Fortin Econ Econometric Review 1. 1 Panel Data Methods Fixed Effects Dummy Variables Regression... 7

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1 Fortin Econ Econometric Review 1 Contents 1 Panel Data Methods Fixed Effects Dummy Variables Regression First Differencing Methods Within Estimator Random Effects Dynamic Panel Data Models

2 Fortin Econ Econometric Review 2 1 Panel Data Methods 1.1 Fixed Effects Fixed effects can be seen as a generalization of Differences-in-Differences (DD) in the case of more than two periods (say T periods) and more than 2 groups (say J groups). In the OLS regression, Y it = β 0 + β 1 X it + δ 0 W it + ɛ it (1) c Nicole M. Fortin. Not to be copied, used, or revised without explicit permission from the copyright owner.

3 Fortin Econ Econometric Review 3 where the double subscript it indicate that we have observations for each observation unit i in each time period t. The estimate of δ 0 will be biased if treatment W it is correlated with ɛ it, for example if ɛ it = α i + u it, where α i is the individual fixed effect Consider the estimation of the impact of unions on log wages. If we have panel data on workers: W it = 1 if a worker has a union job and W it = 0 if the worker is non-unionized. If workers self-select into unionized jobs or if employers screen higher ability workers for unionized jobs, we will expect the α i and thus ɛ it will be positively correlated with W it, and thus the OLS estimate of the union effect on wages will be overestimated.

4 Fortin Econ Econometric Review 4 From Wooldridge s example 14.4 which uses Vella and Verbreek (1998) NLSY data on 545 men from 1980 to reg lwage union educ exper expersq hisp black married Source SS df MS Number of obs = F( 7, 4352) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] union educ exper expersq hisp black

5 Fortin Econ Econometric Review 5 married _cons reg lwage union educ exper expersq hisp black married, vce(cluster nr) Linear regression Number of obs = 4360 F( 7, 544) = Prob > F = R-squared = Root MSE = (Std. Err. adjusted for 545 clusters in nr) Robust lwage Coef. Std. Err. t P> t [95% Conf. Interval] union educ exper expersq

6 Fortin Econ Econometric Review 6 hisp black married _cons Also the OLS standard errors underestimate the true standard errors because of serial correlation. By comparison, the clustered standard errors are much larger. Note that the individual effect α i does not change over time. It is said to be fixed (for historical reasons) if it is tought to be correlated with the X s.

7 Fortin Econ Econometric Review 7 On the other hand, if α i can be assumed to be uncorrelated with the X s, α i is said to be a random effect, and we will see later how to estimate the model in this case Dummy Variables Regression One way to solve this problem is to put group dummies and time dummies in the regression: Y it = β 0 + β 1 X i + δ 0 W it + γ i D i + γ t D t + ɛ jt where D i and D t are individual and time dummies respectively The identification then comes from the within group-time variation, that is, group specific changes over time.

8 Fortin Econ Econometric Review 8 In the case of the effect of union on log wages, the identification comes from the relationship between individual changes in union status and wages over time, that is for switchers.. xi: reg lwage union educ exper expersq hisp black married union i.nr i.year i.nr _Inr_ (naturally coded; _Inr_13 omitted) i.year _Iyear_ (naturally coded; _Iyear_1980 omitted) Source SS df MS Number of obs = F(554, 3805) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] union educ

9 Fortin Econ Econometric Review 9 exper expersq hisp black married Inr_ _Inr_ _Inr_ _Inr_ _Inr_ _Inr_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _cons

10 Fortin Econ Econometric Review 10 Changes common to all groups are captured by the time dummies and thus are not a source of variation that identification of fixed effects. When there are a large number of observations units, such as this example, you may want to use the following command. areg lwage union educ exper expersq hisp black married yrd*, absorb(nr) Linear regression, absorbing indicators Number of obs = 4360 F( 10, 3805) = Prob > F = R-squared = Adj R-squared = Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval]

11 Fortin Econ Econometric Review 11 union educ (dropped) exper expersq hisp (dropped) black (dropped) married yrd1 (dropped) yrd yrd yrd yrd yrd yrd yrd8 (dropped) _cons nr F(544, 3805) = (545 categories)

12 Fortin Econ Econometric Review First Differencing Methods Another way to eliminate the individual effects α i is by first differencing. Since the lagged value of equation (1) is Y it 1 = β 0 + β 1 X it 1 + δ 0 W it 1 + α i + u it 1 (2) Subtracting (2) from (1) (Y it Y it 1 ) = β 1 (X it X it 1 ) + δ 0 (W it W it 1 ) + (u it u it 1 ) or Y it = β 1 X it + δ 0 W it + u it where we now have T 1 time periods for each unit i. The δ F D obtained by first differencing gives us consistent estimates of δ 0

13 Fortin Econ Econometric Review 13 sort nr. by nr:gen dwage=lwage-lwage[_n-1] (545 missing values generated). by nr:gen dexpe=exper-exper[_n-1] (545 missing values generated). by nr:gen dexp2=expersq-expersq[_n-1] (545 missing values generated). by nr:gen dmarr=married-married[_n-1] (545 missing values generated). by nr:gen dunion=union-union[_n-1] (545 missing values generated). tab year, gen(yrd). reg dwage dunion dexpe dexp2 dmarr yrd*, cluster(nr)

14 Fortin Econ Econometric Review 14 Linear regression Number of obs = 3815 F( 9, 544) = 5.46 Prob > F = R-squared = Number of clusters (nr) = 545 Root MSE = Robust dwage Coef. Std. Err. t P> t [95% Conf. Interval] dunion dexpe (dropped) dexp dmarr yrd1 (dropped) yrd yrd yrd yrd yrd yrd

15 Fortin Econ Econometric Review 15 yrd8 (dropped) _cons But differencing introduces spurious negative correlation across observations since cov( u it 1, u it ) = cov[(u it 1 u it 2 ), (u it u it 1 )] = var(u it 1 ). Sometimes, this problem can be solved by clustering across observations units, that is allowing the error terms for each observation unit to be correlated In our union example, it is best to use longer differences since differencing can make measurement error worse!

16 Fortin Econ Econometric Review 16. reg d2wage d2union d2expe d2exp2 d2marr yrd*, cluster(nr) Linear regression Number of obs = 3270 F( 8, 544) = 9.01 Prob > F = R-squared = Number of clusters (nr) = 545 Root MSE = Robust d2wage Coef. Std. Err. t P> t [95% Conf. Interval] d2union d2expe (dropped) d2exp d2marr yrd1 (dropped) yrd2 (dropped) yrd3 (dropped) yrd yrd

17 Fortin Econ Econometric Review 17 yrd yrd yrd _cons In applying first differencing, we should difference all variables appearing in the structural equation, including any binary treatment indicator. In the case of 2 time periods, however, it generally does not matter whether the policy variable is differenced, If W i1 = 0 and W i2 = 1, then W it = W i2 and the differenced equation is just Y it = β 1 X it + δ 0 W i2 + u it

18 Fortin Econ Econometric Review 18 This is a direct extension of DD where there are 2 groups that experience different changes in policy over 2 periods Within Estimator There is yet another way to get rid of the individual effects α i. For each individual, let s average equation (1) over time Ȳ i = β 1 X i + δ 0 W i + α i + ū i (3) where Ȳ it = 1/T T i=1 Y it and so on, but where ᾱ i = α i since α i is constant over time. Substracting equation (3) from (1) gives (Y it Ȳ i ) = β 1 (X it X i ) + δ 0 (W it W i ) + (u it ū i ) (4)

19 Fortin Econ Econometric Review 19 Under strict exogeneity assumptions, the within estimator and the dummy variables estimator are equivalent But within estimators easily implemented for unbalanced panels, not just balanced panels If we were to do this estimation by hand, the standard errors would be incorrect, because the software would think that df = NT k, but really is N(T 1) k because we used up dfs calculating means. global v1list lwage union educ exper expersq hisp black married. foreach var of varlist $v1list { 2. by nr: egen var bar= mean ( var ) 3. gen var _dev= var - var bar

20 Fortin Econ Econometric Review }. reg lwage_dev union_dev educ_dev exper_dev expersq_dev hisp_dev black_dev married_dev, noconstant Source SS df MS Number of obs = F( 4, 4356) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lwage_dev Coef. Std. Err. t P> t [95% Conf. Interval] union_dev educ_dev (dropped) exper_dev expersq_dev hisp_dev (dropped) black_dev (dropped)

21 Fortin Econ Econometric Review 21 married_dev xtset nr year panel variable: nr (strongly balanced) time variable: year, 1980 to 1987 delta: 1 unit. xtdescribe nr: 13, 17,..., n = 545 year: 1980, 1981,..., 1987 T = 8 Delta(year) = 1 unit Span(year) = 8 periods (nr*year uniquely identifies each observation) Distribution of T_i: min 5% 25% 50% 75% 95% max

22 Fortin Econ Econometric Review 22 Freq. Percent Cum. Pattern XXXXXXXX. xtreg lwage union educ exper expersq hisp black married, fe Fixed-effects (within) regression Number of obs = 4360 Group variable (i): nr Number of groups = 545 R-sq: within = Obs per group: min = 8 between = avg = 8.0 overall = max = 8 F(4,3811) = corr(u_i, Xb) = Prob > F =

23 Fortin Econ Econometric Review 23 lwage Coef. Std. Err. t P> t [95% Conf. Interval] union educ (dropped) exper expersq hisp (dropped) black (dropped) married _cons sigma_u sigma_e rho (fraction of variance due to u_i) F test that all u_i=0: F(544, 3811) = 7.98 Prob > F = xtsum married Variable Mean Std. Dev. Min Max Observations

24 Fortin Econ Econometric Review married overall N = 4360 between n = 545 within T = 8. xtsum educ Variable Mean Std. Dev. Min Max Observations educ overall N = 4360 between n = 545 within T = 8 If we anticipate unobserved heterogeneity to persist over time for individuals, we can use clustered standard errors.. xtreg lwage union educ exper expersq hisp black married, fe vce(cluster nr)

25 Fortin Econ Econometric Review 25 Fixed-effects (within) regression Number of obs = 4360 Group variable: nr Number of groups = 545 R-sq: within = Obs per group: min = 8 between = avg = 8.0 overall = max = 8 F(4,544) = corr(u_i, Xb) = Prob > F = (Std. Err. adjusted for 545 clusters in nr) Robust lwage Coef. Std. Err. t P> t [95% Conf. Interval] union educ (dropped) exper expersq hisp (dropped) black (dropped)

26 Fortin Econ Econometric Review 26 married _cons sigma_u sigma_e rho (fraction of variance due to u_i). est store fixed_effects Now the dummy variable estimate also includes that time fixed effects, we can add those to the individual fixed effects model. xi: xtreg lwage union educ exper expersq hisp black married i.year, fe i.year _Iyear_ (naturally coded; _Iyear_1980 omitted) note: educ omitted because of collinearity note: hisp omitted because of collinearity note: black omitted because of collinearity note: _Iyear_1987 omitted because of collinearity

27 Fortin Econ Econometric Review 27 Fixed-effects (within) regression Number of obs = 4360 Group variable: nr Number of groups = 545 R-sq: within = Obs per group: min = 8 between = avg = 8.0 overall = max = 8 F(10,3805) = corr(u_i, Xb) = Prob > F = lwage Coef. Std. Err. t P> t [95% Conf. Interval] union educ 0 (omitted) exper expersq hisp 0 (omitted) black 0 (omitted) married

28 Fortin Econ Econometric Review 28 _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ (omitted) _cons sigma_u sigma_e rho (fraction of variance due to u_i) F test that all u_i=0: F(544, 3805) = 9.16 Prob > F = testparm _Iyear* ( 1) _Iyear_1981 = 0 ( 2) _Iyear_1982 = 0 ( 3) _Iyear_1983 = 0 ( 4) _Iyear_1984 = 0

29 Fortin Econ Econometric Review 29 ( 5) _Iyear_1985 = 0 ( 6) _Iyear_1986 = 0 F( 6, 3805) = 1.96 Prob > F = The Prob > F is > 0.05, so we failed to reject the null that the coefficients for all years are jointly equal to zero, therefore no time fixed effects are needed in this case. The dummy variable estimate δ D =0.080 (0.019) of the union effect is now identical to the fixed effects estimate δ F EF T =0.080 (0.019).

30 Fortin Econ Econometric Review Random Effects Up to now we have assumed that the α i were correlated with the X s, but what if they are not? If Cov(X it, α i ) = 0, t = 1,..., T and if the explanatory variables are strictly exogenous, OLS would be consistent in that case The composite error ɛ it = α i + u it has variance σ ɛ = var(α i + u it ) = (σ 2 α + σ2 u ) For individual i, cov(ɛ it, ɛ is ) = σ 2 α, so the error term is serially correlated : Corr(ɛ it, ɛ is ) = σ 2 α/(σ 2 α + σ 2 u), t s

31 Fortin Econ Econometric Review 31 We will need to transform the model and use feasible GLS to perform the estimation and make correct inferences The idea will be to used quasi-differenced variables using λ = 1 [σ 2 u/(σ 2 u + T σ 2 α)] 1/2 where deriving this transformation requires sophisticated matrix algebra (see, Wooldridge, 1999, chap. 10) The intuition behind the formula is that σ 2 u is the regression error variance from the fixed effects estimator and σ 2 α comes out in the computation of the regression error variance from the between estimator (runned on time averaged observations) (see Adkins and Hill, 2012 on how to do this by hand )

32 Fortin Econ Econometric Review 32 The transformed equation will be (Y it λȳ i ) = β 0 (1 λ)+β 1 (X it λ X i )+δ 0 (W it λ W i )+(u it λū i ) That is, the random effects transformation subtracts a fraction of the unit time average, where the fraction depends on σ 2 u, σ2 α and the number of time periods T. If λ = 1, then this is just the within estimator. If λ = 0, then this is just the OLS estimator. So, the bigger the variance of the unobserved effect, the closer it is to within estimator. The smaller the variance of the unobserved effect, the closer it is to OLS Since it involves quasi-demeaning of the data, a variable without time variation within the individual may be included in the random effects model.

33 Fortin Econ Econometric Review 33 Note that STATA calls the individual fixed effect u i and the unobserved effect e it (I use the same terminology as Wooldridge) and the coefficient rho = σ 2 u /(σ2 u + σ2 e ) corresponds to Corr(ɛ it, ɛ is ) above.. xtreg lwage union educ exper expersq hisp black married, re Random-effects GLS regression Number of obs = 4360 Group variable (i): nr Number of groups = 545 R-sq: within = Obs per group: min = 8 between = avg = 8.0 overall = max = 8 Random effects u_i ~ Gaussian Wald chi2(7) = corr(u_i, X) = 0 (assumed) Prob > chi2 = lwage Coef. Std. Err. z P> z [95% Conf. Interval] union

34 Fortin Econ Econometric Review 34 educ exper expersq hisp black married _cons sigma_u sigma_e rho (fraction of variance due to u_i) Prob > chi2 = The following LM test helps you decide between a random effects regression and a simple OLS regression. The Breusch-Pagan test statistic consider the null H 0 : σ 2 u = 0 [aka σ 2 a = 0] that variances across entities is zero.

35 Fortin Econ Econometric Review 35. xttest0 Breusch and Pagan Lagrangian multiplier test for random effects lwage[nr,t] = Xb + u[nr] + e[nr,t] Estimated results: Var sd = sqrt(var) lwage e u Test: Var(u) = 0 chi2(1) = Prob > chi2 = Rejecting the null leads us to conclude that random effects are present.

36 Fortin Econ Econometric Review 36 We might consider RE as an appropriate model when applied to a random sample of individuals, such as a sample of workers, or the unemployed, or those who have completed a job training program. In the case of union, it make more sense to think of fixed effects, since we think that the problem comes from something unobserved (such as ability) that is correlated with the Xs But we can perform a Hausman asking whether the random effects coefficients are statistically different from the allegedly consistent fixed effects, which should go first in the test. quietly xtreg lwage union educ exper expersq hisp black married, fe. estimates store fixed

37 Fortin Econ Econometric Review 37. quietly xtreg lwage union educ exper expersq hisp black married, re. estimates store random. hausman fixed random ---- Coefficients ---- (b) (B) (b-b) sqrt(diag(v_b-v_b)) fixed random Difference S.E union exper expersq married 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

38 Fortin Econ Econometric Review 38 chi2(4) = (b-b) [(V_b-V_B)^(-1)](b-B) = Prob>chi2 = Thus we reject the hypothesis of equality of the coefficients estimates, suggesting that the random effects estimator is inconsistent Another testing approach asks if there is a systematic relation between α i [u i ] and the X s. It adds time averages to the other explanatory variables which should not be significant if there is no systematic relationship, which we expect to find here.. xtreg lwage union exper experbar expersq expersqbar married marriedbar, re vce(cluster nr)

39 Fortin Econ Econometric Review 39 Random-effects GLS regression Number of obs = 4360 Group variable: nr Number of groups = 545 R-sq: within = Obs per group: min = 8 between = avg = 8.0 overall = max = 8 Random effects u_i ~ Gaussian Wald chi2(8) = corr(u_i, X) = 0 (assumed) Prob > chi2 = (Std. Err. adjusted for 545 clusters in nr) Robust lwage Coef. Std. Err. z P> z [95% Conf. Interval] union exper experbar expersq expersqbar married

40 Fortin Econ Econometric Review 40 marriedbar _cons sigma_u sigma_e rho (fraction of variance due to u_i). test experbar expersqbar marriedbar ( 1) experbar = 0 ( 2) expersqbar = 0 ( 3) marriedbar = 0 chi2( 3) = Prob > chi2 = Sometimes it may be helpful to use a procedure that allow for the error term u it to be correlated over i: u it = ρ i u it 1 + ε it

41 Fortin Econ Econometric Review 41 The Prais-Winsten procedure is one such GLS procedure (with ρ i = ρ) that is correctly implemented in STATA in the case of panel data with the command xtpcse It uses quasi-differenced data Y it ρy it 1, X it ρx it 1 instead or Y it and X it and a special correction ( 1 ρ 2 ) for the first observation (see Wooldridge, chap. 12 or Hill et al. chap. 12). xtpcse lwage union educ exper expersq hisp black married, correlation(ar1) hetonly (note: estimates of rho outside [-1,1] bounded to be in the range [-1,1]) Prais-Winsten regression, heteroskedastic panels corrected standard errors Group variable: nr Number of obs = 4360 Time variable: year Number of groups = 545 Panels: heteroskedastic (balanced) Obs per group: min = 8

42 Fortin Econ Econometric Review 42 Autocorrelation: common AR(1) avg = 8 max = 8 Estimated covariances = 545 R-squared = Estimated autocorrelations = 1 Wald chi2(7) = Estimated coefficients = 8 Prob > chi2 = Het-corrected lwage Coef. Std. Err. z P> z [95% Conf. Interval] union educ exper expersq hisp black married _cons rho

43 Fortin Econ Econometric Review 43 So which estimates of union effects should we trust? Here all specialized estimates are much closer to each other than to the OLS, δ OLS =0.180 (0.017). Whereas δ LSDV =δ F EF T = (0.019) (least squares dummy variables, fixed effects fixed time ) and δ F E =0.082 (0.019) (fixed effects, within estimator), even δ RE = (0.018) (random effects) and δ P rais W insten = 0.099(0.018) (random effects, Prais-Winsten correction) We think that the OLS estimates of the union effect indeed overestimate the impact of unions on wages! Another idea would be to control for occupations!

44 Fortin Econ Econometric Review Dynamic Panel Data Models Another serious difficulty, distinct from the autocorrelation of the error term, arises with the fixed effects model in the context of a dynamic panel data (DPD) model: one containing a lagged dependent variable (and possibly other regressors). In the simplest setup of a pure AR(1) model without additional regressors: Y it = β 0 + ρy it 1 + α i + u it (5) (Y it Ȳ i ) = ρ(y it 1 Ȳ i, 1 ) + α i + (u it ū i ) (6) where Ȳ it = 1/T T i=1 Y it and so on,

45 Fortin Econ Econometric Review 45 Because u it enters every value of Y it by assumption, the lagged dependent variable cannot be independent of the composite error process The demeaning process which subtracts the individual s mean value of Y and each X from the respective variable creates a correlation between regressor and error. If ρ > 0, the bias is invariably negative, so that the persistence of Y will be underestimated. If the regressors are correlated with the lagged dependent variable to some degree, their coefficients may be seriously biased as well. A solution to this problem involves taking first differences of the original model.

46 Fortin Econ Econometric Review 46 Consider a model containing a lagged dependent variable and a single regressor X: Y it = β 0 + ρy it 1 + β 1 X it + α i + u it (7) The first difference transformation removes both the constant term and the individual effect: Y it = ρ Y it 1 + β 1 X it + u it (8) There is still correlation between the differenced lagged dependent variable and the error process: the former contains Y it 1 and the latter contains u it 1. But we can instrument the lagged dependent variable with the second and third lags of Y it, either in the form of differences or lagged levels.

47 Fortin Econ Econometric Review 47 This approach first proposed by Anderson and Hsiao (1982) provides consistent estimates; it is implemented in STATA by xtivreg, fd. The example below uses the Arellano and Bond (1991) panel dataset, which is based on a sample of 140 UK firms surveyed annually from 1976 to 1984, and considers the determinants of firm s employment adjustment. The panel is unbalanced, with some firms having more observations than others. Because hiring and firing workers is costly, we expect employment to adjust with delay to changes in factors such as firms capital stock, wages, and demand for the firms output.

48 Fortin Econ Econometric Review 48 The variables names are as follows: n is firm employment, w is the firms wage level, k is the firms gross capital, and ys is the aggregate output in the firms sector, as a proxy for demand; all variables are in logarithms. Variable names beginning with D indicate differenced variables. Variable names ending in L1 or L2 indicate lagged copies. For example D.nL1 = n it 1. In their model, Arellano and Bond include the current and first lags of wages, the first two lags of employment, the current and first two lags of capital and sector-level output, and a set of time dummies.

49 Fortin Econ Econometric Review 49. webuse abdata. xtivreg n (nl1=nl2) w wl1 k kl1 kl2 ys ysl1 ysl2 yr*, fd note: yr1984 omitted because of collinearity First-differenced IV regression Group variable: id Number of obs = 611 Time variable: year Number of groups = 140 R-sq: within = Obs per group: min = 4 between = avg = 4.4 overall = max = 6 Wald chi2(14) = corr(u_i, Xb) = Prob > chi2 = D.n Coef. Std. Err. z P> z [95% Conf. Interval] nl1

50 Fortin Econ Econometric Review 50 D w D wl1 D k D kl1 D kl2 D ys D ysl1

51 Fortin Econ Econometric Review 51 D ysl2 D yr1979 D yr1980 D yr1981 D yr1982 D yr1983 D

52 Fortin Econ Econometric Review 52 yr1984 D1. 0 (omitted) _cons sigma_u sigma_e rho (fraction of variance due to u_i) Instrumented: nl1 Instruments: w wl1 k kl1 kl2 ys ysl1 ysl2 yr1978 yr1979 yr1980 yr1981 yr1982 yr1983 nl2 Arellano and Bond (1991) actually proposed an alternative GMM approach based on the notion that the instrumental variables approach above does not exploit all of the information available in the sample.

53 Fortin Econ Econometric Review 53 Their original version xtabond was superseeded by xtabond2, which offers a much more flexible syntax and allows different specification of instrument sets, including differences of lagged dependent variables and correction for standard errors of the two-step estimates.