Deck 3: Sample Exam for Chapter 6-7

Question 1

There are at least two different possible approaches to the problem of building a model of the costs of production of electric power.
I: Model I hypothesizes that per-unit costs (C) as a function of the number of kilowatt-hours produced (Q) continually and smoothly falls as production is increased, but it falls at a decreasing rate.
II: Model II hypothesizes that per-unit costs (C) decrease fairly steadily as production (Q) increases, but costs decrease at a much faster rate for hydroelectric plants than for other
kinds of facilities.
Given this information,
(a) What functional form would you recommend for estimating Model I? Be sure to write out a specific equation.
(b) What functional form would you recommend for estimating Model II? Be sure to write out a specific equation.
(c) Would $\overline { \mathrm { R } } ^ { 2 }$ be a reasonable way to compare the overall fits of the two equations? Why or why not?
Note: In the following question, you may want to change the absolute size of the coefficients, depending on the size of your school, but remember to change the size of the standard errors proportionally.

Answer 1

(a) A number of forms are possible, but a reciprocal form would be perhaps the most appropriate:
$\mathrm { C } _ { \mathrm { t } } = \rho _ { 0 } + \rho _ { 1 } / \mathrm { Q } _ { \mathrm { t } } + \varepsilon _ { \mathrm { t } }$
(b) Such a hypothesis calls for the use of a slope dummy defined (for instance) as D= 1 if the plant is hydroelectric and 0 otherwise. The resulting equation would be:
$\mathrm { C } _ { \mathrm { t } } = \beta _ { 0 } + \beta _ { 1 } \mathrm { Q } _ { \mathrm { t } } + \beta _ { 2 } \mathrm { D } _ { \mathrm { t } } + \beta _ { 3 } \mathrm { D } _ { \mathrm { t } } \mathrm { Q } _ { \mathrm { t } } + \varepsilon _ { \mathrm { t } }$
(c) $\overline { \mathrm { R } } ^ { 2 }$ is perfectly appropriate for comparing the overall fits of the two equations because the number of independent variables changes but the functional form of the dependent variable does not.

Question 2

Briefly identify the following in words or equations as appropriate:
(a) The major consequence of including an irrelevant variable in a regression equation.
(b) Four valid criteria for determining whether a given variable belongs in an equation.
(c) The problem with sequential specification searches.
(d) The elasticity of Y with respect to X in:
$\ln Y = \beta _ { 0 } + \beta _ { 1 } \ln X + \varepsilon$ (e) The sign of the bias on the coefficient of age caused by omitting experience in an equation explaining the salaries of various workers.

Answer 2

(a) See Section 6.2.
(b) See Section 6.2.
(c) See Section 6.4.
(d) Constant = $\beta$ ₁.
(e) Positive, since the expected sign of the coefficient of experience is positive and since the simple correlation coefficient between age and experience is positive. See Section 6.1.3.

Question 3

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.

Answer 3

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

Question 4

Most of Chapter 6 is concerned with specification bias.
(a) What exactly is specification bias?
(b) What is known to cause specification bias?
(c) Are unbiased estimates always better than biased estimates? Why or why not?
(d) What's the best way to avoid specification bias?