Future economics Nobel laureate (2011) Christopher A. Sims was a 26 year old assistant professor at Harvard tasked in the fall term of 1968 to teach a graduate level introduction to time-series econometrics. He had been awarded a Harvard economics Ph.D. earlier that year. His dissertation supervisor was Hendrik Houthakker.
A copy of Sims’ initial list of reading assignments and topics can be found in the papers of Zvi Griliches in the Harvard Archives. Sims does appear to have offered a rather heavy dose of time-series econometrics for that time. Perhaps it was too much of a good thing, at least too much to swallow for most of the department’s graduate students. In any event Econometric Methods I was transferred to / taken over by Zvi Griliches in the following years when the topic of time series was reduced to an amuse-bouche of serial correlation.
In the previous year the course had been taught by Marc Nerlove (Yale University) with the following brief description provided in the course catalogue: “An introduction to the construction and testing of econometric models with special emphasis on the analysis of economic time series.”
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Course Announcement
Fall Term, 1968
Economics 224a. Econometric Methods
Half course (fall term). Tu., Th., S., at 9. Assistant Professor C. A. Sims
The theory of stochastic processes with applications to the construction and testing of dynamic economic models. Analysis in the time domain and in the frequency domain, in discrete time and in continuous time.
Prerequisite: Economics 221b [Multiple regression and the analysis of variance with economic applications] or equivalent preparation in statistics.
Source: Harvard University, Faculty of Arts and Sciences. Courses of Instruction, 1968-69, p. 133.
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Fall 1968
Economics 224a
Asst. Prof. C. Sims
Course Description
The accompanying Course Outline gives a detailed description of topics 0 through III which will (hopefully) occupy the first third of the semester. These topics include most of the mathematical tools which will be given econometric application in the later sections. The list of topics in the outline, even under the main headings 0 through III, is not exhaustive; and the topics listed are not all of equivalent importance.
Many of the references listed overlap substantially. In the first, theoretical, section of the course (except for Section 0) the references are chosen to duplicate as nearly as possible what will be covered in lectures. They should provide alternative explanations when you find the lectures obscure or, in some cases, provide more elegant and rigorous discussion when you find the lectures too pedestrian.
The primary emphasis of this course will be on the stationarity, or linear process, approach to dynamic models. The Markov process, control theory, or state space approach which is currently prominent in the engineering literature will be discussed briefly under topics V and VII.
The latter parts of the course will apply the theory developed in the first parts to formulating and testing dynamic economic models or hypotheses. Some background in economics is therefore essential to participation in the course. The mathematical prerequisites are a solid grasp of calculus, a course in statistics, and an ability to absorb new mathematical notions fairly quickly.
The course text is Spectral Methods in Econometrics by Gilbert Fishman. Spectral Analysis by Gwilyn M. Jenkins and Donald G. Watts is more complete in some respects, but it is less thorough in its treatment of some points important in econometrics and it costs three times what Fishman costs. A list of other texts which may be referred to in the accompanying course outline or in future outlines and reading assignments follows. Some of these texts are at a higher mathematical level than is required for this course or cover topics we will not cover in detail. Those texts which should be on library reserve are marked with a “*”, and those which are priced below the usual high prices for technical texts are marked with a “$”.
List of Text References
* Ahlfors, Lars, Complex Analysis, McGraw-Hill, New York, 1953.
Acki, Max., Optimization of Stochastic Systems, Academic Press, 1967.
* Deutsch, Ralph, Estimation Theory, Prentice Hall, 1965.
* Fellner, et.al., Ten Economic Studies in the Tradition of Irving Fisher, Wiley, 1967.
* Freeman, H., Introduction to Statistical Inference, Addison-Wesley, 1963.
Granger, C.W.J., and M. Hatanaka, Spectral Analysis of Economic Time Series, Princeton University Press, 1964.
Grenander, U., and M. Rosenblatt, Statistical Analysis of Stationary Time Series, Wiley, 1957.
Grenander, U., and G. Szego, Toeplitz Forms and Their Applications, University of California Press, 1958.
*$ Hannan, E.J., Time Series Analysis, Methuen, London, 1960.
$ Lighthill, Introduction to Fourier Analysis and Generalized Functions, Cambridge University Press.
Rozanov, Yu. A., Stationary Random Processes, Holden-Day, 1967.
*$ Whittle, P., Prediction and Regulation by Linear Least-Square Methods, English Universities Press, 1963.
* * * * * * * * * * * * * *
Preliminary Course Outline
Fall 1968
Economics 224a
Asst. Prof. C. Sims
0. Elementary Preliminaries.
Complex numbers and analytic functions, definitions and elementary facts. Manipulation of multi-dimensional probability distributions.
The material in this section will not be covered in lectures. A set of exercises aimed at testing your facility in these areas (for your information and mine) will be handed out at the first meeting.
References: Ahlfors, I.1, I.2.1-2.4, II.1; Jenkins and Watts, Chapters 3 and 4 or the sections on probability in a mathematical statistics text, e.g. Freeman, part I.
I. Stochastic Processes: Fundamental definitions and properties.
- Definitions:
stochastic process;
normal (stochastic) process;
stationary process;
linear process; — autoregressive and moving average processes;
covariance stationary process.
autocovariance and autocorrelation functions
stochastic convergence — in probability, almost sure, and in the (quadratic) mean or mean square;
ergodic process — n’th order ergodicity, sufficient conditions for first and second order ergodicity.
process with stationary n’th difference
Markov process
- Extensions to multivariate case.
References: Fishman, 2.1-2.5; Jenkins and Watts, 5.1-5.2.
II. Background from Mathematical Analysis
- Function spaces.
- Linear operator on function spaces; their interpretation as limits of sequences of ordinary weighted averages.
- Convolution of functions with functions, of operators with functions; discrete versus continuous time.
- Measure functions; Lebesgue-Stieltjes measures on the real line.
- Integration; the Lebesgue integral, the Cauchy-Riemann integral, and the Cauchy principal value; inverting the order of integration.
- Fourier transforms; of functions; of operators; continuous, discrete, and finite-discrete time parameters; the inverse transform and Parseval’s theorem.
- Applications to some simple deterministic models.
References: Jenkins and Watts, Chapter 2. For more rigor, see Lighthill. No reference I know of covers topics 4 and 5 in as brief and heuristic a way as we shall.
III. The spectral representation of covariance-stationary processes and its theoretical applications.
- Random measures; the random spectral measure of a covariance stationary process; characteristics of the random spectral measure in the normal and non-normal cases.
- The spectral density; relation to autocovariance function; positive definiteness.
- Wold’s decomposition; regular, mixed, and linearly deterministic processes; discrete and continuous component in the spectral measure; example of non-linearly deterministic process; the criterion for regularity with continuous spectral density.
- The moving average representation; criteria for existence of autoregressive representation.
- Optimal least squares forecasting and filtering.
- Generalized random processes.
- The multivariate case; cross spectra.
- Applications to econometric models.
References: Fishman, 2.6-2.30; Jenkins and Watts, 6.2 and 8.3: For a much more abstract approach, see Rozanov, chapters I – III.
IV. Statistical analysis using spectral and cross-spectral techniques.
V. Regression in time series.
VI. Seasonality.
VII. Estimation in distributed lag models.
Source: Harvard University Archives. Papers of Zvi Griliches, Box 123. Folder “Econometric Methods 1968-1982.”
Image Source: Christopher A. Sims ’63 in Harvard Class Album 1963. From the Harvard Crimson article “Harvard and the Atomic Bomb,” by Matt B. Hoisch and Luke W. Xu (March 22, 2018). Sims was a member of the Harvard/Radcliffe group “Tocsin” that advocated nuclear disarmament.
