Testing for Multivariate Normality

The assumption that multivariate data are (multivariate) normally distributed is central to many statistical techniques. The need to test the validity of this assumption is of paramount importance, and a number of tests are available.

A recently released R package, MVN, by Korkmaz et al. (2014) brings together several of these procedures in a friendly and accessible way. Included are the tests proposed by Mardia, Henze-Zirkler, and Royston, as well as a number of useful graphical procedures.

If for some inexplicable reason you're not a user of R, the authors have thoughtfully created a web-based application just for you!

Reference

Korkmaz, S., D. Goksuluk, and G. Zarasiz, 2014. An R package for assessing multivariate normality. The R Journal, 6/2, 151-162.


© 2015, David E. Giles

New Canadian Provincial Data

I was delighted to see the release, last week, of a new Statistics Canada research paper, "Provincial Convergence and Divergence in Canada, 1926 to 2011". Co-authored by W. Mark Brown and (UVic grad.) Ryan McDonald.

It's a very interesting paper that makes use of some equally interesting new data that Statistics Canada has released recently. For once, Statistics Canada has gone to considerable efforts to assemble a really useful long-run data-set (see Cansim Table 384-5000 This is a far cry from the myriad of "Series Discontinued" flags that we're used to seeing in the Cansim database, and it's great to see that Ryan has been instrumental in its development (McDonald, 2015).

As he notes, "More advanced statistical methods, and models with greater breadth and depth, are difficult to apply to existing fragmented Canadian data. The longitudinal nature of the new provincial dataset remedies this shortcoming."

Data is an econometrician's life-blood. We need to see more of this from Statistics Canada.

Reference

McDonald, R., 2015. Constructing Provincial Time Series: A Discussion of Data Sources and Methods. Income and Expenditure Accounts Technical Series, no. 77 Statistics Canada Catalogue no. 13-604-M. Ottawa: Statistics Canada.

© 2015, David E. Giles

Four Different Types of Regression Residuals

When we estimate a regression model, the differences between the actual and "predicted" values for the dependent variable (over the sample) are termed the "residuals". Specifically, if the model is of the form:

                     y = Xβ + ε ,                                                         (1)

and the OLS estimator of β is b, then the vector of residuals is

                    e = y - Xb .                                                           (2)

Any econometrics student will be totally familiar with this.

The elements of e (the n residuals) are extremely important statistics. We use them, of course, to construct other statistics - e.g., test statistics to be used for testing the validity of the underlying assumptions associated with our regression model. For instance, we want to check, are the errors (the elements of the ε vector) serially independent; are the errors homoskedastic; are they normally distributed; etc.?

What a lot of students don't learn is that these residuals - let's call them "Ordinary Residuals" - are just one type of residuals that are used when analysing the regression model. Lets take a look at this.

Some Suggested Reading

• Bahoc, F., H. Leeb, and B. M. Potscher, 2014. Valid confidence intervals for post-model-selection predictors. Working Paper, Department of Statistics, University of Vienna.
• Baumeister, C. and J. D. Hamilton, 2014. Sign restrictions, structural vector autoregressions, and useful prior information. NBER Working Paper No. 20741.
• Bjerkholt, O., 2015. Fellowship elections in the Econometric Society 1933-1948. Working Paper, Department of Economics, University of Oslo.
• Deuchert, E. and M. Huber, 2014. A cautionary tale about control variables in IV estimation. Discussion Paper No. 2014-39, School of Economics and Political Science, University of St. Gallen.
• Doornik, J. A. and D. F. Hendry, 2014. Statistical model selection with 'Big Data'. Discussion Paper 735, Department of Economics, University of Oxford.
• Duendack, M., R. W. Palmer-Jones, and W. R. Reed, 2014. Replications in economics: A progress report. Working Paper No. 26/2014, Department of Economics and Finance, University of Canterbury.

© 2015, David E. Giles

Extreme Value Modelling in Stata

David Roodman wrote to me today, saying:

"I don’t know if you use Stata, but I’ve just released a Stata package for extreme value theory. It is strongly influenced by Coles’s book on EVT and the associated ismev package for R. Using maximum likelihood, it fits the generalized Pareto distribution and the generalized extreme value distribution, the latter including the extension to multiple order statistics. It also offers various diagnostic plots. There are already many sophisticated R packages for EVT. I suppose mine offers accessibility…and small-sample bias corrections. It can do the Cox-Snell correction for all the models, including with covariates (citing you for GPD, and promising a write-up for the rest). It also offers bias correction based on a parametric bootstrap. I’ve confirmed the efficacy of both bias corrections through simulations, for the GPD and GEV. I’m still tweaking the simulations, and they take time, but I hope to soon post some graphs based on them. The GPD results closely match yours.

Comments welcome. Please circulate to others who might be interested. To install the package in Stata, type “ssc install extreme”.  The help file contains clickable examples that reproduce most results in the Coles book. The web page is https://ideas.repec.org/c/boc/bocode/s457953.html."

The work on the Cox-Snell bias correction for the Generalized Pareto Distribution that David is referring to is Giles et al. (2015). You can find an earlier post about this work here, and you can download the paper here.

Update, 28/1/2015: See more at David's blog.

Reference

Giles, D. E., H. Feng, and R. T. Godwin, 2015. Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution. Communications in Statistics - Theory and Methods, in press.

© 2015, David E. Giles

Econometrics vs. Ad Hoc Empiricism

In a post in 2013, titled "Let's Put the "ECON" Back Into Microeconometrics", I complained about some of the nonsense that is passed off as "applied econometrics". Specifically, I was upset about the disconnect between the economic model (if there is one) and the empirical relationships that are actually estimated, in many "applied" papers.

I urge you to look back at the post before reading further.

Here's a passage from that post:

"In particular, how often have you been presented with an empirical application that's based on just a reduced-form model that essentially ignores the nuances of the theoretical model?

I'm not picking on applied microeconomic papers - really, I'm not! The same thing happens with some applied macroeconomics papers too. It's just that in the micro. case, there's often a much more detailed and rich theoretical model that just lends itself to some nice structural modelling. And then all we see is a regression of the logarithm of some variable on a couple of interesting covariates, and a bunch of controls - the details of which are frequently not even reported."

Well, things certainly haven't improved since I wrote that. In fact, it seems that I'm encountering more and more of this nonsense. This isn't "econometrics", and the purveyors of this rubbish aren't "econometricians". 

My real concern is that students who are exposed to these papers and seminars may not recognize it for what it is - just ad hoc empiricism. 

© 2015, David E. Giles

ARDL Modelling in EViews 9

My previous posts relating to ARDL models (here and here) have drawn a lot of hits. So, it's great to see that EViews 9 (now in Beta release - see the details here) incorporates an ARDL modelling option, together with the associated "bounds testing".

This is a great feature, and I just know that it's going to be a "winner" for EViews.

It certainly deserves a post, so here goes!

First, it's important to note that although there was previously an EViews "add-in" for ARDL models (see here and here), this was quite limited in its capabilities. What's now available is a full-blown ARDL estimation option, together with bounds testing and an analysis of the long-run relationship between the variables being modelled.

Here, I'll take you through another example of ARDL modelling - this one involves the relationship between the retail price of gasoline, and the price of crude oil. More specifically, the crude oil price is for Canadian Par at Edmonton; and the gasoline price is that for the Canadian city of Vancouver. Although crude oil prices are recorded daily, the gasoline prices are available only weekly. So, the price data that we'll use are weekly (end-of-week), for the 4 January 2000 to 16 July 2013, inclusive.

The oil prices are measured in Candian dollars per cubic meter. The gasoline prices are in Canadian cents per litre, and they exclude taxes. Here's a plot of the raw data:

New Year Reading List

Happy New Year - and happy reading!

• Arlot, S. and A. Celisse, 2010. A survey of cross-validation procedures for model selection. Statistics Surveys, 4, 40-79. (HT Rob)
• Marsilli, C., 2014. Variable selection in predictive MIDAS models.Working Paper No. 520, Bank of France.
• Kulaksizoglu, T., 2014. Lag order and critical values of the augmented Dickey-Fuller test: A replication. Journal of Applied Econometrics, forthcoming.
• Mooij, J. M., J. Peters, D. Janzing, J. Zscheischler, and B. Scholkopf, 2014. Distinguishing cause from effect using observational data: Methods and benchmarks. Working Paper. (HT Roger)
• Polak, J., M. L. King, and X. Zhang, 2014. A model validation procedure. Working Paper 21/14, Department of Econometrics and Business Statistics, Monash University.
• Reed, W. R., 2014. Unit root tests, size distortions and cointegrated data. Working Paper 28/2014, Department of Economics and Finance, University of Canterbury. (HT Bob)

© 2015, David E. Giles

Econometricians' Debt to Alan Turing

The other day, Carol and I went with friends to see the movie, The Imitation Game. I definitely recommend it.

I was previously aware of many of Alan Turing's contributions, especially in relation to the Turing Machine, cryptography, computing, and artificial intelligence. However, I hadn't realized the extent of Turing's use of, and contributions to, a range of important statistical tools. Some of these tools have a direct bearing on Econometrics.

For example:

• (HT to Lief Bluck for this one.) In 1935, at the tender age of 22, Turing was appointed a Fellow at King's College, Cambridge, on the basis of his 1934 (undergraduate) thesis in which he proved the Central Limit Theorem. More specifically, he derived a proof of what we now call the Lindeberg-Lévy Central Limit Theorem.  He was not aware of Lindeberg's earlier work (1920-1922) on this problem. Lindeberg, in turn, was unaware of Lyapunov's earlier results. (Hint: there was no internet back then!). How many times has your econometrics instructor waved her/his arms and muttered ".......as a result of the central limit theorem....."?
• In 1939, Turing developed what Wald and his collaborators would later call "sequential analysis". Yes, that's Abraham Wald who's associated with the Wald tests that you use all of the time.Turing's wartime work on this subject remained classified until the 1980's. Wald's work became well-established in the literature by the late 1940's, and was included in the statistics courses that I took as a student in the 1960's. Did I mention that Wald's wartime associates included some familiar names from economics? Namely, Trygve Haavelmo, Harold Hotelling, Jacob Marschak, Milton Friedman, W. Allen Wallis, and Kenneth Arrow.
• The mathematician/statistician I. J. ("Jack") Good was a member of Turing's team at Bletchley Park that cracked the Enigma code. Good was hugely influential in the development of modern Bayesian methods, many of which have found their way into econometrics. He described the use of Bayesian inference in the Enigma project in his "conversation" with Banks (1996). (This work also gave us the Good-Turing estimator - e.g., see Good, 1953.)
• Turing (1948) devised the LU ("Lower and Upper") Decomposition that is widely used for matrix inversion and for solving systems of linear equations. Just think how many times you invert matrices when you're doing your econometrics, and how important it is that the calculations are both fast and accurate!

Added, 20 February, 2015: I have recently become aware of Good (1979)

References

Banks, D. L., 1996. A conversation with I. J. Good. Statistical Science, 11, 1-19.

Good, I. J., 1953.The population frequencies of species and the estimation of population parameters. Biometrika, 40, 237-264.

Good, I. J., 1979. A. M. Turing's statistical work in World War II. Biometrika, 66, 393-396.

Turing, A. M., 1948. Rounding-off errors in matrix processes. Quarterly Journal of Mechanics and Applied Mathematics, 1, 287-308.

© 2014, David E. Giles

Multivariate Medians

I'll bet that in the very first "descriptive statistics" course you ever took, you learned about measures of "central tendency" for samples or populations, and these measures included the median. You no doubt learned that one useful feature of the median is that, unlike the (arithmetic, geometric, harmonic) mean, it is relatively "robust" to outliers in the data.

(You probably weren't told that J. M. Keynes provided the first modern treatment of the relationship between the median and the minimization of the sum of absolute deviations. See Keynes (1911) - this paper was based on his thesis work of 1907 and 1908. See this earlier post for more details.)

At some later stage you would have encountered the arithmetic mean again, in the context of multivariate data. Think of the mean vector, for instance.

However, unless you took a stats. course in Multivariate Analysis, most of you probably didn't get to meet the median in a multivariate setting. Did you ever wonder why not?

One reason may have been that while the concept of the mean generalizes very simply from the scalar case to the multivariate case, the same is not true for the humble median. Indeed, there isn't even a single, universally accepted definition of the median for a set of multivariate data!

Let's take a closer look at this.