Dynamic Panel Data Ch 1. Reminder on Linear Non Dynamic Models

Transcription

1 Dynamic Panel Data Ch 1. Reminder on Linear Non Dynamic Models Pr. Philippe Polomé, Université Lumière Lyon M EcoFi

2 Overview of Ch. 1 Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

3 Data Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

4 Data Panel Data i = 1,...,N: agent (individual, firm, country...) t = 1,...,T :time Generally Ti : number of periods differs from agent to agent Unbalanced Panel (this is the norm) Attrition, the property that agents drop out of the sample To simplify notation, theore uses T But all computer packages manage T i So that you should balance your sample y it one obs. of the dependant variable y x it one obs. of K 1vectoroftheindependant variables regressors Possibly endogenous Ch.

5 Data Data management obs agent i time t y x 1... x K y 11 x 111 x K11.. t 1 t y 1t x 11t x K1t.. T 1 T y 1T x 11T x K1T T+1 1 y 1 x 11 x K1.. it i t y it x 1it x Kit.. NT N T y NT x 1NT x KNT

6 Panel Data Models Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

7 Panel Data Models Typical Linear Panel Data Model The typical panel data model where uit scalar disturbance term ntercepts i vary across agents ntercepts i vary over time Slopes are constant y it = i + t + x 0 it + u it (1)

8 Panel Data Models Typical Linear Panel Data Model Amathematicallyproperwaytowritethismodelis y it = NX TX j d j,it + j=1 s= sd s,it + x 0 it + u it where the N individual dummies d j,it = 1ifi = j and = 0otherwise the T 1timedummiesd s,it = 1ift = s and = 0otherwise x it does not include an intercept f an intercept is included then one of the N individual dummies must be dropped Many packages do that automatically

9 Panel Data Models Time dummies Focus on short panels where N!1but T does not Then (time intercept) can be consistently estimated n the sense that there is a finite number of them T 1 time dummies are simply incorporated into the regressors x it We do not discuss them anymore Long panels are treated using time-series methods The panel dimension is abandonned

10 Panel Data Models ndividual dummies f we inserted the full set of N individual intercepts d j,it t would cause problems as N!1 We cannot estimate consistently an 1 number of parameters nformation does not increase on the i as N increases Challenge : estimating the parameters consistently controlling for the N individual intercepts i n this sense, the i are not the focus of the regression They represent individual unobservables that do not not have much interpretation They are nuisance parameters we are not intrested in them but we must find a way to deal with them

11 Panel Data Models ndividual-specific Effects Model ndividual-specific effects model y it = i + x 0 it + it () where it is iid over i and t =amoreparsimoniouswaytoexpressthepreviousmodel(1) with all the dummies Time dummies may be included in regressors x it standard linear non-dynamic panel data model no y i(t s) in x it i random variables Capture unobserved heterogeneity = unobserved time-invariant individual characteristics n effect: a random parameter model

12 Panel Data Models Reminder : Unobserved Heterogeneity The correct model is Y = x 1 + x + But the estimated model is Y = x 1 + The effect of the missing regressor on Y is implied in the error of the estimated model : = x + = unobserved heterogeneity : Unobserved (individual) factors influence the LHS variable f the missing regressor is correlated with an included regressor Then correlated with at least one included regressor LS inconsistent Furthermore, possibly : Heteroscedasticity if var (x t) 6= var (x s), t 6= s Autocorrelation if corr (x t, x s) 6= 0, t 6= s

13 Panel Data Models Reminder : Unobserved Heterogeneity Same slopes

14 Panel Data Models Exogeneity Throughout this chapter: assume strong/strict exogeneity E[e it a i, x i1,...,x it ]=0, t = 1,...,T (3) So that it is assumed to have mean zero conditional on past, current, and future values of the regressors Zero covariance Nothing is said between the random term i and x i Strong exogeneity rules out models with lagged dependent variables or with endogenous variables as regressors (Ch. ) Take y it = i + x 0 it + y t 1 + it Thus yit 1 = i + x 0 it 1 + y t + it 1 it is often hard to maintain that E ( it it 1 )=0 Strong exogeneity does not hold in dynamic models

15 Panel Data Models Fixed Effects Model variantstomodel()accordinglywithhypotheseson i Both are models with errors i and it Error component models Both variants treat i as an unobserved random variable Variant 1 of model (): fixed effects (FE) model i is potentially correlated with the (time-invariant part of the) observed regressors x it A form of unobserved heterogeneity fixed because early treatments treated i as (non-random) parameters to be estimated (hence fixed )

16 Panel Data Models Random Effects Model Variant of model () : Random effects (RE) model i distributed independently of x Usually makes the additional assumptions that both the random effects i and the error term it in () are iid : i, (4) it 0, No distribution has been specified in model (4) it may show autocorrelation Often it is assumed cov ( it, is ) 6= 0 While both cov ( it, jt )=0andcov ( i, j )=0areassumed Except in spatial models can be treated as the intercept of the model

17 Panel Data Models Other names for the Random Effects Model One-way individual-specific effects model Two-way = inclusion of time-dummies or time-specific random effects Random intercept model To distinguish the model with more general random effects models e.g. random slopes Random components model because the error term is i + it

18 Panel Data Models Equicorrelated Random Effects Model RE model y it = i + x 0 it + it can be viewed as regression of yit on x it with composite error term u it = i + " it The RE hypothesis (4) ( i and it iid) implies that sv Cov[(a i + e it ), (a i + e is )] = a, t 6= s sv a + sv e, t = s (5) RE model thus imposes the constraint that the composite error u it is equicorrelated Since Cor[uit, u is ]= /[ + "] for t 6= s does not vary with the time difference t s RE model is also called the equicorrelated model or exchangeable errors model

19 Panel Data Models Synthesis of Panel Data Models Fixed-effects model Random-effects model y it = i + x 0 it + it () Cov ( i, x it ) 6= 0 i, it 0, (4)

20 Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

21 3commonlyusedpaneldataestimatorsof n this non-dynamic, no endogeneity context : LS variants Differ in the extent to which cross-section and time-series variation in the data are used their properties vary according to what model is appropriate Aregressorx it may be time-invariant xit = x i for t = 1,...,T P so that x i = 1 T t x it = x i For some estimators only the coefficients of time-varying regressors are identified

22 Variance Matrix For a given i we expect correlation in y over time : Cor[yit, y is ] is high Even after inclusion of regressors, Cor[uit, u is ] may remain 6= 0 Call Cor[u it, u is ]= its When t = s, its = it

23 Panel Block-Diagonal Var-Cov Matrix of the Errors 0 sv 11 sv11 sv11t sv )T.... sv 1(T SYM sv1t svn1 svn1 svn1t. sv..... N sv N(T 1)T SYM sv NT 1 C A

24 Variance Matrix The RE model accommodates (partly) this correlation From (5): Cov[(a i + e it ), (a i + e is )] = sv a, t 6= s sv a + sv e, t = s OLS output treats each of the T years as independent information, but The information content is less than this given the positive error correlation Tends to overstate estimator precision Always use panel-corrected standard errors when OLS is applied in a panel Many possible corrections, depending on assumed correlation and heteroskedasticity and whether short or long panel The default is not panel-corrected

25 Within Estimator Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

26 Within Estimator Within Model Principle: ndividual-specific deviations of the dependent variable from its time-averaged value are explained by individual-specific deviations of regressors from their time-averaged values ndividual-specific effects model y it = i + x 0 it + it Average over time : ȳ i = i + x i 0 + " i Subtract: the i terms cancel = the within model y it ȳ i =(x it x i ) 0 +( it i ) 1,...,N, t = 1,...,T (6)

27 Within Estimator Within / Fixed Effects Estimator Within estimator = OLS estimator on y it ȳ i =(x it x i ) 0 +( it i ) Consistent for in the FE model Called the fixed effects estimator by analogy with the FE model does not imply that i are fixed Each i must be observed at least twice in the sample Else x it x i = 0

28 Within Estimator Consistency of Fixed Effects Estimator FE treats i as nuisance parameters can be ignored when interest lies in do not need to be consistently estimated to obtain consistent estimates of the slope parameters Consistency further requires in the within model E ( it i x it x i )=0 y it ȳ i =(x it x i ) 0 +( it i ) Because of the averages, that requires more than E ( it x it )=0 Requires the strict exogeneity assumption (3) E[e it a i, x i1,...,x it ]=0, t = 1,...,T

29 Within Estimator Fixed Effects Estimates f the fixed effects i are of interest they can also be estimated f N is not too large an alternative way to compute Within is Least-Squares Dummy variable estimation Directly estimates y it = i + x 0 it + it by OLS of y it on x it and N individual dummy variables Yields Within estimator for, along with estimates of the N fixed effects: ˆ i =ȳ i x 0 ˆ unbiased estimator of i But in short (small T )panelsˆ i are always inconsistent because information never accumulate for them Their distribution or their variation with a key variable may be informative i

30 Within Estimator Time-nvariant Regressors Major limitation of Within the coefficients of time-invariant regressors are not identified Since if xit = x i then x i = x i so (x it x i )=0 Many studies seek to estimate the effect of time-invariant regressors For example, in panel wage regressions : the effect of gender or race For this reason many practitioners prefer not to use the within estimator RE estimator permits estimation of coefficients of time-invariant regressors but are inconsistent if the FE model is the correct model

31 First-Differences Estimator Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

32 First-Differences Estimator First-Differences Model Principle: ndividual-specific one-period changes in the dependent variable are explained by individual-specific one-period changes in regressors ndividual-specific effects model () y it = i + x 0 it Lag one period yi,t 1 = i + xi,t " i,t 1 Subtract = the first-differences model y it y i,t 1 =(x it x i,t 1 ) 0 +( it i,t 1 ) i = 1,...,N, t =,...,T + it (7)

33 First-Differences Estimator First-Differences Estimator The First-differences estimator D1 is OLS in the first differences model (7) Consistent estimates of in the FE model The coefficients of time-invariant regressors are not identified D1 is less efficient than within if " it is iid (for T > ) However, it may safeguard against (1) / unit root variables That would otherwise lead to inconsistency

34 Random Effects Estimator Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

35 Random Effects Estimator Random Effects Model ndividual-specific effects model () y it = i + x 0 it + it Assume RE model with iid i and it as in RE hyp (4) i, it 0, OLS would be consistent But GLS will be more efficient

36 Random Effects Estimator Reminder : GLS in a cross-section When all the hypotheses of the linear model are satisfied but the errors covariance matrix is not the identity, then OLS is consistent but it is not efficient if we know Let the classical linear (cross-section) model y = x 0 E 0 = 6= Let P 0 P = 1 Unique Cholesky decomposition for real definite positive matrix 1 Premultiply the linear model by P : Py = Px + P y = x + Then Var ( )=E P 0 P 0 = PE 0 P 0 1 = P P 0 = P P 0 0 P P = PP 1 P P = + with

37 Random Effects Estimator Reminder : GLS in a cross-section So the transformed model has spherical disturbances Applying OLS to the transformed data is an efficient estimator That is GLS Since is unknown in practice, we need an estimate Any consistent estimate of, ˆ, yields a Feasible (consistent) GLS estimator

38 Random Effects Estimator RE Panel Block-Diagonal Var-Cov Matrix of the Errors 0 sva + sv sv a e sv a sv sva + sv e a sv a sv a sva sva + sv 0. e sva + sve sva sv sva sv a + sv e a sv a sva sv a sva + sv e 1 C A

39 Random Effects Estimator Random Effects Estimator The feasible GLS estimator of the RE model can be calculated from OLS estimation of the transformed model : 0 y it ˆȳ i = 1 ˆ µ + x it ˆ x i + it (8) where it =(1 ˆ) i +(" it ˆ " i ) is asymptotically iid, and ˆ is consistent for = 1 p + T (9) Called the RE estimator

40 Random Effects Estimator Random Effects Estimator The nonrandom scalar intercept µ is added to normalize the random effects i to have zero mean as in the RE hypothesis Cameron & Trivedi provide a derivation of (8) and ways to estimate and " and hence to estimate Not detailed here Note ˆ = 0 corresponds to pooled OLS ˆ = 1 corresponds to within estimation ˆ! 1asT!1(look at the formula) This is a two-step estimator of

41 Random Effects Estimator Random Effects Estimator Properties RE estimator is Fully efficient under the RE model The efficiency gain compared to Pooled OLS (applied to the RE model) need not be great Might still be inefficient if the equicorrelation hypothesis is not true n particular, under AR (1) processes nconsistent if the FE model is correct since then i is correlated with x it

42 Random Effects Estimator RE Discussion Most disciplines in applied statistics, other than microeconometrics, treat any unobserved individual heterogeneity as being distributed independently of the regressors Then the effects are random effects rather : purely random effects Compared to FE models, this stronger assumption has the advantage of permitting consistent estimation of all parameters ncluding coefficients of time-invariant regressors However, RE and Pooled OLS are inconsistent if the true model is FE Economists often view the assumptions for the RE model as being unsupported by the data

43 Fixed vs. Random Effects Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

44 Fixed vs. Random Effects dentification of the ndividual-specific Effects n y it = i + x 0 it + it the individual effect is a random variable (random coefficient) in both fixed and random effects models Both models assume that E[y it i, x it ]= i + x 0 it i is unknown and cannot be consistently estimated Unless T!1 So we cannot estimate E[y it i, x it ] Prediction is therefore not possible Contrarily to what we usually do with OLS That is reasonnable as i includes unobserved individual characteristics Possibly with a non-zero mean But, take the expectation wrt x it : E[y it x it ]=E[ i x it ]+xit 0 That is, what is the (conditional) expected value of i? FE and RE have different takes on this expectation

45 Random Effects vs. Fixed Effects RE : it is assumed that E[ i x it ]=, soe[y it x it ]= + x 0 it Hence E[yit x it ] is identified Since we estimate consistently a single intercept as NT!1 But the key RE assumption that E[ i x it ] is constant across i might not hold in many microeconometrics applications FE : E[ i x it ] varies with x it and it is not known how it varies So we cannot identify E[yit x it ] Nonetheless Within & First-Diff estimators consistently estimate with short panels Thus identify the marginal effect it i, x it ]/@x it e.g. identify effect on earnings of 1 additional year of schooling But only for time-varying regressors so the marginal effect of race or gender, for example, is not identified And not the expected individual y it as we do not know the individual effect i

46 Fixed vs. Random Effects Random Effects vs. Fixed Effects Both models have different focuses RE Time-series structure Efficiency FE Endogeneity of unobserved heterogeneity Consistency

47 Fixed vs. Random Effects Summary Models & Estimators Table: Linear Panel Model: Common Estimators and Models Model Estimator of Rnd Effects () & (4) Fixed Effects () Within (Fixed Effects) (6) Consistent Consistent First Differences (7) Consistent Consistent Random Effects (8) Consistent & efficient nconsistent This table considers only consistency of estimators of. For correct computation of standard errors see next Section. The only fully efficient estimator is RE under the RE model

48 Fixed vs. Random Effects Example Arellano-Bond Unbalanced panel of 140 U.K. manufacturing companies over the period Download in webuse abdata Year = t, n = log of employment, w = log of real wage, k = log of gross capital, ys = log of industry output, id = firm index (i) Panel structure in xtset id year, yearly Arellano & Bond are interested in a dynamic employment equation (labour demand) n it = 1 n i,t 1 + n i,t + 0 (L) x it + t + i + it where (L) indicates a vector of polynomials in the lag operator so that various lags of x might be used AB use wt, w t 1, k t, k t 1, ys t, ys t 1, ys t And time dummies for all years

49 Fixed vs. Random Effects Example Arellano-Bond AB model is dynamic n this chapter, we estimate without the lags of n in the regressors with them by FE, D1 and RE! AB.do All this is in principle known

50 Fixed vs. Random Effects First-difference in First-Differences estimator is not readily available Define the first differences first, then apply the OLS This is fairly unsatisfactory as there is no real account of the error term panel structure Lag 1 period : by id: gen xl1 = x[_n-1] n indexes observations by id indicates to lag by group defined on the id variable Then by id: gen xd1 = x-xl1 for the 1st diff

51 Fixed vs. Random Effects First-differencing time dummies Take d t atime-dummy Recall that a lag one period of x indicates at time t+1 the value that x had at t By construction L1d t must be one at t + 1andzeroelsewhere with a missing value at t=1 (at the 1st obs period) Thus, e.g. yr1980l1=1 in 1981, 0 in other years so yr1980d1=yr1980-yr1980l1=-1 in 1981, 1 in 1980, 0 in other years, missing in 1976 Also yr1984l1 is zero everywhere since it is the last obs. year (missing in 1976) So yr1984d1 cannot be used as it is identical to yr1984 nterpretation of the 1st diff. of a time dummy is hard

52 Example Arellano-Bond Results Table: Coef. Estimates no lags of n Variable OLS FE D1 RE w wl k kl ys ysl ysl yr yr yr yr yr yr omitted ntercept

53 Example Arellano-Bond Results Table: Coef. Estimates with lags of n; time dummies not presented Variable OLS FE D1 RE nl nl w wl k kl ys ysl ysl ntercept t is interesting to compare parameter estimates, but we postpone to next chapter

54 Panel Data nference Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

55 Panel Data nference Panel-Robust nference Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

56 Panel Data nference Panel-Robust nference Panel-Robust Statistical nference The various panel models include error terms : u it, " it, i n many microeconometrics applications : Reasonable to assume independence over i The errors are potentially 1. serially correlated (correlated over t for given i ). heteroskedastic (at least across i) Valid statistical inference requires controlling for both of these factors

57 Het. & Autoc. Block-Diagonal Errors Var-Cov Matrix 0 sv 11 sv11 sv11t sv )T.... sv 1(T SYM sv1t svn1 svn1 svn1t. sv..... N sv N(T 1)T SYM sv Not enough structure NT 1 C A

58 RE Panel Block-Diagonal Var-Cov Matrix of the Errors 0 sva + sv sv a e sv a sv sva + sv e a sv a sv a sva sva + sv 0. e 0 0 Equicorrelation implies Homoskedasticity A limited form of autocorrelation sva + sve sva sv sva sv a + sv e a sv a sva sv a sva + sv e 1 C A

59 Heteroskedastic RE Block-Diagonal Errors Var-Cov Matrix 0 sv a + sv e 1 sv a sv a sv a sv a + sv e sv a sv a sv a sv a + sv e sv a + sv e N sv a sv a sv a sv a + sv en sv a sv a sv a sv a + sv en Small generalisation of RE for Heteroskedasticity The White heteroskedastic consistent estimator can be extended to short panels since for the i th observation the error variance matrix is of finite dimension T T while N!1

60 Panel Data nference Panel-Robust nference Reminder : The White heteroskedastic-consistent estimator Classical linear model y = x 0 OLS unbiased and consistent X 0 X Var ˆOLS = 1 X 0 X + with E 0 = 6= 1 1 X 0 X 6= X 0 X For pure heteroskedasticity, White (1980) shows that S = 1 N NX i=1 ˆ i X i X 0 i where ˆ i is the OLS residual is a consistent estimate of 1 N X 0 X under general conditions The formula can be extended for Autocorrelation But often autocorrelation reveals time-series properties That need to be investigated in more details

61 Panel Data nference Panel-Robust nference Panel-Robust Statistical nference Panel-robust standard errors can thus be obtained following White s principle Called sandwich or robust estimators without assuming specific functional forms for within-individual error correlation or heteroskedasticity However, we assume a constant covariance as in RE So we use inefficient estimators but at least we get their variance better than with OLS formulas f there is AR(1) or (1) errors, we might still be very wrong Only RE estimator in RE model is efficient More efficient estimators using GMM : Chap FE or RE tend to reduce the serial correlation in errors but not eliminate it The panel commands in many computer packages calculate default se assuming iid errors erroneous inference gnoring it can lead to underestimated se

62 Panel Data nference Panel-Robust nference commands Robust estimator assumes independence over i and N!1 but permits V [uit ] and Cov[u it, u is ] to vary with i, t, ands the case for short panels Panel-robust standard errors based on White can be computed by use of a regular panel command if the command has a cluster-robust standard error option in,clusterontheindividuali Common error : use the standard robust se option Only adjusts for heteroskedasticity n practice in a panel : more important to correct for serial correlation n, in a panel estimator, robust automatically accounts for cluster Bootstrap, computes panel-robust standard errors based on bootstrap Fewer hypotheses Slower, depends on the number of replications Do not specify a cluster variable when in a panel model

63 Example Arellano-Bond Results Table: p-values FE models w/ lags of n; time dummies not presented Variable Standard (Cluster-) Robust Bootstrap (500 rep) nl nl w wl k kl ys ysl ysl ntercept Robust is interpreted as Cluster robust, clustering var. is id, the panel i

64 Panel Data nference Panel-Robust nference Note: Variance Decomposition The total variance s of a series x it can be decomposed as NX TX NX TX 1 NT (x it x) = 1 NT [(x it x i )+( x i x)] i=1 t=1 = 1 NT N i=1 t=1 NX TX (x it x i ) + 1 i=1 t=1 as the cross-product term sums to zero. Total variance s = NX TX ( x i x) N 1 i=1 t=1 s w within variance [sum across individuals of individual deviations around the individual means] + sb between variance [deviations of individual means around the grand mean] The between and within R are defined similarly R often small with panel data

65 Fixed Effects vs. Random Effects Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

66 Fixed Effects vs. Random Effects Non-Test Elements of Choice Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

67 Fixed Effects vs. Random Effects Non-Test Elements of Choice Causation The FE model can establish causation under weaker assumptions than those needed with cross-section data panel data models without fixed effects : pooled & RE models n some studies causation is clear, so RE may be appropriate For example, in a controlled experiment, causation is clear crop yield from different amounts of fertilizers applied to different fields in a laboratory x i is assigned randomly to cases, thus uncorrelated to i n other cases it may be sufficient to use a RE analysis to measure the extent of correlation determination of causation is left to other approaches e.g. effect of smoking on lung cancer

68 Fixed Effects vs. Random Effects Non-Test Elements of Choice Causation Economists are unusual in preferring a FE approach because of adesiretomeasure causation with observational instead of experimental data There is the possibility that instead of measuring causation, we measure only a spurious correlation due to the effect of unobserved variables that are correlated with the variables included in the regression FE eliminates those unobserved variables that are time-invariant by differencing, so that The causative effect of x on y is measured by the association between individual changes in y and in x

69 Fixed Effects vs. Random Effects Non-Test Elements of Choice Fixed Effects Weaknesses in Practice Estimation of the coefficient of any time-invariant regressor is not possible with FE Coefficients of time-varying regressors are estimable, but may be imprecise if most of the variation in a regressor is cross sectional rather than over time As then the within transformation will greatly remove this variation Prediction of the conditional mean is not consistent since the indiv. effects are not consistently estimated Only changes in the conditional mean caused by changes in time-varying regressors can be predicted Still requires the assumption that the unobservables i are time-invariant (no it )

70 Fixed Effects vs. Random Effects Hausman Test Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

71 Fixed Effects vs. Random Effects Hausman Test Reminder : Hausman Test Principle : if two estimators are consistent, then their difference should not be statistically different from zero, asymptotically Consider two estimators ˆ and (in the same model) We test H0 : plim ˆ = 0,H a : plim ˆ 6= 0 Under H 0,thedifferencebetweentheestimatorsconverges to a normal with zero mean : p N ˆ! N [0, V H ] where V H is the variance matrix in the limiting distribution Hausman test statistic H = ˆ 0 1 ˆV 1 N H ˆ asymptotically (q) under H 0 reject H0 at level if H > (q) The question in practice is to find an estimate of V H : ˆVH

72 Fixed Effects vs. Random Effects Hausman Test Hausman Test for Panel Data f individual effects are fixed within estimator ˆW is consistent RE estimator RE is inconsistent vector of coefficients of just the time-varying regressors Hausman test on presence of fixed effects H0 : No systematic difference between the coefficients estimates f holds, prefer RE as it is more efficient n principle, maybe not if errors are (1) Works on any pair of estimators with similar properties e.g first differences versus pooled OLS

73 Fixed Effects vs. Random Effects Hausman Test Hausman Test for Panel Data Large value of H leads to rejection of the null hypothesis We infer that since ˆW is consistent, if RE is much different, it must be inconsistent So that the individual-specific effects are correlated with the regressors t may still be possible to avoid using a FE estimator f regressors are correlated with individual-specific effects because of omitted variables then maybe add further regressors t may be possible to estimate a RE model using instrumental variables methods (Ch. )

74 Fixed Effects vs. Random Effects Hausman Test Hausman Test Computation When RE S Fully Efficient Assume the true model is the RE model with i iid 0, uncorrelated with regressors error "it iid 0, " Then RE fully efficient, thehausmanteststatisticsimplifies apple 0 \ h i \ h i 1 H = 1,RE ˆ1,W V ˆ1,W V 1,RE b1,re ˆb 1,W where 1 denotes the subcomponent of corresponding to time-varying regressors since only that component can be estimated by the within estimator This test stastistic is asymptotically (dim [ 1 ]) under H 0 Very easy since then the ˆV matrices are regular outputs of the estimation

75 Fixed Effects vs. Random Effects Hausman Test Hausman Test When RE S NOT Fully Efficient The above simple form of the Hausman test is invalid if i or " it are not iid e.g with heteroskedasticity inherent in much microeconometrics data Then the RE estimator is not fully efficient under the null hypothesis \ i \ i The expression V hˆb1,w V h b1,re in the formula for H needs to be replaced by the more general V h b1,re \ ˆb1,W i That is NOT implemented in For short panels this variance matrix can be consistently estimated by bootstrap resampling over i

76 Fixed Effects vs. Random Effects Hausman Test Hausman Test When RE S NOT Fully Efficient Apanel-robustHausmanteststatisticis apple H Robust = b1,re ˆb 1,W \ V boot h b1,re ˆb1,W i 1 b1,re ˆb 1,W where \ V boot h b1,re ˆb1,W i = 1 B 1 b=1 BX ˆb ˆ ˆb b is the b th of B bootstrap replications and ˆ = b 1,RE ˆb1,W This test statistic can be applied to subcomponents of 1 use other estimators such as 1,POLS in place of 1,RE and ˆ1,FD in place of ˆ1,W There are user-implementations over the nternet ˆ 0

77 Fixed Effects vs. Random Effects Hausman Test Example Arellano-Bond Results How it works in e.g. to compare FE & RE do xtreg..., fe estimates store EstimEF do xtreg..., re hausman EstimEF. Take care to insert the final dot. that means last estimates computed Stat!Postestimation!Tests!Hausman f you try to use vce(robust) or any other than the default an error message results That is fair as only does the fully efficient version of Hausman

78 Fixed Effects vs. Random Effects Hausman Test Example Arellano-Bond Results Output is fairly complete Test: Ho: difference in coefficients not systematic chi(15) = (b-b) [(V_b-V_B)^(-1)](b-B) = Prob>chi = (V_b-V_B is not positive definite) The last probably because the difference between some variances are machine-zero So what conclusion? The estimators must have the same number of coef estimates t may be necessary to remove time-invariant regressors from FE

79 Fixed Effects vs. Random Effects Hausman Test A Note on Model Selection Often students proceeds as follows 1. Estimate a pooled (OLS) model. Estimate the RE model 3. Test (there are several tests) the RE against the pooled 4. f R RE, then estimate FE 5. Test (Hausman) FE against RE Problem with this approach f correct model is FE then both RE & OLS (pooled) estimators are inconsistent The test(s) in step 3 cannot be relied upon More generally, in applied work Start from the more general model (here : FE) And see whether it can be simplified

80 Unbalanced Panel Data Outline Data Panel Data Models Within Estimator First-Differences Estimator Random Effects Estimator Fixed vs. Random Effects Panel Data nference Panel-Robust nference Fixed Effects vs. Random Effects Non-Test Elements of Choice Hausman Test Unbalanced Panel Data

81 Unbalanced Panel Data Attrition Balanced panel :dataareavailableforeveryi in every t e.g. countries Panel surveys of individuals attrition over time Different individuals appear in different years unbalanced or incomplete panels T becomes T i Sometime purposefully Rotating panel Generally unavoidable

82 Unbalanced Panel Data Consistency The FE estimators is consistent only if presence or absence in the sample is not correlated with individual-specific effects i with regressors x it RE is consistent if additionally i is independent of the regressors x it Non-Randomly Missing Data if the reason for individuals dropping out of the sample is correlated with the error term The panel becomes unrepresentative The panel estimators that we have seen may be inconsistent called attrition bias e.g. individuals with unusually low wages may be more likely to drop out of the sample attrition bias if wage is the dependent variable Consistent estimation requires sample selection methods extended to panel data

83 Unbalanced Panel Data Balancing Convert an unbalanced panel into a balanced panel By including in the sample only those individuals with data available in all years Or by rejecting some years for some individuals Can greatly reduce efficiency One reason for removing obs is because only one variable is not observed Nonresponse rate to income questions can be high Data mputation methods