1) When the estimated slope coefficient in the simple regression model,^1, is zero, then A) R2 = Y .
B) 0 < R2 < 1. C) R2 = 0.
D) R2 > (SSR/TSS). Answer: C
2) The regression R2 is defined as follows: A) ESS TSS B) RSS TSS C) n i=1 (Yi - Y)(Xi - X) n i=1(Yi - Y)2 n i=1(Xi - X)2 D) SSR n-2 Answer: A
3) The standard error of the regression (SER) is defined as follows A) 1 n-2 n i=1 u^2i B) SSR C) 1-R2 D) 1 n-1 n i=1 u^2i Answer: A
4) (Requires Appendix material) Which of the following statements is correct? A) TSS= ESS + SSR B) ESS = SSR + TSS C) ESS> TSS D) R2= 1 - (ESS/TSS) Answer: A 5) Binary variables
A) are generally used to control for outliers in your sample. B) can take on more than two values.
C) exclude certain individuals from your sample. D) can take on only two values.
6) The following are all least squares assumptions with the exception of: A) The conditional distribution of ui given Xi has a mean of zero.
B) The explanatory variable in regression model is normally distributed. C) (Xi, Yi), i = 1,..., n are independently and identically distributed. D) Large outliers are unlikely.
Answer: B
7) The reason why estimators have a sampling distribution is that A) economics is not a precise science.
B) individuals respond differently to incentives. C) in real life you typically get to sample many times.
D) the values of the explanatory variable and the error term differ across samples. Answer: D
8) In the simple linear regression model, the regression slope
A) indicates by how many percent Y increases, given a one percent increase in X. B) when multiplied with the explanatory variable will give you the predicted Y. C) indicates by how many units Y increases, given a one unit increase in X. D) represents the elasticity of Y on X.
Answer: C
9) The OLS estimator is derived by
A) connecting the Yi corresponding to the lowest Xi observation with the Yi corresponding to the highest Xi observation.
B) making sure that the standard error of the regression equals the standard error of the slope estimator. C) minimizing the sum of absolute residuals.
D) minimizing the sum of squared residuals. Answer: D
10) Interpreting the intercept in a sample regression function is
A) not reasonable because you never observe values of the explanatory variables around the origin. B) reasonable because under certain conditions the estimator is BLUE.
C) reasonable if your sample contains values of Xi around the origin.
D) not reasonable because economists are interested in the effect of a change in X on the change in Y. Answer: C
11) The variance of Yi is given by A) 20 + 21 var(Xi) + var(ui).
B) the variance of ui. C) 21 var(Xi) + var(ui). D) the variance of the residuals.
13) The OLS residuals, u^i, are defined as follows: A) Y^i -^0 -^1Xi B) Yi - 0 - 1Xi C) Yi - Y^i D) (Yi - Y)2 Answer: C
14) The slope estimator, 1, has a smaller standard error, other things equal, if A) there is more variation in the explanatory variable, X.
B) there is a large variance of the error term, u. C) the sample size is smaller.
D) the intercept, 0, is small. Answer: A
15) The regression R2 is a measure of A) whether or not X causes Y.
B) the goodness of fit of your regression line. C) whether or not ESS> TSS.
D) the square of the determinant of R. Answer: B
16) (Requires Appendix) The sample regression line estimated by OLS A) will always have a slope smaller than the intercept.
B) is exactly the same as the population regression line. C) cannot have a slope of zero.
D) will always run through the point (X, Y). Answer: D
17) The OLS residuals
A) can be calculated using the errors from the regression function.
B) can be calculated by subtracting the fitted values from the actual values. C) are unknown since we do not know the population regression function.
D) should not be used in practice since they indicate that your regression does not run through all your observations.
Answer: B
18) The normal approximation to the sampling distribution of^1 is powerful because A) many explanatory variables in real life are normally distributed.
B) it allows econometricians to develop methods for statistical inference. C) many other distributions are not symmetric.
D) is implies that OLS is the BLUE estimator for 1. Answer: B
19) If the three least squares assumptions hold, then the large sample normal distribution of ^1 is A) N(0, 1 n var[Xi - X)ui] [var(Xi)]2 ). B) N( 1, 1n var(ui)]2 [var(Xi)]2). C) N( 1, 2 u n i=1(Xi - X)2 . D) N( 1, 1n var(ui)] [var(Xi)]2). Answer: B
20) In the simple linear regression model Yi = 0 + 1Xi + ui, A) the intercept is typically small and unimportant.
B) 0 + 1Xi represents the population regression function. C) the absolute value of the slope is typically between 0 and 1. D) 0 + 1Xi represents the sample regression function. Answer: B
21) To obtain the slope estimator using the least squares principle, you divide the A) sample variance of X by the sample variance of Y.
B) sample covariance of X and Y by the sample variance of Y. C) sample covariance of X and Y by the sample variance of X. D) sample variance of X by the sample covariance of X and Y. Answer: C
22) To decide whether or not the slope coefficient is large or small,
A) you should analyze the economic importance of a given increase in X. B) the slope coefficient must be larger than one.
C) the slope coefficient must be statistically significant.
D) you should change the scale of the X variable if the coefficient appears to be too small. Answer: A
23) E(ui Xi) = 0 says that
A) dividing the error by the explanatory variable results in a zero (on average). B) the sample regression function residuals are unrelated to the explanatory variable. C) the sample mean of the Xs is much larger than the sample mean of the errors.
25) Multiplying the dependent variable by 100 and the explanatory variable by 100,000 leaves the A) OLS estimate of the slope the same.
B) OLS estimate of the intercept the same. C) regression R2 the same.
D) variance of the OLS estimators the same. Answer: C
26) Assume that you have collected a sample of observations from over 100 households and their consumption and income patterns. Using these observations, you estimate the following regression Ci = 0+ 1Yi+ ui where C is consumption and Y is disposable income. The estimate of 1 will tell you
A) Income
Consumption
B) The amount you need to consume to survive C) Income
Consumption
D) Consumption
Income
Answer: D
27) In which of the following relationships does the intercept have a real-world interpretation?
A) the relationship between the change in the unemployment rate and the growth rate of real GDP (“Okun’s Law”)
B) the demand for coffee and its price C) test scores and class-size
D) weight and height of individuals Answer: A
28) The OLS residuals, u^i, are sample counterparts of the population A) regression function slope
B) errors
C) regression function’s predicted vlaues D) regression function intercept
Answer: B
29) Changing the units of measurement, e.g. measuring testscores in 100s, will do all of the following EXCEPT for changing the
A) residuals
B) numerical value of the slope estimate
C) interpretation of the effect that a change in X has on the change in Y D) numerical value of the intercept
Answer: C
30) To decide whether the slope coefficient indicates a “large” effect of X on Y, you look at the A) size of the slope coefficient
B) regression
C) economic importance implied by the slope coefficient D) value of the intercept
4.2 Essays and Longer Questions
1) Sir Francis Galton, a cousin of James Darwin, examined the relationship between the height of children and their parents towards the end of the 19th century. It is from this study that the name “regression” originated. You decide to update his findings by collecting data from 110 college students, and estimate the following relationship:
Studenth = 19.6 + 0.73 × Midparh, R2 = 0.45, SER = 2.0
where Studenth is the height of students in inches, and Midparh is the average of the parental heights.
(Following Galton’s methodology, both variables were adjusted so that the average female height was equal to the average male height.)
(a) Interpret the estimated coefficients.
(b) What is the meaning of the regression R2 ?
(c) What is the prediction for the height of a child whose parents have an average height of 70.06 inches? (d) What is the interpretation of the SER here?
(e) Given the positive intercept and the fact that the slope lies between zero and one, what can you say about the height of students who have quite tall parents? Those who have quite short parents?
(f) Galton was concerned about the height of the English aristocracy and referred to the above result as “regression towards mediocrity.” Can you figure out what his concern was? Why do you think that we refer to this result today as “Galton’s Fallacy ?
Answer: (a) For every one inch increase in the average height of their parents, the student’s height increases by 0.73 of an inch. There is no reasonable interpretation for the intercept.
(b) The model explains 45 percent of the variation in the height of students. (c) 19.6+ 0.73 × 70.06 = 70.74.
(d) The SER is a measure of the spread of the observations around the regression line. The magnitude of the typical deviation from the regression line or the typical regression error here is two inches.
(e) Tall parents will have, on average, tall students, but they will not be as tall as their parents. Short parents will have short students, although on average, they will be somewhat taller than their parents. (f) This is an example of mean reversion. Since the aristocracy was, on average, taller, he was concerned that their children would be shorter and resemble more the rest of the population. If this conclusion were true, then eventually everyone would be of the same height. However, we have not observed a decrease in the variance in height over time.
2) (Requires Appendix material) At a recent county fair, you observed that at one stand people’s weight was forecasted, and were surprised by the accuracy (within a range). Thinking about how the person could have predicted your weight fairly accurately (despite the fact that she did not know about your “heavy bones”), you think about how this could have been accomplished. You remember that medical charts for children contain 5%, 25%, 50%, 75% and 95% lines for a weight/height relationship and decide to conduct an experiment with 110 of your peers. You collect the data and calculate the following sums:
n i=1 Yi = 17,375, n i=1 Xi = 7,665.5, n i=1 y2 i = 94,228.8, n i=1 x 2i = 1,248.9, n i=1 xiyi = 7,625.9
where the height is measured in inches and weight in pounds. (Small letters refer to deviations from means as in zi = Zi – Z.)
(a) Calculate the slope and intercept of the regression and interpret these.
(b) Find the regression R2 and explain its meaning. What other factors can you think of that might have an influence on the weight of an individual?
Answer: (a)^1 = 7625.9
1,248.9 = 6.11,
^
0 = 157.95 - 6.11 × 69.69 = -267.86. For every additional inch in height, students weigh roughly 6 pounds more, on average.
(b) R2= ESS TSS = ^2 1 n i=1 x2i n i=1 y 2i = 46,624.1
94,228.8 = 0.495. Roughly half of the weight variation in the 110 students
is explained by the single explanatory variable, height. Answers will vary by student for the other factors, but calorie intake and amount of exercise typically appear as part of the list.
3) You have obtained a sub-sample of 1744 individuals from the Current Population Survey (CPS) and are interested in the relationship between weekly earnings and age. The regression, using
heteroskedasticity-robust standard errors, yielded the following result:
Earn= 239.16 + 5.20 × Age, R2 = 0.05, SER = 287.21.,
where Earn and Age are measured in dollars and years respectively. (a) Interpret the results.
(b) Is the effect of age on earnings large?
(c) Why should age matter in the determination of earnings? Do the results suggest that there is a guarantee for earnings to rise for everyone as they become older? Do you think that the relationship between age and earnings is linear?
(d) The average age in this sample is 37.5 years. What is annual income in the sample? (e) Interpret the measures of fit.
Answer: (a) A person who is one year older increases her weekly earnings by $5.20. There is no meaning attached to the intercept. The regression explains 5 percent of the variation in earnings.
(b) Assuming that people worked 52 weeks a year, the effect of being one year older translates into an additional $270.40 a year. This does not seem particularly large in 2002 dollars, but may have been earlier.
(c) In general, age-earnings profiles take on an inverted U-shape. Hence it is not linear and the linear approximation may not be good at all. Age may be a proxy for “experience,” which in itself can approximate “on the job training.” Hence the positive effect between age and earnings. The results do not suggest that there is a guarantee for earnings to rise for everyone as they become older since the regression R2 does not equal 1. Instead the result holds “on average.”
(d) Since 0 = Y - 1 X Y= 0 + 1 X. Substituting the estimates for the slope and the intercept then
results in average weekly earnings of $434.16 or annual average earnings of $22,576.32.
(e) The regression R2 indicates that five percent of the variation in earnings is explained by the model. The typical error is $287.21.
4) The baseball team nearest to your home town is, once again, not doing well. Given that your knowledge of what it takes to win in baseball is vastly superior to that of management, you want to find out what it takes to win in Major League Baseball (MLB). You therefore collect the winning percentage of all 30 baseball teams in MLB for 1999 and regress the winning percentage on what you consider the primary determinant for wins, which is quality pitching (team earned run average). You find the following information on team performance:
Summary of the Distribution of Winning 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
would have Winpct = 0.50. Interpret your regression results.
(c) It is typically sufficient to win 90 games to be in the playoffs and/or to win a division. Winning over 100 games a season is exceptional: the Atlanta Braves had the most wins in 1999 with 103. Teams play a total of 162 games a year. Given this information, do you consider the slope coefficient to be large or small?
(d) What would be the effect on the slope, the intercept, and the regression R2 if you measured Winpct in percentage points, i.e., as (Wins/Games) × 100?
(e) Are you impressed with the size of the regression R2? Given that there is 51% of unexplained variation in the winning percentage, what might some of these factors be?
Answer: (a) You expect a negative relationship, since a higher team ERA implies a lower quality of the input. No team comes close to a zero team ERA, and therefore it does not make sense to interpret the intercept. Forcing the regression through the origin is a false implication from this insight. Instead the intercept fixes the level of the regression.
(b) For every one point increase in Team ERA, the winning percentage decreases by 10 percentage points, or 0.10. Roughly half of the variation in winning percentage is explained by the quality of team pitching.
(c) The coefficient is large, since increasing the winning percentage by 0.10 is the equivalent of winning 16 more games per year. Since it is typically sufficient to win 56 percent of the games to qualify for the playoffs, this difference of 0.10 in winning percentage turns can easily turn a loosing team into a winning team.
(d) Clearly the regression R2 will not be affected by a change in scale, since a descriptive measure of the quality of the regression would depend on whim otherwise. The slope of the regression will compensate in such a way that the interpretation of the result is unaffected, i.e., it will become 10 in the above example. The intercept will also change to reflect the fact that if X were 0, then the dependent variable would now be measured in percentage, i.e., it will become 94.0 in the above example.
(e) It is impressive that a single variable can explain roughly half of the variation in winning percentage. Answers to the second question will vary by student, but will typically include the quality of hitting, fielding, and management. Salaries could be included, but should be reflected in the inputs.
5) You have learned in one of your economics courses that one of the determinants of per capita income (the “Wealth of Nations”) is the population growth rate. Furthermore you also found out that the Penn World Tables contain income and population data for 104 countries of the world. To test this theory, you regress the GDP per worker (relative to the United States) in 1990 ( RelPersInc) on the difference between the average population growth rate of that country (n) to the U.S. average population growth rate (nus ) for the years 1980 to 1990. This results in the following regression output:
RelPersInc= 0.518 – 18.831 × 18.831 × (n – nus), R2 = 0.522, SER = 0.197 (a) Interpret the results carefully. Is this relationship economically important?
(b) What would happen to the slope, intercept, and regression R2 if you ran another regression where the above explanatory variable was replaced by n only, i.e., the average population growth rate of the country? (The population growth rate of the United States from 1980 to 1990 was 0.009.) Should this have any effect on the t-statistic of the slope?
(c) 31 of the 104 countries have a dependent variable of less than 0.10. Does it therefore make sense to interpret the intercept?
Answer: (a) A relative increase in the population rate of one percentage point, from 0.01 to 0.02, say, lowers relative per-capita income by almost 20 percentage points (0.188). This is a quantitatively important and large effect. Nations which have the same population growth rate as the United States have, on average, roughly half as much per capita income.
(b) The interpretation of the partial derivative is unaffected, in that the slope still indicates the effect of a one percentage point increase in the population growth rate. The regression R2 will remain the same since only a constant was removed from the explanatory variable. The intercept will change as a result of the change in X.
6) The neoclassical growth model predicts that for identical savings rates and population growth rates, countries should converge to the per capita income level. This is referred to as the convergence hypothesis. One way to test for the presence of convergence is to compare the growth rates over time to the initial starting level. (a) If you regressed the average growth rate over a time period (1960-1990) on the initial level of per capita income, what would the sign of the slope have to be to indicate this type of convergence? Explain. Would this result confirm or reject the prediction of the neoclassical growth model?
(b) The results of the regression for 104 countries were as follows:
g6090 = 0.019 – 0.0006 × RelProd60 , R2 = 0.00007, SER = 0.016,
where g6090 is the average annual growth rate of GDP per worker for the 1960-1990 sample period, and
RelProd60 is GDP per worker relative to the United States in 1960.
Interpret the results. Is there any evidence of unconditional convergence between the countries of the world? Is this result surprising? What other concept could you think about to test for convergence between countries? (c) You decide to restrict yourself to the 24 OECD countries in the sample. This changes your regression output as follows:
g6090 = 0.048 – 0.0404 RelProd60 , R2 = 0.82 , SER = 0.0046
How does this result affect your conclusions from above?
Answer: (a) You would require a negative sign. Countries that are far ahead of others at the beginning of the period would have to grow relatively slower for the others to catch up. This represents unconditional convergence, whereas the neoclassical growth model predicts conditional convergence, i.e., there will only be convergence if countries have identical savings, population growth rates, and production technology.
(b) An increase in 10 percentage points in RelProd60 results in a decrease of 0.00006 in the growth rate from 1960 to 1990, i.e., countries that were further ahead in 1960 do grow by less. There are some countries in the sample that have a value of RelProd60 close to zero (China, Uganda, Togo, Guinea) and you would expect these countries to grow roughly by 2 percent per year over the sample period. The regression R2 indicates that the regression has virtually no explanatory power. The result is not surprising given that there are not many theories that predict unconditional convergence between the countries of the world.
(c) Judging by the size of the slope coefficient, there is strong evidence of unconditional convergence for the OECD countries. The regression R2 is quite high, given that there is only a single explanatory