Category: Econometrics

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Econometrics Exam Questions Harvard Suggested Reading Syllabus

Harvard. Introduction to Econometrics, Syllabus and Final Exam. Sims, 1967-68

Christopher A. Sims, portrait from his senior year

 

Christopher Sims was awarded a Ph.D. in economics from Harvard in February 1968 and in the Spring term of 1967-68 he still taught at the rank of “Instructor in Economics” (Note: he was promoted to assistant professor of economics beginning with the Fall term of 1968-69). We see from the official course announcement for the 1967-68 academic year that the staffing of the undergraduate introduction to econometrics course had not been determined until some time after the printing of the course announcements. 

An earlier post provides the course syllabus and partial reading list for Sims’ graduate course on time series econometrics.

_______________________________

Course Announcement

Economics 195. Introduction to Econometrics.

Half course (spring term). Tu., Th., S., at 11. Dr. ——

Statistics applied to testing of economic hypotheses and estimation of economic parameters. Will center on multiple regression and analysis of variance techniques and their application to tests on time series and cross-section data and to estimation of simultaneous equation models.

Prerequisite: Statistics 123 [Statistics in the Social Sciences] or equivalent.

Source: Harvard University, Faculty of Arts and Sciences. Courses of Instruction, Harvard and Radcliffe, 1967-1968. Pp. 127-128.

_______________________________

Course Outline and Reading List

Dr. Sims
Spring 1968

Economics 195
Reading List and Outline

Alternative texts

Carl F. Christ, Econometric Models and Methods, Wiley, 1967
or
J. Johnston, Econometric Methods, McGraw-Hill, 1963

Other references not required for purchase:

Arthur S. Goldberger, Econometric Theory, Wiley, 1964

E. Malinvaud, Statistical Methods of Econometrics, Rand-McNally, 1966

Other required purchases:

National Income and Product of the U.S., 1929-65, U.S. G.P.O.

Economic Report of the President, 1968, U.S. G.P.O. (This may not be available until a few weeks after the semester begins.)

Some source of statistical tables — the F, t, normal and chi-squared distributions — will be necessary. Christ includes such tables. If you buy Johnston instead and you own no other source of tables, adequate sources are:

Tables for Statisticians, Barnes and Noble, or

Standard Mathematical Tables, Chemical Rubber Publishing Co.

Course Outline

Unstarred readings are passages which should parallel part of what is covered in class.

(*) Readings which are required and which may not be covered in class.
(**) optional readings.

  1. Least squares in econometrics.
    1. Abstract models which justify LS.
    2. How such Models arise in economics: structural models and conditional forecasts.

Johnston, p. 13-29, 106-112

Goldberger, p. 156-165, 179-80, 266-74, 278-80, 288-304.

Christ, Ch. VIII, IX.1-4, IV.4-6

  1. Tests of linear hypotheses in regression models

Johnston, p. 112-138

Goldberger, p. 172-180

Christ, p. 495-514

  1. Constructing models: Principles, gimmicks, and pitfalls.
    1. Models in general: Data transformations; altering specifications in the light of results.
    2. Time series forecasting models: Multicollinearity; distributed lags; lagged endogenous variables.

Christ, Ch. V

Johnston, p. 201-207, p. 212-221

Goldberger, p. 192-4

(*) Christ, p. 579-606

(*) Friend, Irwin and Robert C. Jones, “Short-Run Forecasting Models” in Models of Income Determination.

(*) Orcutt, G.H., and S.F. James, “Testing the Significance of Correlations between Time Series”, Biometrika 12/48

(**) National Bureau of Economic Research, The Quality and Significance of Economic Anticipations Data, 1960.

(**) National Bureau of Economic Research, Short Term Economic Forecasting (Studies in Income and Wealth, v. 27)

(**) National Bureau of Economic Research, Models of Income Determination (Studies in Income and Wealth, v. 28)

  1. Refinements of multiple regression
    1. Nonspherical disturbances and generalized least squares.

Christ, IX.5, IX.13

Goldberger, p. 201-212, 231-243

Johnston, Ch. 7

Zellner, A. “An Efficient Method for Estimating Seemingly Unrelated Regressions”, J. Amer. Stat. Assoc., 6/62

(**) Balestra, P. and M. Nerlove, “Pooling Cross-Section and Time Series Data in the Estimation of a Dynamic Model”, Econometrica, 7/66

    1. Errors in variables and instrumental variable estimation.

Goldberger, p. 282-287

    1. Dummy variables and analysis of variance

Goldberger, p. 218-231

Johnston, p.221-228

    1. Distributed lag estimation

Johnston, Ch.8.3

Goldberger, p. 274-8

Almon, S. “The Distributed Lag between Capital Appropriations and Expenditures”, Econometrica, 1/65

(**) Mundlak, Y., “Aggregation over Time in Distributed Leg Models”, Int. Econ. Rev. 5/61

(**) Jorgenson, D., “Rational Distributed Lag Functions”, Econometrica, 1/66

  1. Testing Revisited
    1. Serial correlation; the Durbin-Watson Statistic

Johnston, Ch.7.4

Christ, Ch.X.4

Goldberger, p. 243-244

Nerlove, M., and Wallis, K., “Use of the Durbin-Watson Statistic in inappropriate situations”, Econometrica, 1/66

    1. Tests of structural stability and forecast accuracy

Johnston, Ch.4.4

Christ, Ch. X.6, 7, 9

Goldberger, p. 169-70, 210-11

(**) Theil, H., Applied Economic Forecasting.

(**) Sims, C. “Evaluating Short-term Macro-economic Forecasts: The Dutch Experience”, Rev. Econ. and Stat. 5/67

  1. Estimation and testing in simultaneous equations models
    (Reading assignments for this section will be made when it is reached in class).
  1. Possible Additional. Topics
    (Readings for topics in this section will be assigned when and if they are reached in class.)

    1. Bayesian regression
    2. Nonlinear regression
    3. Principal components and discriminant analysis
    4. Spectral analysis

Source: Harvard University Archives. Syllabi, course outlines and reading lists in economics, 1895-2003. Box 9. Folder “Economics, 1967-68”.

_______________________________

Final Examination
Economics 195

Spring 1968
Dr. C. Sims

Do all questions on the exam. Do not spend more than the suggested amount of time on any one question, unless you have time left over at the end. Most questions include at least some quite difficult parts, so you need not finish every question to get a good grade. Formulas and a table appear on the last page.

  1. (Time: 25 minutes)
    1. Define the terms “consistent estimator” and “unbiased estimator”.
    2. We wish to estimate a scalar parameter β, and we have available four estimators, bi(n), 1=1, … 4, where n is sample size.
      Suppose we know that these estimators have the forms

b1(n) = β + (1 + ν) / n

b2(n) = β + 2 ν /n

b3(n) = β + ν /200

b4(n) = β (1 + (ν / n)),

where ν ∼ N(0,1). For each of the four estimators, tell whether it is consistent or inconsistent, biased or unbiased.

    1. Among the three estimators b1(n), b2(n) and b3(n), which is the minimum variance unbiased estimator for a sample of size 10? Would your answer change if b4(n) were included in the comparison?
  1. (Time: 40 minutes)
    We are interested in measuring the relationship between I.Q. and income using the relationship:

yi(t) = α + β xi(t) + νi(t)

where yi(t) is the income and xi(t) the I.Q. of the i’th individual at time t. The variable νi(t) is an unobserved random term assumed to satisfy E [νi(t) | xi(t)] = 0.

    1. Suppose we have observations on yi(t0) and xi(t0) for a large cross-section of individuals at a single time (t=t0). What additional assumptions are necessary to guarantee that least squares regression of y on x in this sample will yield unbiased estimates of α and β? Comment briefly on the reasonableness of these assumptions in this context.
    2. Suppose we have observations on y0(t) and x0(t) for a single individual (i=0) for a large number of time periods. What additional assumptions are necessary to guarantee that least squares regression of y on x in this sample will yield unbiased estimates of α and β? Comment briefly on the reasonableness of these assumptions in this context.
    3. For a given sample size, which would you expect to yield more reliable estimates for this model — a cross-section as in part (a) or a time series on an individual as in part (b)? Why?
    4. Give sufficient conditions for the regression in part (b) to yield consistent estimates of α and β. Comment briefly on the reasonableness of these assumptions in this context.
  1. (Time: 35 minutes)

Economist A believes an individual’s savings in a given year depend only on his mean income over the current year and the preceding year, i.e.,

\bar{y}_{i} =\left( 1/2 \right) \left( y_{i}+y_{i}\left( -1 \right) \right).

Economist B believes savings depend only on the change in income between the current and the preceding year, i.e.,

\Delta y_{i}=y_{i}-y_{i}\left( -1 \right) .

They take a sample of 20 randomly selected individuals and regress savings (si) on current and lagged income
(yi and yi (-1)) with no constant term to obtain

1)      si = .08 yi – .02 yi(-1) + residual

as an estimated equation. Their computer printout contains in addition the following information:

\sum y_{i}^{2}=5;     \sum y_{i}\left( -1 \right)^{2} =5;     \sum y_{i}y_{i}\left( -1 \right) =4;

\sum s_{i}y_{i}=.32;     \sum s_{i}y_{i}\left( -1 \right) =.22;     \sum s_{i}^{2}=.0374;

\hat{\sigma}^{2} =\ .0009

(\hat{\sigma}^{2} =\ \text{equation residual variance, unbiased estimate).}

Formulate A’s and B’s theories as hypotheses about the coefficients in (1) and compute a test of each theory at the 5% level of significance in a two-tail test. Can either hypothesis be rejected at this significance level? State the assumptions about the distribution of the residuals in the model which are necessary to justify the test you use here.

(See table on last page)

  1. (Time: 40 minutes)
    1. Two of the following cannot be covariance matrices. Which two? (Point out what’s wrong with each of the two). (7 minutes)
      1. \left[ \begin{matrix}4&1\\ 1&2\end{matrix} \right]
      2. \left[ \begin{matrix}4&-1\\ -1&4\end{matrix} \right]
      3. \left[ \begin{matrix}4&1\\ -1&4\end{matrix} \right]
      4. \left[ \begin{matrix}1&-3\\ -3&3\end{matrix} \right]
      5. \left[ \begin{matrix}1&0\\ 0&1\end{matrix} \right]
    1. What is multicollinearity? (7 minutes)
    2. What is a Koyck distributed lag relationship? (7 minutes)
    3. What is an Almon polynomial distributed lag relationship? (7 minutes)
    4. What are some advantages and disadvantages of the Koyck as compared to the Almon distributed lag relationship for purposes of econometric model-building? (12 minutes)
  1. (Time: 45 minutes)

Consider the following model of income and employment determination in New England:

[demand for output] (A) Y = a1 + b1 Yus + c1W + ν1

[demand for labor] (B) E = a2 + b2 Y + ν2

[wage determination] (C) W = a3 + b3 (L – E) + ν3

[labor supply] (D) L = a4 + b4 t + c4 W + d4 E + ν4

where Y is aggregate income in New England

L is labor force (number at work or looking for work in New England)

E is employment (number actually at work in New England)

W is the ratio of New England wages to the national average.

Yus is aggregate U.S. income.

t is the current year.

νi, i = 1, . . ., 4 are random disturbances.

    1. Which variables are most reasonably treated as endogenous, which as exogenous in this model?
    2. Would any of the variables you specify as exogenous possibly be better treated as endogenous in a larger model? Why?
    3. Using the order criterion, which equations in the model are under-identified? over-identified?
    4. Describe briefly in words the simplest way to obtain consistent estimates of equation (A) (under the usual assumptions about distribution of residuals).
    5. Suppose you discovered that the residuals from equation (A) were highly correlated with federal defense expenditures. How would you modify the model? How would this modify your answer to part (c)?
Table and Formulae t-statistic
P[ |t| > x]
Degrees of Freedom .1 .05 .01
2 2.920 4.303 6.965
8 1.860 2.306 2.896
14 1.761 2.145 2.624
15 1.753 2.131 2.602
16 1.746 2.120 2.583
17 1.740 2.110 2.567
18 1.734 2.101 2.552
19 1.729 2.093 2.539
20 1.725 2.086 2.528

Ordinary least squares

\hat{b} =\left( X^{\prime}X \right)^{-1} X^{\prime}Y

V\left( \hat{b} \right) =\sigma^{2} \left( X^{\prime}X \right)^{-1}

\hat{\sigma}^{2} =\left( Y^{\prime}Y-\hat{b}^{\prime} X^{\prime}Y \right) /\left( n-k \right)

t-test on H_{0}:\ c^{\prime}b = M


t=\frac{c^{\prime}b-M}{\sqrt{\hat{\sigma}^{2} \left( X^{\prime}X \right)^{-1}}}

Source: Harvard University, Faculty of Arts and Sciences. Papers Printed for Final Examinations [for] History, History of Religions, Government, Economics, … (June 1968).

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.

Categories
Econometrics Exam Questions Johns Hopkins

Johns Hopkins. Final Exams for “Econometrics”. Christ and Harberger, 1951-1952

 If you have ever wondered why the journal Econometrica has always published much content with next to no “econometrics” (in the sense of mathematical statistics with special application to economics), the final exams for the Johns Hopkins graduate course “Econometrics” taught by Carl Christ and Arnold Harberger in 1951-52 provide us with a ready explanation. We can see that their course offered a combination of mathematical modeling and econometrics, narrowly defined. At mid-20th century economists regarded “econometrics” as the union of mathematical economics and mathematical statistics rather than as the intersection of the two fields.

Fun fact: Marc Nerlove, who entered the Johns Hopkins graduate program in economics in 1952, was in Carl Christ’s econometrics course. This fact and the photo of Christ and Harberger come from Nerlove’s note included on the In Memoriam page for Carl Christ (1923-2017).

________________________

EXAMINATION
ECONOMETRICS

Friday, January 25, 1952 — 2-5 p.m.

Dr. Christ
Dr. Harberger

  1. A monopolist produces Z goods, X1 and X2, under constant unit costs C1 and C2 respectively. The demands for his products are

x1 = x10 – a11 (P1 – C1) – a12 (P2 – C2)
x2 = x20 – a21 (P1 – C1) – a22 (P2 – C2)

Find the Outputs of X1 and X2 which the monopolist will produce in order to maximize profits. What condition on the a’s must be satisfied if your solution is to reflect a true maximum?

  1. Prove Euler’s theorem for homogeneous functions of the first degree.
  2. Consider the utility functions

(1) U1 = ху
(2) U2 = logex + logey

For each function state:

      1. whether the marginal utility of each good is increasing, decreasing, or constant.
      2. whether the marginal utility of one good is independent of the amount of the other good consumed.
      3. the demand functions of a person having a fixed income.

What conclusions do your results suggest?

  1. Two countries. A and B, produce export commodities XA and XB at constant cost in local currency. Income in each country is stabilized by government policy, and the demand for imports depends solely on the local-currency price of imports. The exchange rate is normally fixed, but is subject to change by policy action. Assume Country A, in an initial equilibrium of the system, does not receive as much foreign currency as it has to pay for the imports its citizens demand. What are the conditions under which the gap between its receipts and expenditures of foreign currency can be decreased by devaluation? Do these same conditions apply to the gap between receipts and expenditures expressed in its own currency?
  2. Factor A is the only factor used by a monopolist, who produces good X. The suppliers of factor A always demand a constant percentage of the product price p as their unit price. At, this price they are willing to supply unlimited amounts of A.
      1. Assume returns to scale are constant. What output will the monopolist produce? Is thin output any different from that he would produce if A were free good.
      2. Assume returns to scale are decreasing. What output will the monopolist produce? Compare your present result with your answer to (a).

*  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *

ECONOMETRICS 633-34
Final Examination

Thursday, May 22, 1952
  1. Define
      1. exogenous variable
      2. overidentified equation
      3. consistency
      4. likelihood function
      5. condensed likelihood function
  2. Suppose the actual supply and demand equations for 2 goods X1 and X2 are as follows (where p1 and p2 are their respective prices, and income Y and wage rate w are exogenous):

S1:           X1 = 3p1 – 6w + 1

D1:           X1 = 2p1 – 5p2 + Y + 2

S2:           X2 = 7p2 – 8w +3

D2:           X2 = 4p1 – 4p2 + 2Y +4

State whether each equation is identified.

  1. Given that y = ax + b + u, where a is an independent variable, u is a random normal disturbance with mean 0 and constant variance σ2, and a and b are parameters. Derive the maximum likelihood estimates of a, b, and σ2 based on N observations on the pair of variables x and y.
  2. What assumptions must you make and what data do you need in order to obtain limited-information maximum-likelihood estimates of the following equation:

C= α Y + β C-1 + γ

where C and Y are real consumption and disposable income, respectively.

  1. The output of each of n industries (excluding households) is produced by a given process requiring fixed proportions of inputs of the other n-1 commodities. If these proportions are known and if a final-demand bill of goods is specified, how are the total outputs of the n industries determined?
  2. It has been asserted that the materials restrictions imposed on durable goods manufactures after Korea, while limiting the output of durable goods well below the level of 1950, did not reduce the quantities sold to a point below what they would have been in the absence of the restrictions. This assertion is supported by empirical evidence is the form of the observed accumulation of manufacturers’ and dealers’ inventories and of some price-cutting in 1951-52. Can you think of any way whereby back in 1950 you could have anticipated these developments? To answer this question, what empirical data would you seek and how would you use it, with respect to consumer durables generally or to any particular durable good?

Source: Johns Hopkins University. Eisenhower Library, Ferdinand hamburger, Jr. Archives. Department of Political Economy, Series 6, Series 7, Subseries 1, Box 3/1, Folder “Department of Political Economy, Graduate Exams 1933-1965.”

Image Source: Department of Economics, Johns Hopkins University. Webpage “In Memoriam – Carl Christ (1923-2017).” Carl Christ and Arnold Harberger at the Johns Hopkins conference in honor of Marc Nerlove, 2014.

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Econometrics Harvard Statistics Suggested Reading Syllabus

Harvard. Syllabus and partial reading list for graduate time-series econometrics. Sims, 1968-1969

Christopher A. Sims, portrait from his senior year

 

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.” 

_______________________

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.

_______________________

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.
  1. 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

  1. Extensions to multivariate case.

References: Fishman, 2.1-2.5; Jenkins and Watts, 5.1-5.2.

II. Background from Mathematical Analysis
  1. Function spaces.
  2. Linear operator on function spaces; their interpretation as limits of sequences of ordinary weighted averages.
  3. Convolution of functions with functions, of operators with functions; discrete versus continuous time.
  4. Measure functions; Lebesgue-Stieltjes measures on the real line.
  5. Integration; the Lebesgue integral, the Cauchy-Riemann integral, and the Cauchy principal value; inverting the order of integration.
  6. Fourier transforms; of functions; of operators; continuous, discrete, and finite-discrete time parameters; the inverse transform and Parseval’s theorem.
  7. 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.
  1. Random measures; the random spectral measure of a covariance stationary process; characteristics of the random spectral measure in the normal and non-normal cases.
  2. The spectral density; relation to autocovariance function; positive definiteness.
  3. 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.
  4. The moving average representation; criteria for existence of autoregressive representation.
  5. Optimal least squares forecasting and filtering.
  6. Generalized random processes.
  7. The multivariate case; cross spectra.
  8. 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.