydn Shaughnnessy wrote a piece for Forbes titled, Who are the Top 20 Influencers in Big Data?

Fans of R will be delighted to see David Smith of Revolution Analytics up there at number 2!

Congratulations!

Minimizing the Length of a Confidence Interval

2012-02-04T09:50:00.000-08:00

Right now I'm teaching an introductory course on statistical inference for Economics students. We've been dealing with confidence intervals, starting off (as usual) with one for the mean of a Normal population.

For a given confidence level, the shorter the interval is, the more "informative" it is. The question that then arises is how to make the interval as short as possible, everything else being equal? Good question!

First, let's review some basic results. Say I want to construct a 95% confidence interval for the mean of a Normal population. If the population variance, σ² is known, then this interval will be constructed by using the properties of a standard Normal random variable. If the variance is unknown, then the interval will be constructed using the properties of a Student-t random variable.

Traditionally, in each case we center the (random) interval at the sample mean, and get the lower and upper limits by adding and subtracting an equal amount. When the population variance is known, this amount depends on the standard deviation of the population, the sample size, and the quantile for the Standard Normal distribution that assigns 2.5% to the right tail of the density. That quantile is roughly 1.96.

When the population variance is unknown, this amount depends on the estimated standard deviation of the population, the sample size, and the quantile for Student's t distribution that assigns 2.5% to the right tail of the density. That quantile depends on the degrees of freedom, (n - 1). For instance if n = 10, then this quantile is 2.262; while if n = 15, this quantile is 2.145

Now, why do we build the confidence interval so that it is symmetric about the sample mean, in each of these cases? That is, why do construct "equal tails" confidence intervals?

After all, for the example where the population variance is known, and I'm using quantiles from the Standard Normal distribution, I don't have to use -1.96σn^-1/2 and +1.96σn^-1/2 to get the lower and upper limits of the 95% confidence. You can easily check that I could subtract -1.6693σn^-1/2 from the sample mean, and add 2.81σn^-1/2, and I'd still get a confidence level (coverage probability) of 95%.

Indeed, given the continuity of the Normal density, there's actually an infinity of different asymmetric intervals that I could construct, each of which would be 95% confidence intervals! For any fixed degrees of freedom, the same point applies when we're using quantiles from Student's t distribution in case where the population variance is unknown.

So, the question remains - why do we traditionally construct "equal tails" intervals here?

Well, it's not just because of tradition! For the problem we're talking about, it can be shown that the "equal tails" confidence interval is also the shortest (and hence, most informative) interval. That's for a chosen confidence level, a fixed value of n, and a particular (actual or estimated) population standard deviation.

To illustrate (but certainly not prove) this point, take the example above based on the quantiles of the Normal distribution. The symmetric confidence interval has length 3.92σn^-1/2, while the asymmetric interval I suggested has length 4.4793σn^-1/2.

One thing that you'll have noticed about the particular interval estimation problem that I've been discussing so far is that the Standard Normal and Student's t distributions have density functions that are both uni-modal and symmetric about their mode. Moreover, this mode is zero.

That's actually the key feature of the problem. If the density of the (pivotal) statistic that's being used as the starting point for the interval is uni-modal at zero, and symmetric about this point, then the "equal tails" confidence interval will be the shortest possible interval. In our problem, the statistic in question is either the standardized sample mean, which is pivotal whether we use the true or estimated population standard deviation. ("Pivotal" means that the statistic's distribution doesn't depend on the unknown parameters.)

You'll find this discussed, for example, by Casella and Berger (2002), and taken up in detail by Ferentinos and Karakostas (2006).

Another sufficient condition for the "equal tails" and shortest confidence intervals to coincide is given by Kirmani (1990). If the distribution of the (pivotal) statistic is symmetric, and is concave on the right side of the point of symmetry, then the "equal tails" confidence interval will again be shortest.

Interesting! But the conditions given above are only sufficient conditions. Are they also necessary?

In other words, if the shortest confidence interval for a particular problem is also the "equal tails" confidence interval, does this mean that the distribution of the pivotal statistic has to be symmetric and uni-modal?

Surprisingly, this remains an open question! Ferentinos and Karakostas (2006) note that no formal proof is available, though there are informal reasons to believe that the conditions may also be necessary.

Some other interesting problems can arise, even when the distribution of the pivotal statistic is symmetric (but not uni-modal). For instance, the"shortest" confidence interval may not even exist! An example is given by Kirmani (1990).

Of course, once we move to situations where the distribution of the pivotal statistic is asymmetric, we open up a different can of worms.

The obvious example is when we want to construct a confidence interval for the variance of a Normal population. In this case, the pivotal statistic that we use to build the interval is (n - 1)s²/σ². This statistic has a χ² distribution, with (n - 1) degrees of freedom, and this distribution is asymmetric. It's skewed to the right.

In cases such as this, it's still traditional to construct an "equal tails" confidence interval. Now, of course, if we want a confidence level of (say) 95%, we determine the lower limit of the interval by using the quantile of the χ² distribution that gives a 2.5% left-tail area for the density; and we determine the upper limit by using the quantile that gives a 2.5% right-tail area for the density.

For example, for 10 degrees of freedom, these quantiles are 3.247 and 20.483 respectively. As a result, the confidence interval that we construct will be asymmetric about s² (the usual unbiased estimator of the population variance).

How do we construct a "shortest" confidence interval in situations like this?

Well, as long as the distribution of the pivotal statistic is uni-modal (as the χ² distribution is), there is a result from Casella and Berger (2002) and Ferentinos and Karakostas (2007) that will help us. We choose the upper and lower quantiles from the distribution so as to ensure that the height to the density function is the same in each case, while still ensuring that our choice gives us the desired confidence level. As long as the chosen quantiles "straddle" the median of the distribution, we'll have the shortest confidence interval.

In practice, this is going to take quite a lot of effort! The mathematical problem that we face is one of solving two complicated equations for two unknowns - the latter being the two quantiles. One of the equations says that the value of the density function has to be the same when evaluated at the two unknowns. The other equation says that the sum of two areas under the density (two integrals) has to equal one minus the desired confidence level. When you consider the expression for the density function of the χ² distribution with v degrees of freedom:

p(x) = [2^(v/2)Γ(v/2)]^-1x^(v/2)-1exp(-x/2) ; x > 0 ,

you can see that we have an interesting problem.

That's why the "equal tails" confidence interval remains popular in cases such as this.

Note: The links to the following references will be helpful only if your computer's IP address gives you access to the electronic versions of the publications in question. That's why a written References section is provided.

References

Casella, G. and R. Berger, 2001 Statistical Inference, 2nd. ed., Duxbury.

Ferentinos, K. K. and K. X. Karakostas, 2006. More on shortest and equal tails confidence intervals. Communications in Statistics - Theory and Methods, 35, 821-829.

Kirmani, S., 1990. On minimum length confidence intervals. International Journal of Mathematical Education in Science and Technology, 21, 791-793.

Take Comfort From This

2012-01-29T10:32:00.000-08:00

If we knew what it was we were doing, it would not be called research, would it?

– Albert Einstein

Asking for What you Don't Really Want

2012-01-27T12:27:00.000-08:00

Sometimes, asking for something you don't really want can be an indirect way of getting something you actually do want or need. A million dollars? Not quite what I had in mind, actually. Nice thought, though!

Actually, what I had in mind was something a little more mundane. In particular, getting an econometric package to save you a lot of work by delivering up some information you need, when it's not at all apparent that the package is able to do so. It's a matter of asking it for something else, and getting what you really want as a by-product. A bonus, if you will!

The situation that I was thinking of is as follows.We've estimated a model, say by Maximum Likelihood estimation, and there is some nonlinear function of the parameters that we want an estimate for. Moreover, we want an interval estimate, so we need a standard error for the estimate of the nonlinear function.

More specifically, suppose we've estimated a CES production function, of the following form:

Q_i = γ [δ K_i^-ρ + (1 - δ)L_i^-ρ]^-ν/ρ exp(ε) ; i = 1, 2, ....., n ;

where is ε is a well-behaved, Normally distributed, error term, and γ, δ, ν, and ρ are unknown parameters, such that γ > 0; 0 < δ < 1; ν > 0; and ρ ≥ -1. The parameter γ is the “efficiency parameter”, δ is the “distribution parameter”, ν is the “returns to scale parameter”, and ρ is the “substitution parameter”.

It's easily shown that the elasticity of substitution between capital and labour in this model is given by the parameter, s = 1 / (1+r). This is a nonlinear function of ρ that I may want an interval estimate for.

Now, many econometrics packages make it easy construct such an interval estimate. Unfortunately, EViews doesn't appear to be one of these.

By the invariance of maximum likelihood, the MLE for σ is obtained immediately as 1 / (1 + r), where r is the MLE for ρ. But this is just a point estimate. To get an interval estimate what's needed, of course, is the complete estimated (asymptotic) covariance matrix for the estimated coefficients. Then, the Delta method can be used to get an asymptotic standard error for the estimate of the function of interest. The asymptotic normality of MLE's then enables us to construct an asymptotically valid confidence interval for σ.

Getting the asymptotic standard error by the Delta method would involve a bit of work in this example, but fear not! This is where we ask EViews for something we don't really want.

Once we've estimated the model, we pretend that we want to test some hypothesis involving the nonlinear function of interest. While we could easily, and legitimately test the hypothesis that ρ = 0 by using the asymptotic standard error for r, and a z-test, instead let's ask EViews to test the hypothesis that [1 / (1 + ρ)] = 1. As well as computing the associated Wald test statistic (which we're not really interested in), EViews will also report the MLE of the function [1 / ( 1 + ρ)], namely [1 / (1 + r)], as well as the asymptotic standard error of the latter, computed using the Delta method!

Let's illustrate this using some actual data. These data are available on the Data page that accompanies this blog, and the EViews file is available on the associated Code page.

I've taken the logarithm of both sides of the CES production function in (1), and the parameters I'm going to estimate are log(γ), ν, ρ, δ, and the standard deviation of the Normally distributed errors (coefficients C(1) to C(5) in the EViews code). I've set up a "LOGL" object in my EViews workfile, and used the following starting values for the parameters (in the order above): 1.0, 0.8, 0.6, 0.4 and 1.0 respectively.

Here are the MLE results:

(You get the same results if you estimate the model by nonlinear least squares, of course, given the assumed normality of the errors.)

Now, if we select the "VIEW" tab, and then select "Wald coefficient tests...", we can formulate the restriction (nonlinear function) of interest to us:

And this is what we get:

Indirectly, this gives us the information that we wanted all along. The (asymptotic) standard error of the estimated normalized restriction is 0.116633. The fact that we've subtracted unity (or any other constant)value from the function we're interested in doesn't alter the standard error, so 0.116633 is the asymptotic standard error for [1 /(1 + r)] itself. Then, an asymptotically valid 95% confidence interval for the elasticity of substitution can be constructed as [1 /(1 + r)] +/- 1.96(0.11633), where [1 /(1 + r)] = 0.6623.

So the 95% confidence interval for the elasticity of substitution is [0.3943 ; 0.8503].

Notice that when we set up the restriction in order to "trick EViews", although we used [1 / (1 + ρ)] = 1, we could equally have used [1 / (1 + ρ)] = k, for any value of k. This would not have altered the standard error. (Try it!)

By the way, I know that when testing nonlinear restrictions, the value of the Wald test statistic not invariant to the way in which the restrictions are expressed. However, that doesn't affect anything here - we're not interested in the Wald test statistic per se - just in the asymptotic standard error associated with a particular nonlinear function.

So, sometimes asking for what we don't want will result in us getting something that we're actually hoping for. Of course, I wouldn't recommend doing this just before your next birthday!

Hot Topics in Econometrics

2012-01-26T20:26:00.000-08:00

Last week, Takamitsu Kurita asked me "What do you think will be the big developments in Econometrics over the next decade". We were having a drink following his seminar, and I really didn't have a good answer. I think those of us present ducked the question by saying that, as econometricians, we know only too well the pitfalls associated with forecasting! But Taka's question was a good one, and it certainly deserved a better response than I had at the time.

I don't think I'm at all qualified to provide a decent answer, but the question got me thinking. And we all know where that leads to!

So, here are some thoughts about some topics that are hot, even if they don't remain that way for a decade:

Modelling extremely large data-sets.
Modelling ultra-high frequency financial data.
Information theory and entropy econometrics.
The econometrics of networks.
Treatment effect models.
An increasing acceptance of Bayesian methods in econometrics.
Nonlinear time-series analysis.

Perhaps you'd like to add to this very short list?

Cointegration Analysis With I(2) & I(1) Data

2012-01-22T22:09:00.000-08:00

Last Friday I went to a great seminar given by Takamitsu Kurita (Fukuoka University, Japan). Taka is currently a visiting scholar in our department, and his paper (here) dealt with an interesting application of cointegration analysis when we have both I(2) and I(1) data to contend with.

This is a topic in time-series econometrics that's of great practical importance, and (quite rightly) is currently attracting quite a bit of attention.

One of the interesting features of the analysis that Taka presented was that it clearly reflected the difference between the broad approaches to cointegration analysis, in general, that tend to be used by European econometricians on the one hand, and their North American counterparts on the other. This is a point that was raised by Georg in a comment on an earlier post in this blog.

A lot of us approach cointegration analysis in a two-step manner - and I'm not referring to the Engle-Granger two-step cointegration test. What I mean is this. We first test each of the time-series for their order of integration, perhaps using the Augmented Dickey-Fuller test, and/or the KPSS test. Then, if we find that the series are integrated of the same order (greater than zero) we proceed at the second stage to test for the number of cointegrating relationships among them. Here, we'd generally use Johansen's framework.

On the other hand, a lot of our European colleagues avoid this two-step approach, and the "preliminary test" distortions that can be associated with it. Instead, they move directly to Johansen's set-up, and out of this they can infer the order of integration of the time-series, as well as the number of cointegrating vectors.

Although a good deal of progress has been made when it comes to modelling with a mixture of both I(2) and I(1) data, there are still lots of important questions waiting to be addressed. This, of course, is good news for young econometricians looking for an interesting field of research!

Some of the seminal contributions to date include those of Johansen (1992, 1995, 1997, 2006), Paruolo (1996), Haldrup (1998), Paruolo and Rahbek (1999), and Rahbek et al. (1999), among others.

And what of those issues yet to be addressed? I believe they include:

An extension of the I(2)/I(1) cointegration analysis to the case where there are unit roots at the seasonal frequencies. Obviously, this is an important issue in the case of quarterly or monthly data.
A treatment of the (commonly occurring) situation where the I(2) and I(1) data exhibit structural breaks; thereby extending the analysis discussed in this earlier post.

I think that we're going to see a lot more research on the modelling of (possibly cointegrated) I(2) and I(1) economic time-series data.

Note: The links to the following references will be helpful only if your computer's IP address gives you access to the electronic versions of the publications in question. That's why a written References section is provided.

References

Haldrup, N., 1998. An econometric analysis of I(2) variables. Journal of Economic Surveys, 12, 595-650.

Johansen, S., 1992. A representation of vector autoregressive processes integrated of order 2. Econometric Theory, 8, 188–202.

Johansen, S., 1995. A statistical analysis of cointegration for I(2) variables. Econometric Theory, 11, 25–59.

Johansen, S., 1997. Likelihood analysis of the I(2) model. Scandinavian Journal of Statistics, 24, 433–62.

Johansen, S., 2006. Statistical analysis of hypotheses on the cointegrating relations in the I(2) model. Journal of Econometrics, 132, 81–115.

Kurita, T., 2012. Modelling time series data of monetary aggregates using I(2) and I(1) cointegration analysis. Bulletin of Economic Research, in press.

Pauolo, P., 1996. On the determination of integration indices in I(2) systems. Journal of Econometrics, 72, 313–56.

Paruolo, P. and A. Rahbek, 1999. Weak exogeneity in I(2) VAR systems. Journal of Econometrics, 93, 281–308.

Rahbek, A., H. C. Kongsted, and C. Jørgensen, 1999. Trend stationarity in the I(2) cointegration model. Journal of Econometrics, 90, 265–289.

The Dynamic Stability of AR Models - Tricking EViews

2012-01-21T09:32:00.000-08:00

In this post I'm going to focus on understanding the extent to which there's an equivalence between two different ways of estimating an AR(p) model for a time-series, Y_t, using EViews, and to see what information is generated in each case.

In particular, I want to show you how you can "trick" EViews into showing you if your estimated dynamic regression model is "dynamically stable". That is, if the estimated coefficients for the lagged values of Y are such that the model is stationary. If the lag-order is above 2, this isn't something that's always easy to do by just looking at the estimated coefficient values.

By way of example, consider the case of an AR(2) model, with an intercept (or “drift”) term included to allow for a non-zero mean in the series. The results apply equally to the general AR(p) case, as will become apparent in the empirical example that's given below.

AR(2) Model:

Y_t = α + φ₁Y_t-1 + φ₂Y_t-2 + ε_t , (1)

where ε_t is a “white noise” series.

In the case of EViews, we could estimate the model using OLS with the specification:

Y C Y(-1) Y(-2)

Consider an alternative model for Y:

Y_t = α + u_t ; u_t = φ₁u_t-1 + ε_t . (2)

That is, we just explain Y in term of a level and a error term that is itself AR(2). From (2), notice that:

φ₁Y_t-1 = φ₁ α + φ₁u_t-1 (3)

and

φ₂Y_t-2 = φ₂ α + φ₂u_t-2. (4)

So, subtracting (3) and (4) from (2), we have:

Y_t - φ₁Y_t-1 - φ₂Y_t-2 = α (1- φ₁ - φ₂) + (u_t - φ₁u_t-1 - φ₂u_t-2), (5)

or, from the definition of in (2):

Y_t = α (1 - φ₁ - φ₂) + φ₁Y_t-1 + φ₂Y_t-2 + ε_t. (6)

We see that (1) and (6) (and hence (1) and (2)) are the same, except for the intercept term. To estimate model (2) in EViews we'd use the specification:

Y C AR(1) AR(2)

We see that if we do this then the estimates of phi1 and phi2 will be same as if used the EViews specification for (1), above. Of course, we could also recover the estimate of α in this second case by dividing the estimated intercept coefficient by the estimate of (1 - φ₁ - φ₂).

What would be the point of using this second specification? We should certainly use it only with care, given that the estimated intercept coefficient has to re-interpreted. However, if we set up this specification with the AR(.) terms in EViews, the package provides us with an extra item in the regression output that we can VIEW – an item that's not available otherwise.

And that's what we're looking for!

Specifically, if there are (any) AR terms in the model specification (even not all successive ones, as in AR(1), AR(4)), then we can VIEW the ARMA STRUCTURE. This then provides information about the dynamic stability of the estimated AR model, in terms of whether or not the inverted roots of the characteristic polynomial lie within the unit circle. In addition, we have the ability to see how the estimated model responds, dynamically, to various types of “shocks” to the data.

The mathematical details relating to the derivation of the stationarity (dynamic stability) conditions for the AR(2) can be found here.

One important thing to notice. We're really interested in a model that is autoregressive in the variable, Y. The second version of the model appears to lack this feature, but have an autoregressive error term. The point is, though, that this second version can be re-written as the first version, except for the intercept - and this intercept doesn't affect the dynamic stability.

An Example:

Consider the following simple autoregressive model for quarterly Japanese consumption data. These data are available in the Data page that goes with this blog; and the EViews file that I have used for the results below is available on the Code page. The model is not intended to be particularly sophisticated - I just want to illustrate simple point here.

The following transformations have been applied to the data:

W_t = log(CONS_t) ; to linearize the trend

Y_t = (W_t - W_t-4) ; to remove the seasonality.

The second transformation also has the effect of removing most of the trend, as we can see:

Now let’s model the {Y_t} series using an AR(4) model, along the lines of equation (1):

Then, consider the alternative approach, along the lines of equation (2):

Note that, apart from the intercept coefficient, the various estimated coefficients and their standard errors are the same in each output. Also, note the additional information that is provided about the (complex) inverse roots of the characteristic equation at the bottom of this output. Let’s explore this information in more detail. This is what we see if we select VIEW / ARMA Structure:

The estimated AR(4) process is stationary – and that's something that may not be altogether obvious just by looking at the estimated coefficients in the first OLS output.

Returning to the ARMA Structure option, we might also choose the impulse response function. In this case we are going to “shock” the Y_t series by an amount equal to one sample standard deviation, and then see how the predicted series “settle down” over a 24-quarter (6-year) period:

Because the estimated specification is stationary, things settle down to their original state after about 5 years.

So, sometimes with a bit of thought we can "trick" out favourite econometrics package into providing us with information that it might otherwise be reluctant to yield up. This can save us a lot of work!

New Page on Blog

2012-01-18T12:50:00.000-08:00

I've added a new page to the blog site - Econometrics Jobs.

On this page you'll find a small selection of advertisements for interesting jobs in Econometrics, around the world. These have been chosen to provide information about the wide-ranging opportunities for Econometricians.

The list will change frequently, as new jobs are posted, and others expire.

Are Those Conditions Necessary, or Just Sufficient?

2012-01-17T14:09:00.000-08:00

We all know the difference between conditions that are necessary, and ones that are sufficient, for some result to hold. However, it's not uncommon for us to lose track of which is which when it comes to certain econometric results. I'm going to focus on just one example of this, and in doing so I'll try and clear up a common misconception.

Specifically, we're going to take a look at the OLS estimator of the coefficient vector in a standard linear regression model, and focus on the conditions that are usually mentioned in the context of this estimator being (weakly) consistent.

Here's the model we'll be dealing with:

y = Xβ + ε ; ε ~ [0 , σ²I] .

The model has k regressors, and we have a sample of size n.

I'll assume that rank(X) = k, so the OLS estimator of is defined, and is

b = (X 'X)^-1X 'y.

I'm going to assume either that the columns of the X matrix are non-random; or, if they are random then any correlation that exists between these random regressors and the errors of the model disappears if n is large enough.

That's to say, if the regressors are non-random, then as n → ∞, [X 'X / n] tends to a finite and non-singular matrix (say) Q. Note that this matrix will be unobservable, as we can't ever see what happens if n is infinitely large. So, what we're assuming is that the full rank assumption about X (and hence X 'X) continues to hold if the sample size is allowed to grow without limit.

If, on the other hand, the regressors are random, I'm going to assume that plim[X 'X / n] = Q. As before, Q is assumed to be a finite, non-singular, but unobservable matrix.

Now, if we think about the dimensions of the (X 'X) matrix, you might be a bit puzzled by this. After all, the (X 'X) matrix is (k x k). These dimensions don't change as n increases. However, if we focus on the X matrix itself, we see that as n grows, so does the number of rows in this matrix. So, the elements that are involved in the calculation of the (X 'X) matrix are indeed changing as n increases, and so this matrix changes in terms of the values of its elements.

Now I'm going to make a second, crucial, assumption. I'm going to assume, again, that either the regressors are non-random - in which case they can't possibly be correlated with the errors in the regression model (no matter what the sample size is) - or else, if the regressors are random, then they eventually become uncorrelated with the errors if the sample size grows without limit.

That is, I'm going to assume that plim[X 'ε / n] = 0 ( a null vector).

Let's now take a look at our OLS estimator, which we can write as:

b = [X 'X / n]^-1[X 'y / n] = [X 'X / n]^-1[X '(Xβ + ε) / n] ;

or,

b = β + [X 'X / n]^-1[X 'ε / n].

Then, under the assumptions that I've made, as the sample size grows without limit,

plim(b) = β + (Q^-1)(0) = β ,

as Q^-1 is finite, by assumption. That is, the OLS estimator of the coefficient vector is weakly consistent.

Incidentally, notice that if the regressors are non-random, then plim[X 'X / n] = limit[X 'X / n] = Q, so this result still holds.

Now, let's think about a simple, but rather special, regression model - one that has an intercept and just one other regressor - a linear time-trend. That is:

y_t = β₁ + β₂ t + ε_t ; ε_t ~ i.i.d. [0, σ²] ; t = 1, 2, ...., n.

For this model, the (1 , 1) and (2 , 2) elements of [X 'X] are n and [n (n + 1)(2n + 1) / 6], respectively; and the (1 , 2) and (2 , 1) elements are each [n (n + 1) / 2]. If we then consider plim[X 'X / n] (which equals the limit[X 'X / n] in this case), we see that three of the four elements become infinitely large as n → ∞, and so Q is not a finite matrix. One of the assumptions above isn't satisfied.

Because the regressor, t, is non-random, it can't be correlated with the error term, no matter what the sample size is. Given this, and the zero mean for the errors, the OLS estimators of the two coefficients are definitely unbiased. Now, if I can show that the OLS estimators of the coefficients are mean-square consistent then, by Chebyshev's Inequality, they will also be weakly consistent.

Given this estimator's unbiasedness all I have to show, then, is that the covariance matrix of b = (b₁ , b₂)' tends to a null matrix, as n tends to infinity. This covariance matrix has the form σ²[X 'X]^-1, as usual.

The determinant of [X ' X] is [n²(n + 1)(n -1) / 12], and so the elements of the [X 'X]^-1 matrix are as follows:

(1 , 1) : [2 (2n + 1)] / [n (n -1)]
(2 , 2) : 12 / [n (n + 1)(n - 1)]
(1 , 2) = (2 , 1) : -6 / [n (n - 1)] .

Clearly, as n → ∞, each of these elements tends to zero, and so the covariance matrix of b becomes a null matrix (as σ²is positive, but finite).

Accordingly, in this time-trend model, the OLS estimator of the coefficient vector is mean-square consistent, and hence it is also weakly consistent (i.e., plim(b) = β ).

So, what is going on here? The assumption that plim[X 'X / n] = Q, a finite and positive definite matrix, isn't satisfied, but the OLS estimator is still weakly consistent. Obviously, what we've shown is that this particular assumption is sufficient, but not necessary, for consistency to hold. You might ask yourself if the other assumption, that plim[X 'ε / n], is necessary or sufficient.

The take-home message:

When looking at the assumptions that are made in establishing results in Econometrics (or elsewhere), be careful to determine which assumptions are sufficient, which ones are necessary, and which ones are both.

Different Types of "Asymptotics"

2012-01-16T16:23:00.000-08:00

When econometricians talk about the "asymptotic" properties of their estimators or tests, they're usually referring to their properties when the sample size becomes infinitely large. However, there are other types of "asymptotics" that are also interesting and important. It's worth being aware of this, and of the way they arise in econometric analysis.

Large-sample asymptotics focus on the "limiting distribution" of a suitably scaled statistic (estimator or test statistic) of interest, as the sample size (n) becomes infinitely large. We're all familiar with the estimator properties of (weak) consistency, asymptotic efficiency, etc. Most of the time, Maximum Likelihood estimators enjoy these properties, for instance. When it comes to testing, we often have to rely on the asymptotic distribution of the test statistic to get critical values, because the finite-sample distribution is intractable.

Of course, we can use the bootstrap to learn about the finite-sample distribution of a statistic.

Large-sample asymptotic results can provide very helpful, but approximate, results in many cases. In others, they can be quite misleading when the sample size is modest. Estimators may be substantially biased and imprecise, and the "size" of our tests can be distorted badly.

I like to think of "weak consistency" as being a particularly well-named estimator property! If an estimator is inconsistent, then it's probably best to avoid it altogether. That's because if an estimator is inconsistent then it will give the wrong answer, with probability one, even when we have entire population of data in front of us, and we actually know the true answer!

Who would want to use such an estimator? Not me!

A second type of "asymptotics" that we use when evaluating estimators and tests is the so-called "Small-Sigma Asymptotics", suggested originally by Kadane (1971). You may not have encountered this, but it's been used to good effect by various econometricians.

The motivation for large-sample asymptotics is that we'd like our inferences to eventually become very good indeed if the sample size grows without limit. Small-sigma asymptotics are motivated by the idea that we'd like these inferences to become extremely good when the variability in the population (and hence in the sample) becomes less and less.

Think of the extreme case. If the population has zero variance, then all of the population items will take the same value, and so will all of the values in any sample. In that case, any "sensible" statistic that we construct from the sample should be able to give us an exact result when we use it to draw inferences about the population. For instance, in this extreme case, the sample mean will equal the population mean exactly, and it will provide a "perfect" estimator of the latter parameter.

So, with "Small-Sigma" asymptotics, we're asking what happens to the sampling distribution of a statistic when the population variance goes to zero. This provides us with an alternative set of asymptotic results - a different type of "consistency" than we're perhaps used to.

Let's see how this works out in the case of the OLS estimator for the usual k-regressor linear regression model:

y = Xβ + ε ; ε ~ [0, σ²I].

We can re-write this equivalently as:

y = Xβ + σv ; v ~ [0, I].

Now, the OLS estimator of β is:

b = (X'X)^-1 X'y = (X'X)^-1X'(Xβ + σv),

and we can write the "estimation error" as:

(b - β) = (X'X)^-1 X'βσv.

Clearly, as σ → 0, b → β.

Not only is the OLS estimator of β (large-n) consistent under certain conditions, but it is also small-σ consistent. Notice that the latter result holds provided that:

The OLS estimator is defined - that is, as long as X has full column rank; and
X does not depend on σ.

Just as random regressors will render the OLS estimator (large-n) inconsistent unless the regressors are uncorrelated with the errors in the limit; so too, this estimator may be (small-σ) inconsistent in the unlikely event that the regressors are random and their variation is a function of σ.

As a second, related, example consider the usual unbiased estimator of σ2 in this model, namely:

                                   s² = (y - Xb)'(y - Xb) / (n - k).

It's easy to prove, using Khintchine's Theorem, that this estimator is (large-n) consistent for σ². So, by Slutsky's Theorem s is (large-n) consistent for σ.

Notice that we can write:

                                 s² = (Xβ + σv - Xb)'(Xβ + σv - Xb) / (n - k),

or,
                                s² = (σv)² / (n - k).

Then, as σ → 0, so does s², and we see that in a rather trivial sense, s²is also (small-σ) consistent for σ². Equally, s itself is (small-σ) consistent for σ.

Small-σ asymptotics have been used to analyze the properties of various econometric estimators and test statistics. Some interesting examples are provided by Kadane (1971), Inder (1986), Ullah et al. (1995), Srivastava and Ullah (1995), and Ullah (2004, pp. 36-45).

Finally, there's an interesting interview with the man who started all of this - "Jay" Kadane - here.

References

Inder, B., 1986. An approximation to the null distribution of the Durbin-Watson statistic in models containing lagged dependent variables. Econometric Theory

Kadane, J. B., 1971. Comparison of k-class estimators when the disturbances are small. Econometrica, 39, 723-737

Kiviet, J. F. and G. D. A. Philips, 1993. Alternative bias approximations in regressions with a lagged dependent variable. Econometric Theory, 9, 62-80.

Srivastava, V. K. and A. Ullah, 1995. Stein-rule estimation in models with a lagged dependent variable. Communications in Statistics - Theory and Methods, 24, 1343-1353.

Ullah, A., 2004. Finite Sample Econometrics, Oxford University Press, Oxford.

Ullah, A., V. K. Srivastava, and N. Roy, 1995. Moments of the function of non-normal random vector with applications to econometric estimators and test statistics. Econometric Reviews, 4, 459-471.

Solving Mathematical Problems - the Tricki

2012-01-15T10:04:00.000-08:00

Hat-tip to Sean Brocklebank (Economics, University of Edinburgh), through the Subgame Equilibrium blog, for pointing us to The Tricki.

This is a Wiki site devoted to discussing and explaining the methods of proof that are used in various areas of mathematics. Probability and Statistics are among the fields covered, although as yet there are no entries for the second of these two particular sections.

In the Elementary Probability section I especially liked the entry on "Bounding Probabilities by Expectations", where there are some examples of using Markov's Inequality to good effect.

In short, there's a wealth of great information for students and teachers of Econometrics alike. I'll certainly be using it, and I'll be looking forward to seeing some entries in the Statistics section.

Gastronometrica

2012-01-11T19:40:00.000-08:00

If you were planning to participate in the Choconomics Conference this September, a useful warm-up might be the 2012 Conference of the Society for Quantification in Gastronomy (Gastronometrica).

Apparently Coimbra & Viseu (Portugal) is the place to be from 30 May to 2 June this year.

Too many temptations.....too little time!

Extracting the Correct Mean(ing) From the Data

2012-01-10T22:15:00.000-08:00

We've all taken, and/or taught, an introductory course in descriptive statistics where we encounter measures of "central tendency", variability, summarizing grouped data, and so on. In such courses students are usually told about three ways of calculating the mean, or average, of a sample. These are the Arithmetic Mean, Geometric Mean, and Harmonic Mean. In my experience, economists often fail to use the most appropriate of these three measures. I think this is because often we don't provide enough motivation and explanation in those introductory courses.

Let's begin by recalling how these three averages are computed:

Arithmetic mean: AM = (1/n) Σ_i(x_i)
Geometric mean: GM = [Π_i(x_i)]^1/n
Harmonic mean: HM = [(1/n) Σ_i(1/x_i)]^-1

They're obviously related to one another - in fact they're the three so-called "Pythagorean means", as studied originally by Pythagoras and his followers. Pythagoras developed them from geometric principles, and chose them because they each exhibit four (desirable) properties. Letting "M" denote any one of the three means above, these properties are:

Value preservation: M[x, x, x, ....x] = x.
First-order homogeneity: M[bx₁, bx₂, ...., bx_n] = b M[x₁, x₂, ...., x_n].
Exchange invariance: M[....., x_i, ..., x_j, .....] = M[...., x_j, ...,x_i,...]; for all i and j.
Averaging: Min{x₁, x₂, ...., x_n} ≤ M[x₁, x₂, ...., x_n] ≤ Max.{x₁, x₂, ...., x_n}.

We can see immediately that the Harmonic Mean is just the reciprocal of the Arithmetic Mean of the reciprocals of the data. (What a mouth-full!) We can also see that the logarithm of the Geometric Mean is the Arithmetic Mean of the logarithms of the data.

Applying these three formulae to the sample of data ${1, 4, 7, 10}, for example, we get the results:

AM = $5.5 ; GM = $4.091 ; and HM = $2.679 .

Notice that these averages are ranked HM < GM < AM, and this is no accident. If the data are non-negative then this ranking must always hold, as is discussed towards the end this post; and the three measures will be equal if and only if every item in the sample takes the same value (They'll all equal this single value, of course, by the Value Preservation property above.) For this particular sample, the Harmonic Mean takes a value that really doesn't seem to be "representative" - at least not in the way that the geometric or arithmetic means are - and here, the latter happens to equal the sample median.

One thing to notice about these three different averages is that they differ in their robustness to "outliers" in the data. Just as we might think of using the sample median (rather than a sample mean) to reduce sensitivity to extreme values in the sample, so too we might think carefully of our choice between AM, GM, and HM.

To see this, suppose that we change the sample above so that now it is ${1, 4, 7, 100}. The sample median is unchanged at $5.5 [ = (4 + 7) / 2], but now we have AM = $28 ; GM = $7.274; and HM = $2.851. In this particular case, the Geometric Mean is more robust to the outlier than is the Arithmetic Mean. The Harmonic Mean is particularly robust (and its value changes only to $2.870 if the last sample value is increased to $1000), but hence still not visibly "representative".

The two samples considered so far involve observations which have simple units of measurement - namely, dollars. In order to get further insight into the appropriate way of constructing a sample average, it's instructive to "look behind the numbers", and ask what they are actually measuring.

Let's suppose that in our sample ${1, 4, 7, 10} the values are the price of the same item in four different locations. The numbers represent levels, and (putting outlier issues to one side) the AM is quite appropriate.

Suppose, however, that we had these four prices together with the prices for a different good, at the same four locations: ${2, 12, 14, 24}. Now consider the relative price of good 2 to good 1, at the four locations. These are {2, 3, 2, 2.4}, and note that these ratios are unit-less. What's the average of these relative prices?

We can easily see that AM = (9.4 / 4) = 2.35. But is this the most appropriate measure in this case? One way to think about this question is as follows. In the original (or second) sample none of the values would change if we added an amount of zero dollars to them. We could view zero as being a benchmark value. However, when we look at relative prices, things are rather different. A ratio of unity is now the more appropriate benchmark, and note that multiplying numbers by unity also leaves them unchanged.

So, when the data are measuring ratios, it`s generally accepted that the Geometric Mean is more appropriate than the Arithmetic Mean. In our example, the answer is GM = 2.317 (c.f. AM = 2.35).

There's another really important situation that arises with economic / financial data where the GM is the appropriate way to average the data. Consider an investment of $100 which yields returns of 5% p.a., 10% p.a., and 15% p.a. in three successive years. The Arithmetic Mean of these three values is 10% p.a., but as we'll see, this is not the appropriate way to calculate the average in this case.

Compounding the returns, we find that our $100 is worth $105 after one year; $115.5 at the end of the second year; and $132.825 at the end of the three years. Notice that if we compounded the investment using the average return AM = 10%, the implied value at the end of three years would be $133.1 [= $100 (1.1)³]. This overstates the correct answer of $132.825.

On the other hand, consider computing the Geometric Mean of the growth "multipliers":

GM = (1.05*1.1*1.15)^1/3= 1.099242 .

Then, the value of our investment, compounded over three years using this average is $100*(1.099242)³ = $132.825. This is the correct answer!

Further economic examples of where the Geometric Mean arises are with the United Nations "Human Development Index", which has been constructed on the basis of Geometric (rather than Arithmetic) Means since 2010; and with necessary conditions for stochastic dominance (Jean, 1980).

The Harmonic Mean is encountered less frequently then the other two averages when describing economic data, but there are some important instances where it arises or should be used. In particular, the Harmonic Mean is the appropriate average to use when dealing with data that are "rates". The classic example is fuel economy (miles per gallon, or liters per 100 Km), but economic examples also abound.

For instance, consider data on "hours worked per week" (a rate). Suppose that we have four people (sample observations), each of whom work a total of 2,000 hours. However, they work for different numbers of hours per week, as follows:

Person Total Hours Hours per Weeks

Week Taken

1 2,000 40 50

2 2,000 45 44.4444

3 2,000 35 57.142857

4 2,000 50 40

Total: 8,000 191.587297

The Arithmetic Mean of the values in the third column is AM = 42.5 hours per week. However, notice what this value implies. Dividing the total number of weeks worked by the sample members (8,000) by this average value yields a value of 188.2353 as the total number of weeks worked by all four people.

Now look at the last column in the table above. In fact the correct value for the total number of weeks worked by sample members is 191.5873 weeks.

If we compute the Harmonic Mean for the values for Hours per Week in the third column of the table we get HM = 41.75642 hours (< AM), and dividing this number into the 8,000 hours gives us the correct result of 191.5873 for the total number of weeks worked. Here is a case where the Harmonic Mean provides the appropriate measure for the sample average.

The Harmonic Mean also arises in the stochastic dominance literature (e.g., Jean, 1984).

For simplicity, the discussion so far has been restricted to "simple", or "unweighted" mean values. Just as we're familiar with the concept of a "weighted" Arithmetic Mean, we can also construct weighted Geometric or Harmonic Means, so as to give different emphasis to different items in the sample. Letting w_i denote the weight for i^th sample value, we have:

AM_w = [Σ_i (w_ix_i)] / [Σ_i (w_i)]
GM_w = [Π_i (x_i^w_i)]^{1/Σ_i (w_i)}
HM_w = [Σ_i (w_i)] / [Σ_i (w_i / x_i)]

(Obviously, if all of the weights are equal, then we just get the simple AM, GM, and HM formulae.)

Additional economic examples of the occurrence of the Geometric and Harmonic Mean arise, in these "weighted" forms, in the construction of  index numbers (such as price indices). Let p_it and q_it be the price and quantity of the i^th good in period "t", and let period "0" be the base period for the index.

Then, one sensible price index can be constructed as a Geometric Mean of "price-relatives", with base-period expenditures as the weights. The value of the index in period "t" would be:

                                P^G_t = [Π_i (p_it / p_i0)^p_i0q_i0 ]^{1/Σ_i ((p_i0q_i0)}.

You'll also recall that Fisher's "ideal" price index is computed by taking the Geometric Mean of Laspeyres' price index, and Paasche's price index.

When we consider the Laspeyres' price index itself, it can be considered either as an weighted aggregative index, with base-period quantities as the weights, or (equivalently) as an arithmetic weighted average of "price-relatives", with base-period expenditures as the weights. Similarly, Paasche's price index is not only a weighted aggregative index, with current-period quantities as the weights, but it also a weighted arithmetic average of price-relatives, with the "mixed" expenditures, (p_i0q_it) as the weights.

These last two results tell us immediately that the Laspeyres' and Paasche's price indices are likely to exhibit some sort of "distortion" - as indeed, they do. Price-relatives are ratios, so Geometric means, rather than Arithmetic means should be used.

Further, Paasche's price index can also be written in the form of an Harmonic Mean. The usual (aggregative) form of the index is:

                              P^P_t = [Σ_i(p_itq_it)] / [Σ_i(p_i0q_it)],

which can also be written as the Harmonic Mean of price-relatives, with current-period expenditures as the weights:

                        P^P_t = {[Σ_i((p_itq_it)(p_i0/ p_it))] / [Σ_i(p_itq_it)]}^-1 .


Now, to round things up, let's return to the matter of the rankings of the values of each of three means, when applied to the same set of positive data.

First, consider the ranking of the AM and GM, and for simplicity let's just take the case where there are only two different values, x₁ and x₂. (We know already that if x₁ = x₂, then AM = GM.)

So, we have

(x₁ - x₂) ≠ 0,

which implies that

(x₁ - x₂)² > 0,

or,

x₁² - 2x₁x₂ + x₂² > 0.

Adding 4x₁x₂ to both sides, we have

x₁² + 2x₁x₂ + x₂² > 4x₁x₂

or,

(x₁ + x₂)² > 4x₁x₂

or,

[(x₁ + x₂) / 2]² > x₁x₂.

Finally, this implies that [(x₁ + x₂) /2] > (x₁x₂)^(1/2).

That is, AM > GM.

You'll find various proofs that AM > GM in the general case, here.

Now, what about the ranking of the Harmonic and Geometric Means? In this case we can deal with the general case on n sample values, not all equal in value quite easily. We begin by using the result that we just established, namely that AM > GM, or

[(x₁ + x₂ + .... + x_n) / n] > (x₁x₂......x_n)^(1/n).

Applying this result to the reciprocals of the data, we immediately have:

[(1/x₁) + (1/x₂) + ... + (1/x_n)] / n > [(1/x₁)(1/x₂)....(1/x_n)]^(1/n),

or,

[1 / HM] > [(1)^(1/n)] / [(x₁x₂....x_n)^(1/n)].

In other words, we have [1 / HM] > [ 1/ GM], implying that HM < GM.

So, what's the take-away message here? It's simple enough. While there are various ways of calculating the "average" of a sample of economic data, we need to think about the context and the form of the data before we leap in. Failure to do so could result in some very misleading results.

References

Jean, W. H., 1980. The geometric mean and stochastic dominance. Journal of Finance, XXXV, 157-151-158.

Jean, W. H., 1984. The harmonic mean and other necessary conditions for stochastic dominance. Journal of Finance, XXXIX, 527-534.