Panel Data. STAT-S-301 Exercise session 5. November 10th, vary across entities but not over time. could cause omitted variable bias if omitted

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1 Panel Data STAT-S-301 Exercise session 5 November 10th, 2016 Panel data consist of observations on the same n entities at two or mor time periods (T). If two variables Y, and X are observed, the data is denoted as (X it, Y it ) where i = 1,..., n is the entity subscript and t = 1,..., T refers to the date at which the observations are collected. Panel data can also be referred to as longitudinal data. The main goal of panel data is to control for Z factors, which are factors that: vary across entities but not over time could cause omitted variable bias if omitted are unobserved or unmeasured Panel data with two time periods Let's write a model with one regressor X and one unobserved variable, Z, that determines Y in the ith entity but does not change over time: Y it = β 0 + β 1 X it + β 2 Z i + u it Z is not observed but could still cause omitted variable bias if it is not included in the regression. We can solve this issue by eliminating its inuence as such: Y it+1 = β 0 + β 1 X it+1 + β 2 Z i + u it+1 Y it+1 Y it = β 1 (X it+1 X it ) + u it+1 u it Since the Z factor does not change over time, it does not inuce the dierence variable Y it+1 Y it. Thus its eects can be eliminated by looking at the change in the dependent variable between two time periods. This dierence equation can be estimated via OLS. Hence we can say that analyzing changes in Y and X has the eect of controlling for variables that are constant over time (xed factors), thereby eliminating this source of omitted variable bias. TA: Elise Petit (elise.lb.petit@ulb.ac.be) 1

2 Fixed eects regression Entity xed eects Fixed eects regression is a method for controlling for omitted variables in panel data when the omitted variables vary across entities but do not change over time. This method can be used when there are two or more time observations for each entity. Y it = β 0 + β 1 X it + β 2 Z i + u it What interests us is β 1, the eect on Y of a change in X, holding constant the unobserved entity characteristics Z. One option is to interpret the population regression model as having n intercepts, one for each entity. Specically, let α i = β 0 + β 2 Z i, the xed eects regression model becomes Y it = β 1 X it + α i + u it where α 1, α 2,..., α n are the unknown intercepts to be estimated for each entity and the slope β 1 is the same for all entities. Hence α i represent the entity xed eects. Another way to express the entity xed eects is to use binary variables to denote each entity: D1 i = 1 when i = 1, D2 i = 1 when i = 2,..., Dn i = 1 when i = n To avoid the dummy variable trap we can arbitrarily omit D1 and write the xed eects regression model as Y it = β 0 + β 1 X it + γ 2 D2 i + γ 3 D3 i γ n Dn i + u it where β 0, β 1, γ 2, γ 3,..., γ n are unknown parameters to estimate. The two methods are equivalent with α 1 = β 0 ; α i = β 0 + γ i (for i 2). Time xed eects The same methodology can be used to control for omitted variables that vary across time but are constant across entities, namely S t. Y it = β 0 + β 1 X it + β 2 S t + u it The rst xed eects regression model is thus Y it = β 1 X it + λ t + u it ; λ t = β 0 + β 2 S t where λ t are the time xed eects and T intercepts need to be estimated (one for each period). The binary regression form is D1 t = Y it = β 0 + β 1 X it + δ 2 D2 t + δ 3 D3 t δ T DT t + u it 1 when t = 2, D2 t =,..., DT t = 1 when t = when t = T

3 Entity and time xed eects There could be both entity and time xed eects. Hence we can use a combined model: Y it = β 1 X it + α i + λ t + u it Y it = β 0 +β 1 X it +γ 2 D2 i +γ 3 D3 i +...+γ n Dn i +δ 2 D2 t +δ 3 D3 t +...+δ T DT t +u it Exercices 1 Trac fatality rates were observed for 48 U.S. states annually for 1982 through Here is the data that was collected to construct a regression model explaining the factors inuencing fatility rates. Fatality rate death/10,000 Beer tax tax (%) on beer Drinking age 18 binary variable =1 if drinking age is 18 Drinking age 19 binary variable =1 if drinking age is 19 Drinking age 20 binary variable =1 if drinking age is 20 Drinking age in years Minimum Punishment binary variable =1 if jail time or community services Vehicle miles average vehice miles traveled annually Unemployment rate in % Real income log(real income per capita) 3

4 (a) New Jersey has a population of 8.85 million people. Suppose that New Jersey increased the tax on a case of beer by $2 (in 1988 dollars). Use the results in column (5) to predict the number of lives that would be saved over the next year. Construct a 99% condence interval for your answer. (b) The drinking age in New Jersey is 21. Suppose that New Jersey lowered its drinking age to 19. Use the results in column (5) to predict the change in the number of trac fatalities in the next year. Construct a 95% condence interval for your answer. (c) Suppose that real income per capita in New Jersey increases by 3% in the next year. Use the results in column (6) to predict the change in the number of trac fatalities in the next year. Construct a 95% condence interval for your answer. (d) How should minimum drinking age be included in the regressions? Should it enter as a continuous variable or a series of indicator variables? Be specic. 2 Suppose a researcher believes that trac fatalities increase when roads are icy, so that states with more snow will have more fatalities than other 4

5 states. Comment on the following methods designed to estimate the eect of snow on fatalities. (a) The researcher collects data on the average snowfall for each state and adds this regressor to the regression given in the previous exercise. (b) The researcher collects data on the snowfall in each state for each year in the sample and adds this regressor to the regressions. 3 Suppose a researcher believes that the occurence of natural disasters, such as earthquakes, leads to increased activity n the construction industry. The researcher decides to collect province-level data on employment in the construction industry of an earthquake-prone country, and regress this variable on an indicator variable that equals 1 if an earthquake took place in that province in the last ve years. (a) Should the researcher include province-xed eects in order to control for location-specic characteristics of the labor market? (b) What can the researcher do to control for location eects? 4 Using the following regression Y it = β 0 + β 1 X it + γ 1 D1 i + γ 2 D2 i + γ 3 D3 i γ n Dn i + u it (a) Suppose n = 3. Show that the binary regressors and the constant regressor are perfectly multicollinear; that is, express one of the variables D1 i, D2 i, D3 i as a perfect linear function of the others. (b) Show the result in (a) for general n 5

Solutions to Odd-Numbered End-of-Chapter Exercises: Chapter 10

Introduction to Econometrics (3 rd Updated Edition) by James H. Stock and Mark W. Watson Solutions to Odd-Numbered End-of-Chapter Exercises: Chapter 10 (This version July 20, 2014) Stock/Watson - Introduction