Introduction to econometrics update 3rd edition stock test bank

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Introduction to Econometrics, 3e (Stock) Chapter 10 Regression with Panel Data 10.1 Multiple Choice 1) The notation for panel data is (Xit, Yit), i = 1, ..., n and t = 1, ..., T because A) we take into account that the entities included in the panel change over time and are replaced by others.

B) the X's represent the observed effects and the Y the omitted fixed effects.

C) there are n entities and T time periods.

D) n has to be larger than T for the OLS estimator to exist.

Answer: C 2) The difference between an unbalanced and a balanced panel is that A) you cannot have both fixed time effects and fixed entity effects regressions.

B) an unbalanced panel contains missing observations for at least one time period or one entity.

C) the impact of different regressors are roughly the same for balanced but not for unbalanced panels.

D) in the former you may not include drivers who have been drinking in the fatality rate/beer tax study.

Answer: B 3) Consider the special panel case where T = 2. If some of the omitted variables, which you hope to capture in the changes analysis, in fact change over time, then the estimator on the included change regressor A) will be unbiased only when allowing for heteroskedastic-robust standard errors.

B) may still be unbiased.

C) will only be unbiased in large samples.

D) will always be unbiased.

Answer: B 4) The Fixed Effects regression model A) has n different intercepts.

B) the slope coefficients are allowed to differ across entities, but the intercept is "fixed" (remains unchanged).

C) has "fixed" (repaired) the effect of heteroskedasticity.

D) in a log-log model may include logs of the binary variables, which control for the fixed effects.

Answer: A 5) In the Fixed Effects regression model, you should exclude one of the binary variables for the entities when an intercept is present in the equation A) because one of the entities is always excluded.

B) because there are already too many coefficients to estimate.

C) to allow for some changes between entities to take place.

D) to avoid perfect multicollinearity.

Answer: D

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6) In the Fixed Effects regression model, using (n – 1) binary variables for the entities, the coefficient of the binary variable indicates A) the level of the fixed effect of the ith entity.

B) will be either 0 or 1.

C) the difference in fixed effects between the ith and the first entity.

D) the response in the dependent variable to a percentage change in the binary variable.

Answer: C 7) cov (uit, uis Xit, Xis = 0 for t ≠ s means that A) there is no perfect multicollinearity in the errors.

B) division of errors by regressors in different time periods is always zero.

C) there is no correlation over time in the residuals.

D) conditional on the regressors, the errors are uncorrelated over time.

Answer: D 8) With Panel Data, regression software typically uses an "entity-demeaned" algorithm because A) the OLS formula for the slope in the linear regression model contains deviations from means already.

B) there are typically too many time periods for the regression package too handle.

C) the number of estimates to calculate can become extremely large when there are a large number of entities.

D) deviations from means sum up to zero.

Answer: C 9) The "before and after" specification, binary variable specification, and "entity-demeaned" specification produce identical OLS estimates A) as long as there are observations for more than two time periods.

B) if you use the heteroskedasticity-robust option in your regression program.

C) for the case of more than 100 observations.

D) as long as T = 2 and the intercept is excluded from the "before and after" specification.

Answer: D 10) In the Fixed Time Effects regression model, you should exclude one of the binary variables for the time periods when an intercept is present in the equation A) because the first time period must always excluded from your data set.

B) because there are already too many coefficients to estimate.

C) to avoid perfect multicollinearity.

D) to allow for some changes between time periods to take place.

Answer: C 11) If you included both time and entity fixed effects in the regression model which includes a constant, then A) one of the explanatory variables needs to be excluded to avoid perfect multicollinearity.

B) you can use the "before and after" specification even for T > 2.

C) you must exclude one of the entity binary variables and one of the time binary variables for the OLS estimator to exist.

D) the OLS estimator no longer exists.

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12) Consider estimating the effect of the beer tax on the fatality rate, using time and state fixed effect for the Northeast Region of the United States (Maine, Vermont, New Hampshire, Massachusetts, Connecticut and Rhode Island) for the period 1991-2001. If Beer Tax was the only explanatory variable, how many coefficients would you need to estimate, excluding the constant?

A) 18 B) 17 C) 7 D) 11 Answer: B 13) Consider the regression example from your textbook, which estimates the effect of beer taxes on fatality rates across the 48 contiguous U.S. states. If beer taxes were set nationally by the federal government rather than by the states, then A) it would not make sense to use state fixed effect.

B) you can test state fixed effects using homoskedastic-only standard errors.

C) the OLS estimator will be biased.

D) you should not use time fixed effects since beer taxes are the same at a point in time across states.

Answer: D 14) In the panel regression analysis of beer taxes on traffic deaths, the estimation period is 1982-1988 for the 48 contiguous U.S. states. To test for the significance of time fixed effects, you should calculate the Fstatistic and compare it to the critical value from your Fq,∞ distribution, where q equals A) 6.

B) 7.

C) 48.

D) 53.

Answer: A 15) When you add state fixed effects to a simple regression model for U.S. states over a certain time period, and the regression R2 increases significantly, then it is safe to assume that A) the included explanatory variables, other than the state fixed effects, are unimportant.

B) state fixed effects account for a large amount of the variation in the data.

C) the coefficients on the other included explanatory variables will not change.

D) time fixed effects are unimportant.

Answer: B 16) Time Fixed Effects regression are useful in dealing with omitted variables A) even if you only have a cross-section of data available.

B) if these omitted variables are constant across entities but vary over time.

C) when there are more than 100 observations.

D) if these omitted variables are constant across entities but not over time.

Answer: B

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17) Indicate for which of the following examples you cannot use Entity and Time Fixed Effects: a regression of A) OECD unemployment rates on unemployment insurance generosity for the period 1980-2006 (annual data).

B) the (log of) earnings on the number of years of education, using the Current Population Survey of 60,000 households for March 2006.

C) the per capita income level in Canadian Provinces on provincial population growth rates, using decade averages for 1960, 1970, and 1980.

D) the risk premium of 75 stocks on the market premium for the years 1998-2006.

Answer: B 18) Panel data is also called A) longitudinal data.

B) cross-sectional data.

C) time series data.

D) experimental data.

Answer: A 19) (Requires Appendix material) When the fifth assumption in the Fixed Effects regression (cov (uit, uis Xit, Xis) = 0 for t ≠ s) is violated, then A) using heteroskedastic-robust standard errors is not sufficient for correct statistical inference when using OLS. B) the OLS estimator does not exist.

C) you can use the simple homoskedasticity-only standard errors calculated in your regression package.

D) you cannot use fixed time effects in your estimation.

Answer: A 20) In the panel regression analysis of beer taxes on traffic deaths, the estimation period is 1982-1988 for the 48 contiguous U.S. states. To test for the significance of entity fixed effects, you should calculate the Fstatistic and compare it to the critical value from your Fq,∞ distribution, where q equals A) 48.

B) 54.

C) 7.

D) 47.

Answer: D 21) The main advantage of using panel data over cross sectional data is that it A) gives you more observations.

B) allows you to analyze behavior across time but not across entities.

C) allows you to control for some types of omitted variables without actually observing them.

D) allows you to look up critical values in the standard normal distribution.

Answer: C

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22) One of the following is a regression example for which Entity and Time Fixed Effects could be used: a study of the effect of A) minimum wages on teenage employment using annual data from the 48 contiguous states in 2006 .

B) various performance statistics on the (log of) salaries of baseball pitchers in the American League and the National League in 2005 and 2006.

C) inflation and inflationary expectations on unemployment rates in the United States, using quarterly data from 1960-2006.

D) drinking alcohol on the GPA of 150 students at your university, controlling for incoming SAT scores.

Answer: B 23) Consider a panel regression of unemployment rates for the G7 countries (United States, Canada, France, Germany, Italy, United Kingdom, Japan) on a set of explanatory variables for the time period 1980-2000 (annual data). If you included entity and time fixed effects, you would need to specify the following number of binary variables: A) 21.

B) 6.

C) 28.

D) 26.

Answer: D 24) A pattern in the coefficients of the time fixed effects binary variables may reveal the following in a study of the determinants of state unemployment rates using panel data: A) macroeconomic effects, which affect all states equally in a given year.

B) attitude differences towards unemployment between states.

C) there is no economic information that can be retrieved from these coefficients.

D) regional effects, which affect all states equally, as long as they are a member of that region.

Answer: A 25) In the panel regression analysis of beer taxes on traffic deaths, the estimation period is 1982-1988 for the 48 contiguous U.S. states. To test for the significance of time fixed effects, you should calculate the Fstatistic and compare it to the critical value from your Fq,∞ distribution, which equals (at the 5% level) A) 2.01.

B) 2.10.

C) 2.80.

D) 2.64.

Answer: B 26) Assume that for the T = 2 time periods case, you have estimated a simple regression in changes model and found a statistically significant positive intercept. This implies A) a negative mean change in the LHS variable in the absence of a change in the RHS variable since you subtract the earlier period from the later period B) that the panel estimation approach is flawed since differencing the data eliminates the constant (intercept) in a regression C) a positive mean change in the LHS variable in the absence of a change in the RHS variable D) that the RHS variable changed between the two subperiods Answer: C

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27) HAC standard errors and clustered standard errors are related as follows: A) they are the same B) clustered standard errors are one type of HAC standard error C) they are the same if the data is differenced D) clustered standard errors are the square root of HAC standard errors Answer: B 28) In panel data, the regression error A) is likely to be correlated over time within an entity B) should be calculated taking into account heteroskedasticity but not autocorrelation C) only exists for the case of T > 2 D) fits all of the three descriptions above Answer: A 29) It is advisable to use clustered standard errors in panel regressions because A) without clustered standard errors, the OLS estimator is biased B) hypothesis testing can proceed in a standard way even if there are few entities (n is small) C) they are easier to calculate than homoskedasticity-only standard errors D) the fixed effects estimator is asymptotically normally distributed when n is large Answer: D 30) If Xit is correlated with Xis for different values of s and t, then A) Xit is said to be autocorrelated B) the OLS estimator cannot be computed C) statistical inference cannot proceed in a standard way even if clustered standard errors are used D) this is not of practical importance since these correlations are typically weak in applications Answer: A

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10.2 Essays and Longer Questions 1) A study, published in 1993, used U.S. state panel data to investigate the relationship between minimum wages and employment of teenagers. The sample period was 1977 to 1989 for all 50 states. The author estimated a model of the following type:

where E is the employment to population ratio of teenagers, M is the nominal minimum wage, and W is average hourly earnings in manufacturing. In addition, other explanatory variables, such as the adult unemployment rate, the teenage population share, and the teenage enrollment rate in school, were included. (a) Name some of the factors that might be picked up by time and state fixed effects. (b) The author decided to use eight regional dummy variables instead of the 49 state dummy variables.

What is the implicit assumption made by the author? Could you test for its validity? How? (c) The results, using time and region fixed effects only, were as follows: = -0.182 × ln(Mit /Wit ) + ...; R2= 0.727 (0.036) Interpret the result briefly. (d) State minimum wages do not exceed federal minimum wages often. As a result, the author decided to choose the federal minimum wage in his specification above. How does this change your interpretation?

How is the original equation

affected by this?

Answer: (a) Time effects will pick up the effect of omitted variables that are common to all 50 states at a given point in time. Federal fiscal and monetary variables, exchange rate and U.S. terms of trade movements, aggregate business cycle developments, etc., are candidates here. State fixed effects will include variables that are slowly changing over time within a specific state such as attitudes toward employment or labor force participation, state specific labor market policies, industrial and labor force composition, etc. (b) The implicit assumption by the author is that the coefficients on the state fixed effects are identical within a region but differ between regions. Since these coefficients imply linear restrictions, they can be tested using the F-test. (c) Consider a ten percent increase in minimum wages, say from $5 to $5.50 with constant average hourly earnings. This corresponds to a ten percent increase in relative minimum wages. The resulting decrease in the teenage to population ratio is 1.8 or almost 2 percent. The regression explains roughly 73 percent of the employment to population ratio of teenagers during the period of 1977 to 1989 for the 50 U.S. states. (d) This choice in effect drops the i subscript from the minimum wage, since there is no variation by state.

The original equation then reads

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Furthermore, since the federal minimum wage is constant across the nine regions at a point in time, it is absorbed by the time effects. The coefficient on the relative minimum wage therefore reflects regional variations in average hourly earning in manufacturing. The minimum wage only enters indirectly as changes in the federal minimum wage since there are different relative levels to average hourly earnings in each region.

2) You want to find the determinants of suicide rates in the United States. To investigate the issue, you collect state level data for ten years. Your first idea, suggested to you by one of your peers from Southern California, is that the annual amount of sunshine must be important. Stacking the data and using no fixed effects, you find no significant relationship between suicide rates and this variable. (This is good news for the people of Seattle.) However, sorting the suicide rate data from highest to lowest, you notice that those states with the lowest population density are dominating in the highest suicide rate category. You run another regression, without fixed effect, and find a highly significant relationship between the two variables. Even adding some economic variables, such as state per capita income or the state unemployment rate, does not lower the t-statistic for the population density by much. Adding fixed entity and time effects, however, results in an insignificant coefficient for population density. (a) What do you think is the cause for this change in significance? Which fixed effect is primarily responsible? Does this result imply that population density does not matter? (b) Speculate as to what happens to the coefficients of the economic variables when the fixed effects are included. Use this example to make clear what factors entity and time fixed effects pick up. (c) What other factors might play a role?

Answer: (a) Population density only changes slowly over time, hence state effects will pick up the influence of this variable. This does not imply that population is of no relevance. However, there are other omitted variables in this regression, such as religious and cultural attitudes towards suicide, that are also captured by the state effects, and these may also be correlated with population density. (b) Since there is sufficient variation of state unemployment rates and state per capita income both over time and across states, the coefficients on these variables are likely to remain statistically significant.

However, there may be multicollinearity between the two variables, and the standard errors may therefore be large. (c) Answers will vary by student. Cultural and institutional factors, such as attitudes towards suicide and religion, and social services, are frequently mentioned.

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3) Two authors published a study in 1992 of the effect of minimum wages on teenage employment using a U.S. state panel. The paper used annual observations for the years 1977-1989 and included all 50 states plus the District of Columbia. The estimated equation is of the following type (Eit )= β0 + β1 (Mit /Wit ) +

D2i + ... +

D51i +

B2t + ... +

B13t + uit,

where E is the employment to population ratio of teenagers, M is the nominal minimum wage, and W is average wage in the state. In addition, other explanatory variables, such as the prime-age male unemployment rate, and the teenage population share were included. (a) Briefly discuss the advantage of using panel data in this situation rather than pure cross sections or time series. (b) Estimating the model by OLS but including only time fixed effects results in the following output it = 0 - 0.33 × (Mit /Wit ) + 0.35(SHYit) – 1.53 × uramit; R2 = 0.20 (0.08) (0.28) (0.13) where SHY is the proportion of teenagers in the population, and uram is the prime-age male unemployment rate. Coefficients for the time fixed effects are not reported. Numbers in parenthesis are homoskedasticity-only standard errors.

Comment on the above results. Are the coefficients statistically significant? Since these are level regressions, how would you calculate elasticities? (c) Adding state fixed effects changed the above equation as follows: it = 0 + 0.07 × (Mit /Wit ) – 0.19 × (SHYit) – 0.54 × uramit; 2 = 0.69 (0.10) (0.22) (0.11) Compare the two results. Why would the inclusion of state fixed effects change the coefficients in this way? (d) The significance of each coefficient decreased, yet 2 increased. How is that possible? What does this result tell you about testing the hypothesis that all of the state fixed effects can be restricted to have the same coefficient? How would you test for such a hypothesis?

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Answer: (a) There are likely to be omitted variables in the above regression. One way to deal with some of these is to introduce state and time effects. State effects will capture the influence of omitted variables that are state specific and do not vary over time, while time effects capture those of country wide variables that are common to all states at a point in time. Furthermore, there are more observations when using panel data, resulting in more variation. (b) There is negative relationship between minimum wages and the employment to population ratio.

Increases in the share of teenagers in the population result in a higher employment to population ratio, and increases in the prime-age male unemployment rate lower the employment to population ratio. 20 percent of employment to population of teenagers variation is explained by the above regression. The relative minimum wage and the prime-age male unemployment rate are significant using a 1% significance level, while the proportion of teenagers in the population is not. Elasticities vary with levels here. One possibility is to report elasticities at sample means. (c) The parameter of interest here is the coefficient on the relative minimum wage. While it was highly significant in the previous regression, it now has changed signs and is statistically insignificant. The explanatory power of the equation has increased substantially. The size of the other two coefficients has also decreased. The results suggest that omitted variables, which are now captured by state fixed effects, were correlated with the regressors and caused omitted variable bias. (d) The influence of the state effects is large. These are bound to be statistically significant and the hypothesis to restrict these coefficients to zero is bound to fail. Since these are linear hypothesis that are supposed to hold simultaneously, an F-test is appropriate here.

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4) You learned in intermediate macroeconomics that certain macroeconomic growth models predict conditional convergence or a catch up effect in per capita GDP between the countries of the world. That is, countries which are further behind initially in per-capita GDP will grow faster than the leader. You gather data from the Penn World Tables to test this theory. (a) By limiting your sample to 24 OECD countries, you hope to have a more homogeneous set of countries in your sample, i.e., countries that are not too different with respect to their institutions. To simplify matters, you decide to only test for unconditional convergence. In that case, the laggards catch up even without taking into account differences in some of the driving variables. Your scatter plot and regression for the time period 1975-1989 are as follows:

= 0.024 – 0.005 PCGDP75_US; R2= 0.025, SER = 0.006 (0.06) (0.008) where

is the average annual growth rate of per capita GDP from 1975-1989, and PCGDP75_US is

per capita GDP relative to the United States in 1975. Numbers in parenthesis are heteroskedasticityrobust standard errors.

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Interpret the results. Is there indication of unconditional convergence? What critical value did you use? (b) Although you are quite discouraged by the result, you think that it might be due to the specific time period used. During this period, there were two OPEC oil price shocks with varying degrees of exposure for the OECD countries. You therefore repeat the exercise for the period 1960-1974, with the following results:

= 0.061 – 0.043 PCGDP60_US; R2= 0.613, SER = 0.008 (0.004) (0.007) where

is the average annual growth rate of per capita GDP from 1960-1974, and PCGDP60_US is

per capita GDP relative to the United States in 1960.

Compare this regression to the previous one. (c) You decide to run one more regression in differences. The dependent variable is now the change in the growth rate of per capita GDP from 1960-1974 to 1975-1989 (diffg) and the regressor the difference in the initial conditions (diffinit). This produces the following graph and regression:

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= -0.006 – 0.096 × diffinit; R2 = 0.468; SER = 0.009 (0.03) (0.021) Interpret these results. Explain what has happened to unobservable omitted variables that are constant over time. Suggest what some of these variables might be. (d) Given that there are only two time periods, what other methods could you have employed to generate the identical results? Why do you think that the slope coefficient in this regression is significant given the results over the sub-periods?

Answer: (a) Although the slope coefficient is negative, thereby indicating unconditional convergence, the t-statistic does not exceed the critical value. However, using the standard normal distribution here is not really justified here since there are only 24 observations. (b) The explanatory power of the regression is much higher and there is a larger t-statistic for the slope coefficient. If a standard normal distribution could be used here, then the absolute value of the t-statistic would easily exceed the critical value of 1.64. This suggests unconditional convergence over the sample period. However, the same comment regarding the sample size as in (a) applies here. (c) The slope coefficient suggests that countries which are further behind initially with respect to the United States, will grow relatively faster. Almost 50 percent of the relative growth difference variation is explained by the regression. Decreasing the initial per capita income ratio to the United States by 10 percentage points will decrease the relative growth performance by 1 percentage point. Omitted variables that remain constant over time are picked up by focusing on changes in the variables. Some of these may be cultural and institutional variables such as the level of educational attainment, saving rates, population growth rates, independence of the central bank, etc. (d) Using either a fixed effects regression or entity-demeaned OLS would have resulted in identical estimates. In general, it is possible for included coefficients to be statistically insignificant as a result of omitted variable bias. This result depends, among other factors, on the relationship between the omitted variables and the included variables. Using the differencing method has eliminated at least some of the omitted variables.

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5) A researcher investigating the determinants of crime in the United Kingdom has data for 42 police regions over 22 years. She estimates by OLS the following regression ln(cmrt)it = αi + φt + β1unrtmit + β2proythit + β3 ln(pp)it + uit; i = 1,..., t = 1,..., 22 where cmrt is the crime rate per head of population, unrtm is the unemployment rate of males, proyth is the proportion of youths, pp is the probability of punishment measured as (number of convictions)/(number of crimes reported). α and φ are area and year fixed effects, where αi equals one for area i and is zero otherwise for all i, and φt is one in year t and zero for all other years for t = 2, …, 22. φ1 is not included. (a) What is the purpose of excluding φ1? What are the terms α and φ likely to pick up? Discuss the advantages of using panel data for this type of investigation. (b) Estimation by OLS using heteroskedasticity and autocorrelation-consistent standard errors results in the following output, where the coefficients of the fixed effects are not reported: = 0.063 × unrtmit + 3.739 × proythit – 0.588 × ln(pp)it ; R2 = 0.904 (0.109)

(0.179)

(0.024)

Comment on the results. In particular, what is the effect of a ten percent increase in the probability of punishment? (c) To test for the relevance of the area fixed effects, your restrict the regression by dropping all entity fixed effects and add single constant is added. The relevant F-statistic is 135.28. What are the degrees of freedom? What is the critical value from your F table? (d) Although the test rejects the hypothesis of eliminating the fixed effects from the regression, you want to analyze what happens to the coefficients and their standard errors when the equation is re-estimated without fixed effects. In the resulting regression, standard errors roughly double. However,

and

do not change by much, although their

is now 1.340 with a standard error of 0.234. Why do you

think that is?

Answer: (a) Since there is no constant in addition to the entity and time fixed effects, setting φt to one in year t and zero for all other years for t = 1, …, 22 would result in perfect multicollinearity. α picks up omitted variables that are specific to police regions and do not vary over time. φ picks up effects that are common to all police regions in a given year. Attitudes toward crime may vary between rural regions and metropolitan areas. These would be hard to capture through measurable variables. Common macroeconomic shocks that affect all regions equally will be captured by the time fixed effects. Although some of these variables could be explicitly introduced, the list of possible variables is long. By introducing time fixed effects, the effect is captured all in one variable. (b) A higher male unemployment rate and a higher proportion of youths increase the crime rate, while a higher probability of punishment decreases the crime rate. The coefficients on the probability of punishment and the proportion of youths is statistically significant, while the male unemployment rate is not. The regression explains roughly 90 percent of the variation in crime rates in the sample. A ten percent increase in the number of convictions over the number of crimes reported decreases the crime rate by roughly six percent.

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(c) The coefficients of the three regressors other than the entity coefficients would have been unaffected, had there been a constant in the regression and (n-1) police region specific entity variables. In this case, the entity coefficients on the police regions would have indicated deviations from the constant for the first police region. Hence there are 41 restrictions imposed by eliminating the entity fixed effects and adding a constant. Since there are over 100 observations (900 degrees of freedom), the critical value for F41,∞ ≈ F30,∞ = 1.70 at the 1% level. Hence the restrictions are rejected. (d) This result would make the male unemployment rate coefficient significant. It suggests that male unemployment rates change slowly over the years in a given police district and that this effect is picked up by the entity fixed effects. Of course, there are other slowly changing variables, such as attitudes towards crime, that are captured by these fixed effects.

6) You want to investigate the relationship between cumulative GPA scores at graduation and incoming SAT scores of students. For this purpose, you have collected data from a balanced panel of 120 undergraduate colleges and universities in the United States over a ten year period. Discuss some of the entity fixed effects which you potentially capture by allowing for a binary variable for each of the colleges.

Answer: Students will come up with various possible entity fixed effects. These should include differences between educational institutions that have • made it a policy not to fight grade inflation • a different degree of selectivity (if you admit only students with an SAT score of 2400, then there will be no relationship) • gender or religion specific requirements • an admission process that is need-blind • large varsity sports programs and so forth.

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7) You want to study the relationship between weight and height of young children (4 th grade to 7th grade). You collect data for more than 400 students and track the progress of these students over the following four years, where you end up with a balanced panel of 400 students (you discard the observations for the students who moved away). Discuss some of the entity fixed effects which you potentially capture by allowing for a binary variable for each of the students. Do you expect significant time fixed effects if you allowed for them?

Answer: Students will come up with various possible entity fixed effects. These will reflect differences between students potentially depending on • • • • • •

gender ethnicity degree of participation in exercises/athletic programs growth spurts during these years nutrition genes

and so forth. It is hard to think of time fixed effects. Potentially there could be an effect if all students went to a different school in 7th grade (e.g. middle school) and this school had a less/more healthy lunch diet.

8) You first encountered growth regression in your intermediate macroeconomics course ("betaconvergence regressions"), that is, conditionally on some initial condition in per capita income, different authors tried to find the determinants of growth. Since growth is a long-run phenomenon, various studies collected data for a panel of numerous countries using 10-year averages, over a time period stretching from 1960 to 2005. For example, a balanced panel might consist of 50 or so odd countries for the time periods 1960-1970, 1971-1980, …, 2000-2005. Instead of using two-way fixed effects (entity fixed effects and time fixed) authors often only employed time fixed effects. Why do you think that is? What sort of information would be lost if these authors employed entity fixed effects as well?

Answer: Time fixed effects will eliminate common growth phenomenon experienced by all countries during the same decade (say). These could include productivity slow-downs due to the oil crisis of the ‘70s, effects of the Great Moderation of the ‘90s, etc. However, most of these studies were interested in determining the effect of institutional differences between countries. These effects, such as the degree of democracy, law and order, openness of the economy, size of government, civil wars, geography, religion, etc., are typically slowly changing, and by including entity fixed effects, you would lose the effects you are interested in studying.

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10.3 Mathematical and Graphical Problems 1) Your textbook suggests an "entity-demeaned" procedure to avoid having to specify a potentially large number of binary variables. While it is somewhat tedious to specify a binary variable for each entity, this can still be handled relatively easily in the case of the 48 contiguous states. Give a few examples where it might be close to impossible to implement specifying such large number of entity binary variables. The idea of the "entity-demeaned" procedure was introduced as a computationally convenient and simplifying procedure. Since there are also time fixed effects, why is there no discussion of using a "timedemeaned" procedure? Using the following equation Yit = β0 + β1Xit + β3St + uit, Show how β1 can be estimated by the OLS regression using "time-demeaned" variables.

Answer: Answers will vary by student with regard to the examples. A panel containing tens of thousands of individuals would make it impractical to specify entity fixed effects. The same would hold for a large number of firms. Regression software typically does not estimate panel regressions using "time-demeaned" variables, since there are not that many observations across time. In the textbook example, there were seven years of data for the 48 contiguous U.S. states. There maybe observations for 10,000 individuals over a few years. Still, in principle, you could use a "time-demeaning" procedure.

Taking averages on both sides of the above equation results in t = β1 t + β0 + β3St + t where Y t =

1 n 1 n 1 n Yit , X t =  X it and u t =  uit .  n i =1 n i =1 n i =1

Subtracting the averaged equation from the original one yields Yit - t = β1 (Xit - t) + (uit - t) or Y it = β1 X it + u it, where Y it = Yit - i, and X it and u it are defined similarly. The "time-demeaned" regression can then be estimated by OLS.

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2) Consider the case of time fixed effects only, i.e., Yit = β0 + β1Xit + β3St + uit, First replace β0 + β3St with φt. Next show the relationship between the φt and δt in the following equation Yit = β0 + β1Xit + δ2B2t + ... + δTBTt + uit, where each of the binary variables B2, …, BT indicates a different time period. Explain in words why the two equations are the same. Finally show why there is perfect multicollinearity if you add another binary variable B1. What is the intuition behind the fact that the OLS estimator does not exist in this case? Would that also be the case if you dropped the intercept?

Answer: Yit = β1Xit + φt + uit. The relationship is φ1 = β0, and φt = β0 + δt for t ≥ 2. Consider time period t, then the population regression line for that period is φt + β1Xit, with β1 being the same for all time periods, but the intercept varying from time period to time period. The variation of the intercept comes from factors which are common to all entities in a given time period, i.e., the St. The same role is played by δ2, … δT, since B2t …BTt are only different from zero during one period. There is perfect multicollinearity if one of the regressors can be expressed as a linear combination of the other regressors.

Define B0t as a variable that equals one for all period. In that case, the previous regression can be rewritten as Yit = β0 B0t + β1Xit + δ2B2t + ... + δTBTt + uit.

Adding B1 with a coefficient here results in Yit = β0 B0t + β1Xit + δ1B1t + δ2B2t ... + δTBTt + uit.

But B0t = B1t + B2t + ... + BTt, and hence there is perfect multicollinearity. Intuitively, whenever any one of the binary variable equals one in a given period, so does the constant. Hence the coefficient of that variable cannot pick up a separate effect from the data. Dropping the intercept from the regression eliminates the problem.

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3) Consider the following panel data regression with a single explanatory variable Yit = β0 + β1Xit + uit.

In each of the examples below, you will be adding entity and time fixed effects. Indicate the total number of coefficients that need to be estimated. (a) The effect of beer taxes on the fatality rate, annual data, 1982-1988, nine U.S. regions (New England, Pacific, Mid-Atlantic, East North Central, etc.). (b) The effect of the minimum wage on teenage employment, annual data, 1963-2000, five Canadian Regions (Atlantic Provinces, Quebec, Ontario, Prairies, British Columbia). (c) The effect of savings rates on per capita income, data for three decades (1960-1969, 1970-1979, 19801989; one observation per decade), 104 countries of the world. (d) The effect of pitching quality in baseball (as measured by the Team ERA) on the winning percentage, annual data, 1998-1999 season, 1999-2000 season, 30 teams.

Answer: (a) 16 coefficients (6 time fixed effects, 8 entity fixed effects, intercept, slope). (b) 43 coefficients (37 time fixed effects, 5 entity fixed effects, intercept, slope). (c) 107 coefficients (3 time fixed effects, 103 entity fixed effects, intercept, slope). (d) 32 coefficients (1 time fixed effect, 29 entity fixed effects, intercept, slope).

4) Your textbook modifies the four assumptions for the multiple regression model by adding a new assumption. This represents an extension of the cross-sectional data case, where errors are uncorrelated across entities. The new assumption requires the errors to be uncorrelated across time, conditional on the regressors as well (cov(uit, uis Xit, Xis) = 0 for t ≠ s.). (a) Discuss why there might be correlation over time in the errors when you use U.S. state panel data.

Does this mean that you should not use OLS as an estimator? (b) Now consider pairs of adjacent states such as Indiana and Michigan, Texas and Arkansas, New York and Connecticut, etc. Is it likely that the fifth assumption will hold here, even though the "contemporaneous" errors are correlated? If not, can you still use OLS for estimation?

Answer: (a) The error term may contain omitted variables. If these change slowly from one period to the next, then the error term will be correlated over time. In that case (cov(uit, uis Xit, Xis) = 0 for t ≠ s will be violated.

The OLS estimator is still unbiased, but valid statistical inference cannot be conducted, even when using heteroskedasticity-robust standard errors. However, heteroskedasticity- and autocorrelation- consistent standard errors can be used in this situation. (b) The fifth assumption deals with observations that do not occur during the same time period. It does not address the problems of errors of one entity being affected by errors in another entity during the same period. While potentially there are more efficient estimators available in such a situation, OLS can still be used for estimation.

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5) In Sports Economics, production functions are often estimated by relating the winning percentage of teams (Y) to inputs indicating performance in certain aspects of the game. However, this omits the quality of management. Assume that you could measure the quality of pitching and hitting by a single index L, and that managerial ability is represented by M, which is assumed to be constant over time. The production function would then be specified as follows: Yit = β0 +β1 Lit + β2Mi + uit where i is an index for the baseball team, and t indexes time and all variables are in logs. (a) Assume that managerial ability is unobservable but is positively related, in a linear way, to L. Explain why the OLS estimator 1 is inconsistent in the case of a single cross-section, i.e., if you attempt to estimate the above regression for a single year. Do you expect this coefficient to over- or under-estimate β1? (b) If you had data for two years, indicate the transformation, which allows you to obtain a consistent estimator for β1.

Answer: (a) Regressing Y on L alone will result in omitted variable bias. An increase in the pitching and hitting index will increase managerial ability, which in return increases the winning percentage. Hence 1 will be expected to overestimate the effect of pitching and hitting on the winning percentage. Said differently, OLS will attribute more to pitching and hitting quality and it deserves. (b) Since managerial ability is assumed to be constant over time, then differencing the data over the two time-periods will eliminate this effect for all teams. This can be shown as follows: Yi2 = β0 +β1 Li2 + β2Mi + ui2 Yi1 = β0 +β1 Li1 + β2Mi + ui1 Subtracting the second equation from the first results in Yi2 - Yi1 = β1(Li2 - Li1) + ui2 - ui1.

Alternatively, the binary variable specification or the "entity-demeaned" specification could have been used with identical estimation results.

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6) A study attempts to investigate the role of the various determinants of regional Canadian unemployment rates in order to get a better picture of Canadian aggregate unemployment rate behavior.

The annual data (1967-1991) is for five regions (Atlantic region, Quebec, Ontario, Prairies, and British Columbia), and four age-gender groups (female and male, adult and young). Focusing on young females, the authors find significant effects for the following variables: the regional relative minimum wage rate (minimum wages divided by average hourly earnings), the regional share of youth in the labor force, the regional share of adult females in the labor force, United States activity shocks (deviations of United States GDP from trend), an indicator of the degree of monetary tightness in Canada, regional union density, and a regional index of unemployment insurance generosity. Explain why the authors only used region fixed effects. How would their specification have to change if they also employed time fixed effects?

Answer: Since the study used Canada-wide effects (United States activity shocks, and monetary tightness), these are identical for all regions at a point in time. Using time fixed effects in addition to these two variables would have generated perfect multicollinearity among the regressors, and hence the OLS estimator would not exist. An alternative specification would include time fixed effects, but eliminate the two variables which are constant across all regions at a given point in time.

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7) (Requires Matrix Algebra) Consider the time and entity fixed effect model with a single explanatory variable Yit = β0 + β1Xit +

D2i + ... +

Dni + δ2B2t + ... + δTBTt + uit,

For the case of n = 4 and T = 3, write this model in the form Y = Xβ + U, where, in general,

Y=

,U=

,X=

=

, and β =

How would the X matrix change if you added two binary variables, D1 and B1? Demonstrate that in this case the columns of the X matrix are not independent. Finally show that elimination of one of the two variables is not sufficient to get rid of the multicollinearity problem. In terms of the OLS estimator, ( X)-1 Y, why does perfect multicollinearity create a problem?

Answer: For the case of n = 4 and T = 3, the general model would look as follows:

=

+

Adding the two binary variable would change the X matrix in this way:

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=

X=

Adding columns 6, 7, and 9 results in column 1. Also adding columns 3, 4, 5, and 8 results in column 1.

Hence the columns are not linearly independent and there is perfect multicollinearity among the columns of the matrix. Eliminating column 9, say, is not sufficient to get rid of this problem, since adding columns 3, 4, 5, and 8 still equals column 1. In case of perfect multicollinearity, the X matrix will not have full rank, and hence ( X)-1 will also not have full rank (it is singular). In this case, ( X)-1 cannot be inverted, and hence the OLS estimator does not exist.

8) Consider the time and entity fixed effect model with a single explanatory variable Yit = β0 + β1Xit +

D2i + ... +

Dni + δ2B2t + ... + δTBTt + uit,

Assume that you had estimated the above equation by OLS. Typically the coefficients for the entity and time binary variables are not reported. Can you think of situations where the pattern of these coefficients might be of interest? What could you do, for example, if you had a strong theoretical justification for believing that a few macroeconomic variables had an effect on Yit? Answer: The coefficients pick up the effects of omitted variables that are common to all entities at a point in time (time fixed effects), or that are constant across time for entities (entity fixed effect). If data is available on slowly changing variables across time, say population density or average educational attainment by U.S. state, or on macroeconomic variables, then you could perform a regression of the binary variable coefficients on these variables to determine the degree of correlation. Obviously, the correlation will be less than perfect, and unless these variables bear coefficients of interest, then there is little to be gained from these auxiliary regressions.

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9) "Empirical studies of economic growth are flawed because many of the truly important underlying determinants, such as culture and institutions, are very hard to measure." Discuss this statement paying particular attention to simple cross-section data and panel data models. Use equations whenever possible to underscore your argument.

Answer: Although some cultural and institutional variables, such as corruption, black market activity, central bank independence, trust, etc., are hard to measure, authors have developed such series for the countries of the world. Still, either these variables are measure with error or not all cultural and institutional aspects are bound to be captured. Hence you would expect omitted variable bias to be present in cross-sectional studies. However, if you could argue that these effects are constant across time or at least slowly changing, then introducing country fixed effects in panel studies goes some way to alleviate the omitted variable problem. Similarly by using time fixed effects, common world business cycle effects can be largely eliminated. For an empirical study of economic growth using U.S. states, time fixed effects would eliminate common effects of monetary policy and inflation.

The above argument can be made using equations along the theoretical arguments presented in sections 8.3 and 8.4 of the textbook.

10) Give at least three examples from macroeconomics and five from microeconomics that involve specified equations in a panel data analysis framework. Indicate in each case what the role of the entity and time fixed effects in terms of omitted variables might be.

Answer: Answers will vary by student. Given the textbook example, you can expect a study of fatality rates and beer taxes to appear. Other examples mentioned may be minimum wage studies using data from U.S. states or Canadian provinces, panel data in earnings studies, empirical studies of economic growth across the countries of the world or regions within a country, determinants of unemployment rates using data from geographical units (countries, regions, states), degree of democratization of the countries of the world, etc. Students should point out in the various examples how entity and time fixed effects pick up variables that are constant across entities at a point in time, or constant over time for specific entities. For geographical units, these typically involve cultural and institutional factors, and common macroeconomic effects.

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11) Your textbook specifies a simple regression problem for two time periods for the years 1982 and 1988 as follows: FatalityRatei,1982 = β0 + β1BeerTaxi,1982 + ui,1982 FatalityRatei,1988 = β0 + β1BeerTaxi,1988 + ui,1988 After subtracting the first equation from the second equation, the authors estimate the model and find a negative intercept. a. Show how you would have to modify the two equations to allow for the presence of an intercept in the differenced model. b. What would the relative magnitude of the modified model have to be for you to find a negative intercept?

Answer: a. FatalityRatei,1982 = β0 + β1BeerTaxi,1982 + ui,1982 FatalityRatei,1988 = α0 + β1BeerTaxi,1988 + ui,1988 b. α0 < β0 12) Your textbook reports the following result from an two-way fixed effects (entity and time fixed effects) regression model: = -0.66 BeerTax + StateFixedEffects + TimeFixedEffects (0.36) Where the number in parenthesis is the heteroskedasticity- and autocorrelation-consistent (HAC) standard error. a. Calculate the t-statistic. Can you reject the null hypothesis that the slope coefficient is zero in the population, using a two-sided test and a 5% significance level? b. Given that economic theory suggests that the population slope is negative under the alternative hypothesis, is it possible to use a one-sided test here? In that case, does your conclusion change? c. Using only heteroskedasticity-robust standard errors, but not HAC standard errors, the value in parenthesis becomes 0.25. Repeat the calculations in (a) and report your decision based on a two-sided test. d. Since the coefficient becomes more statistically significant in (d), should this influence your choice of standard errors? Why or why not?

Answer: a. t =

= -1.78 < -1.96. Hence you cannot reject the null hypothesis that the coefficient is zero in the

population.

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b. The beer tax represents part of the cost (price) of alcohol consumption and an increase in price should reduce the demand for alcohol. Hence economic theory suggests a negative price coefficient. It therefore seems reasonable to use a one-sided test. Since the critical value is -1.64 in that case, you can reject the null hypothesis at the 5% significance level. c. The t-statistic is now -2.56 and you can reject the null hypothesis at the 5% level, and almost at the 1% level. d. It is better to use the clustered standard errors, since these are valid whether or not there is heteroskedasticity, autocorrelation, or both. Using heteroskedasticity-robust standard errors only will result in invalid statistical inference, since they were derived under the assumption of no serial correlation in the error term.

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