a. What can you say about the nature of the bias in fi ?
b. If the sample size increases indefinitely, will the estimated f tend to equality with the true f?
13.9. Capital asset pricing model. The capital asset pricing model (CAPM) of modern investment theory postulates the following relationship between the average rate of return of a security (common stock), measured over a certain period, and the volatility of the security, called the beta coefficient (volatility is measure of risk):
where Ri = average rate of return of security i pi = true beta coefficient of security i ui = stochastic disturbance term
The true pi is not directly observable but is measured as follows:
where rit = rate of return of security i for time t rm = market rate of return for time t (this rate is the rate of return on some broad market index, such as the S&P index of industrial securities) et = residual term and where p * is an estimate of the "true" beta coefficient. In practice, therefore, instead of estimating (1), one estimates
CHAPTER THIRTEEN: ECONOMETRIC MODELING 551
where 0* are obtained from the regression (2). But since 0* are estimated, the relationship between true 0 and 0' can be written as
where vj can be called the error of measurement.
a. What will be the effect of this error of measurement on the estimate of a2?
b. Will the a2 estimated from (3) provide an unbiased estimate of true
2? If not, is it a consistent estimate of 2? If not, what remedial measures do you suggest?
13.10. Consider the model
To find out whether this model is mis-specified because it omits the variable X3 from the model, you decide to regress the residuals obtained from model (1) on the variable X3 only (Note: There is an intercept in this regression). The Lagrange multiplier (LM) test, however, requires you to regress the residuals from (1) on both X2 and X3 and a constant. Why is your procedure likely to be inappropriate?*
13.11. Consider the model
In practice we measure Xi* by Xi such that a. Xi = X* + 5
c. Xi = (X* + ei), where ei is a purely random term with the usual properties
What will be the effect of these measurement errors on estimates of true 01 and 02?
13.12. Refer to the regression Eqs. (13.3.1) and (13.3.2). In a manner similar to (13.3.3) show that
where b32 is the slope coefficient in the regression of the omitted variable X3 on the included variable X2.
13.13. Critically evaluate the following view expressed by Learner^:
My interest in metastatistics [i.e., theory of inference actually drawn from data] stems from my observations of economists at work. The opinion that econometric theory is irrelevant is held by an embarrassingly large share of the economic profession. The wide gap between econometric theory and econometric practice might be expected to cause professional tension. In fact, a calm equilibrium permeates our
^Edward E. Leamer, Specification Searches: Ad Hoc Inference with Nonexperimental Data, John Wiley & Sons, New York, 1978, p. vi.
552 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
journals and our [professional] meetings. We comfortably divide ourselves into a celibate priesthood of statistical theorists, on the one hand, and a legion of inveterate sinner-data analysts, on the other. The priests are empowered to draw up lists of sins and are revered for the special talents they display. Sinners are not expected to avoid sins; they need only confess their errors openly.
13.14. Evaluate the following statement made by Henry Theil*:
Given the present state of the art, the most sensible procedure is to interpret confidence coefficients and significance limits liberally when confidence intervals and test statistics are computed from the final regression of a regression strategy in the conventional way. That is, a 95 percent confidence coefficient may actually be an 80 percent confidence coefficient and a 1 percent significance level may actually be a 10 percent level.
13.15. Commenting on the econometric methodology practiced in the 1950s and early 1960s, Blaug stated1':
. . . much of it [i.e., empirical research] is like playing tennis with the net down: instead of attempting to refute testable predictions, modern economists all too frequently are satisfied to demonstrate that the real world conforms to their predictions, thus replacing falsification [a la Popper], which is difficult, with verification, which is easy.
Do you agree with this view? You may want to peruse Blaug's book to learn more about his views.
13.16. According to Blaug, "There is no logic of proof but there is logic of disproof."* What does he mean by this?
13.17. Refer to the St. Louis model discussed in the text. Keeping in mind the problems associated with the nested F test, critically evaluate the results presented in regression (13.8.4).
If you use observations of Yat X = -3, -2, -1, 0, 1, 2, 3, and estimate the "incorrect" model, what bias will result in these estimates?5 13.19. To see if the variable Xf belongs in the model Y = fa1 + fa2X* + u*, Ramsey's RESET test would estimate the linear model, obtaining the estimated Y values from this model [i.e., Y = fa1 + fa2X*] and then
Henry Theil, Principles of Econometrics, John Wiley & Sons, New York, 1971, pp. 605-606. ^M. Blaug, The Methodology of Economics. Or How Economists Explain, Cambridge University Press, New York, 1980, p. 256. ilbid., p. 14.
§AdaptedfromG. A. F., Linear Regression Analysis, John Wiley & Sons, New York, 1977, p. 176.
Yi = ßi + ß2 Xi + ß2 X2 + ß3 X3 + Ui but you estimate
CHAPTER THIRTEEN: ECONOMETRIC MODELING 553
estimating the model Yi = a1 + a2 Xi + a3 Yf + vi and testing the significance of a3. Prove that, if a3 turns out to be statistically significant in the preceding (RESET) equation, it is the same thing as estimating the following model directly: Yi = fi1 + fi2 Xi + fi3 Xf + ui. (Hint: Substitute for Yi in the RESET regression*).
13.20. State with reason whether the following statements are true or false.1'
a. An observation can be influential but not an outlier.
b. An observation can be an outlier but not influential.
c. An observation can be both influential and an outlier.
d. If in the model Yi = fi1 + fi2 Xi + fi3 X;2 + ui ¡3 turns out to be statistically significant, we should retain the linear term Xi even if ¡2 is statistically insignificant.
e. If you estimate the model Yi = ¡1 + ¡2 X2i + ¡3 X3i + ui or Yi = a1 + ¡2x2i + ¡3x3i + ui by OLS, the estimated regression line is the same, where x2i = (X2i — X2) and x3i = (X3i — X3).
13.21. Use the data for the demand for chicken given in exercise 7.19. Suppose you are told that the true demand function is ln Yt = ¡1 + ¡2 ln X2t + ¡3 ln X3t + ¡6 ln X& + u (1)
but you think differently and estimate the following demand function:
where Y = per capita consumption of chickens (lb) X2 = real disposable per capita income X3 = real retail price of chickens X6 = composite real price of chicken substitutes a. Carry out RESET and LM tests of specification errors, assuming the demand function (1) just given is the truth.
b. Suppose ¡6 in (1) turns out to be statistically insignificant. Does that mean there is no specification error if we fit (2) to the data?
c. If ¡6 turns out to be insignificant, does that mean one should not introduce the price of a substitute product(s) as an argument in the demand function?
13.22. Continue with exercise 13.21. Strictly for pedagogical purposes, assume that model (2) is the true demand function.
a. If we now estimate model (1), what type of specification error is committed in this instance?
b. What are the theoretical consequences of this specification error? Illustrate with the data at hand.
13.23. The true model is
Adapted from Kerry Peterson, op. cit., pp. 184-185.
^Adapted from Norman R. Draper and Harry Smith, op. cit., pp. 606-607.
554 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
but because of errors of measurement you estimate
where Yi = Y* + ei and Xi = X* + wi, where ei and wi are measurement errors.
Using the data given in Table 13.2, document the consequences of estimating (2) instead of the true model (1). 13.24. In exercise 6.14 you were asked to estimate the elasticity of substitution between labor and capital using the CES (constant elasticity of substitution) production function. But the function shown there is based on the assumption that there is perfect competition in the labor market. If competition is imperfect, the correct formulation of the model is a. What kind of specification error is involved in the original CES estimation of the elasticity of substitution if in fact the labor market is imperfect?
b. What are the theoretical consequences of this error for fa2, the elasticity of substitution parameter?
c. Assume that the labor supply elasticities in the industries shown in exercise 6.23 were as follows: 2.0, 1.8, 2.5, 2.3, 1.9, 2.1, 1.7, 2.7, 2.2, 2.1, 2.9, 2.8, 3.2, 2.9, and 3.1. Using these data along with those given in exercise 6.14, estimate the foregoing model and comment on your results in light of the theory of specification errors.
13.25. Monte Carlo experiment*: Ten individuals had weekly permanent income as follows: $200, 220, 240, 260, 280, 300, 320, 340, 380, and 400. Permanent consumption (Yi* ) was related to permanent income Xi* as
Each of these individuals had transitory income equal to 100 times a random number ui drawn from a normal population with mean = 0 and a2 = 1 (i.e., standard normal variable). Assume that there is no transitory component in consumption. Thus, measured consumption and permanent consumption are the same.
a. Draw 10 random numbers from a normal population with zero mean and unit variance and obtain 10 numbers for measured income Xi (= X* + 100 ui).
b. Regress permanent ( = measured) consumption on measured income using the data obtained in a and compare your results with those where (V/L) = value added per unit of labor
L = labor input W = real wage rate E = elasticity of supply of labor
Adapted from Christopher Dougherty, Introduction to Econometrics, Oxford University Press, New York, 1992, pp. 253-256.
CHAPTER THIRTEEN: ECONOMETRIC MODELING 555
shown in (1). A priori, the intercept should be zero (why?). Is that the case? Why or why not? c. Repeat a 100 times and obtain 100 regressions as shown in b and compare your results with the true regression (1). What general conclusions do you draw?
13.26. Refer to exercise 8.26. With the definitions of the variables given there, consider the following two models to explain Y:
Using the nested F test, how will you choose between the two models?
13.27. Continue with exercise 13.26. Using the J test, how would you decide between the two models?
13.28. Refer to exercise 7.19, which is concerned with the demand for chicken in the United States. There you were given five models.
a. What is the difference between model 1 and model 2? If model 2 is correct and you estimate model 1, what kind of error is committed? Which test would you apply—equation specification error or model selection error? Show the necessary calculations.
b. Between models 1 and 5, which would you choose? Which test(s) do you use and why?
13.29. Refer to Table 8.9, which gives data on personal savings (Y) and personal disposable income (X) for the period 1970-1995. Now consider the following models:
How would you choose between these two models? State clearly the test procedure(s) you use and show all the calculations. Suppose someone contends that the interest rate variable belongs in the savings function. How would you test this? Collect data on 3-month treasury bill rate as a proxy for the interest and demonstrate your answer.
13.30. Use the data in exercise 13.29. To familiarize yourself with recursive least squares, estimate the savings functions for 1970-1981, 1970-1985, 1970-1990, and 1970-1995. Comment on the stability of estimated coefficients in the savings functions.
13.31. Continue with exercise 13.30. Suppose you estimate the savings function for 1970-1981. Using the parameters thus estimated and the personal disposable income data from 1982-1995, estimate the predicted savings for the latter period and use Chow's prediction failure test to find out if it rejects the hypothesis that the savings function between the two time periods has not changed.
13.32. Omission of a variable in the K-variable regression model. Refer to Eq. (13.3.3), which shows the bias in omitting the variable X3 from the model Y = 0i + 02X2i + 03X3i + u. This can be generalized as follows: In the k-variable model Y = 01 + 02X2i + • • • + 0kXki + u, suppose we omit the variable Xk. Then it can be shown that the omitted variable bias
556 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
of the slope coefficient of included variable Xj is:
where bkj is the (partial) slope coefficient of Xj in the auxiliary regression of the excluded variable Xk on all the explanatory variables included in the model.*
Refer to exercise 13.21. Find out the bias of the coefficients in Eq. (1) if we excluded the variable ln X6 from the model. Is this exclusion serious? Show the necessary calculations.
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