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

 

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.

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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.

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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. 1. Errors in variables and instrumental variable estimation.

Goldberger, p. 282-287

1. 1. Dummy variables and analysis of variance

Goldberger, p. 218-231

Johnston, p.221-228

1. 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. 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”.

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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. 1. Define the terms “consistent estimator” and “unbiased estimator”.
   2. We wish to estimate a scalar parameter β, and we have available four estimators, b_i(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. 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:

y_i(t) = α + β x_i(t) + ν_i(t)

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

1. 1. Suppose we have observations on y_i(t₀) and x_i(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.,

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

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

1)      s_i = .08 y_i – .02 y_i(-1) + residual

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

        

       

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. 
      2. 
      3. 
      4. 
      5. 

1. 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 Y_us + 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.

Y_us is aggregate U.S. income.

t is the current year.

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

1. 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

t-test on

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.