Linear Panel Data Models Michael R. Roberts Department of Finance - PowerPoint PPT Presentation

First Difference Estimators Fixed Effects Estimation Random Effects (RE) Estimation Dynamic Linear Panel Data Models (DLPDM) Linear Panel Data Models Michael R. Roberts Department of Finance The Wharton School University of Pennsylvania October 5, 2009 Michael R. Roberts Linear Panel Data Models 1/56

First Difference Estimators Two Period Model Fixed Effects Estimation Policy Analysis Random Effects (RE) Estimation Three Period Panel Dynamic Linear Panel Data Models (DLPDM) General Period Panel Example Link between crime and unemployment. Data for 46 cities in 1982 and 1987. Consider CS regression using 1987 data � R 2 = 0 . 033 crimeRate = 128 . 38 − 4 . 16 unem , (20 . 76) (3 . 42) Higher unemployment decreases the crime rate (insignificantly)?!?!?! Problem = omitted variables Solution = add more variables (age distribution, gender distribution, education levels, law enforcement, etc.) Use like lagged crime rate to control for unobservables Michael R. Roberts Linear Panel Data Models 2/56

First Difference Estimators Two Period Model Fixed Effects Estimation Policy Analysis Random Effects (RE) Estimation Three Period Panel Dynamic Linear Panel Data Models (DLPDM) General Period Panel Panel Data Approach Panel data approach to unobserved factors. 2 types: constant across time 1 vary across time 2 y it = β 0 + δ 0 d 2 t + β 1 x it + a i + u it , t = 1 , 2 where d 2 = 1 when t = 2 and 0 when t = 1 Intercept for period 1 is β 0 , for period 2 β 0 + δ 0 Allowing intercept to change over time is important to capture secular trends. a i captures all variables that are constant over time but different across cross-sectional units. (a.k.a. unobserved effect , unobserved heterogeneity ) u it is idiosyncratic error or time-varying error and represents unobserved factors that change over time and effect y it Michael R. Roberts Linear Panel Data Models 3/56

First Difference Estimators Two Period Model Fixed Effects Estimation Policy Analysis Random Effects (RE) Estimation Three Period Panel Dynamic Linear Panel Data Models (DLPDM) General Period Panel Example (Cont) Panel approach to link between crime and unemployment. crimeRate it = β 0 + δ 0 d 78 t + β 1 unem it + a i + u it where d 87 = 1 if year is 1987, 0 otherwise, and a i is an unobserved city effect that doesn’t change over time or are roughly constant over the 5-year window. Examples: Geographical features of city 1 Demographics (race, age, education) 2 Crime reporting methods 3 Michael R. Roberts Linear Panel Data Models 4/56

First Difference Estimators Two Period Model Fixed Effects Estimation Policy Analysis Random Effects (RE) Estimation Three Period Panel Dynamic Linear Panel Data Models (DLPDM) General Period Panel Pooled OLS Estimation How do we estimate β 1 on the variable of interest? Pooled OLS. Ignore a i . But we have to assume that a i is ⊥ to unem since it would fall in the error term. crimeRate it = β 0 + δ 0 d 78 t + β 1 unem it + v it where v it = a i + u it . SRF: � R 2 = 0 . 012 crimeRate = 93 . 42 + 7 . 94 d 87 + 0 . 427 unem , (12 . 74) (7 . 98) (1 . 188) Positive coef on unem but insignificant Michael R. Roberts Linear Panel Data Models 5/56

First Difference Estimators Two Period Model Fixed Effects Estimation Policy Analysis Random Effects (RE) Estimation Three Period Panel Dynamic Linear Panel Data Models (DLPDM) General Period Panel First Difference Estimation Difference the regression equation across time to get rid of fixed effect and estimate differenced equation via OLS. y i 2 = ( β 0 + δ 0 ) + β 1 x i 2 + a i + u i 2 , ( t = 2) y i 1 = β 0 + β 1 x i 1 + a i + u i 1 , ( t = 1) Differencing yields ∆ y i = δ 0 + β 1 ∆ x i + ∆ u i where ∆ denotes period 2 minus period 1. Key assumption: ∆ x i ⊥ ∆ u i , which holds if at each time t , u it ⊥ x it ∀ t . (i.e., strict exogeneity). This rules out lagged dependent variables. Key assumption: ∆ x i must vary across some i Michael R. Roberts Linear Panel Data Models 6/56

First Difference Estimators Two Period Model Fixed Effects Estimation Policy Analysis Random Effects (RE) Estimation Three Period Panel Dynamic Linear Panel Data Models (DLPDM) General Period Panel First Difference Example Reconsider crime example: � R 2 = 0 . 012 crimeRate = 15 . 40 + 2 . 22∆ unem , (4 . 70) (0 . 88) Now positive and significant effect of unemployment on crime Intercept = ⇒ crime expected to increase even if unemployment doesn’t change! This reflects secular increase in crime rate from 1982 to 1987 Michael R. Roberts Linear Panel Data Models 7/56

First Difference Estimators Two Period Model Fixed Effects Estimation Policy Analysis Random Effects (RE) Estimation Three Period Panel Dynamic Linear Panel Data Models (DLPDM) General Period Panel Practical Issues Differencing can really reduce variation in x x may vary greatly in cross-section but ∆ x may not Less variation in explanatory variable means larger standard errors on corresponding coefficient Can combat by either Increasing size of cross-section (if possible) 1 Taking longer differences (over several periods as opposed to adjacent 2 periods) Michael R. Roberts Linear Panel Data Models 8/56

First Difference Estimators Two Period Model Fixed Effects Estimation Policy Analysis Random Effects (RE) Estimation Three Period Panel Dynamic Linear Panel Data Models (DLPDM) General Period Panel Example Michigan job training program on worker productivity of manufacturing firms in 1987 and 1988 scrap i t = β 0 + δ 0 y 88 t + β 1 grant i t + a i + u i t where i , t index firm-year, scrap = scrap rate = # of items per 100 that must be tossed due to defects, grant = 1 if firm i in year t received job training grant. a i is firm fixed effect and captures average employee ability, capital, and managerial skill...things constant over 2-year period. Difference to zap a i and run 1st difference (FD) regression N = 54 , R 2 = 0 . 022 ∆ � scrap = − 0 . 564 − 0 . 739∆ grant , (0 . 405) (0 . 683) Job training grant lowered scrap rate but insignificantly Michael R. Roberts Linear Panel Data Models 9/56

First Difference Estimators Two Period Model Fixed Effects Estimation Policy Analysis Random Effects (RE) Estimation Three Period Panel Dynamic Linear Panel Data Models (DLPDM) General Period Panel Example (Cont) Is level-level model correct? � N = 54 , R 2 = 0 . 067 ∆ log ( scrap ) = − 0 . 57 − 0 . 317∆ grant , (0 . 097) (0 . 164) Job training grant lowered scrap rate by 31.7% (or 27.2% = exp(-0.317) - 1). Pooled OLS estimate implies insignificant 5.7% reduction Large difference between pooled OLS and first difference suggests that firms with lower-ability workers (low a i ) are more likely to receive a grant. I.e., Cov ( a i , grant it ) < 0. Pooled OLS ignores a i and we get a downward omitted variables bias Michael R. Roberts Linear Panel Data Models 10/56

First Difference Estimators Two Period Model Fixed Effects Estimation Policy Analysis Random Effects (RE) Estimation Three Period Panel Dynamic Linear Panel Data Models (DLPDM) General Period Panel Program Evaluation Problem Let y = outcome variable, prog = program participation dummy. y it = β 0 + δ 0 d 2 t + β 1 prog it + a i + u it Difference regression ∆ y it = δ 0 + β 1 ∆ prog it + ∆ u it If program participation only occurs in the 2nd period then OLS estimator of β 1 in the differenced equation is just: ˆ β 1 = ∆ y treat − ∆ y control (1) Intuition: ∆ prog it = prog i 2 since participation in 2nd period only. (i.e., ∆ prog it 1 is just an indicator identify the treatment group) Omitted group is non-participants. 2 So β 1 measures the average outcome for the participants relative to 3 the average outcome of the nonparticipants Michael R. Roberts Linear Panel Data Models 11/56

First Difference Estimators Two Period Model Fixed Effects Estimation Policy Analysis Random Effects (RE) Estimation Three Period Panel Dynamic Linear Panel Data Models (DLPDM) General Period Panel Program Evaluation Problem (Cont) Note: This is just a difference-in-differences (dif-in-dif) estimator “Equivalent” model: = β 0 + δ 0 d 2 t + β 1 prog it + β 2 d 2 t × prog it + a i + u it y it where β 2 has same interpretation as β 1 from above. If program participation can take place in both periods, we can’t write the estimator as in (1) but it has the same interpretation: change in average value of y due to program participation Adding additional time-varying controls poses no problem. Just difference them as well. This allows us to control for variables that might be correlated with program designation. y it = β 0 + δ 0 d 2 t + β 1 prog it + γ ′ X it + a i + u it Michael R. Roberts Linear Panel Data Models 12/56