Practice Midterm 2—Spring 2021
Prof. Beheshti
April 5, 2021
Define and Discuss
Problem 1. For each of the following terms provide the mathematical definition (if appli-
cable). Then, explain the terms like you were talking to a smart non-economist. For the
explanation, use no more than two sentences. (2 points each).
1. Heteroskedasticity
2. Panel data
3. Multicollinearity
4. Interaction term
1
True/False/Uncertain For each of the following statements, decide whethe
it is true, false, or uncertain. Justify your answer. (2 points each). Note: Your score will be
determined entirely by your justification.
For the next several problems, consider the following population regression model
Y = β0 + β1X + β2Z + � (1)
where Y is a students grade point average, X is class size, and Z is parental income. Suppose
X and Z are negatively co
elated. Further suppose neither β1, β2 are equal to zero.
8. β1 =
cov(X,Y )
var(X)
9. In order for the regression coefficients to be unbiased, must assume � = 0 for all values
of X and Z.
In addition to model (1), also consider the following regression:
Y = α0 + α1X + ν (2)
10. α̂1 > β̂1
2
Application and Interpretation
Figure 1 on the following page presents a screenshot from RStudio. The dataset ”mtcars”
contains cross sectional data on 32 cars in 1981. The variable qsec is the time (in seconds)
that it takes to drive the car 1
4
miles. The variables mpg and hp represent miles per gallon
and horsepower, respectively.
Problem 11. Using the output from Figure 1, sketch a scatter plot of qsec on the vertical
axis against mpg on the horizontal axis. Also include the fitted regression line. Label the
y-intercept and the slope.
Problem 12. Interpret each of the coefficients from the second regression (line 9).
Problem 13. The sign of the coefficient on mpg flipped when we added hp to the regression.
What does this tell you about the co
elation between mpg and hp? Explain.
For the next problem consider the following regression:
Yi = β0 + β1Xi + β2Zi + β3Xi · Zi + �i
where
• Yi =
{
1 if employed
0 if not
• Xi = number of years of education
• Zi =
{
1 Hispanic
0 non-Hispanic
Problem 15. Interpret each β coefficient. (4 points)
3
Figure 1: RStudio Screenshot
4
Derivations For the following questions, you may assume MLR.1-MLR.4. Suppose
the true regression model is
Y = β0 + β1X + β2Z + �
and you estimate the following regression:
Y = α0 + α1X + �
Problem 16. Fill in the following table with (+), (-), and 0, where (+) means α̂1 > β1, (-)
means α̂1 < β1, and 0 means α̂1 = β1. (1 point per cell) Hint: remember the formula fo
omitted variable bias. If you can’t, try to think of real-world examples for each case.
Table 1: Bias Table
Co
(X,Z) > 0 Co
(X,Z) < 0 Co
(X,Z) = 0
β2 > 0
β2 < 0
β2 = 0
Problem 17. Consider the following regression model
Yi = β0 + β1Xi + β2Zi + β3Xi · Zi + �i
where
• Yi = hourly wage for individual i
• Xi = number of years of previous work experience
• Zi =
{
1 Hispanic
0 non-Hispanic
Suppose β0 > 0, β1 > 0, β2 > 0, and β3 > 0. Sketch wages as a function of previous years
work experience for Hispanics and non-Hispanics. Label all axes, intercepts, and slopes. (5
points)
5