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?

Question

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?

Quizplus · Accepted Answer

The Answer is

\begin{array} { l c c c } ext { (a) Coefficient } & \beta _ { Y ^ { * } } & \beta _ { Y } & \beta _ { \mathrm { R } } \ ext { Expected sign } & + & + & - \ ext { Calculated } t ext {-score } & 7.0 & - 0.1 & - 15.0 \ ext { Decision } & ext { reject } & ext { do not } & ext { reject } \& & ext { 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.

In a Study of the Long-Run and Short-Run Demands for Money