Statistics- How does the R-squared value of the reverse regression

| March 31, 2017

Question
ECO 341K
University of Texas

Introduction to Econometrics
Prof. Haiqing Xu

Homework Assignment #3 (due February 16th, end of the class)
Extended to February 18th, end of the class
Written problems:
1.

Show that R-squared (for a simple linear regression) is equal to the square of the
correlation between y and x (i.e., R2 = rxy2). (Hint: Use one of the R-squared
formulas and plug in the formula for the slope estimator.)

2.

Recall the reverse regression that you ran on Assignment #2. Rather than the SLR of
y on x, you ran a SLR of x on y.
a. How does the R-squared value of the reverse regression (x on y) compare to the
R-squared value of the original regression (y on x)? Explain.
b. Do the slope estimates from the two regressions have the same sign? Explain.
c. How are the magnitudes of the two slope estimates related? Explain.

3.

Wooldridge Problem 2.6

4.

Although we focused upon logarithms in class, one could use other non-linear
transformations within a regression model. For the wage and education example that
has been considered in lecture, suppose that we wanted to take the square root of
education and relate wages to that. So the SLR model would be
wage = β0 + β1 educ + u

a. Figure out the formula for the effect of educ on wage by taking the derivative
dE(wage|educ)/d educ. Note that unlike the basic SLR model, this derivative
depends upon the x variable (here, educ).
Here is the Stata for the regression of this model (using wage1.dta):
. gen sqrteduc = sqrt(educ)
. regr wage sqrteduc
Source |
SS
df
MS
————-+—————————–Model | 930.606128
1 930.606128
Residual | 6229.80816
524 11.8889469
————-+—————————–Total | 7160.41429
525 13.6388844

Number of obs
F( 1,
524)
Prob > F
R-squared
Adj R-squared
Root MSE

=
=
=
=
=
=

526
78.27
0.0000
0.1300
0.1283
3.448

—————————————————————————–wage |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
————-+—————————————————————sqrteduc |
2.947042
.3331004
8.85
0.000
2.292666
3.601419
_cons | -4.464346
1.180639
-3.78
0.000
-6.783714
-2.144978
——————————————————————————

b. Estimate the effect of educ on wage at both educ=12 and educ=16 (using your
formula from the previous part). How do these effects compare to the estimated
effects from the original SLR model? For your reference, the original regression
(wage on educ) results were:
. regr wage educ
Source |
SS
df
MS
————-+—————————–Model | 1179.73204
1 1179.73204
Residual | 5980.68225
524 11.4135158
————-+—————————–Total | 7160.41429
525 13.6388844

Number of obs
F( 1,
524)
Prob > F
R-squared
Adj R-squared
Root MSE

=
=
=
=
=
=

526
103.36
0.0000
0.1648
0.1632
3.3784

—————————————————————————–wage |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
————-+—————————————————————educ |
.5413593
.053248
10.17
0.000
.4367534
.6459651
_cons | -.9048516
.6849678
-1.32
0.187
-2.250472
.4407687
——————————————————————————

c. Which regression (wage on educ or wage on sqrteduc) appears to give a better
overall fit? Should overall fit (as measured by R-squared) be the only reason to
choose one model specification over another? Explain.
Computer problems (show any relevant Stata output):

a. Wooldridge C2.4, part (iii)
b. Wooldridge C2.6 (note that log(expend) is in the data as the lexpend variable)
For the same dataset (MEAP93.DTA), also answer the following questions:
(vi)
(vii)

(viii)

Plot math10 versus lexpend with the fitted regression line shown.
To visualize the non-linearity described by this model, create the fitted values
from the regression and then do a scatter plot of the fitted values versus expend
(not lexpend). Explain how the effect of expenditures changes at higher values of
expenditures.
Suppose that we wanted to measure math10 as a fraction (a number between 0
and 1) rather than on a 0-to-100 scale. Specifically, we could do the following in
Stata to re-scale the math10 variable:
. replace math10 = math10 / 100

(This replaces the original math10 values with the values divided by 100.) If you
re-ran the regression (now regressing the re-scaled math10 upon lexpend), how
would the following quantities change (compared to the original results)? Be
specific, and try these on your own before checking your answers in Stata.
(a) R-squared

(b) SST
(c) the slope estimate
(d) the intercept estimate

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