Introduction to Regression Models for Panel Data Analysis. Indiana University Workshop in Methods February 13, Professor Patricia A.

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1 Introduction to Regression Models for Panel Data Analysis Indiana Universy Workshop in Methods February 13, 2015 Professor Patricia A. McManus Panel Data Analysis February 2015

2 What are Panel Data? Panel data are a type of longudinal data, or data collected at different points in time. Three main types of longudinal data: Time series data. Many observations (large t) on as few as one un (small N). Examples: stock price trends, aggregate national statistics. Pooled cross sections. Two or more independent samples of many uns (large N) drawn from the same population at different time periods: o General Social Surveys o US Decennial Census extracts o Current Population Surveys* Panel data. Two or more observations (small t) on many uns (large N). o Panel surveys of households and individuals (PSID, NLSY, ANES) o Data on organizations and firms at different time points o Aggregated regional data over time This workshop is a basic introduction to the analysis of panel data. In particular, I will cover the linear error components model. WIM Panel Data Analysis October 2015 Page 1

3 Why Analyze Panel Data? We are interested in describing change over time o social change, e.g. changing attudes, behaviors, social relationships o individual growth or development, e.g. life-course studies, child development, career trajectories, school achievement o occurrence (or non-occurrence) of events We want superior estimates trends in social phenomena o Panel models can be used to inform policy e.g. health, obesy o Multiple observations on each un can provide superior estimates as compared to cross-sectional models of association We want to estimate causal models o Policy evaluation o Estimation of treatment effects WIM Panel Data Analysis October 2015 Page 2

4 What kind of data are required for panel analysis? Basic panel methods require at least two waves of measurement. Consider student GPAs and job hours during two semesters of college. One way to organize the panel data is to create a single record for each combination of un and time period: StudentID Semester Female HSGPA GPA JobHrs Notice that the data include: o A time-invariant unique identifier for each un (StudentID) o A time-varying outcome (GPA) o An indicator for time (Semester). Panel datasets can include other time-varying or time-invariant variables WIM Panel Data Analysis October 2015 Page 3

5 An alternative way to structure the data is to keep all the measures related to each student in a single record. This is sometimes called wide format. StudentID Female HSGPA GPA5 JobHrs5 GPA6 JobHrs o Why are there two variables for GPA and JobHrs? o Why is there only one variable for gender and high school GPA? o Where is the indicator for time? WIM Panel Data Analysis October 2015 Page 4

6 Estimation Techniques for Panel Models We can wre a simple panel equation predicting GPA from hours worked: GPA = b + TERM b + HSGPA b + JOB b + v 0 T H J General Linear Model is the foundation of linear panel model estimation o Ordinary Least Squares (OLS) o Weighted least squares (WLS) o Generalized least squares (GLS) Least-squares estimation of panel models typically entails three steps: (a) Data transformation or first-stage estimation (b) Estimation of the parameters using Ordinary Least Squares (c) Estimation of the variance-covariance matrix of the estimates (VCE) Parameter estimates are sometimes refined using eratively reweighted least squares (IRLS), a maximum likelihood estimator. WIM Panel Data Analysis October 2015 Page 5

7 Basic Questions for the Panel Analyst What s the story you want to tell? Is this a descriptive analysis? Less worry, fewer controls are usually better. Is this an attempt at causal analysis using observational data? Careful specification AND theory are essential. How does time matter? Some analyses, e.g. difference-in-difference analysis associates time wh an event (before and after) Some analyses may be interested in growth trajectories. Panel analysis may be appropriate even if time is irrelevant. Panel models using cross-sectional data collected at fixed periods of time generally use dummy variables for each time period in a two-way specification wh fixed-effects for time. Are the data up to the demands of the analysis? Panel analysis is data-intensive. Two waves are a bare minimum Can you perform the necessary specification tests? How will you address panel attrion? WIM Panel Data Analysis October 2015 Page 6

8 Review of the Classical Linear Regression Model y = b + x b + x b + + x b + u, i=1,2,3, N i 0 1i 1 2i 2... ki k i Where we assume that the linear model is correct and: E u x, x,.., x = 0 Covariates are Exogenous: ( ) Uncorrelated errors: ( ) i 1i 2i ki Cov u, u = 0 Homoskedastic errors: ( ) ( ) 2 i j Var u = Var y x, x,..., x = s i i 1i 2i ki If assumptions do not hold, OLS estimates are BIASED and/or INEFFICIENT Biased - Expected value of parameter estimate is different from true. o Consistency. If an estimator is unbiased, or if the bias shrinks as the sample size increases, is CONSISTENT Inefficient - An estimator is inefficient if an alternative estimator converges more rapidly on the true coefficients as sample size increases. o Estimators that explo all available information are more efficient WIM Panel Data Analysis October 2015 Page 7

9 OLS Bias Due to Endogeney Omted Variable Bias o Intervening variables, selectivy Measurement Error in the Covariates Simultaney Bias o Feedback loops o Omted variables Conventional regression-based strategies to address endogeney bias Instrumental Variables estimation Structural Equations Models Propensy score estimation Fixed effects panel models WIM Panel Data Analysis October 2015 Page 8

10 Illustration of Whin-un correlation. Peak-flow Measurements Wright Peak Flow measures Subject ID Wright Peak #1 Wright Peak #2 WIM Panel Data Analysis October 2015 Page 9

11 OLS Inefficiency due to Correlated Errors Many data structures are susceptible to error correlation: Hierarchical data sample multiple individuals from each un, e.g. household members, employees in firms, multiple pupils from each school. Multistage probabily samples often incorporate cluster-based sampling designs wh errors that may be correlated whin clusters. Repeated observations data often show whin-un error correlation. Time series data often have errors that are serially correlated, that is, correlated over time. Panel data have errors that can be correlated whin un (e.g. individuals), whin period. Conventional regression-based strategies to address correlated errors Cluster-consistent covariance matrix estimator to adjust standard errors. Generalized Least Squares instead of OLS to explo correlation structure. Generalized Estimation Equations (GEE) Mixed Effects Estimators for multilevel models WIM Panel Data Analysis October 2015 Page 10

12 Linear Panel Data Model (LPM) Suppose the data are on each cross-section un over T time periods: y y = x ' b, u,1,1 t1,1 = x ' b, u,2,2 t2,1 ::: y = x ' b, u it, it, T it,, t=1,2,,t We can express this concisely using y i to represent the vector of individual outcomes for person i across all time periods: y = X b + u i i i, where y ' i = y,1, y,2,..., yit For comparison, begin wh two conventional OLS linear regression models, one for each period. Note that the variables female highgpa (HS GPA) are time-invariant. WIM Panel Data Analysis October 2015 Page 11

13 OLS Results for each term: Term 5 GPA Term 6 GPA Estimate SE t-stat Estimate SE t-stat Intercept jobhrs female highgpa Pooled OLS Results for both terms: Term 5&6 GPA Term 5&6 GPA (Clustered SE) Estimate SE t-stat Estimate SE t-stat Intercept jobhrs female highgpa term WIM Panel Data Analysis October 2015 Page 12

14 Linear Unobserved Effects Panel Data Model Motivation: Unobserved heterogeney Suppose we have a model wh an unobserved, time-constant variable c: y = c + x c + x c + + x c + c + u k k Where u is uncorrelated wh all explanatory variables in x. Because c is unobserved is absorbed into the error term, so we can wre the model as follows: y = c + x c + x c + + x c + v v = c + u k k The error term v consists of two components, an idiosyncratic component u and an unobserved heterogeney component c. WIM Panel Data Analysis October 2015 Page 13

15 OLS Estimation of the Error Components Model If the unobserved heterogeney c i is correlated wh one or more of the explanatory variables, OLS parameter estimates are biased and inconsistent. If the unobserved heterogeney c is uncorrelated wh the explanatory variables in x i, OLS is unbiased even in a single cross-section. If we have more than one observation on any un, the errors will be correlated and OLS estimates will be inefficient y = c, x c, x c,..., x c, v i, k k i,1 i1 i1 i1 y = c, x c, x c,..., x c, v i, k k i,2 v = c, u i,1 i i,1 v = c, u i,2 i i,2 cov(v, v ) ¹ 0 i,1 i,2 i2 i2 i2 WIM Panel Data Analysis October 2015 Page 14

16 Unobserved Heterogeney in Panel Data Suppose the data are on each cross-section un over T time periods. This is an unobserved effects model (UEM), also called the error components model. We can wre the model for each time period: y = x c + c + u i1 i1 i i1 y = x c + c + u i2 i2 i i2 y = x c + c + u it it i it, Where there are T observations on outcome y for person i, x is a vector of explanatory variables measured at time t, c i is unobserved in all periods but constant over time u is a time-varying idiosyncratic error Define v = c i + u as the compose error. WIM Panel Data Analysis October 2015 Page 15

17 Consistent estimation of the Error Components Model wh Pooled OLS If we assume no contemporaneous correlation of the errors and the explanatory variables, pooled OLS estimation is consistent: ' E ( x u ) = 0 and ' i E( x c ) = 0, t=1,2,,t Efficient estimation of the Error Components Model wh Pooled OLS Even if estimation is consistent, pooled OLS may not be efficient. One strategy is to combine pooled OLS wh cluster-consistent standard errors. Panel methods over OLS to explo OR remove unobserved heterogeney. In the next sections, we consider the dominant approaches to estimation of the error components panel model: fixed effects and random effects. WIM Panel Data Analysis October 2015 Page 16

18 Just a few panel data examples: Wage penalty for motherhood Men s wage premium for heterosexual marriage Effect of regulation of nursing pay on hospal qualy Effect of Incarceration on wages and income inequaly Effect of parental divorce on mental health over life-course Determinants of Death Penalty in US states Effect of Democracy on Human Capal and Economic Growth WIM Panel Data Analysis October 2015 Page 17

19 Fixed Effects Methods for Panel Data Suppose the unobserved effect c i is correlated wh the covariates. Example: Motherhood wage penalty We observe that mothers earn less than other women, cet par. b ˆ KIDS = in a log wage model suggests that each addional OLS child reduces mothers hourly wages by about 8% But if women who are less oriented towards work are also more likely to have more children, omting work orientations from the model will bias the coefficient on children. Fixed-effects methods transform the model to remove c i b ˆ KIDS = FE estimates a persistent but much smaller penalty. FE WIM Panel Data Analysis October 2015 Page 18

20 Caution: Fixed effects has some disadvantages FE is not a panacea for all sources of endogeney bias. time-varying unobserved effects time-varying measurement error simultaney or feedback loops All time-constant effects are removed. No estimation of effects of race, gender, birth order, etc. Poor estimates if ltle variation (e.g. education in adulthood) FE trades consistency for efficiency. FE uses only whin-un change, ignores between-un variation. Parameter estimates may be imprecise, standard errors large. Despe limations, FE is an indispensable tool in the panel analyst s toolbox. WIM Panel Data Analysis October 2015 Page 19

21 Fixed Effects Transformation - the Whin Estimator Suppose we have the UEM model: ' = + i + y x c c u, t=1,2,,t For each un, average this equation over all time periods t: ' i = xic + i + i y c u Subtract the whin-un average from each observation on that un: ' ' ( ) ( ) ( ) y - y = x - x c + c - c + u -u, t=1,2,,t i i i i i This is the fixed effects transformation. We can wre as: y, ' = xb + u where ci - ci = 0 and y = y -y, x i = x -xi, u = u -ui and x does not contain an intercept term. WIM Panel Data Analysis October 2015 Page 20

22 The fixed-effects estimator, also called the whin estimator, applies pooled OLS to the transformed equation: -1-1 N N N T N T ' ' ' ' FE = å XX i i å Xy i i = åå xx åå xy çi= 1 çi= 1 çi= 1 t= 1 çi= 1 t= 1 æ ö æ ö æ ö æ ö ˆ b è ø è ø è ø è ø Recall the student GPA Data: StudentID Semester Female HSGPA GPA JobHrs After applying the fixed-effects transform, the demeaned (mean-centered) data: StudentID Semester CFemale CHSGPA CGPA CJobHrs WIM Panel Data Analysis October 2015 Page 21

23 Fixed Effects Dummy Variables Regression Up to now, we ve treated the unobservables c i as random variables: ' = xc + i + y c u An alternative approach is to treat c i as a fixed parameter for each un. In this case, we can use dummy variables regression to estimate c i. Step one: Create a dummy variable for each of sample un i Step two: Substute the vector of N-1 dummies for c i : ' = g g2 + 3 g gn + y x b d d dn u, (where the intercept g 1 estimates the effect when d 1=1) Step three: Estimate the equation using pooled OLS. The fixed effects dummy variables (FEDV) estimator produces precisely the same coefficient vector and standard errors as the FE estimator. WIM Panel Data Analysis October 2015 Page 22

24 Why not Just Use a Lagged Dependent Variable? Source: David Johnson. Journal of Marriage and Family, Vol. 67, No. 4 (Nov., 2005), pp WIM Panel Data Analysis October 2015 Page 23

25 Random Effects Methods If we can assume that the unobserved heterogeney will not bias the estimates: Fixed effects methods are inefficient. They throw away information. Pooled OLS is inefficient because does not explo the autocorrelation in the compose error term. Random effects methods use feasible GLS estimation (RE FGLS) to explo whin-cluster correlation Random effects estimation is more efficient than FE or OLS The random effects assumption of no bias due to c i is more stringent Ec ( x,..., x ) = Ec ( ) = 0 i i1 it i WIM Panel Data Analysis October 2015 Page 24

26 A Conventional FGLS Random Effects Estimator Assume the errors are correlated whin each un Assume the errors are uncorrelated across uns Assume the variance in the compose errors is equal to the sum of the variances in the unobserved effectc i and the idiosyncratic error u i : v = u + c s s s RE strategy: If v u c s = s + s, find estimators such that v = u + c sˆ sˆ sˆ WIM Panel Data Analysis October 2015 Page 25

27 Practical Feature of Random Effects Estimation Recall that the fixed effects whin estimator essentially transforms the data by centering each variable on the un-specific mean. OLS is then performed on the fully demeaned transformed data. The random effects estimator essentially transforms the data by partially demeaning each variable. Instead of subtracting the entire un-specific mean, only part of the mean is subtracted. The demeaning factor lis between 0 and 1, wh the specific value based on the variance components estimation. WIM Panel Data Analysis October 2015 Page 26

28 RE Results compared to pooled OLS Results for two terms: RE Term 5&6 GPA OLS Term 5&6 GPA Estimate SE z-stat Estimate SE t-stat Intercept jobhrs female highgpa term RE Results for six terms: Terms 1-6 GPA (FE, N=400) Estimate SE Intercept jobhrs female highgpa term WIM Panel Data Analysis October 2015 Page 27

29 Random Effects or Fixed Effects - How to decide? Hausman test for the Exogeney of the Unobserved Error Component If the unobserved effects are exogenous, the FE and RE are asymptotically equivalent. This suggests the null hypothesis for the Hausman test: H bˆ = b ˆ, 0 : RE FE where b ˆRE and b ˆFE are coefficient vectors for the time-varying explanatory variables, excluding the time variables. If the null hypothesis is rejected, we conclude that RE is inconsistent, and the FE model is preferred. If the null hypothesis cannot be rejected, random effects is preferred because is a more efficient estimator. WIM Panel Data Analysis October 2015 Page 28

30 Conventional Hausman Test in Stata:. xtreg gpa job sex highgpa,fe. estimates store fe. xtreg gpa job sex highgpa,re. estimates store re. hausman fe re ---- Coefficients ---- (b) (B) (b-b) sqrt(diag(v_b-v_b)) fe re Difference S.E job b = consistent under Ho and Ha; obtained from xtreg B = inconsistent under Ha, efficient under Ho; obtained from xtreg Test: Ho: difference in coefficients not systematic chi2(1) = (b-b)'[(v_b-v_b)^(-1)](b-b) = Prob>chi2 = We reject the null and conclude the fixed effects estimator is appropriate. WIM Panel Data Analysis October 2015 Page 29

31 An Alternative Hausman Test and FE/RE Hybrid Methods The Correlated Random Effects Model (Allison 2005,2009; Wooldridge 2005) Suppose we have both time-varying and time-invariant covariates: ' ' = x + zi + i + y c d c u, t=1,2,,t Add a vector of whin-un means for the time-varying covariates: ' ' ' = xc + zid + xiy + i + y c u Estimate using the random effects estimator and the result is a hybrid Alternative Hausman: If a Wald test for the joint statistical significance of the coefficient estimates in x rejects the null, FE is preferred The coefficient vector b yields the fixed effects estimates The coefficient vector x produces the between estimates The coefficient vector dproduces estimates for the time-invariante covariates. Interpret wh caution: these might still be correlated wh the unobserved error. WIM Panel Data Analysis October 2015 Page 30

32 Interpretation of Results from the Error Components Model Since the UEM model is derived as a levels model, coefficients can be interpreted much the same as interpretations of a conventional OLS model, but there are nuances: For example, suppose we estimate the relationship between marriage and men s wages, b ˆ MARRIED 0.05 in every model. Pooled OLS cross-section coefficients contain information about average differences between uns. Ey [ x ] = x c + c i This is a population-averaged effect. On average, married men earn 5% more than men who are not married. This says nothing about the causal effect of marriage on men s earnings. WIM Panel Data Analysis October 2015 Page 31

33 RE/FE/FD estimate average effects whin uns. If the unobserved effects are exogenous these are asymptotically equivalent to the population averaged effect. Ey [ x, c ] = x c i On average, entering marriage increases men s earnings by 5%. RE coefficients represent average change whin uns, estimated from all uns whether they experience change or not. FE coefficients represent average changes whin uns, only for uns that did experience change This is akin to a treatment effect among the treated. On average, men who married increased their earnings by 5%. WIM Panel Data Analysis October 2015 Page 32

34 Best Practices Theorize the model What exactly does this unobserved heterogeney represent? Why would you expect to be correlated / uncorrelated wh the regressors? Is likely there is endogeney due to time-varying unobserved heterogeney or feedback from the idiosyncratic error to the next wave of covariates? Specification Testing for Panel Analysis - Interval/Continuous Outcomes Always neglected but formal test for unobserved effect can be useful. Optional: Obtain intraclass correlation coefficient (ICC) as indicator of the extent of whin-un clustering. This is a descriptive statistic, not a test. Specification test(s) for strict exogeney Hausman-type specification test for RE vs. FE Test for serial correlation in the idiosyncratic errors WIM Panel Data Analysis October 2015 Page 33

35 Extensions FE Models wh Time-Invariant Predictors Interactions between time and covariate Panel Models for Categorical Outcomes Fixed effects log and random effects log for binary outcomes Fixed and random effects Poisson models can be used for count outcomes. Population averaged models can be estimated using General Estimation Equations (GEE). Dynamic panel models i.e. lagged dependent variable as a covariate: GPA = b, GPA b, TERM b, HSGPA b, JOB b, v 0 i, t-1 GPA T H J GLM models for instrumental variables (IV) estimation Generalized Method of Moments (GMM) is used for some dynamic panel models because allows a flexible specification of the instruments WIM Panel Data Analysis October 2015 Page 34

Applied Econometrics. Lecture 3: Introduction to Linear Panel Data Models

Applied Econometrics. Lecture 3: Introduction to Linear Panel Data Models Applied Econometrics Lecture 3: Introduction to Linear Panel Data Models Måns Söderbom 4 September 2009 Department of Economics, Universy of Gothenburg. Email: mans.soderbom@economics.gu.se. Web: www.economics.gu.se/soderbom,