This course is intended for Master's students with a serious interest in econometrics. 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:
Now that we have had our last class, I can confirm that the material to be covered on the Christmas exam will be the entire revised textbook, with just one exception. That is the subsection entitled "A Simple Example" in section 8.4. The word "Simple" is relative in this context! The subsection "Fixed Effects" in section 3.5 was not explicitly covered when we did Chapter 3, because it had been shifted from where it was in the original. unrevised, textbook. But its contents are taken up again in section 9.10, on Panel Data.
The schedule for exams in graduate courses in the department sets a date and time of Friday December 11 at 13.00 for 662D1. It is explained that this is the time when the exam is due. Undergraduate courses, scheduled by the exam office, allow a 48-hour window for completion of the exam, and I think that is appropriate for us as well. The exam will therefore be made available on the course webpage at 13.00 on Wednesday December 9.
This link is for the Midterm Exam. It is due on Thursday October 22, at or before 13.00 Montreal time.
Click here for the data set to be used for the exam.
The midterm has been scheduled for Thursday October 22. 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 Wednesday the 21st, Montreal time. The completed exam is then to be uploaded by 13.00, again Montreal time, on the 22nd. 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!
The midterm will cover everything covered in class up to but not including section 5.4. The subsection on "Fixed Effects" in section 3.5 is not included, since we haven't as yet looked at it in class.
The class meets on Tuesdays and Thursdays, from 13.00-14.30, via Zoom.
https://mcgill.zoom.us/j/94916348966
Meeting ID: 949 1634 8966
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
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. 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. Our TA uses MatLab, and can answer questions about it.
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 more information about getting Matlab, click on this link.
Log of material covered
On September 3, we looked at some of the basic concepts of econometrics, in particular, data-generating process (DGP) and model, and the relation between scientific models and virtual reality. From the textbook, we covered section 1.1.
On September 8, we covered much of section 1.2, on basic material about random variables and probability distributions. We stopped just before the subsection entitled Conditional Probabilities.
We resumed on September 10 where we left off at the end of the previous class. We went over conditional probability, and concluded section 1.2. In section 1.3, on the specification of regresssion models, we covered the material up to and including the algorithm on page 19, which details how to simulate a simple linear regression model, or, better, a DGP chosen from a simple linear regression model.
We continued work on section 1.3 of the textbook on September 15, finished that, and went on to cover section 1.4, on matrix algebra, completely. In section 1.5, on method-of-moments estimation, we looked at the introductory section only, stopping before the subsection on Estimating the Simple Linear Regression Model.
On September 17, we completed Chapter 1, and started on Chapter 2. We stopped just before the subsection on Subspaces of Euclidean Space.
Work on Chapter 2 continued on September 22. The main topic covered was orthogonal projections. We had not quite finished section 2.3, leaving for next time the example about temperature at the end of that section.
Section 2.3 was completed, and section 2.4, on the Frisch-Waugh-Lovell theorem, was also completed. We had begun section 2.5, and were in the middle of discussing seasonal adjustment.
We managed to finish Chapter 2 on September 29. We looked at the various applications of the FWL theorem, demeaning, deseasonalising, detrending, etc., and the goodness-of-fit measure called R^{2}. The final section deals with the influence of individual observations in a sample and the concept of leverage.
On the first of October, I started the transition to a new improved text, and embarked on the chapter on statistical properties of OLS. We reminded ourselves about models and DGPs, and then defined bias and unbiasedness of estimators and estimating equations. We got started on asymptotic theory, and defined convergence in probability.
We made considerable strides into asymptotic theory on October 6, continuing to use the revised text, in which we are now in Chapter 4. We began with convergence in distribution, something quite different from convergence in probability. Then we defined consistency, using a law of large numbers to show that OLS is consistent without exogeneity if the regressors are predetermined. We then embarked on the study of covariance and precision matrices, developing along the way material about positive (semi-)definite matrices.
The main topic on October 8 was the Gauss-Markov theorem, which states that, under certain conditions, the OLS estimator is BLUE, best unbiased linear estimator. After that, the next section is about estimating the variance of the disturbances. We ended by deriving the unbiased estimator that is usually called s^{2}.
After Thanksgiving weekend, the next class was on October 13. We finished Chapter 4, after discussing misspecified models. In Chapter 5, we embarked on the theory of hypothesis testing, and had finished the part of section 5.2 up to but not including the subsection on P values.
On October 15, we continued working through Chapter 5. We discussed the important concept of P values, and their importance for hypothesis testing. Then we did all of section 5.3, on various probability distributions we will want to use. Although we embarked on section 5.4, the midterm will cover material only up to and not including section 5.4
We started with section 5.4 on October 20, and completed that section, on t and F tests. The threefold orthogonal decomposition of a regressand used by these tests was studied for the intuition it gives about how statistical discrimination is effected by the tests. The well-known Chow test was given as an example. In section 5.5, we return to asymptotic theory. We used the law of large numbers to prove the Fundamental Theorem of Statistics.
We continued with asymptotics on October 22, starting with the central-limit theorem. With this we defined asymptotic normality and root-n convergence, and this let us show that t and F tests work asymptotically with a decent asymptotic construction. Next was section 5.7, on testing several hypotheses at once. The Bonferroni inequality was needed for a good treatment of this. Section 5.8, on the power of tests, came next, and led us to defined various noncentral distributions, t, F, and chi-squared.
The last section in Chapter 5 is on pretesting, and we worked through that on October 27. Then, in Chapter 6, we defined confidence sets, and had just started to look at asymmetric confidence intervals.
On October 29, we completed the first three sections of Chapter 6, the sections on confidence sets. We began with the counter-intuitive flip of upper and lower quantiles into lower and upper limits of a confidence interval. Then came confidence regions, and the reasons for the shapes these may take. Section 6.4 is on heteroskedasticity and the HCCME. We completed all but the last subsection of this section.
Continuing through Chapter 6 on November 3, we first saw when heteroskedasticity matters, and then moved on to autocorrelation and HAC estimators in section 6.5. In section 6.6, we saw the notion of clustering of observations, and the consequences for that. The CRVE attempts to take account of clustering, and does so with more or less success depending on the problem being analysed. We started on section 6.7, which deals with the difference-in-differences method.
Chapter 6 was finished on November 5 by completing the section on DiD, and then section 6.8, on the delta method, which allows asymptotic inference on nonlinear transforms of model parameters. Chapter 7 is devoted to Bootstrap methods. After some generalities about random-number generators in section 7.2, we started section 7.3, and encountered the bootstrap principle, which will underlie what we will see in the rest of the chapter.
On November 10, we started work on bootstrap inference. We began with simulated P values, and saw how to implement a Monte Carlo test based on a pivotal statistic. A bootstrap test works in exactly the same way, but uses a bootstrap DGP that is an estimate of the unknown true DGP. We saw how to choose the number of bootstrap samples in such a way that, with a pivotal statistic, an exact test is possible. Then we saw examples of a parametric bootstrap, with a dynamic model, and a resampling bootstrap, when it is not assumed that the disturbances are normal.
The Golden Rules of bootstrapping were the first things to be studied on November 12. We saw, on the basis of experimental evidence, that, when these rules are respected, the bootstrap can give much more reliable inference than asymptotic theory, while yielding almost as much power as the exact tests of the classical normal linear model. After that, in section 7.6, we looked at bootstrapping in the presence of heteroskedasticity, and saw that the wild bootstrap are very effective. For autocorrelation, the subject of section 7.7, the story is less good. Available techniques include versions of the block bootstrap, and the sieve bootstrap.
On November 17, we completed Chapter 7, beginning with the section on bootstrap confidence sets. This section also mentions bootstrap standard errors, which are not generally recommended. Then on to Chapter 8 on IV (instrumental variables) estimation. Two possible causes of correlation between the disturbances of a regression and some of its regressors were discussed: errors in variables and simultaneity. This led us to define the simple IV estimator.
Much progress was made on November 19 with Chapter 8, on IV estimation. First came the generalised IV estimator, which allows for more instruments than regressors. We established all the usual asymptotic properties for it: consistency and asymptotic normality. There was also a construction that leads to asymptotic identification, which in turn is sufficient for consistency. Two-stage least squares is a way of computing the IV estimator, and study of it led to the formulation of a valuable artificial regression which allows one to compute a consistent covariance matrix estimate, and also serves as a way to compute Wald test statistics. The usual F statistic is not valid.
In section 8.4, we covered only the first page and a half. In section 8.5, we saw that statistics based on the IV criterion function are valid as a replacement for the usual, invalid, F test. In addition, use of the artificial regression allows for tests that are robust to heteroskedasticity and autocorrelation.
The rest of Chapter 8 was covered on November 24. First there was the test of over-identifying restrictions, and then the DWH test. The last section was on the bootstrap. It is necessary to come up with a bootstrap DGP that can generate all of the endogenous variables, and to that end we can use OLS estimates of the reduced-form equations for the endogenous regressors. The other thing to worry about is maintaining the correlations of all the disturbances, and we saw that the best way to do that was to use the wild bootstrap. The wild bootstrap can also be pressed into service with clustered data.
Chapter 9 deals with generalised least squares (GLS). We define the GLS estimator for a linear regression by transforming all the variables in such a way that the transformed disturbances have an identity covariance matrix. This makes the GLS estimator efficient in the class of all instrumental-variables estimators with exogenous instruments. Feasible GLS allows for a covariance matrix that depends on unknown parameters that can be estimated consistently. A good example is feasible weighted least squares with a skedastic function specified as the exponential of something that looks like a regression function.
In section 9.5, which we studied on November 26, 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.
Our last class for this term was on December 1st. We finished off the revised textbook, resuming the study of testing for serial correlation in section 9.7. After developing asymptotic tests, we moved on to much more accurate tests, either Monte Carlo tests or bootstrap tests. 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.
Recordings
For the first class, on September 3, the video/audio recording is available here. You may prefer the shorter file , which has the audio recording only.
The class on September 8 was not recorded. Instead, I have made a second recording which covers the material covered in that class. Click here for the full audio/video recording, and here for audio only.
For September 10, the main recording is here, and the audio-only version is here.
For September 15, the main recording is here, and the audio-only version is here.
For September 17, the main recording is here, and the audio-only version is here.
For September 22, the main recording is here, and the audio-only version is here.
For September 24, the main recording is here, and the audio-only version is here.
For September 29, the main recording is here, and the audio-only version is here.
For October 1st, the main recording is here, and the audio-only version is here.
For October 6, the main recording is here, and the audio-only version is here.
For October 8, the main recording is here, and the audio-only version is here.
For October 13, the main recording is here, and the audio-only version is here.
For October 15, the main recording is here, and the audio-only version is here.
For October 20, the main recording is here, and the audio-only version is here.
For October 22, the main recording is here, and the audio-only version is here.
For October 27, the main recording is here, and the audio-only version is here.
For October 29, the main recording is here, and the audio-only version is here.
For November 3, the main recording is here, and the audio-only version is here.
For November 5, the main recording is here, and the audio-only version is here.
For November 10, the main recording is here, and the audio-only version is here.
For November 12, the main recording is here, and the audio-only version is here.
For November 17, the main recording is here, and the audio-only version is here.
For November 19, the main recording is here, and the audio-only version is here.
For November 24, the main recording is here, and the audio-only version is here.
For November 26, the main recording is here, and the audio-only version is here.
For December 1st, the main recording is here, and the audio-only version is here.
Assignments
The first assignment, dated September 21, can be accessed here. It is due one week later, on Tuesday September 29.
The second assignment, dated November 7, can be accessed here. It is due on November 17. The data to be used are found here.
The third assignment, dated November 19, can be accessed here. It is due on the date of the last class, December 1st. The data to be used are found here.
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
I used this set of slides for discussion of models, data-generating processes, and virtual reality.
URL: https://russell-davidson.arts.mcgill.ca/e468