# The educator is describing a threshold effect in which performance is constant for class sizes less than 15

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```							                                                                               Economics 403
Introduction to Econometrics
Problem Set 5

1. (From 6.10) (yi, X1i, X2i) satisfy the assumptions in Key Concept 6.4; in addition,
var(ui|X1i, X2i) = 6 and var (X1i) =4. A random sample of size n=400 is drawn from the
population.
ˆ
a. Assume that X1 and X2 are uncorrelated. Compute the variance of  1 . (Hint: See
Equation 6.17 in the Appendix 6.2)
b. Assume that cor(X1, X2) =.5 Compute the variance of  1 .    ˆ
c. Comment on the following statements: “When X1 and X2 are correlated, the
ˆ
variance of  1 . is larger than it would be if X1 and X2 were uncorrelated. Thus,
ˆ
if you are interested in  , it is best to leave X2 out of the regression if it is
1
correlated with X1.”

2. Suppose you are interested in estimating a model of average starting pay of MBA
graduates. You have information on each individual’s GPA in business school, GMAT
score (the score for the business school entrance exam), the selectivity of the business
school attended, and tuition.
a. If you include both GPA and GMAT scores in the model, what problem(s) may
you encounter and why?
b. If the coefficient on tuition is positive and statistically significant, does that mean
it pays to go to the most expensive business school? Carefully explain why or
why not.
c. Suppose you regress GMAT score on GPA and find a statistically significant
positive relationship between the two. What does that say about the problem of
multicollinearity?

3. In 1992, there was an increase in the (state) minimum wage in one U.S. state (New
Jersey) but not in a neighboring location (eastern Pennsylvania). A study of the effects of
this law provides you with the following information,

PA                 NJ
FTE Employment before        23.33              20.44
FTE Employment after         21.17              21.03

Where FTE is “full time equivalent” and the table reports the average FTEs per fast food
restaurant. (This means that 2 part time workers working 20 hours a week are counted as
equivalent to 1 full time worker working 40 hours a week.)

a. Calculate the change in the treatment group, the change in the control group, and
ˆ
finally  diff indiff
b. Since minimum wages represent a price floor, did you expect          ˆ
 diff indiff to be
positive or negative?

ˆ
c. The standard error for  diff indiff is 1.36. Test whether or not the coefficient is
statistically significant, given that there are 410 observations.

d. Do you have any concerns about forming a conclusion about the effect of the
minimum wage law using this approach? (Stronger answers will make specific
reference to the information presented.)

4. After reading this chapter’s analysis of test scores and class size, an educator comments:

“In my experience, student performance depends on class size, but not in the way
your regressions say. Rather, students do well when class size is less than 15
students and do very poorly when class size is greater than 19. There are no gains
from reducing class size below 15 students and the relationship is constant in the
intermediate region between 15 and 19 students. There is no additional loss to
increasing class size when it is already greater than 19.”

The educator is describing a “threshold effect” in which performance is constant for class
sizes less than 15, then jumps and is constant for class sizes between 15 and 19, and then
jumps again for class sizes greater than 19. To model these effects, define the binary
(dummy) variables

STRsmall = 1 if STR<15 and STRsmall=0 otherwise
STRmoderate = 1 if 15≤STR≤19 and STRmoderate=0 otherwise
STRlarge = 1 if STR>19 and STRlarge=0 otherwise

a. Consider the following regression:

TestScorei = β0 + β1STRsmalli + β2STRlargei +ui

Sketch the regression function relating TestScore to STR for hypothetical values of
the regression coefficients that are consistent with the educator’s statement.

b. A researcher tries to estimate the regression

TestScorei = β0 + β1STRsmalli + β2STRmoderatei + β3STRlargei +ui

and finds that his computer crashes. Why?

5. A researcher suspects that the effect of % children in school in poverty has a nonlinear
effect on test scores. In particular, she conjectures that increases in this variable from
10% to 15% have little effect on test scores, but that changes from 20% to 50% have a
much larger effect.

a. Describe a nonlinear specification that can be used to model this form of non-
linearity.

b. How would you test whether the researcher’s conjecture was better than a linear
specification?

6. A researcher suspects that the effect of household income is different in districts with
small classes than in districts with large classes.

a. Describe a nonlinear specification that can be used to model this form of
nonlinearity.

b. How would you test whether the researcher’s conjecture was better than a linear
specification?
Problem Set 5 Computer Exercises

Create a do file and corresponding log file to allow you to answer the questions above using the
state education data (education.dta from the class website).

Turn in a copy of your do file with this assignment.

1. After reading this chapter’s analysis of test scores and class size, an educator comments:

“In my experience, student performance depends on class size, but not in the way
your regressions say. Rather, students do well when class size is less than 15
students and do very poorly when class size is greater than 19. There are no gains
from reducing class size below 15 students and the relationship is constant in the
intermediate region between 15 and 19 students. There is no additional loss to
increasing class size when it is already greater than 19.”

The educator is describing a “threshold effect” in which performance is constant for class
sizes less than 15, then jumps and is constant for class sizes between 15 and 19, and then
jumps again for class sizes greater than 19. To model these effects, define the binary
(dummy) variables

STRsmall = 1 if STR<15 and STRsmall=0 otherwise
STRmoderate = 1 if 15≤STR≤19 and STRmoderate=0 otherwise
STRlarge = 1 if STR>19 and STRlarge=0 otherwise

a. Consider the following regression:

TestScorei = β0 + β1STRsmalli + β2STRlargei +ui

In the education.dta data, the variable studteach measures student teacher ratio.
Use this variable to create the threshold variables described above and then
estimate the regression above, using satr as the TestScore variable.

Interpret the estimated coefficients for β1 and β2, commenting on their economic
significance and their statistical significance.

b. A researcher tries to estimate the regression

TestScorei = β0 + β1STRsmalli + β2STRmoderatei + β3STRlargei +ui

and finds that his computer crashes. Why?

Estimate the equation above and explain the result.
2. A researcher suspects that the effect of % children in school in poverty has a nonlinear
effect on test scores. In particular, she conjectures that increases in this variable from
10% to 15% have little effect on test scores, but that changes from 20% to 50% have a
much larger effect.

a.      Describe a nonlinear specification that can be used to model this form of non-
linearity.

The perckidpov variable in the education .dta data set measures the percent of
the children in the school that is in poverty. Use this variable to estimate the
non-linear relationship between satr and perckidpov that you postulate based on
the description above. Interpret the estimated coefficient(s) on the poverty
variable(s) in terms of their economic and statistical significance.

b.      How would you test whether the researcher’s conjecture was better than a
linear specification?

3. A researcher suspects that the effect of household income is different in districts with
small classes than in districts with large classes.

a.      Describe a nonlinear specification that can be used to model this form of
nonlinearity.

The hhincome variable in the data measures mean annual household income in
the state. Use this variable to estimate the specification you postulate. Interpret
the estimated coefficient(s) on the income variable(s) in terms of their economic
and statistical significance.

b.      How would you test whether the researcher’s conjecture was better than a
linear specification?

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