Example Questions for AAE637 Exam
1. In the 1970s, there was an increase in the number of statistical analyses focused on
the factors of production in the aggregate U.S. manufacturing sector. Two of the
more important pieces of research were undertaken by Berndt and Wood (1975,
1979)1, who collected measures of aggregate U.S. manufacturing output as well as
indices of the use of capital (K), labor (L), and other material (M) inputs over the
1947-1971 period. Although they used a different model in the above work, you have
obtained their data and as a first pass you decide to estimate the following CobbDouglas production function.
(1.1)
Y   M 1 K 2 L3 exp( )
Table 1 provides a partial listing of the output you obtain from estimating the above
model after you transform it in the usual manner to enable you to use CRM
estimation techniques. Again assume that all CRM assumptions hold.
(a) (10 pts) In Table 1 there are 12 pieces of information not displayed. Using the
information provided, provide estimates of the missing information.
(b) (5 pts) Let’s refer to the naïve model of production as one where you assume that
ln(Yt) would equal ln(Yt ) for all observations regardless the level of input use.
Undertake a formal hypothesis test that when compared to this naïve model, the
estimation of (1.1) generates a statistically significant increase in explanatory
power.
(c) (5 pts) Test the null hypothesis that the sum of the materials (M) and labor (L)
output elasticities equals 1.0. What are the null and alternate hypotheses and what
is the result of your hypothesis test?
Berndt, E.R. and D. O. Wood, 1975. “Technology, Prices and the Derived Demand for Energy”, Review
of Economics and Statistics, 57:3, August, 259-268; Berndt, E.R. and D.O. Wood, 1979. “Engineering and
Econometric Interpretation of Energy-Capital Complementarity”, American Economic Review, 69:3, June,
342-354.
1
Table 1. Summary of Regression Results
No. of Obs
DF
SSE
TSS
R2
Adjusted R2
σ2 U
σU
Variable
Intercept
LN(Materials)
LN(Capital)
LN(Labor)
Regression Statistics
25
0.997755
0.014493
Estimate
6.8232
0.8292
0.0524
0.3414
Std. Error
T-Value
48.82
0.0777
0.1090
3.13
Partial Listing of the Elements of (X'X)-1
(1st row and column missing)
Materials
Capital
Labor
Materials
28.770
Capital
-6.767
7.456
Labor
-34.659
-0.690
56.609
2. Suppose you and grandmother are both interested in how the sale price of a house is
affected by its distance to the nearest WAL-MART. You both go collect data and
estimate regression models using all of the CRM assumptions. Your grandmother,
having a strong econometrics background, estimates the following model:
(2.1)
yi= β0 + β1xi + εi
where yi = sale price of the house where the sale price is measured in $
xi = distance to nearest WAL-MART where distance is measured in miles
Using the same data you estimate the following model:
(2.2)
yi*= α0 + α1*xi + υi
where yi* = sale price of the house measured in $000
xi* = distance to nearest WAL-MART where distance is measured in
kilometers ( 1 km = .62 miles)
Because you are a good econometrician, you observe that yi* = yi/1000 and xi* =
xi/.62
(a) You want to compare your estimates. What is the mathematical relationship
between β0 and α0? Between β1 and α1? Between V(εi) and V(υi)?
(b) Compare your estimate of the elasticity of price with respect to distance using the
results of your regression model and the elasticity estimate obtained from your
grandmother's model? Give some brief intuition behind this result.
3.
The regression equation Y = 1 + 2 X +  was estimated using 80 cross-sectional
observations on countries via classical regression techniques. To check for
heteroskedasticity related to population, separate regressions were run for the 32
countries with the lowest populations and the 32 countries with the highest
populations. The sum of squared residuals for the low-population countries was
240. The sum of squared residuals for the high-population countries was 90.
(a) Compute unbiased estimates of the variance of the error term in the two
subsamples.
(b) Given these results, which subsample appears to lie closer to the true
regression line: the low-population-countries or the high-population countries?
Explain your answer.
(c) Test the null hypothesis of homoskedasticity, against the (one-sided)
alternative hypothesis that high-population countries have higher error
variance, at 5% significance using a Goldfeld-Quandt test. Give the value of
the test statistic, its distribution under the null hypothesis, the critical point,
and your conclusion (accept or reject the null hypothesis of homoskedasticity).
(d) Suppose you believe that heteroskedasticity was indeed present and that the
variance of the error term were inversely proportional to population (i.e.,
Var(i ) = POPi , where  = an unknown constant and POPi = population of
the ith country). Provide the formulas necessary to transform the data so as to
ensure that the above error terms satisfy the classical assumptions of the linear
model.
(e) Suppose the first observation in the raw data were as shown below:
Obs(i) Xi
Yi
POPi
1
50
60
100
Use the formulas you gave in part (d) to compute the first observation of the
transformed data.
4.
Assume you have a sample of observed values of a random variable, yi (i=1,…,T)
which are distributed iid .
(a) Assume that yi has the following pdf, f(yi): f(y t )=β
Xβt
yβt
. Find the maximum
likelihood estimator given your T sample of observations
(b) Assume that that yi has the following pdf, f(yi):
f ( y) 
 y e
y!
, y  0,1, 2,3,... That is, y is a Poisson random number. y has
the characteristic the its mean and variance equals the parameter λ. Find the
maximum likelihood estimation of λ. Find the mean and variance of the
maximum likelihood estimator of λ. Is this estimator consistent?
5.
You have been hired as a consultant by a firm that manages a large number of egg
laying operations across the U.S. They are interested in obtaining a better
understanding of egg consumption patterns in the U.S. As such you collect monthly
data and formulate the following model where you let Eggt represent the per capita
number of eggs consumed in month t and Bacont to be the per capita pounds of
bacon consumed:
Eggt = α + βBacont + εt
where α and β are unknown coefficients, and εt is an error term where εt ~ N(0,σ2IT).
You suspect there may be a complicating factor determining egg consumption.
That is, Easter, which occurs in April, increases the consumption of eggs. To
recognize this effect you define a dummy variable, Aprilt which is 1 if the month t
is April, 0 otherwise. To analyze consumption patterns, you are given the following
set of results which all come from the same data set consisting of n=81 observations
(estimated coefficient std. errors are in parentheses). Unfortunately, your GAUSS
coding is not what it should be and some pieces of information were not printed out.
Using the information that did print you should be able to answer the following
questions.
Regression Results from Alternative Models of U.S. Egg Consumption
Variable
Model I
Model II
Model III
30.69
(4.46)
2.15
(XXX)
18.89
(XXX)
2.15
(0.08)
23.60
(XXX)
RSS
53,230
XXXXX
21.03
(3.76)
2.08
(XXX)
19.33
(5.32)
0.14
(XXX)
XXXXX
R2
0.7519
0.9132
0.9140
Intercept
Bacont
Aprilt
Aprilt Bacont
a. Using Model III, sketch two regression lines showing the relationship between
egg and bacon consumption, one for April and one for December. Make sure
you explicitly identify the values of the slope and intercept for these two
months.
b. Test at the 0.05 level, the hypothesis that either the intercept, slope or both are
different in April than other months. Write out the null and alternate
hypotheses.
c. Test at the 0.05 level, the hypothesis that the intercept is different in April
given that the slope is the same in April as in non-April months. Write out the
null and alternate hypotheses.
d. Test at the 0.05 level the hypothesis that the slope changes in April, given that
the intercept is different. Write out the null and alternate hypotheses.
e. Using the information provided for Model I, test at the 0.05 level the
hypothesis that Bacon consumption has a significant impact on egg
consumption. Write out your null and alternate hypotheses. (Hint: What is
your restricted model? Look at Question 1(c). )