Economics 231W, Econometrics
University of Rochester
Fall 2008
Homework: Chapter 6
Text Problems: Pages 156 – 165: 6.1, 6.4, 6.7, 6.8,
6.1.
(a) It states how the population
mean value of the dependent
variable is related to one or more explanatory variables.
(b) It is the sample counterpart of the PRF.
(c) It tells how the individual Y are related to the explanatory
variables and the stochastic error term, u, in the population as a
whole.
(d) A model that is linear in the parameters, the Bs.
(e) It is a proxy for all omitted or neglected variables that affect the
dependent variable Y. The individual influence of each of these
variables is random and small so that on average their influence on Y
is zero.
(f) It is the sample counterpart of the stochastic error term.
(g) The expected value of Y conditional upon a given value of X. It is
obtained from the conditional (probability) distribution of Y, given
X.
(h) The expected value of an r.v. regardless of the values taken by
other random variables. It is obtained from the unconditional, or
marginal, probability distributions of the relevant random variables.
(i) The B coefficients in a linear regression model are called
regression coefficients or regression parameters.
(j) The bs, which tell how to compute the Bs, are called the
estimators. Numerical values taken by the bs are known as estimates.
6.4.
(a) False. The residual e i is an approximation (i.e., an estimator) of
the true error term, u i .
(b) False. It gives the mean value of the dependent variable, given
the values of the explanatory variables.
(c) False. A linear regression model is linear in the parameters and
not necessarily linear in the variables.
(d) False, generally. The cause and effect relationship between the
Xs and Y must be justified by theory.
(e) False, unless the “conditioned” and conditioning variables are
independent.
(f) False. It is the other way around.
(g) False. It measures the change in the mean value of Y per unit
change in X.
(h) Uncertain.
There are many a phenomena which can be
explained by the two-variable model. One example is the Market
Model of portfolio theory which regresses the rate of return on a
single security on the rate of return on a market index (e.g., S&P 500
stock index). The slope coefficient in this model, popularly known
as the beta coefficient, is used extensively in portfolio analysis.
(i) True.
6.7.
(a) The answer will depend on how the various components of GDP
(consumption expenditure, investment expenditure, government
expenditure and expenditure on net exports) react to the higher
interest rate. For instance, ceteris paribus, investment expenditure
and the interest rate are inversely related.
(b) Positive. Ceteris paribus, the higher the interest rate is, the
greater will be the incentive to save.
(c) Generally positive.
(d) Positive, to maintain at least the status quo.
(e) Probably positive.
(f) Probably negative; familiarity may breed contempt.
(g) Probably positive.
(h) Positive. Statistics is a major foundation of econometrics.
(i) Positive. As income increases, discretionary income is likely to
increase, leading to an increased demand for more expensive cars. A
large number of Japanese cars are expensive. In general, the income
elasticity of demand for items like cars has been found to be not only
positive but generally greater than 1.
6.8.
(a) Yes
(b) Yes
(c) Yes
(d) Yes
(e) No
(f) No.
Other Problems
1. The following is a random sample of weight, in pounds, and height, in inches over 5 feet.
Height and Weight of a Random Sample
a.
b.
Observation
Y = Weight (Pounds)
X = Height (Inches
over 60)
1
125.0
5.0
2
145.0
6.0
3
225.0
12.0
4
200.0
10.0
5
150.0
6.0
6
185.0
1.0
7
195.0
2.0
8
135.0
5.0
9
140.0
6.0
10
190.0
11.0
What is the SRF function for this data? Show all of your calculations. This problem is to be done
“by hand”.
Interpret your estimated regression equation.
Height and Weight of a Random Sample
Observation
1
2
3
4
5
6
7
8
9
10
Y=
Weight
(Pounds)
125.0
145.0
225.0
200.0
150.0
185.0
195.0
135.0
140.0
190.0
X=
Height
(Inches
over
60)
5.0
6.0
12.0
10.0
6.0
1.0
2.0
5.0
6.0
11.0
(Y(i) Y-bar)
-44.0
-24.0
56.0
31.0
-19.0
16.0
26.0
-34.0
-29.0
21.0
(X(i) X-bar)
-1.4
-0.4
5.6
3.6
-0.4
-5.4
-4.4
-1.4
-0.4
4.6
(X(i) Xbar)^2
2.0
0.2
31.4
13.0
0.2
29.2
19.4
2.0
0.2
21.2
(Y(i) - Ybar)*(X(i)
- X-bar)
61.6
9.6
313.6
111.6
7.6
-86.4
-114.4
47.6
11.6
96.6
Sum
Mean
1690.0
169.0
b1 =
b2 =
144.19
3.877
64.0
6.4
0.0
0.0
0.0
0.0
118.4
11.8
459.0
45.9
The estimated regression equation is given by: Y(i)-hat = 144.19 + 3.88*X(i)
Interpret the coefficient b1: If someone is 5 feet tall, they are predicted to
weigh 144.19 lbs
Interpret the coefficient b2: For each inch over 60 inches, weight increases
by 3.88 pounds.
2. Okun’s Law is one of the more famous two-variable regression models and can be written as
%GDPt  B1  B2 Unemployment t  ut
a. Using the Data “Okun’s Law” from: http://www.geneseo.edu/~annala/Econ_231W.html , estimate the
sample regression function over the period 1978 to 2007.
%GDPt  b1  b2 Unemployment t  et
Note: You will have to calculate both the %ΔGDP and ΔUnemployment from the raw data:
%X 
X t  X t 1
 100
X t 1
X  X t  X t 1
When you calculate the percent change and the absolute change you will lose one observation.
b. Write out the estimated regression equation.
%ΔGDPt = 2.826 – 1.917ΔU
c. How does a one percentage point increase in the unemployment rate effect the growth rate of GDP?
Each 1 point increase in U is associated with a 1.917 decrease in the growth rate of GDP.
d. If the unemployment rate remains unchanged, what is the expected rate of growth of GDP?
If Δ = 0, %ΔGDP = 2.826
SUMMARY
OUTPUT
Regression Statistics
Multiple R
0.885
R Square
Adjusted R Square
Standard Error
Observations
0.783
0.775
0.831
29
ANOVA
df
1
27
28
SS
67.374
18.651
86.025
Coefficients
2.826
-1.917
Standard
Error
0.155
0.194
Regression
Residual
Total
Intercept
Change U
MS
67.374
0.691
F
97.536
t Stat
18.271
-9.876
P-value
0.000
0.000
Significance
F
0.000
Using the data “Compensation and Productivity” we can study the relationship between productivity and
real wages in the business sector, from 1980 – 2008. Both “Compensation” and “Productivity” are index
values calculated by the BLS, and “Compensation” can be considered a measure of the real wage.
a. Estimate the regression equation:
Compensationt  b1  b2 Pr oductivityt  et
b. Interpret your regression results. Does the intercept have an economic interpretation in this case?
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.997
0.994
0.994
2.904
114
ANOVA
df
Regression
Residual
Total
Intercept
X = Productivity Per
Hour
1
112
113
SS
148988.154
944.534
149932.688
MS
148988.154
8.433
F
17666.569
Coefficients
-94.937
Standard
Error
1.557
t Stat
-60.972
P-value
0.000
1.945
0.015
132.916
0.000
Significance
F
0.000