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

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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