Introduction to Econometrics, 2e (Stock/Watson)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: ln(Eit )= β0 + β1ln(Mit /Wit ) + γ2D2i + ... + γnD50i + δ 2B2t + ... + δTB13t + uit, 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: ln Eit = -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 ln(Eit )= β0 + β1ln(Mit /Wit ) + γ2D2i + ... + γnD8i + δ2B2t + ... + δ TB13t + uit, 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 ln(Eit )= β0 + β1ln(Mit /Wit ) + γ2D2i + ... + γnD8i + δ2B2t + ... + δ TB13t + uit. 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. Stock/Watson 2e -- CVC2 8/23/06 -- Page 253 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. 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 and included all 50 states plus the District of Columbia. The estimated equation is of the following type (Eit )= β0 + β1 (Mit /Wit ) + γ2D2i + ... + γnD51i + δ2B2t + ... + δTB13t + 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 E ^ it = β ^ 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: E ^ it = β ^ 0 + 0.07 × (Mit /Wit ) – 0.19 × (SHYit) – 0.54 × uramit; R2 = 0.69 (0.10) (0.22) (0.11) Stock/Watson 2e -- CVC2 8/23/06 -- Page 254