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Introduction to Econometrics, 3e (Stock)
Chapter 7 Hypothesis Tests and Confidence Intervals in Multiple Regression
7.1 Multiple Choice
1) The confidence interval for a single coefficient in a multiple regression
A) makes little sense because the population parameter is unknown.
B) should not be computed because there are other coefficients present in the model.
C) contains information from a large number of hypothesis tests.
D) should only be calculated if the regression R2 is identical to the adjusted R2.
2) The following linear hypothesis can be tested using the F-test with the exception of
A) β2 = 1 and β3= β4/β5.
B) β2 =0.
C) β1 + β2 = 1 and β3 = -2β4.
D) β0 = β1 and β1 = 0.
3) The formula for the standard error of the regression coefficient, when moving from one explanatory
variable to two explanatory variables,
A) stays the same.
B) changes, unless the second explanatory variable is a binary variable.
C) changes.
D) changes, unless you test for a null hypothesis that the addition regression coefficient is zero.
4) All of the following are examples of joint hypotheses on multiple regression coefficients, with the
exception of
A) H0 : β1 + β2 = 1
B) H0 :
3
2
= β1 and β4 = 0
C) H0 : β2 = 0 and β3 = 0
D) H0 : β1 = -β2 and β1 + β2 = 1
5) When testing joint hypothesis, you should
A) use t-statistics for each hypothesis and reject the null hypothesis is all of the restrictions fail.
B) use the F-statistic and reject all the hypothesis if the statistic exceeds the critical value.
C) use t-statistics for each hypothesis and reject the null hypothesis once the statistic exceeds the critical
value for a single hypothesis.
D) use the F-statistics and reject at least one of the hypothesis if the statistic exceeds the critical value.
6) The overall regression F-statistic tests the null hypothesis that
A) all slope coefficients are zero.
B) all slope coefficients and the intercept are zero.
C) the intercept in the regression and at least one, but not all, of the slope coefficients is zero.
D) the slope coefficient of the variable of interest is zero, but that the other slope coefficients are not.
7) For a single restriction (q = 1), the F-statistic
A) is the square root of the t-statistic.
B) has a critical value of 1.96.
C) will be negative.
D) is the square of the t-statistic.
8) The homoskedasticity-only F-statistic is given by the following formula
A)
( )
( )
/
( / 1
restricted unrestricted
unrestricted unrestricted
SSR SSR q
FSSR n k
−
=−−
B)
( )
( )
/
/1
restricted unrestricted
restricted unrestricted
SSR SSR q
FSSR n k
−
=−−
C)
( )
( )
/
/1
restricted unrestricted
unrestricted unrestricted
SSR SSR q
FSSR n k
−
=−−
D)
( )
( )
/ 1)
/
restricted unrestricted
unrestricted unrestricted
SSR SSR q
FSSR n k
−−
=−
9) All of the following are correct formulae for the homoskedasticity-only F-statistic, with the exception of
A)
( )
( )
/
/1
restricted unrestricted
unrestricted unrestricted
SSR SSR q
FSSR n k
−
=−−
B)
( )
( )
/
/1
unrestricted restricted
restricted restricted
SSR SSR q
FSSR n k
−
=−−
C)
( )
1
restricted unrestricted unrestricted
unrestricted
SSR SSR nk
FSSR q
−−−
=
D)
( )
1
1unrestricted
restricted
unrestricted
nk
SSR
FSSR q
−−
= −
10) In the multiple regression model, the t-statistic for testing that the slope is significantly different from
zero is calculated
A) by dividing the estimate by its standard error.
B) from the square root of the F-statistic.
C) by multiplying the p-value by 1.96.
D) using the adjusted R2 and the confidence interval.
11) To test joint linear hypotheses in the multiple regression model, you need to
A) compare the sums of squared residuals from the restricted and unrestricted model.
B) use the heteroskedasticity-robust F-statistic.
C) use several t-statistics and perform tests using the standard normal distribution.
D) compare the adjusted R2 for the model which imposes the restrictions, and the unrestricted model.
12) The homoskedasticity-only F-statistic is given by the following formula
A)
( )
( )
( )
22
2
/
1 / 1
unrestricted restricted
unrestricted unrestricted
R R q
FR n k
−
=− − −
B)
( )
2
2
1 ) /
/1
unrestricted
unrestricted unrestricted
Rq
FR n k
−
=−−
C)
( )
( )
( )
22
2
/
1 / 1
unrestricted restricted
unrestricted restricted
R R q
FR n k
−
=− − −
D)
( )
( )
( )
22
2
/
1 / 1
unrestricted unrestricted
unrestricted restricted
R R q
FR n k
−
=− − −
13) Let R2unrestricted and R2restricted be 0.4366 and 0.4149 respectively. The difference between the
unrestricted and the restricted model is that you have imposed two restrictions. There are 420
observations. The F-statistic in this case is
A) 4.61
B) 8.01
C) 10.34
D) 7.71
14) If you wanted to test, using a 5% significance level, whether or not a specific slope coefficient is equal
to one, then you should
A) subtract 1 from the estimated coefficient, divide the difference by the standard error, and check if the
resulting ratio is larger than 1.96.
B) add and subtract 1.96 from the slope and check if that interval includes 1.
C) see if the slope coefficient is between 0.95 and 1.05.
D) check if the adjusted R2 is close to 1.
15) If the absolute value of your calculated t-statistic exceeds the critical value from the standard normal
distribution you can
A) safely assume that your regression results are significant.
B) reject the null hypothesis.
C) reject the assumption that the error terms are homoskedastic.
D) conclude that most of the actual values are very close to the regression line.
16) If you reject a joint null hypothesis using the F-test in a multiple hypothesis setting, then
A) a series of t-tests may or may not give you the same conclusion.
B) the regression is always significant.
C) all of the hypotheses are always simultaneously rejected.
D) the F-statistic must be negative.
17) When your multiple regression function includes a single omitted variable regressor, then
A) use a two-sided alternative hypothesis to check the influence of all included variables.
B) the estimator for your included regressors will be biased if at least one of the included variables is
correlated with the omitted variable.
C) the estimator for your included regressors will always be biased.
D) lower the critical value to 1.645 from 1.96 in a two-sided alternative hypothesis to test the significance
of the coefficients of the included variables.
18) A 95% confidence set for two or more coefficients is a set that contains
A) the sample values of these coefficients in 95% of randomly drawn samples.
B) integer values only.
C) the same values as the 95% confidence intervals constructed for the coefficients.
D) the population values of these coefficients in 95% of randomly drawn samples.
19) When there are two coefficients, the resulting confidence sets are
A) rectangles.
B) ellipses.
C) squares.
D) trapezoids.
20) When testing the null hypothesis that two regression slopes are zero simultaneously, then you cannot
reject the null hypothesis at the 5% level, if the ellipse contains the point
A) (-1.96, 1.96).
B) .
C) (0,0).
D) (1.962, 1.962).
21) The OLS estimators of the coefficients in multiple regression will have omitted variable bias
A) only if an omitted determinant of Yi is a continuous variable.
B) if an omitted variable is correlated with at least one of the regressors, even though it is not a
determinant of the dependent variable.
C) only if the omitted variable is not normally distributed.
D) if an omitted determinant of Yi is correlated with at least one of the regressors.
22) At a mathematical level, if the two conditions for omitted variable bias are satisfied, then
A) E(ui X1i, X2i,..., Xki) ≠ 0.
B) there is perfect multicollinearity.
C) large outliers are likely: X1i, X2i,..., Xki and Yi and have infinite fourth moments.
D) (X1i, X2i,..., Xki,Yi), i = 1,..., n are not i.i.d. draws from their joint distribution.
23) All of the following are true, with the exception of one condition:
A) a high R2 or does not mean that the regressors are a true cause of the dependent variable.
B) a high R2 or does not mean that there is no omitted variable bias.
C) a high R2 or always means that an added variable is statistically significant.
D) a high R2 or does not necessarily mean that you have the most appropriate set of regressors.
24) The general answer to the question of choosing the scale of the variables is
A) dependent on you whim.
B) to make the regression results easy to read and to interpret.
C) to ensure that the regression coefficients always lie between -1 and 1.
D) irrelevant because regardless of the scale of the variable, the regression coefficient is unaffected.
25) If the estimates of the coefficients of interest change substantially across specifications,
A) then this can be expected from sample variation.
B) then you should change the scale of the variables to make the changes appear to be smaller.
C) then this often provides evidence that the original specification had omitted variable bias.
D) then choose the specification for which your coefficient of interest is most significant.
26) You have estimated the relationship between testscores and the student-teacher ratio under the
assumption of homoskedasticity of the error terms. The regression output is as follows:
= 698.9 - 2.28 × STR, and the standard error on the slope is 0.48. The homoskedasticity-only
"overall" regression F- statistic for the hypothesis that the Regression R2 is zero is approximately
A) 0.96
B) 1.96
C) 22.56
D) 4.75
27) Consider a regression with two variables, in which X1i is the variable of interest and X2i is the control
variable. Conditional mean independence requires
A) E(ui|X1i, X2i) = E(ui|X2i)
B) E(ui|X1i, X2i) = E(ui|X1i)
C) E(ui|X1i) = E(ui|X2i)
D) E(ui) = E(ui|X2i)
28) The homoskedasticity-only F-statistic and the heteroskedasticity-robust F-statistic typically are
A) the same
B) different
C) related by a linear function
D) a multiple of each other (the heteroskedasticity-robust F-statistic is 1.96 times the homoskedasticity-
only F-statistic)
29) Consider the following regression output where the dependent variable is testscores and the two
explanatory variables are the student-teacher ratio and the percent of English learners:
= 698.9 - 1.10×STR - 0.650×PctEL. You are told that the t-statistic on the student-teacher ratio
coefficient is 2.56. The standard error therefore is approximately
A) 0.25
B) 1.96
C) 0.650
D) 0.43
30) The critical value of F4,∞ at the 5% significance level is
A) 3.84
B) 2.37
C) 1.94
D) Cannot be calculated because in practice you will not have infinite number of observations
7.2 Essays and Longer Questions
1) The F-statistic with q = 2 restrictions when testing for the restrictions β1 = 0 and β2 = 0 is given by the
following formula:
Discuss how this formula can be understood intuitively.
2) The cost of attending your college has once again gone up. Although you have been told that education
is investment in human capital, which carries a return of roughly 10% a year, you (and your parents) are
not pleased. One of the administrators at your university/college does not make the situation better by
telling you that you pay more because the reputation of your institution is better than that of others. To
investigate this hypothesis, you collect data randomly for 100 national universities and liberal arts
colleges from the 2000-2001 U.S. News and World Report annual rankings. Next you perform the following
regression
= 7,311.17 + 3,985.20 × Reputation – 0.20 × Size
(2,058.63) (664.58) (0.13)
+ 8,406.79 × Dpriv – 416.38 × Dlibart – 2,376.51 × Dreligion
(2,154.85) (1,121.92) (1,007.86)
R2=0.72, SER = 3,773.35
where Cost is Tuition, Fees, Room and Board in dollars, Reputation is the index used in U.S. News and
World Report (based on a survey of university presidents and chief academic officers), which ranges from
1 ("marginal") to 5 ("distinguished"), Size is the number of undergraduate students, and Dpriv, Dlibart,
and Dreligion are binary variables indicating whether the institution is private, a liberal arts college, and
has a religious affiliation. The numbers in parentheses are heteroskedasticity-robust standard errors.
(a) Indicate whether or not the coefficients are significantly different from zero.
(b) What is the p-value for the null hypothesis that the coefficient on Size is equal to zero? Based on this,
should you eliminate the variable from the regression? Why or why not?
(c) You want to test simultaneously the hypotheses that βsize = 0 and βDilbert = 0. Your regression
package returns the F-statistic of 1.23. Can you reject the null hypothesis?
(d) Eliminating the Size and Dlibart variables from your regression, the estimation regression becomes
= 5,450.35 + 3,538.84 × Reputation + 10,935.70 × Dpriv – 2,783.31 × Dreligion;
(1,772.35) (590.49) (875.51) (1,180.57)
R2=0.72, SER = 3,792.68
Why do you think that the effect of attending a private institution has increased now?
(e) You give a final attempt to bring the effect of Size back into the equation by forcing the assumption of
homoskedasticity onto your estimation. The results are as follows:
= 7,311.17 + 3,985.20 × Reputation – 0.20 × Size
(1,985.17) (593.65) (0.07)
+ 8,406.79 × Dpriv – 416.38 × Dlibart – 2,376.51 × Dreligion
(1,423.59) (1,096.49) (989.23)
R2=0.72, SER = 3,682.02
Calculate the t-statistic on the Size coefficient and perform the hypothesis test that its coefficient is zero. Is
this test reliable? Explain.
3) In the multiple regression model with two explanatory variables
Yi = β0 + β1X1i + β2X2i + ui
the OLS estimators for the three parameters are as follows (small letters refer to deviations from means as
in zi = Zi -
Z
):
12
0 1 2
ˆ ˆ ˆ
Y X X
= − −
2
1 2 2 1 2
1 1 1 1
12
22
1 2 1 2
1 1 1
ˆ
n n n n
i i i i i i i
i i i i
n n n
i i i i
i i i
y x x y x x x
x x x x
= = = =
= = =
−
=
−
2
1 1 1 1 2
1 1 1 1
22
22
1 2 1 2
1 1 1
ˆ
n n n n
i i i i i i i
i i i i
n n n
i i i i
i i i
y x x y x x x
x x x x
= = = =
= = =
−
=
−
You have collected data for 104 countries of the world from the Penn World Tables and want to estimate
the effect of the population growth rate (X1i) and the saving rate (X2i) (average investment share of GDP
from 1980 to 1990) on GDP per worker (relative to the U.S.) in 1990. The various sums needed to calculate
the OLS estimates are given below:
1
n
i
i
Y
=
= 33.33;
1
1
n
i
i
X
=
= 2.025;
2
1
n
i
i
X
=
=17.313
2
1
n
i
i
y
=
= 8.3103;
2
1
1
n
i
i
x
=
= .0122;
2
2
1
n
i
i
x
=
= 0.6422
1
1
n
ii
i
yx
=
= - 0.2304;
2
1
n
ii
i
yx
=
= 1.5676;
12
1
n
ii
i
xx
=
= -0.0520
The heteroskedasticity-robust standard errors of the two slope coefficients are 1.99 (for population
growth) and 0.23 (for the saving rate). Calculate the 95% confidence interval for both coefficients. How
many standard deviations are the coefficients away from zero?
4) A subsample from the Current Population Survey is taken, on weekly earnings of individuals, their
age, and their gender. You have read in the news that women make 70 cents to the $1 that men earn. To
test this hypothesis, you first regress earnings on a constant and a binary variable, which takes on a value
of 1 for females and is 0 otherwise. The results were:
= 570.70 - 170.72 × Female, R2=0.084, SER = 282.12.
(9.44) (13.52)
(a) Perform a difference in means test and indicate whether or not the difference in the mean salaries is
significantly different. Justify your choice of a one-sided or two-sided alternative test. Are these results
evidence enough to argue that there is discrimination against females? Why or why not? Is it likely that
the errors are normally distributed in this case? If not, does that present a problem to your test?
(b) Test for the significance of the age and gender coefficients. Why do you think that age plays a role in
earnings determination?
6). Obviously this is a better proxy for some subsample of individuals than for others.
5) You have collected data from Major League Baseball (MLB) to find the determinants of winning. You
have a general idea that both good pitching and strong hitting are needed to do well. However, you do
not know how much each of these contributes separately. To investigate this problem, you collect data for
all MLB during 1999 season. Your strategy is to first regress the winning percentage on pitching quality
("Team ERA"), second to regress the same variable on some measure of hitting ("OPS – On-base Plus
Slugging percentage"), and third to regress the winning percentage on both.
Summary of the Distribution of Winning Percentage, On Base plus Slugging Percentage,
and Team Earned Run Average for MLB in 1999
Average
Standard
deviation
Percentile
10%
25%
40%
50%
(median)
60%
75%
90%
Team
ERA
4.71
0.53
3.84
4.35
4.72
4.78
4.91
5.06
5.25
OPS
0.778
0.034
0.720
0.754
0.769
0.780
0.790
0.798
0.820
Winning
Percentage
0.50
0.08
0.40
0.43
0.46
0.48
0.49
0.59
0.60
The results are as follows:
= 0.94 – 0.100 × teamera, R2 = 0.49, SER = 0.06.
(0.08) (0.017)
= –0.68 + 1.513 × ops, R2=0.45, SER = 0.06.
(0.17) (0.221)
= –0.19 – 0.099 × teamera + 1.490 × ops, R2=0.92, SER = 0.02.
(0.08) (0.008) (0.126)
(a) Use the t-statistic to test for the statistical significance of the coefficient.
(b) There are 30 teams in MLB. Does the small sample size worry you here when testing for significance?
6) In the process of collecting weight and height data from 29 female and 81 male students at your
university, you also asked the students for the number of siblings they have. Although it was not quite
clear to you initially what you would use that variable for, you construct a new theory that suggests that
children who have more siblings come from poorer families and will have to share the food on the table.
Although a friend tells you that this theory does not pass the "straight-face" test, you decide to
hypothesize that peers with many siblings will weigh less, on average, for a given height. In addition,
you believe that the muscle/fat tissue composition of male bodies suggests that females will weigh less,
on average, for a given height. To test these theories, you perform the following regression:
= –229.92 – 6.52 × Female + 0.51 × Sibs+ 5.58 × Height,
(44.01) (5.52) (2.25) (0.62)
R2=0.50, SER = 21.08
where Studentw is in pounds, Height is in inches, Female takes a value of 1 for females and is 0 otherwise,
Sibs is the number of siblings (heteroskedasticity-robust standard errors in parentheses).
(a) Carrying out hypotheses tests using the relevant t-statistics to test your two claims separately, is there
strong evidence in favor of your hypotheses? Is it appropriate to use two separate tests in this situation?
(b) You also perform an F-test on the joint hypothesis that the two coefficients for females and siblings are
zero. The calculated F-statistic is 0.84. Find the critical value from the F-table. Can you reject the null
hypothesis? Is it possible that one of the two parameters is zero in the population, but not the other?
(c) You are now a bit worried that the entire regression does not make sense and therefore also test for the
height coefficient to be zero. The resulting F-statistic is 57.25. Does that prove that there is a relationship
between weight and height?
7) You have collected data for 104 countries to address the difficult questions of the determinants for
differences in the standard of living among the countries of the world. You recall from your
macroeconomics lectures that the neoclassical growth model suggests that output per worker (per capita
income) levels are determined by, among others, the saving rate and population growth rate. To test the
predictions of this growth model, you run the following regression:
= 0.339 – 12.894 × n + 1.397 × SK, R2=0.621, SER = 0.177
(0.068) (3.177) (0.229)
where RelPersInc is GDP per worker relative to the United States, n is the average population growth rate,
1980-1990, and SK is the average investment share of GDP from 1960 to 1990 (remember investment
equals saving). Numbers in parentheses are for heteroskedasticity-robust standard errors.
(a) Calculate the t-statistics and test whether or not each of the population parameters are significantly
different from zero.
(b) The overall F-statistic for the regression is 79.11. What is the critical value at the 5% and 1% level?
What is your decision on the null hypothesis?
(c) You remember that human capital in addition to physical capital also plays a role in determining the
standard of living of a country. You therefore collect additional data on the average educational
attainment in years for 1985, and add this variable (Educ) to the above regression. This results in the
modified regression output:
= 0.046 – 5.869 × n + 0.738 × SK + 0.055 × Educ, R2=0.775, SER = 0.1377
(0.079) (2.238) (0.294) (0.010)
How has the inclusion of Educ affected your previous results?
(d) Upon checking the regression output, you realize that there are only 86 observations, since data for
Educ is not available for all 104 countries in your sample. Do you have to modify some of your statements
in (d)?
8) Attendance at sports events depends on various factors. Teams typically do not change ticket prices
from game to game to attract more spectators to less attractive games. However, there are other
marketing tools used, such as fireworks, free hats, etc., for this purpose. You work as a consultant for a
sports team, the Los Angeles Dodgers, to help them forecast attendance, so that they can potentially
devise strategies for price discrimination. After collecting data over two years for every one of the 162
home games of the 2000 and 2001 season, you run the following regression:
= 15,005 + 201 × Temperat + 465 × DodgNetWin + 82 × OppNetWin
(8,770) (121) (169) (26)
+ 9647 × DFSaSu + 1328 × Drain + 1609 × D150m + 271 × DDiv – 978 × D2001;
(1505) (3355) (1819) (1,184) (1,143)
R2=0.416, SER = 6983
where Attend is announced stadium attendance, Temperat it the average temperature on game day,
DodgNetWin are the net wins of the Dodgers before the game (wins-losses), OppNetWin is the opposing
team's net wins at the end of the previous season, and DFSaSu, Drain, D150m, Ddiv, and D2001 are binary
variables, taking a value of 1 if the game was played on a weekend, it rained during that day, the
opposing team was within a 150 mile radius, the opposing team plays in the same division as the
Dodgers, and the game was played during 2001, respectively. Numbers in parentheses are
heteroskedasticity- robust standard errors.
(a) Are the slope coefficients statistically significant?
(b) To test whether the effect of the last four binary variables is significant, you have your regression
program calculate the relevant F-statistic, which is 0.295. What is the critical value? What is your decision
about excluding these variables?
16
9) The administration of your university/college is thinking about implementing a policy of coed floors
only in dormitories. Currently there are only single gender floors. One reason behind such a policy might
be to generate an atmosphere of better "understanding" between the sexes. The Dean of Students (DoS)
has decided to investigate if such a behavior results in more "togetherness" by attempting to find the
determinants of the gender composition at the dinner table in your main dining hall, and in that of a
neighboring university, which only allows for coed floors in their dorms. The survey includes 176
students, 63 from your university/college, and 113 from a neighboring institution.
The Dean's first problem is how to define gender composition. To begin with, the survey excludes single
persons' tables, since the study is to focus on group behavior. The Dean also eliminates sports teams from
the analysis, since a large number of single-gender students will sit at the same table. Finally, the Dean
decides to only analyze tables with three or more students, since she worries about "couples" distorting
the results. The Dean finally settles for the following specification of the dependent variable:
GenderComp =
Where " " stands for absolute value of Z. The variable can take on values from zero to fifty.
After considering various explanatory variables, the Dean settles for an initial list of eight, and estimates
the following relationship, using heteroskedasticity-robust standard errors (this Dean obviously has
taken an econometrics course earlier in her career and/or has an able research assistant):
= 30.90 – 3.78 × Size – 8.81 × DCoed + 2.28 × DFemme +2.06 × DRoommate
(7.73) (0.63) (2.66) (2.42) (2.39)
- 0.17 × DAthlete + 1.49 × DCons – 0.81 SAT + 1.74 × SibOther, R2=0.24, SER = 15.50
(3.23) (1.10) (1.20) (1.43)
where Size is the number of persons at the table minus 3; DCoed is a binary variable, which takes on the
value of 1 if you live on a coed floor; DFemme is a binary variable, which is 1 for females and zero
otherwise; DRoommate is a binary variable which equals 1 if the person at the table has a roommate and is
zero otherwise; DAthlete is a binary variable which is 1 if the person at the table is a member of an athletic
varsity team; DCons is a variable which measures the political tendency of the person at the table on a
seven-point scale, ranging from 1 being “liberal” to 7 being “conservative”; SAT is the SAT score of the
person at the table measured on a seven-point scale, ranging from 1 for the category “900-1000” to 7 for
the category “1510 and above”; and increasing by one for 100 point increases; and SibOther is the number
of siblings from the opposite gender in the family the person at the table grew up with.
(a) Indicate which of the coefficients are statistically significant.
(b) Based on the above results, the Dean decides to specify a more parsimonious form by eliminating the
least significant variables. Using the F-statistic for the null hypothesis that there is no relationship
between the gender composition at the table and DFemme, DRoommate, DAthlete, and SAT, the regression
package returns a value of 1.10. What are the degrees of freedom for the statistic? Look up the 1% and 5%
critical values from the F- table and make a decision about the exclusion of these variables based on the
critical values.
(c) The Dean decides to estimate the following specification next:
= 29.07 – 3.80 × Size – 9.75 × DCoed + 1.50 × DCons + 1.97 × SibOther,
(3.75) (0.62) (1.04) (1.04) (1.44)
R2=0.22 SER = 15.44
Calculate the t-statistics for the coefficients and discuss whether or not the Dean should attempt to
simplify the specification further. Based on the results, what might some of the comments be that she will
write up for the other senior administrators of your college? What are some of the potential flaws in her
analysis? What other variables do you think she should have considered as explanatory factors?
10) The Solow growth model suggests that countries with identical saving rates and population growth
rates should converge to the same per capita income level. This result has been extended to include
investment in human capital (education) as well as investment in physical capital. This hypothesis is
referred to as the "conditional convergence hypothesis," since the convergence is dependent on countries
obtaining the same values in the driving variables. To test the hypothesis, you collect data from the Penn
World Tables on the average annual growth rate of GDP per worker (g6090) for the 1960-1990 sample
period, and regress it on the (i) initial starting level of GD-P per worker relative to the United States in
1960 (RelProd60), (ii) average population growth rate of the country (n), (iii) average investment share of
GDP from 1960 to 1990 (SK - remember investment equals savings), and (iv) educational attainment in
years for 1985 (Educ). The results for close to 100 countries is as follows (numbers in parentheses are for
heteroskedasticity-robust standard errors):
= 0.004 - 0.172 × n + 0.133 × SK + 0.002 × Educ – 0.044 × RelProd60,
(0.007) (0.209) (0.015) (0.001) (0.008)
R2=0.537, SER = 0.011
(a) Is the coefficient on this variable significantly different from zero at the 5% level? At the 1% level?
(b) Test for the significance of the other slope coefficients. Should you use a one-sided alternative
hypothesis or a two-sided test? Will the decision for one or the other influence the decision about the
significance of the parameters? Should you always eliminate variables which carry insignificant
coefficients?
