MLR Problem Set
1. In the equation, y = β0 + β1 x1 + β2 x2 + u, β2 is a(n) _____.
a. independent variable
b. dependent variable
c. slope parameter
d. intercept parameter
2. If the explained sum of squares is 35 and the total sum of squares is 49, what is the residual
sum of squares?
a. 10
b. 12
c. 18
d. 14
3. Exclusion of a relevant variable from a multiple linear regression model leads to the problem
of _____.
a. omitted variable bias
b. multicollinearity
c. perfect collinearity
d. homoskedasticity
4. The normality assumption implies that:
a. the population error u is dependent on the explanatory variables and is normally distributed
with mean equal to one and variance σ2.
b. the population error u is independent of the explanatory variables and is normally
distributed with mean equal to one and variance σ.
c. the population error u is dependent on the explanatory variables and is normally distributed
with mean zero and variance σ.
d. the population error u is independent of the explanatory variables and is normally
distributed with mean zero and variance σ2.
5. Which of the following is true of heteroskedasticity?
a. Heteroskedasticty causes bias in the Ordinary Least Squares estimators.
b. Population R2 is affected by the presence of heteroskedasticty.
c. The Ordinary Least Square estimators are not the best linear unbiased estimators (BLUE)
if heteroskedasticity is present.
d. It is not possible to estimate a MLRM if errors are heteroskedastic.
6. Consider the following regression model: yi=β0+β1 xi+ui. If the first four Gauss-Markov
assumptions hold true, and the error term is homoskedastic, then _____.
a. Var(u|x) =0
b. Var(u|x) =1
c. Var(u|x) = σ2
d. Var(u|x) =σ
7. When the estimated slope coefficient in the simple regression model, ̂1 , is zero, then
R2 = Y .
0 < R2 < 1.
R2 = 0.
R2 > (SSR/TSS).
a.
b.
c.
d.
8. The sample average of the OLS residuals is
a. some positive number since OLS uses squares.
b. zero.
c. unobservable since the population regression function is unknown.
d. dependent on whether the explanatory variable is mostly positive or negative.
9. E(u | X) =
a.
b.
c.
d.
0 says that
dividing the error by the explanatory variable results in a zero (on average).
the sample regression function residuals are unrelated to the explanatory variable.
the sample mean of the Xs is much larger than the sample mean of the errors.
the conditional distribution of the error given the explanatory variable has a zero mean.
10. In a SLRM, multiplying the dependent variable by 100 and the explanatory variable by 100,000
a. leaves the OLS estimate of the slope the same.
b. leaves the OLS estimate of the intercept the same.
c. leaves the regression R2 the same.
d. leaves the variance of the OLS estimators the same.
True or False
1) When you have an omitted variable problem, the assumption that E(ui | Xi) = 0 is violated. This
implies that the sum of the residuals is no longer zero.
2)
3)
4)
5)
6)
7)
8)
2
2
A high R or R means that the model explains much of the variation in y and that there is no
omitted variable bias.
Adding more regressors can cause R2 to decrease if the regressors are irrelevant.
Adding irrelevant regressors will not bias the relevant coefficients (the s), but may increase the
standard errors on those coefficients.
R2 is usually very high for models using time series data.
The normality assumption is required for OLS estimators to be BLUE.
The conditional mean assumption is required for OLS to be BLUE.
The B in BLUE stands for Best and refers to the consistency of OLS estimators.
Problems
1. Concerned about the lack of sleep of 5 college students, a caring professor estimated the
following model:
̂ = 3,840.83 − .163𝑠𝑡𝑢𝑑𝑦 − 11.71𝑐𝑙𝑎𝑠𝑠 − 8.70𝑎𝑔𝑒 + 87.75𝑀𝐻𝐶
𝑠𝑙𝑒𝑒𝑝
n=706; R2=0.123
where sleep is the total minutes of sleep per week, study is the total weekly minutes spent
working on assignments, class is years of education (12 if first year, 13 if second year, etc.), age
is in years, and MHC is a dummy equal to 1 if the student is a Mount Holyoke College student
and 0 if she is from another campus.
Interpret each slope coefficient. Does the intercept have a useful interpretation? Explain the R2.
2. Specify a model that would allow you to estimate the price elasticity of demand for a good (be
sure to label the variables and indicate which coefficient(s) would give you the price elasticity
(you do not have to be concerned with any variables other than price and quantity)).
3. One reason that “town-gown” relationships (the relationship between non-student inhabitants
of the college town and the college students) can be stressed is that inhabitants feel that a large
student population drives up apartment rents. The following model tests whether or not rents are
influenced by the student population of a college town:
log(𝑟𝑒𝑛𝑡) = 𝛽0 + 𝛽1 log(𝑝𝑜𝑝) + 𝛽2 log(𝑎𝑣𝑒𝑖𝑛𝑐) + 𝛽3 𝑝𝑐𝑡𝑠𝑡𝑢 + 𝑢
where rent is the average monthly rent in dollars, pop is the total city population, aveinc is the
average city income, and pctstu is the student population as a percentage of total population.
a. What results would you expect to see if student populations drive rents up?
b. The equation was estimated for 64 college towns, yielding the following results:
Number of obs = 64
R-squared = 0.4579
----------------------------------------log(rent) |
Coef.
Std. Err.
-------------+--------------------------log(pop) |
.0658678
.038826
log(avginc) |
.507015
.0808356
pctstu |
.0056297
.0017421
_cons |
.0427803
.8438753
-----------------------------------------
Interpret each coefficient.
c. Name a variable that you think could be included in this model, why you would include
it, and the expected sign of its coefficient.
4. Using the formula below, briefly describe why standard errors become very large when
explanatory variables are highly correlated.
Var(bˆ j ) =
s2
SSTj (1- R2j )
5. The output below describes determinants of hourly wages:
. reg wage union female married exper educ
Source |
SS
df
MS
-------------+-----------------------------Model | 5374.76485
5 1074.95297
Residual | 21972.6743 1254 17.5220688
-------------+-----------------------------Total | 27347.4392 1259 21.7215561
Number of obs =
1260
--------------------------------------wage |
Coef.
Std. Err.
-------------+------------------------union |
.7781225
.2678531
female | -2.327294
.2670039
married |
.6629434
.2745184
exper |
.0783962
.0105638
educ |
.4656917
.0459519
_cons | -.8361084
.7000405
---------------------------------------
a)
b)
What is the predicted wage for an unmarried male who does not belong to a union with one year of
experience and 12 years of education? How much would his predicted wage change if he were in a
labor union?
Calculate the R2 and interpret its meaning.