Exam 10: Regression With Panel Data

Panel data is also called

Your textbook reports the following result from an two-way fixed effects (entity and time fixed effects)regression model: $\widehat{\text { FatalityRate }}$ = -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?

You first encountered growth regression in your intermediate macroeconomics course ("beta-convergence 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?

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?

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(u_it, u_is | X_it, X_is)= 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?

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

In panel data, the regression error

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 (E_it )= β₀ + β₁ (M_it /W_it )+ $\gamma _ { 2 }$ D2_i + ... + $\gamma _ { n }$ D51_i + $\delta _ { 2 }$ B2_t + ... + $\delta _ { \mathrm { T } }$ B13_t + u_it, 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 $\hat { E }$ _it = $\hat { \beta }$ ₀ - 0.33 × (M_it /W_it )+ 0.35(SHY_it)- 1.53 × uram_it; 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: $\hat { E }$ _it = $\hat { \beta }$ ₀ + 0.07 × (M_it /W_it )- 0.19 × (SHY_it)- 0.54 × uram_it; $\bar { R }$ 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 $\bar { R }$ 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?

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?

cov (u_it, u_is | X_it, X_is = 0 for t ? s means that