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### ch10-1

Course: ECONOMICS 220:322, Fall 2010
School: Rutgers
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Word Count: 707

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10 Regression Chapter with Panel Data (Part 1) Trafc Deaths and Alcohol Taxes There are approximately 40,000 highway trafc fatalities each year in US. Approximately one-third of fatal crashes involve a driver who was drinking. One study estimates that as many as 25% of drivers on the road between 1AM and 3AM have been drinking. A driver who is legally drunk is at least 13 times as likely to cause a fatal crash as...

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10 Regression Chapter with Panel Data (Part 1) Trafc Deaths and Alcohol Taxes There are approximately 40,000 highway trafc fatalities each year in US. Approximately one-third of fatal crashes involve a driver who was drinking. One study estimates that as many as 25% of drivers on the road between 1AM and 3AM have been drinking. A driver who is legally drunk is at least 13 times as likely to cause a fatal crash as a driver who has not been drinking. We study how effective government policies designed to discourage drunk driving actually are in reducing trafc deaths. Panel Data The state trafc fatality data we will study are panel data. Data are for n = 48 states, where each state is observed in T = 7 time periods (1982, ..., 1988). Panel data consist of observation on the same n entities at two or more time periods T. If the data set contains observations on the variables X and Y, then the data are denoted (Xit , Yit ) , i = 1, ..., n and t = 1, ..., T, where the rst subscript, i, referers to the entity being observed, and the second subscript, t, refers to the date at which it is observed. Trafc Deaths and Alcohol Taxes The OLS regression lines obtained by regressing the fatality rate on the beer tax are: \ FatalityRate = 2.01 + 0.15BeerTax (1982 data). (0.15) (0.13) \ FatalityRate = 1.86 + 0.44BeerTax (1988 data). (0.11) (0.13) Should we conclude that an increase in the tax on beer leads more trafc deaths? Consider the omitted variable bias. Some variables, such as the cultural acceptance of drinking and driving might be hard to measure. Trafc Deaths and Alcohol Taxes Panel Data with Two Time Periods: "Before and After" Comparisons Let Zi be a variable that determines the fatality rate in the ith state, but does not change over time. FatalityRateit = 0 + 1 BeerTaxit + 2 Zi + uit , where uit is the error term and i = 1, ..., n and t = 1, ...T. The inuence of Zi can be eliminated: FatalityRatei1982 = 0 + 1 BeerTaxi1982 + 2 Zi + ui1982 FatalityRatei1988 = 0 + 1 BeerTaxi1988 + 2 Zi + ui1988 Subtracting the rst from second eliminates the effect of Zi : FatalityRatei1988 = 1 (BeerTaxi1988 FatalityRatei1982 BeerTaxi1982 ) + ui1988 ui1982 . Panel Data with Two Time Periods: "Before and After" Comparisons Figure presents 10.2 a scatterplot of the change in the fatality rate between 1982 and 1988 against the change in the beer tax. The estimated OLS regression line is \ FatalityRatei1988 FatalityRatei1982 = 0.072 1.04 (BeerTaxi1988 BeerTaxi1982 ) . (0.065) (0.36) An increase in the beer tax by \$1 per case reduces the trafc fatality rate by 1.04 deaths per 10,000 people. Consider that the avarage fatality rate is 2 in the data, this effect is very large. Panel Data with Two Time Periods: "Before and After" Comparisons Fixed Effects Regression Unlike the "before and after" comparisons, xed effects regression can be used with more than two time observations for each entity. Yit = 0 + 1 Xit + 2 Zi + uit , where Zt is an unobserved variable that varies from one state to the next but does not change over time. We want to estimate the 1 , the effect on Y on X controlling the unobserved Zi . Because Zi varies across states but is constant over time, can be interpreted as having n intercepts, one for each state. Fixed Effects Regression Let i = 0 + 2 Zi , Yit = 1 Xit + i + uit , is the xed effects regression model, in which 1 , ..., n are treated as unknown intercepts to be estimated, one for each state. 1 , ..., n are known as entity xed effects. Equivalently, the xed regression model can be written using binary variables: Yit = 0 + 1 Xit + 2 D2i + 3 D3i + + n Dni + uit , where D2i = 1 if i = 2 and D2i = 0 otherwise, and so forth. Fixed Effects Regression In general, the xed effects regression model is Yit = 1 Xit + + k Xk,it + i + uit , where i = 1, ..., n and t = 1, ..., T, where X1,it is the value of the rst regressor for entity i in time period t, X2,it is the value of the second regressor, and so forth, and 1 , ..., n are entity specic intercepts. Equivalently, the xed regression model can be written using binary variables: Yit = 0 + 1 Xit + +3 D3i + + k Xk,it + 2 D2i + n Dni + uit , where D2i = 1 if i = 2 and D2i = 0 otherwise, and so forth. Fixed Effects Regression The OLS estimate of the xed effects regression line relating the beer tax to the fatality rate is \ FatalityRate = 0.66BeerTax + StateFixedEffects (0.20) where the estimated state xed intercepts are not listed to save space.
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