Economics 4818 - Introduction to Econometrics
Midterm 1 - Answers
1. Discuss the e¤ect of the following phenomena on the unbiasedness and the variance
of the OLS estimators, cj :
1. heteroskedasticity
Answer: Heteroskedasticity means that the error variance is not a constant and so
Assumption 5 of the Gauss-Markov assumptions is violated, that is V ar (u) 6= 2 .
(3 points) E¤ ect on the unbiasedness of cj : Heteroskedasticity does not a¤ect the
unbiasedness of the OLS estimators, cj .
(3 points) E¤ ect on the variance of cj : Under heteroskedasticity, the usual formula
2
for the variance of OLS estimators, V ar cj = SST 1 R2 , is no longer valid.
(
j
j)
2. perfect collinearity
Answer: Perfect collinearity means that some independant variable, xj , is a perfect
linear combination of other independent variables.
(3 points) E¤ ect on the unbiasedness of cj : Perfect collinearity violates Assumption
3 for the unbiasedness of OLS estimators, cj . In fact, under perfect collinearity, the
OLS estimators, cj , practically cannot be computed. That’s why we have to drop
one of the variables in the perfect linear combination.
(3 points) E¤ ect on the variance of cj : If the independant variable, xj , is a perfect
linear combination of other independent variables, then Rj2 = 1 and the variance of
2
c, which is given by V ar c =
, is not de…ined (or it is in…nitely large).
j
j
SSTj (1 Rj2 )
3. multicollinearity
Answer: Multicollinearity means that some independant variables are higly correlated with each other.
(3 points) E¤ ect on the unbiasedness of cj : Multicollinearity does not a¤ect the
unbiasedness of the OLS estimators, cj .
(3 points) E¤ ect on the variance of cj : If the independant variable, xj , is highly
correlated with other independent variables, then Rj2 is "close to 1". Thus, the variance
2
of cj , given by V ar cj = SST 1 R2 , is large and this decreases the precision of
j(
j)
c.
j
4. including an irrelevant variable
Answer:
1
(3 points) E¤ ect on the unbiasedness of cj : Including an irelevant variable does
not a¤ect the unbiasedness of the OLS estimators, cj .
(3 points) E¤ ect on the variance of cj : Including an irelevant variable increases the
variances of the OLS estimators of the relevant independent variables. In particular,
2
Rj2 increases and thus, the variance of cj , given by V ar cj = SST 1 R2 , increases.
(
j
j)
5. omitting a relevant variable
Answer:
(3 points) E¤ ect on the unbiasedness of cj : If the omitted relevant variable is
correlated with some of the other independent variables in the model, omitting the
relevant variable biases the OLS estimators cj . In particular, the omitted relevant
variable will be included in the error term u, and will violate the zero conditional mean
assumption (Assumption 4). If the omitted relevant variable is not correlated with
any of the other independent variables in the model, omitting the relevant variable
will not bias the OLS estimators cj .
(3 points) E¤ ect on the variance of cj : Omitting a relevant variable actually decreases the variances of the OLS estimators of the remaining independent variables. In
2
particular, Rj2 decreases and thus, the variance of cj , given by V ar cj = SST 1 R2 ,
j(
j)
decreases.
2.
A) What does R2 measure?
Answer: (5 points) R2 measures how well the OLS regression line …ts the data in
the sample. More speci…cally, it measures what portion of the variation in y is explained
by the variation in the independent variables x1 ; x2 ; ::::; xk . Mathematically, R2 = SSE
SST .
B) Is a regression useless if its R2 is low? Explain.
Answer: (5 points) Not necessarily. However, it means that the independent variables in the regression, x1 ; x2 ; ::::; xk , are not very important factors of y.
C) If R2 = 1, how close is the OLS regression line to the actual data points?
Answer: (5 points) All data points are on the OLS regression line.
3. Suppose you have a simple model explaining monthly salary in dollars (wage) in
terms of IQ score (IQ):
wage =
0
+
The OLS regression line for this model is:
2
1 IQ
+u
wage
[ = 116:9 + 8:30IQ
n = 936; R2 = :096
A) Interpret the OLS estimates in this regression.
Answer: (5 points) The interpretation of the intercept estimate is not very meaningfull: an individual with IQ = 0 has a predicted wage of $116:9. The interpretation of
the slope estmate is that, other things equal, an increase in IQ of 1 point is predicted to
increase an individual’s wage by $8:30.
B) What kinds of factors are contained in the eror term u? List at least two factors
that are included in u and are correlated with IQ.
Answer: (5 points) Education and innate ability, for example, are included in u and
are correlated with IQ. Thus the zero conditional mean assumption is violated.
C) Does this simple regression model uncover the ceteris paribus e¤ect of IQ on wage?
Explain.
Answer: (5 points) No, it doesn’t. The OLS estimates c0 and c1 are biased since
the zero conditional mean assumption is violated.
D) Does IQ explain most of the variation in wage in the above regression?
Answer: (5 points) No, IQ explains only 9:6% of the variation in wage. Thus, there
are other important factors of wage, which explain more of its variation.
E) Suppose you have estimated an alternative speci…cation of the wage
ship:
IQ relation-
log\
(wage) = 5:89 + :0088IQ
n = 936; R2 = :099
Interpret the slope estimate in this model?
Answer: (5 points) Other things equal, an increase in IQ of 1 point is predicted to
increase an individual’s wage by 100 (:0088) = 0:8%.
F) Which regression model is preferred - the one with dependent variable wage or the
one with dependent variable log (wage)? Explain.
Answer: (5 points) The second model is preferred because it explains more of the
variation in wage.
3
4. Suppose you have the model
price =
0
+
1 sqrf t
+
2 bdrms
+u
where price is the house price measured in thousands of dollars, sqrf t is the size of
the house in square feet and bdrms is the number of bedrooms in the house. The OLS
regression line for this model is:
[ =
price
19:32 + :128sqrf t + 15:20bdrms
n = 88; R2 = :632
A) Interpret the estimated coe¢ cient on sqrf t.
Answer: (5 points) Holding bdrms …xed, a 1 square foot increase in the square
footage of a house is predicted to increase the price of the house by $128.
B) What s the predicted increase in price for a house having one more bedroom, holding
square footage constant?
Answer: (5 points) $15; 200.
C) Are the variances of the slope estimators c1 and c2 likely to be large or small?
Explain.
Answer: (5 points) The slope estimators c1 and c2 are likely to have large variances
because sqrf t and bdrms are highly correlated. In particular, the terms R12 and R22 are
2
2
large and so V ar c1 = SST 1 R2 and V ar c1 = SST 1 R2 are large.
)
(
1(
2
1
2)
D) If you run the regression
] = f0 + f1 sqrf t;
price
what would be the likely sign and magnitude of f1 relative to c1 = :128? Explain.
Answer: (5 points) f1 is biased since the relevant variable bdrms is omitted from the
second regression model. The bias is given by the formula Bias f1 = 1 e1 . Here, 1
has been estimated to be positive, c1 = :128, and sqrf t and bdrms are positively correlated,
e1 > 0. So, Bias f > 0. Thus, f is likely to have a positive sign and to be larger in
1
1
magnitude than c1 = :128.
4