Panel data methods for policy analysis

Transcription

1 IAPRI Quantitative Analysis Capacity Building Series Panel data methods for policy analysis Part I: Linear panel data models

2 Outline 1. Independently pooled cross sectional data vs. panel/longitudinal data 2. Analysis of independently pooled cross sectional data 3. Analysis of panel data (linear models) a. Unobserved effects panel data model b. Pooled ordinary least squares c. First-differenced estimator d. Fixed effects estimator e. Random effects estimator 4. Intro to correlated random effects (CRE) approach à will revisit in future session on non-linear panel data models 5. Difference-in-differences estimator 6. General program evaluation model 7. Attrition (definition, testing, correcting w/ IPW)

3 Independently pooled cross sectional data vs. panel/longitudinal data What is independently pooled cross sectional (IPCS) data? o Zambia examples? o Are observations i.i.d? What is panel/longitudinal data? o Zambia examples? o Are observations i.i.d?

4 Source: Wooldridge (2002)

5 Source: Wooldridge (2002)

6 Analysis of independently pooled cross sectional data Why pool independent cross sections? Distribution may not be identical over time. What to do? How to allow effect of a given explanatory variable to change over time? Estimate linear model with pooled OLS (POLS), possibly correcting for heteroskedasticity. (Serial correlation is not an issue why?)

7 Chow Test for Structural Change (can do w/ IPCS or panel or time series data) Do all slope coefficients change over time? Suppose we have 2 years of data (2008 & 2012) and our base model is: y = β 0 + δ 0 d β 1 x 1 + β 2 x 2 + u à How to test for structural change?

8 Unobserved effects panel data model y it = β 0 + β 1 x it + c i + u it, t = 1, 2,...,T Two-part error term: o c i time invariant unobserved effects ( unobserved heterogeneity ) o u it idiosyncratic (or time-varying) error Want to estimate β 1. What do we need to assume about c i and x it for POLS to be consistent?

9 If c i and x it are correlated, transform the data to eliminate c i before using OLS 2 options: first differencing or time-demeaning EX) First differencing w/ 2 years of data y i2 = β 0 + β 1 x i2 + c i + u i2 (t = 2) y i1 = β 0 + β 1 x i1 + c i + u i1 (t = 1) ( y i2 y i1 ) = β 1 (x i2 x i1 ) + (u i2 u i1 ) or Δy i = β 1 Δx i + Δu i OLS on differenced equation = firstdifferenced estimator

10 If c i and x it are correlated, transform the data to eliminate c i before using OLS 2 options: first differencing or time-demeaning EX) Time-demeaning w/ 2+ years of data y it = β 0 + β 1 x it + c i + u it, y i = β 0 + β 1 x i + c i + u i t = 1, 2,..., T y it y i = β 1 (x it x i ) + (u it u i ), t = 1, 2,..., T or y it = β 1 x it + u it, t = 1, 2,..., T OLS on timedemeaned equation = fixed effects estimator

11 First differencing vs. Fixed effects FD and FE are identical if T=2 and include year dummy for 2 nd time period Not identical if T>2 o Both unbiased & consistent o FE more efficient than FD if u it serially uncorrelated. Why?

12 FD vs. FE (cont d) FE more efficient if u it serially uncorrelated b/c differencing introduces serial correlation (in Δu it ) Δu it = u it u it-1 ; Δu it-1 = u it-1 u it-2 Is Cov(Δu it, Δu it-1 )=0??? FD more efficient if u it is a random walk Random walk: u it = a + u it-1 + e it (where e it is white noise) Δu it = u it u it-1 = a + e it à implies Δu it is white noise

13 Key assumption for FD & FE estimators to be consistent: strict exogeneity Strict exogeneity: explanatory variables (x it ) are uncorrelated with the idiosyncratic error (u it ) in ALL time periods o Rules out including lagged dependent variable (y it-1 ) as explanatory variable. Why? o Also rules out feedback from y it to x it+1 See Wooldridge texts for details on all assumptions for consistency, efficiency

14 If assume c i and x it are uncorrelated è random effects (RE) estimator Like FD and FE, RE requires strict exogeneity to be consistent Assumption that c i and x it are uncorrelated is strong but if really uncorrelated, then efficiency gain if use RE instead of FD/FE o Efficiency gain: explicitly modeling serial correlation in composite error term (v it =c i +u it ) via feasible GLS

15 Random effects transformation y it ˆλ y i = β 0 (1 ˆλ) + β 1 (x it ˆλx i ) +(v it ˆλv i ) where ˆλ = 1 [ σˆ 2 / ( σˆ 2 + Tσˆ 2 )] 1/2 u u a and σˆ 2 = Var u (uit ), σˆ 2 = Var a (ai ) Key point is that RE transformation quasi-time demeans the data, and the POLS & FE are special cases of FE Note that if ˆλ = 0 POLS ˆλ = 1 FE

16 Stata commands FD: generate change variables then <reg> FE: xtreg y x1 x2 xk, fe RE: xtreg y x1 x2 xk, re Robust standard errors (put at end of commands above): o Heteroskedasticity only: <robust> o Hetero. and serial correlation: <cluster(clustervar)>

17 Hausman test of FE vs. RE Basically tests if c i and x it are uncorrelated Stata code: xtreg y x1 x2 xk, fe estimates store fixed xtreg y x1 x2 xk, re hausman fixed., sigmamore

18 Correlated random effects (CRE) Fixed effects-like approaches to control for c i generally inconsistent in non-linear models (e.g., probit, Tobit, hurdle models, etc.; incidental parameters problem) Alternative: CRE approach a.k.a. Mundlak- Chamberlain device Make assumption about distribution of c i : c i =ψ + x i ξ + a i where a i x i Normal(0,σ a 2 )

19 Intro to CRE approach (cont.) Like FE & FD, need to assume strict exogeneity CRE: include time averages of time-varying variables ( x i ) as additional regressors in probit or Tobit Unlike FD & FE, can include time-constant regressors in CRE models CRE-POLS slope parameters identical to FE (see Stata example). Why?

20 Summary of linear panel data estimators (FD, FE, CRE-POLS, RE) Note: All 4 require strict exogeneity (so can t have lagged dependent variable or feedback from y it to x it+1 ) c i correlated w/ x it? Yes Hausman test RE vs. FE No All 4 are consistent but RE is most efficient FD, FE, CRE-POLS consistent Key explanatory variable of interest is time-constant? No Yes CRE-POLS FD, FE Original error term (u it ) serially Yes uncorrelated (i.e., no autocorrelation)? No FE more efficient FD more efficient if u it ~ random walk

21 General program evaluation model y it = α + γ prog it + x it β + dyear it δ + c i + u it prog = binary or continuous measure of program participation or policy Key parameter estimate of interest? prog correlated w/ c i but not u it è How to consistently estimate γ? prog correlated w/ u it è endogeneity problem (future session)

22 Attrition in panel datasets What is attrition? Why does it happen? If reason for not being re-interviewed is correlated with error term, then have attrition bias o If only correlated with c i, then can control for attrition bias with FD, FE, CRE. (Not an issue w/ RE why?) o If correlated with u it, then have a problem

23 Testing for attrition bias No regression-based test when T=2 and using FE, FD, or CRE. Why? If T=2 and not using FE/FD/CRE OR T>2 and using FE/FD/CRE, can test for attrition bias by: 1. Create re-interview dummy variable, reinterview it =1 if HH is re-interviewed in next wave =0 o.w. 2. Include as additional regressor in model 3. If p 0.10 (or your cutoff) è attrition bias. Why? This test assumes that attrition is an absorbing state.

24 Inverse Probability Weighting (IPW) to control for attrition bias Compatible w/ POLS, FD, & non-panel estimators NOT compatible w/ FE, RE, CRE IPW approach: 2-period panel (ex: UCS) 1. Estimate reduced form probit/logit with dep. var. =1 if HH was re-interviewed in 2 nd wave =0 o.w. Use all observations from 1 st wave in probit/logit. Regressors are everything from structural model and anything else you think might affect attrition (e.g., enumerator/supervisor dummies). à Get predicted probability of re-interview. 2. Compute IPW= 1/(predicted probability of re-interview) 3. Weight 2 nd period observations by IPW. (If using sampling weights, use sampling weight*1 in 1 st period; sampling weight*ipw in 2 nd period.)

25 IPW (cont d) 3-period panel (ex: SS) **Assume attrition is an absorbing state. 1. Same as 2-period panel to get IPW for t=2 (call it IPW i,2 ) 2. To get IPW for t=3: a. Estimate reduced form probit/logit with dep. var. =1 if HH was re-interviewed in 3 rd wave =0 o.w. Use all observations from 2 nd wave. Same regressors as previous slide.à Get predicted probability of re-interview. b. Compute IPW i,3 = IPW i,2 *[1/(predicted probability of re-interview)] 3. Weight 2 nd period observations by IPW i,2, 3 rd period observations by IPW i,3. (If using sampling weights, use sampling weight*1 in 1 st period; sampling weight*ipw i,2 in 2 nd period; sampling weight*ipw i,3 in 3 rd period.) Similar approach for >3 period panel.

26 IPW i,1 = 1 IPW i,2 = IPW i,1 IPW i,3 = IPW i,2... IPW i,t = IPW i,t 1 1 ˆp i,2 = 1 1 ˆp i,3 = 1 IPW (cont d) 1 ˆp i,2 1 ˆp i,t = 1 1 ˆp i,2 1 ˆp i,3 1 ˆp i,2 1 ˆp i,3... where ˆp i,t is the pred. prob. of being re-interviewed in round t. 1 ˆp i,t If using sampling weights, then in period t, multiply weight by IPW i,t.

27 See handout on Econometric & other impact assessment approaches to policy analysis: the problem of causality in policy analysis, and internal vs. external validity

28 Difference-in-differences (DD) In general: estimator y = β 0 + β 1 dtreat + δ 0 dafter + δ 1 dafter dtreat + u ˆ δ 1 = ( y treat,after y treat,before ) ( y control,after y control,before ) δˆ 1 is the average treatment effect (ATE) Important to control for other factors If do, simple expression for δˆ 1 not the same but interpretation is similar Useful for analysis of natural/quasiexperiments

29 Key assumption for DD: time invariant selection bias (parallel trend) Treatment effect Same Selection bias DD estimate = Treatment effect Source: Ravallion (2008)

30 Time-varying selection bias (nonparallel trend) causes DD to be biased Treatment effect Different Selection bias DD estimate Treatment effect (here < ) Source: Ravallion (2008)

31 Difference-in-differences (DD) estimator w/ IPCS data (can also do w/ panel data) Useful for policy analysis if collected before & after policy change/event EX) Effect of new farm block on land values in nearby areas o Data for 2008 & 2012 (before & after) o rvland: real value of land per ha (ZMK, 2012=100) o dnearfb: =1 if land near farm block; =0 o.w. rvland = β 0 + β 1 dnearfb + δ 0 d δ 1 d2012idnearfb + u o Estimate via pooled OLS (POLS) ˆ δ 1 = (rvland 12,nr rvland 12, fr ) (rvland 08,nr rvland 08, fr ) = difference over time in average difference of land values in the 2 locations = estimated parameter of interest (average treatment effect)

32 References Ravallion, M Evaluating Anti-Poverty Programs. In T.P. Schultz and J. Strauss, eds. Handbook of Development Economics, Volume 4. Amsterdam: North- Holland, pp Wooldridge, J. M Introductory Econometrics: A Modern Approach, 4 th Edition. Mason, OH: South-Western Cengage Learning. Wooldridge, J.M Econometric Analysis of Cross Section and Panel Data, 2 nd Edition. Cambridge, MA: MIT Press. Wooldridge, J.M. 2009/2010. Class notes from EC 821, Econometrics IIA & IIB. Michigan State University Department of Economics. East Lansing, MI. Slides prepared by Nicole Mason, October Updated January masonn@msu.edu