Econ444: Homework No. 1 1) Calculate the OLS estimates from the ...

Econ444: Homework No. 1
1) Calculate the OLS estimates from the following data (on per captia income in thousands of U.S.
dollars as a function of the percent of the labor force in agriculture in ten developed countries) using your
calculator. The data are
Country A B C D E F G H I J
Income 6. 8. 8. 7. 7. 12. 9. 8. 9. 10.
Percent on farms 9. 10. 8. 7. 10. 4. 5. 5. 6. 7.
a. Calculuate ˆβ 0 and ˆβ 1 .
b. Calculate R 2 , ¯R 2 ,andther (sample correlation coefficient of income and percent on farms).
c. If the percentage of the labor force in agriculture in another developed country was 8 percent, what
level of per capita income (in thousands of US dollars) would you guess that country had?
2) Consider the following two least-squares estimates of the relationship between interest rates and the
federal budget deÞcit in the U.S.,
Model A : Ŷ1 =0.103 − 0.079X 1 , ¯R2 =0.00,
where Y 1 = the interest rate on Aaa corporate bonds, X 1 = the federal budget deÞcit as a percentage of
GNP (quarterly model: n =56).
Model B : Ŷ2 =0.089 + 0.369X 2 +0.887X 3 , R 2 =0.40,
where Y 2 = the interest rate on 3-month Treasury bills, X 2 = the federal budget deÞcit in billions of dollars,
and X 3 =therateofinßation (in percent) (quarterly model: n =38)
a. What does “least-squares estimates” mean? What is being estimated? What is being squared in each
model? In what sense are the squares “least”?
b. What does it mean to have an ¯R 2 of 0.00? Is it possible for an R 2 to be negative? Calculate R 2 for
each model.
c. Calculate ¯R 2 for both equations. Is it possible for ¯R 2 to be negative?
3) Suppose that you have been asked to estimate an econometric model to explain the number of people
jogging a mile or more on the school track to help decide whether to build a second track to handle all the
joggers. You collect data by living in the press box for the spring term, and you run two possible explanatory
equations.
A: Ŷ =125.0 − 15.0X 1 − 1.0X 2 +1.5X 3 , ¯R2 =0.75;
B: Ŷ =123.0 − 14.0X 1 +5.5X 2 − 3.7X 4 , ¯R2 =0.73,
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where Y = the number of joggers on a given day, X 1 = inches of rain that day, X 2 = hours of sunshine that
day, X 3 = the high temperature for that day (in degrees F), X 4 = the number of classes with term papers
due the next day.
a) Which of the two equations do you prefer? Why?
b) How is it possible to get different estimated signs for the coefficient of the same variable using the same
data?
4) David Katz studied faculty salaries as a function of their “productivity” and estimated a regression
equation with the following coefficients:
Ŝ i =11, 155 + 230B i +18A i +102E i +489D i +189Y i + ···,
where S i =thesalaryoftheithe professor in 1969-70 in dollars per year; B i = the number of books published,
lifetime; E i = the number of “excellent” articles published, lifetime; D i = the number of dissertations
supervised since 1964; and Y i = the number of years teaching experience.
a. Do the signs of the coefficients match your prior expectations?
b. Dotherelativesizesofthecoefficients seem reasonable?
c. Suppose a professor had just enough time (after teaching, etc.) to write a book, write two excellent
articles, or supervise three dissertations, which would you recommend? Why?
d. Would you like to reconsider your answer to part b above? Which coefficient seems out of line? What
explanation can you give for that result? Is the equation in some sense invalid? Why or why not?
5) Consider the least squares estimation of the multiple linear regression model:
y i = β 0 + β 1 x 1i + ···+ β k x ki + ² i ,
where i =1, ···,n.Letˆβs be the OLSE’s, ŷ i = ˆβ 0 + ˆβ 1 x 1i + ···+ ˆβ k x ki and e i = y i − ŷ i be the residual.
Show that P n
i=1 (ŷ i − Ȳ )e i =0.
(Hint: use the Þrst order conditions for the least squares minimization problem which characterizes e i ).
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