On your way to the cashier's office (to pay yet another dorm damage bill), you overhear U. R. Accepted (the Dean of Admissions) having a violent argument with I. M. Smart (the director of the Computer Center) about an equation that Smart built to understand the number of applications that the school received from high school seniors. In need of an outside opinion, they turn to you to help them evaluate the following regression results (standard errors in parentheses):
$\begin{array} { c } \hat { \mathrm { N } } _ { \mathrm { t } } = 150 + 180 \mathrm {~A} _ { \mathrm { t } } + 1.50 \ln \mathrm { T } _ { \mathrm { t } } + 30.0 \mathrm { P } _ { \mathrm { t } } \\( 90 ) \quad ( 1.50 ) \quad ( 60.0 ) \\\overline { \mathrm { R } } ^ { 2 } = 0.50 \quad \mathrm {~N} = 22 \quad \text { (annual) }\end{array}$ where: N_t =the number of high school seniors who apply for admission in year t
A_t=the number of people on the admission staff who visit high schools full time
spreading information about the school in year t
T_t =dollars of tuition in year t
P_t =the percent of the faculty in year t that had PhDs in year t
How would you respond if they asked you to:
(a) Discuss the expected signs of the coefficients.
(b) Compare these expectations with the estimated coefficients by using the appropriate tests.
(c) Evaluate the possible econometric problems that could have caused any observed differences between the estimated coefficients and what you expected.
(d) Determine whether the semilog function for T makes theoretical sense.
(e) Make any suggestions you feel are appropriate for another run of the equation.

Question

On your way to the cashier's office (to pay yet another dorm damage bill), you overhear U. R. Accepted (the Dean of Admissions) having a violent argument with I. M. Smart (the director of the Computer Center) about an equation that Smart built to understand the number of applications that the school received from high school seniors. In need of an outside opinion, they turn to you to help them evaluate the following regression results (standard errors in parentheses):

\begin{array} { c } \hat { \mathrm { N } } _ { \mathrm { t } } = 150 + 180 \mathrm {~A} _ { \mathrm { t } } + 1.50 \ln \mathrm { T } _ { \mathrm { t } } + 30.0 \mathrm { P } _ { \mathrm { t } } \\( 90 ) \quad ( 1.50 ) \quad ( 60.0 ) \\\overline { \mathrm { R } } ^ { 2 } = 0.50 \quad \mathrm {~N} = 22 \quad \text { (annual) }\end{array}

where: N_t =the number of high school seniors who apply for admission in year t
A_t=the number of people on the admission staff who visit high schools full time
spreading information about the school in year t
T_t =dollars of tuition in year t
P_t =the percent of the faculty in year t that had PhDs in year t
How would you respond if they asked you to:
(a) Discuss the expected signs of the coefficients.
(b) Compare these expectations with the estimated coefficients by using the appropriate tests.
(c) Evaluate the possible econometric problems that could have caused any observed differences between the estimated coefficients and what you expected.
(d) Determine whether the semilog function for T makes theoretical sense.
(e) Make any suggestions you feel are appropriate for another run of the equation.

Quizplus · Accepted Answer

The Answer is

\begin{array} { l c c c } ext { (a) } / ext { (b) Coefficient } & \beta _ { 1 } & \beta _ { 2 } & \beta _ { 3 } \ ext { Expected sign } & + & - & + \t ext {-score } & + 2.0 & + 1.0 & + 0.5 \ ext { Decision } & ext { reject } & ext { do not } & ext { do not } \& & ext { reject } & ext { reject }\end{array}

(given the 5% level and a resulting critical t-score of 1.734)
(c)

\hat { \beta { } } _ { 2 }

has an insignificant unexpected sign, and

\hat { \beta } _ { 3 }

is not significant, so an irrelevant variable and an omitted variable (s) are both possible.
(d) Many would expect that the impact of an extra dollar of tuition (on applications) decreases as tuition increases. If this is so, the semilog functional form makes sense; if not, then we should use the linear form as a default.
(e) Percent PhDs appears to be intended as a proxy for quality of the school. If we could find a better proxy for quality, then it would make sense to substitute it for X₃, and that would be a possible answer. If the unexpected sign for lnT is being caused by an omitted variable, that omitted variable must be causing positive bias. Thus for a suggested omitted variable to be a good suggestion, it would have to be likely to have caused positive bias. Therefore it must be either positively correlated with tuition and have a positive expected coefficient (like the number of high school seniors in the country or in the region) or else have a negative correlation with tuition and have a negative expected coefficient (like the number of inadequate facilities, etc.).

On Your Way to the Cashier's Office (To Pay Yet