Deck 4: Sample Exam for Chapters 8-10

Question 1

A model of the number of cars sold in the United States from 1980 through 2004 produced the following results (standard errors in parentheses):
$$\begin{array} { l } \hat { \mathrm { C } } _ { \mathrm { t } } = 3738 - 48.0 \mathrm { P } _ { \mathrm { t } } + 10.0 \mathrm { Y } _ { \mathrm { t } } + 6.0 \mathrm {~A} _ { \mathrm { t } } - 360.0 \mathrm { R } _ { \mathrm { t } } \\\begin{array} { l l l l } ( 12.0 ) \quad ( 2.0 ) \quad ( 2.0 ) \quad ( 120.0 )\end{array} \\\overline { \mathrm { R } } ^ { 2 } = 0.85 \quad \text { DW } = 1.86 \quad \mathrm {~N} = 25 \text { (annual) } \\\end{array}$$ where: C_t = thousands of cars sold in year t
P_t= price index for domestic cars in year t
Y_t = disposable income (billions of dollars) in year t
A_t = billions of dollars of auto industry advertising expenditures in year t
R_t = the interest rate in year t
(a) Hypothesize the expected signs of the coefficients and test the appropriate null hypotheses at the 1% level.
(b) What econometric problems appear to be present in this equation? Why?
(c) Suppose you were now told that the simple correlation coefficients between P, A, and Y were all between 0.88 and 0.94 and that a Park test with Y as Z produced a t-score of 0.50. Would your answer to part (b) above change? Why or why not? How would it change?
(d) What suggestions would you have for another run of this regression?

Answer 1

$$\begin{array} { l c c c c } \text { (a) Coefficient } & \beta _ { \mathrm { p } } & \beta _ { \mathrm { Y } } & \beta _ { \mathrm { A } } & \beta _ { \mathrm { R } } \\\text { Expected sign } & - & + & + & - \\\text { Calculated } t \text {-score } & - 4.0 & 5.0 & 3.0 & - 3.0 \\\text { Decision } & \text { reject } & \text { reject } & \text { reject } & \text { reject }\end{array}$$ (given a 1% critical t-score of 2.528)
In addition, the DW =1.86 is higher than the d_U of 1.77, so there is no evidence of positive serial correlation.
(b) There may be theoretically sound variables that have been omitted, for instance some measure of competition from foreign-made cars, but the results themselves give no indication of any problems.
(c) These results are clear indications of multicollinearity but not of heteroskedasticity.
(d) Unless a theoretically sound variable can be added measuring competition, the regression need not be changed at all.

Question 2

In a study of the long-run and short-run demands for money, Chow estimated the following demand equation (standard errors in parentheses) for the United States from 1947:1 through 1965:4: $$\begin{array} { l } \hat { \mathrm { M } } _ { \mathrm { t } } = 0.14 + 1.05 \mathrm { Y } _ { \mathrm { t } } ^ { * } - 0.01 \mathrm { Y } _ { \mathrm { t } } - 0.75 \mathrm { R } _ { \mathrm { t } } \\\text { (0.15) (0.10) (0.05) } \\\overline { \mathrm { R } } ^ { 2 } = 0.996 \quad \mathrm { DW } = 0.88 \quad \mathrm {~N} = 76 \text { (quarterly) } \\\end{array}$$ where: M_t = the log of the money stock in quarter t
$\mathrm { Y } _ { \mathrm { t } } ^ { * }$ =the log of permanent income (a moving average of previous quarters' current
income) in quarter t
Y_t =the log of current income in quarter t
r_T = the log of the rate of interest in quarter t
(a) Hypothesize signs and test the appropriate null hypotheses at the 5% level of significance.
(b) What econometric problems seem likely to be in this equation?
(c) In particular, are there are any problems related to the coefficient of Y? If so, are these problems more likely to have been caused by multicollinearity, serial correlation, or heteroskedasticity?
(d) What suggestions would you have for another estimation of this equation? Why?

Answer 2

$$\begin{array} { l c c c } \text { (a) Coefficient } & \beta _ { Y ^ { * } } & \beta _ { Y } & \beta _ { \mathrm { R } } \\\text { Expected sign } & + & + & - \\\text { Calculated } t \text {-score } & 7.0 & - 0.1 & - 15.0 \\\text { Decision } & \text { reject } & \text { do not } & \text { reject } \\& & \text { reject } &\end{array}$$ (given a 5% critical t-score of about 1.67)
In addition, the DW = 0.88 is less than the d_L of 1.54, so we can reject the null hypothesis of no positive serial correlation.
(b) The poor significance of could have been caused by multicollinearity, an omitted variable, or by an irrelevant variable (Y) and in addition, as mentioned above, we have serial correlation.
$\hat { \beta } _ { \mathrm { Y } }$ (c) Of the three, only multicollinearity between Y ^* and Y could have caused the problem, since serial correlation and heteroskedasticty would cause the t-scores to appear higher than they actually should be.
(d) This is not an easy question, but we would not suggest dropping Y, nor would we consider GLS right now. It turns out that making the equation a distributed lag by adding M_t-1 to the right-hand side switches the insignificance to Y* (and raises the DW, naturally), giving another indication of multicollinearity. The best action right now would seem to be to review the theory in search of an omitted variable and failing to find one, do nothing. This problem was adapted from Edward Tower, Econometrics Exams, Puzzles and Problems (Durham, NC: Eno River Press, 1985) p. 167.

Question 3

Briefly identify the following in words or equations as appropriate:
(a) Impure serial correlation
(b) Dominant variable
(c) Variance inflation factor
(d) Generalized least squares
(e) Given a calculated Durbin-Watson d statistic of 2.58, a d_L of 1.21, and a d_U of 1.55, what would you conclude?

Answer 3

(a) See Section 9.1.
(b) See Section 8.1.
(c) See Section 8.3.
(d) See Section 9.4.
(e) The answer depends on whether you encourage your students to use one-sided or two-sided tests. For a one-sided test, the correct answer is that we can reject the null hypothesis of positive serial correlation. For a two-sided test, the correct answer is that the test is inconclusive.

Question 4

Carefully outline (be brief!) a description of the problem typically referred to as pure heteroskedasticity.
(a) What is it?
(b) What are its consequences?
(c) How do you diagnose it?
(d) What do you do to get rid of it?