Economics 102
Problem Set 4
Due in Canvas on SATURDAY, July 21 before 10 p.m.
Department of Economics Professor Siegler
UC Davis Summer 2018
1. Ordinary Least Squares by Hand
Consider the following random sample of five 30-year old females (g1866 = 5), where
the dependent variable, g1851, is each person’s hourly wage and the explanatory
variable, g1850, is each person’s years of education.
Hourly wages in dollars
g4666g2181g2191g4667
Years of education
g4666g2180g2191g4667
30 15
20 12
20 14
40 16
40 18
A. Use ordinary least squares (OLS) to estimate the following linear
regression model by hand:
g1851g3552g3036 = g1854g2869 +g1854g2870g1850g3036
B. Compute the coefficient of determination (g1844g2870) by hand.
C. Use R to replicate the results from Parts A and B using the lm()
command as shown in lecture. Report your results in tabular form. using
the package stargazer.
D. Is the slope coefficient statistically different from zero at the 10-percent
level of significance? Is it statistically different from zero at the 5-percent
level of significance? Show your work and explain.
E. Precisely interpret the exact meaning of the estimated slope coefficient in
this particular case?
F. What is the predicted hourly wages of someone with 15 years of
education, based on the regression results? Show your work and explain.
2. Simple Regression and the Convergence Hypothesis
Many theories of economic growth predict that poorer economies should
experience faster rates of economic growth. That is, poor economies should
converge in per capita income levels to previously richer economies.
Convergence is based on three main channels: (1) because of diminishing
marginal product of capital, capital should flow from economies with more
capital (and a lower marginal return) to countries with lower levels of capital
since the marginal impact of an additional unit of capital should be higher in
poorer economies, (2) labor should flow from low wage countries to high wage
countries helping to equalize wages and reduce differences in per capita
incomes, and (3) technology should flow from rich to poor over time, which also
allows for convergence.
This problem asks you to test the convergence hypothesis using data from 48
U.S. states (each U.S. state is considered as a separate “economy”). Go to the U.S.
Bureau of Economic Analysis website (www.bea.gov). On the main page, as part
of “Regional” data, click on “State and Local Area Personal Income.”
Next, scroll down below “Interactive Tables: Personal Income and Employment
and click on “State Annual Personal Income Employment.” Then, on the next
page, go down to STATE ANNUAL PERSONAL INCOME AND
EMPLOYMENT and then proceed to download the data series “SA1 - Personal
Income Summary: Personal Income, Population, Per Capita Personal Income”
You will want to download the data “Per capita personal income (dollars)” for
“All Areas” measured in “Levels” for the years 1929 to the most recent year
available. You can hold down the Control key (Ctrl) to select only the two years
of data needed.
Note that the only data you need for this problem are per capita personal income
in 1929 and the most recent year available for each of the 48 states (exclude the
United States as a whole, Alaska, Hawaii, Washington, D.C., and the regions of
the U.S. such as New England, Mideast, etc.). To make sure you’ve downloaded
the correct data series, in 1929, per capita personal income in Alabama was $319.
A. Attach your Excel or similar dataset showing the values of personal per
capita income for each of the 48 states in both 1929 and the most recent
year available.
B. Create a well-labeled and well-formatted, two-variable scatter diagram in
R for all 48 U.S. states. The level of per capita personal income in 1929 (in
dollars) by state should be on the horizontal axis while the average annual
growth rate of per capita income by state from 1929 to 2017 on the vertical
axis. You can compute the average annual growth rate from 1929 to 2017,
for example, for each state using the following formula. It is easier to
compute growth rates in Excel before you import the data set into R, but
you can do it in R too:
g1859g1870g1867g1875g1872ℎ = g4686g3436g1868g1857g1870uni0009g1855g1853g1868g1861g1872g1853uni0009g1868g1857g1870g1871g1867g1866g1853g1864uni0009g1861g1866g1855g1867g1865g1857uni0009g1861g1866uni00092017g1868g1857g1870uni0009g1855g1853g1868g1861g1872g1853uni0009g1868g1857g1870g1871g1867g1866g1853g1864uni0009g1861g1866g1855g1867g1865g1857uni0009g1861g1866uni00091929g3440
g2869
g2876g2876 −1g4687 ∙100
The formula above computes the average annual growth rate in percent.
For example, you should get ≈ 5.643 percent for Alabama. That is,
nominal per capita income increased at approximately 5.643 percent per
year from 1929 to 2017.
The data set in R should consist of 48 cross-sectional observations for the
variable growth for each state and the variable per capita personal income
in 1929 for each state. Is your scatter diagram consistent with the
convergence hypothesis? Briefly explain.
C. Estimate and report the following OLS regression model using R:
g1859g1870g1867g1875g1872ℎg3555 = g1854g2869 +g1854g2870g1868g1857g1870uni0009g1855g1853g1868g1861g1872g1853uni0009g1868g1857g1870g1871g1867g1866g1853g1864uni0009g1861g1866g1855g1867g1865g1857uni0009g1861g1866uni00091929
D. Formally test the convergence hypothesis, using a t-test at the 1-percent
level of significance. What are the null and alternative hypotheses? Can
you reject the null? Are your results consistent with the convergence
hypothesis at the 1-percent level of significance? Explain.
3. Interpreting Multiple Regression
Consider the following OLS multiple regression results from Table 4 of “The
Effect of Education on Cognitive Ability” [Economic Inquiry 49(3), July 2011, 838-
856].
The dependent variable in the table below is an individual’s IQ score at the age
of 20 (IQ20). Note that the authors do not report the constant, but the regressions
do include a constant, which you need to take into account in determining the
degrees of freedom.
The coefficient estimates (with the standard errors in parentheses) are reported
below for one of the regression equations from the article, denoted as Equation 6:
Dependent Variable: IQ20
Explanatory Variables Equation 6
IQ10 0.42 (0.04)
Born early -4.40 (1.56)
Years of schooling 2.87 (0.29)
Father has higher education 3.21 (1.24)
Month of birth 0.36 (0.11)
GPA in third grade 2.70 (1.16)
Rating of third grade teacher 0.89 (0.58)
n 566
R2 0.731
A. What explanatory variables above are dummy variables in Equation 6?
Briefly explain.
B. In Equation 6, which of the coefficients above are individually statistically
different from zero at the 5-percent level of significance? Briefly explain.
C. Using Equation 6, can you reject the null hypothesis that all of the
reported coefficients are jointly equal to zero at the 1-percent level of
significance. Show your work and explain.