This course in econometrics is intended for all Honours students in Economics. Its aim is to make you conversant with modern econometric theory and practice. Mathematical prerequisites are reasonable, just enough to grasp the theoretical underpinnings of the subject. In particular, only a little calculus is used, but there is quite heavy emphasis on matrix algebra. The practical side of the course involves working with real data, and also conducting simulation experiments on the computer. Since simulation plays an increasingly important role in both econometric theory and practice, a good deal of emphasis is placed on it.
Course Outline
Course details and announcements:
The class meets on Wednesdays and those Mondays that are not public holidays. Towards the end of the term, there will be makeup classes at times that only Minerva knows about. The class time is 11.30-13.00, and of course we meet via Zoom.
https://mcgill.zoom.us/j/99184789475
Meeting ID: 991 8478 9475
Now that we have had our last class, I can tell you definitively that the material to be covered for the final exam is all of the revised textbook, with one small exception. The subsection entitled "A Simple Example" in section 8.4 is omitted.
This link is for the Midterm Exam. It is due on Monday October 19, at or before 13.00 Montreal time.
Click here for the data set to be used for the exam.
The exam office has by now issued the definitive schedule for exams this December. The slightly updated information is that the exam will be made available at 18.30 on December 10, and must be submitted to myCourses by 18.30 on December 12. These are both Montreal times, and so be careful to take account of your own timezone if it is different.
The midterm has been scheduled for Monday October 19. I had thought that McGill had some guidelines for dealing with exams when people are in different time zones, but it seems not. I have therefore completely revised my plans for the exam. To start at the end: completed exams should be uploaded to myCourses as if the exam were another assignment. Whatever worked for you for the assignment will work as well for the midterm. The exam itself, and a link to the data to be used, will appear on the webpage around 13.00 on Sunday the 18th, Montreal time. The completed exam is then to be uploaded by 13.00, again Montreal time, on the 19th. This gives you a full 24 hours to complete the exam, and I think that should be possible for everyone, wherever they may be. (If not, please send me an email.) It's obviously impossible to impose a time limit in these circumstances, and so I won't try. I remind you, though, that the exam you submit must be all your own work. You must not seek help from anyone else, a fellow student or otherwise. This reminder will appear at the beginning of the exam as well. Work hard, and good luck to everyone!
Since you will not be writing the exam in class time, class will meet as usual on Monday the 19th. We have already lost two Mondays, with Labour Day and Thanksgiving Day, and by having a class on the 19th we will have less catching up to do at the end of the term.
Our TA is Jean-François Fournel. His virtual office hours are as follows:
Monday, 9:30am - 10:30am - Wednesday, 10:30am - 11:30am
https://mcgill.zoom.us/j/5956336119
Meeting ID: 595 633 6119
The midterm exam will cover the material treated in class up to and including October 7. This means up to and including all but the last subsection of Section 4.3 in the revised textbook.
Follow this link to see last year's midterm. But note that this year the format will be quite different, since we are never physically together in a classroom.
Textbooks:
The principal textbook for the course is Econometric Theory and Methods, Russell Davidson and James G. MacKinnon, Oxford University Press. An older, and more advanced, book by the same authors and the same publisher is Estimation and Inference in Econometrics.
The URL of the website for Econometric Theory and Methods is not what is given in the book itself at the end of the Preface. Instead it is https://qed.econ.queensu.ca/ETM/.
Although the plan for a second edition of the textbook has been abandoned, I am in the process of updating those chapters of the book that are covered in this course. The first few chapters of this revision are available here. Note that the revision is under active development, and will change very regularly. Be sure to get the latest revision.
Software:
This year, at least, the recommended software packages for econometrics are, first, MatLab, a commercial product, but freely available to people at McGill on account of a site licence, and, second, Python, a general-purpose interpreted programming language. Our TA uses MatLab, and can answer questions about it. Python is free software (prefer Python 3 to Python 2) and has a vast number of libraries available to it for many things, including econometric calculations.
A venerable software package for econometrics, and statistics more generally, is Stata. This file gives you information on how to get it for yourself.
For those of you who may be having trouble with available software for
running regressions, simulations, etc.,
you might like to try my own software, Ects. The documentation is available, not all but most of
it in English, all of it in French. For ease, you can find the first volume here (in English), and the second volume here.
Log of material covered
On September 2, we went through some material of a philosophical nature on scientific models, data-generating processes, random-number generators, virtual reality, and causation. We finished by looking at section 1.1 of the main textbook.
On September 9, the material covered was taken from section 1.2 of the book, up to but not including the subsection on Conditional Probability.
On September 14, we completed section 1.2, with the subsections on conditional probability and conditional expectations. In section 1.3, we went as far as the subsection on simulating econometric models.
We completed section 1.3 on September 16, and then rushed through section 1.4, on matrix algebra. We managed only the introductory part of Section 1.5, on estimation by the method of moments.
Chapter 1 was finished on September 21, along with a note that introduces a lot of terminology related to estimation, and includes the essential notion of an estimating function. Chapter 2 was barely begun. We stopped after seeing the parallelogram of vectors.
More geometry on September 23. We completed the first two sections of Chapter 2, and, in section 2.3, on the geometry of OLS estimation, we got as far as seeing that a projection must be idempotent.
On September 28, we finished section 2.3 on OLS geometry, and then went all the way through section 2.4 on the important Frisch-Waugh-Lovell theorem.
Most of the class on September 30 was devoted to section 2.5, on applications of the FWL theorem. We spoke about seasonal adjustment, deseasonalisation, detrending. There was also discussion of R^{2} as a measure of goodness of fit. We ended after a brief introduction to section 2.6, on influential observations and leverage.
Completing section 2.6 on October 5 brought us to the end of Chapter 2. But this is Chapter 3 of the revision now in use instead of the original textbook. The section on R^{2} in the old book is now in a later chapter in the revision, while a new subsection, on Fixed Effects, replaces it. It should be possible to go back and cover it later. Meanwhile, we started on the new Chapter 4, and concluded the first subsection of section 4.2, on Bias and Unbiasedness.
On October 7, we resumed working through Chapter 4 of the new revised textbook. We saw that the OLS estimator and the OLS estimating equation are unbiased with exogenous regressors, but, if exogeneity is replaced by predeterminedness, the estimator is biased in general, but the estimating equation remains unbiased. In section 4.3, asymptotic theory is the main concern. We looked at two main types of stochastic convergence, convergence in probability and convergence in distribution. This led on to the law of large numbers (LLN). We ended by defining "big-O" notation for both nonrandom and stochastic sequences.
We continued to work through Chapter 4 on October 14. We defined the concept of consistency of an estimator, and saw how it depends on the asymptotic construction. Section 4.4 deals with covariance matrices and precision matrices, both of which are usually positive definite matrices. We had just started Section 4.5, on the precision of the OLS estimator.
On October 19, we managed to complete our study of Chapter 4. The high point was the Gauss-Markov theorem, reached after looking at the determinants of precision, and how comparisons of precision can be made using positive semi-definite matrices. The next section dealt with misspecification, overspecification, and underspecification, and introduced the notion of the mean-squared-error (MSE) matrix. The last section of the Chapter, on R^{2}, was covered earlier.
Section 5.2 deals with the main ideas of hypothesis testing, and is full of definitions. On October 21, we made our way through that, and, in section 5.3, studied the normal family of distributions, including the multivariate normal family, and also the chi-squared distribution. Next time, we will start by completing section 5.3.
We did indeed start class on October 26 by completing section 5.3, where we studied the t and F distributions. In section 5.4, we worked through t tests and F tests with the classical normal linear model, and looked at the threefold orthogonal decomposition involved with these tests. The main example of an F test was the Chow test. In section 5.5, we got as far as the Fundamental Theorem of Statistics. Next will be Central Limit Theorems.
Central Limit Theorems got treated on October 28. We were then able to use such theorems along with laws of large numbers in order to get asymptotic results that mirror the exact results for the classical normal linear model. In particular, we defined what is meant by a consistent estimator of the covariance matrix of a root-n consistent estimator. We then covered most of section 5.7 on multiple testing, using the Bonferroni inequality.
On November 2, we finished Chapter 5. In section 5.8, we introduced non-central versions of the t, F, and chi-squared distributions, and this let us examine power functions. Section 5.9 deals with the important issue of pre-testing, and makes use of the mean-squared error (MSE) matrix. In Chapter 6, we defined confidence sets, and saw how to construct confidence intervals for scalar parameters based on a squared t statistic.
We completed the material in Chapter 5 on confidence sets on November 4, discussing asymmetric intervals and confidence regions for more than one parameter. Then we moved on to section 6.4, on heteroskedasticity and the HCCME. We stopped after defining the variants HC_{0}, HC_{1}, HC_{2}, and HC_{3}.
On November 9, we finished section 6.4 on the HCCME, and then did section 6.5 on HAC covariance matrix estimators, and also section 6.6, on the CRVE, and the whole business of clustering. We also completed our study of section 6.7, on the difference-in-differences method.
We were able to complete Chapter 6 on November 11, with the section on the delta method, used to get standard errors and confidence intervals for nonlinear transformations of parameters. Chapter 7 deals with the bootstrap. After seeing what a random-number generator does, we saw how to use one to perform simulations. It was possible to get P values by simulation, and we explored different ways of doing that. We finished in the middle of section 7.3 just before the subsection on Monte Carlo tests.
Monte Carlo and bootstrap tests were the first things on the agenda for November 16. It is possible to get an exact test with an exact pivot, even with a small value of B. Next came section 7.4, on bootstrap tests for regression models. We discussed the parametric bootstrap, recursive sampling, resampling, and saw how resampling works. In section 7.5, on the Golden Rules of bootstrapping, we first saw that the bootstrap can work much better than asymptotics, and then looked at the golden rules themselves.
We almost managed to get to the end of Chapter 7 on the bootstrap on November 18. After the Golden Rules, it was seen that the power of bootstrap tests increases with B, the number of bootstrap repetitions, and tends to what one would get if an exact test happened to be available. Then heteroskedasticity and autocorrelation were successively considered in sections 7.6 and 7.7 respectively. The so-called wild bootstrap works very well with heteroskedasticity, but there is nothing as reliable for autocorrelation. The next topic was bootstrap confidence sets. It was found necessary to play around with Golden Rule 1 to avoid very long computation times, and there was a slightly tricky calculation needed to compute the critical values of bootstrap tests.
After just a few minutes on bootstrap standard errors, we embarked on November 23 on Chapter 8, which deals with instrumental variables. We covered the first three sections, looking at the classic problem of errors in variables, and models defined by a set of simultaneous equations. In both cases, consistent and asymptotically normal estimators can be obtained by instrumental variable (IV) estimators. Problems studied included under- and over-identification, and the efficient choice of instruments in the over-identified setup. We mentioned two-stage least squares (2SLS) and an artificial regression used to compute a consistent covariance estimator.
In section 8.4, we covered only the first few paragraphs, on reduced-form and structural equations, and skipped the rest of the section; this on November 25. Much of the class was taken up with section 8.5, on hypothesis testing. It was seen that an artificial regression made it possible to perform testing with IV in almost the same way as with OLS. A version of the Wald statistic can be easily obtained from the artificial regression as n times the uncentred R^{2}. The next section, 8.6, is on over-identifying restrictions. A suitable test statistic can be obtained either by use of the artificial regression or by the minimised IV criterion function. The latter approach is called the Sargan test.
We completed Chapter 8 on November 30. This took us through the DWH test and then a section on bootstrapping IV models. The main point there is that the bootstrap DGP has to generate all the endogenous variables. Among three possibilities for the bootstrap, parametric, resampling, and wild, the last is almost always the best.
Chapter 9 starts with generalised least squares (GLS). If the covariance matrix of the disturbances is known, it can be used to transform a linear regression into a form that satisfies the requirements of the Gauss-Markov theorem. The GLS estimator is thus more efficient than OLS, and indeed than any linear unbiased estimator. Feasible GLS involves being able to estimate the parameters of the covariance matrix consistently. The easiest example is weighted least squares, or feasible weighted least squares, where one has to estimate the skedastic function.
In section 9.5, which we studied on December 2, the topic is heteroskedasticity. A test is described for the null hypothesis of homoskedasticity against a particular parametric alternative for which the skedastic function is some nonlinear function of some explanatory variables. It turns out that it is unnecessary to specify exactly which nonlinear function. The next few sections consider autocorrelation. in section 9.6, autoregressive and moving-average stochastic processes are defined. Then, in section 9.7, we are shown tests for a null of no serial correlation against an alternative of an AR(p) process. The test statistics have known asymptotic distributions, but their properties are greatly improved by bootstrapping.
Our last class for this term was on December 3. We finished the revised textbook. In section 9.8 we saw how to use feasible GLS to estimate a regression model with AR(1) disturbances. However, in section 9.9, it was pointed out that it might be better to try to find a more complicated, possibly non-linear, dynamic model with white-noise disturbances. Finally, we had a preliminary look at panel data in section 9.10, and studied the fixed-effects estimator and the more constrained random-effects estimator. We also noted that there are many similarities between panel-data models an models with clustered data.
Recordings
Click here for the video/audio recording of the first class, on September 2. If all you need is the audio recording (smaller file, shorter download time) click here.
In order to replace the non-existent recording for September 9, I have made another recording. For the full audio/video version, click here, and for audio only, click here.
The recording of the class of September 14 is available here. Audio only here.
The recording of the class of September 16 is available here. Audio only here.
The recording of the class of September 21 is available here. Audio only here.
The recording of the class of September 23 is available here. Audio only here.
The recording of the class of September 28 is available here. Audio only here.
The recording of the class of September 30 is available here. Audio only here.
The recording of the class of October 5 is available here. Audio only here.
The recording of the class of October 7 is available here. Audio only here.
The recording of the class of October 14 is available here. Audio only here.
The recording of the class of October 19 is available here. Audio only here.
The recording of the class of October 21 is available here. Audio only here.
The recording of the class of October 26 is available here. Audio only here.
The recording of the class of October 28 is available here. Audio only here.
The recording of the class of November 2 is available here. Audio only here.
The recording of the class of November 4 is available here. Audio only here.
The recording of the class of November 9 is available here. Audio only here.
The recording of the class of November 11 is available here. Audio only here.
The recording of the class of November 16 is available here. Audio only here.
The recording of the class of November 18 is available here. Audio only here.
The recording of the class of November 23 is available here. Audio only here.
The recording of the class of November 25 is available here. Audio only here.
The recording of the class of November 30 is available here. Audio only here.
The recording of the class of December 2 is available here. Audio only here.
The recording of the class of December 3 is available here. Audio only here.
Assignments:
Follow this link to access the first assignment, dated September 21. It is due on Wednesday September 30. In answer to a question posed in class, I confirm that it is the seasonally adjusted series that I ask you to get from the Statcan site.
The second assignment can be accessed by following this link, and the data to be used are found here. The assignment is dated November 7, and is due on November 16.
The third assignment can be accessed by following this link, and the data to be used are found here. The assignment is dated November 19, and is due on November 30.
If you wish to take advantage of it, it has been decided to extend the period for submitting Assignment 3 until Thursday December 3.
Notes
Follow this link for material intended to supplement the textbook. The first note gives definitions related to zero functions and estimating equations.
Other teaching materials
This link is to the paper that was the basis for my presidential address to the CEA in 2015. The first couple of sections were also the basis for much of what I said in the first lecture. And this link takes you to the slides I used for the presentation - perhaps enough for our purposes.
URL: https://russell-davidson.arts.mcgill.ca/e468