Efficient Estimation of Time-Invariant and Rarely Changing Variables in Finite Sample Panel Analyses with Unit Fixed Effects

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1 Effcent Estmaton of Tme-Invarant and Rarely Changng Varables n Fnte Sample Panel Analyses wth Unt Fxed Effects Thomas Plümper and Vera E. Troeger Date: Verson: trc_80 Unversty of Essex Department of Government Wvenhoe Park Colchester CO4 3SQ UK contact: tpluem@essex.ac.uk, vtroe@essex.ac.uk Abstract: Earler versons of ths paper have been presented at the 21st Polmeth conference at Stanford Unversty, Palo Alto, July 2004, the 2005 MPSA conference n Chcago, Aprl and the APSA annual conference 2005 n Washngton, September We thank the edtor and the referees of Poltcal Analyss and Neal Beck, Greg Wawro, Donald Green, Jay Goodlffe, Rodrgo Alfaro, Rob Franzese, Jörg Bretung and Patrck Brandt for helpful comments on prevous drafts. The usual dsclamer apples.

2 2 Effcent Estmaton of Tme-Invarant and Rarely Changng Varables n Fnte Sample Panel Analyses wth Unt Fxed Effects Abstract: Ths paper suggests a three-stage procedure for the estmaton of tme-nvarant and rarely changng varables n panel data models wth unteffects. The frst stage of the proposed estmator runs a fxed-effects model to obtan the unt effects, the second stage breaks down the unt-effects nto a part explaned by the tme-nvarant and/or rarely changng varables and an error term, and the thrd stage re-estmates the frst stage by pooled-ols (wth or wthout autocorrelaton correcton and wth or wthout panel-corrected standard errors) ncludng the tme nvarant varables plus the error term of stage 2, whch then accounts for the unexplaned part of the unt effects. Snce the estmator decomposes the unt effects nto an explaned and an unexplaned part, we call t fxed effects vector decomposton (fevd). We use Monte Carlo smulatons to compare the fnte sample propertes of our estmator to the fnte sample propertes of competng estmators. Specfcally, we set the vector decomposton technque aganst the random effects model, pooled OLS and the Hausman-Taylor procedure when estmatng tme-nvarant varables and juxtapose fevd, the random effects model, pooled OLS, and the fxed effects model when estmatng rarely changng varables. In dong so, we demonstrate that our proposed technque provdes the most relable estmates under a wde varety of specfcatons common to real world data. 1. Introducton The analyss of panel data has mportant advantages over pure tme-seres or cross-sectonal estmates advantages that may easly justfy the extra costs of collectng nformaton n both the cross-sectonal and the longtudnal dmenson. Many appled researchers rank the ablty to deal wth unobserved heterogenety across unts most promnently. They pool data just for the purpose of

3 3 controllng for the potentally large number of unmeasured explanatory varables by estmatng a fxed effects (FE) model. Yet, these clear advantages of the fxed effects model come at a certan prce. One of ts drawbacks, the problem of estmatng tme-nvarant varables n panel data analyses wth unt effects, has wdely been recognzed: Snce the FE model uses only the wthn varance for the estmaton and dsregards the between varance, t does not allow the estmaton of tme-nvarant varables (Baltag 2001, Hsao 2003, Wooldrdge 2002). A second drawback of the FE model (and by far the less recognzed one) s ts neffcency n estmatng the effect of varables that have very lttle wthn varance. Typcal examples n poltcal scence nclude nsttutons, but poltcal scentsts have used numerous varables that show much more varaton across unts than over tme. An neffcent estmaton s not merely a nusance leadng to somewhat hgher standard errors. Ineffcency leads to hghly unrelable pont estmates and may thus cause wrong nferences n the same way a based estmator could. Therefore, the neffcency of the FE model n estmatng varables wth low wthn varance needs to be taken serously. Ths artcle dscusses a remedy to the related problems of estmatng tmenvarant and rarely changng varables n fxed effects model wth unt effects. We suggest an alternatve estmator that allows estmatng tme-nvarant varables and that s more effcent than the FE model n estmatng varables that have very lttle longtudnal varance. We call ths superor alternatve fxed effects vector decomposton (fevd) model, because the estmator decomposes the unt fxed effects n an unexplaned part and a part explaned by the tme-nvarant or the rarely changng varables. The fxed effects vector decomposton technque nvolves the followng three steps: Frst, estmaton of the unt fxed effects by the baselne panel fxed effects model excludng the tme-nvarant but not the rarely changng rght hand sde varables. Second, regresson of the fxed effects vector on the tme nvarant and/or rarely changng explanatory varables of the orgnal model (by OLS) to decompose the unt specfc effects nto a part explaned by the tme nvarant varables and an unexplaned part. And thrd, estmaton of a pooled OLS model by ncludng all explanatory tme-varant varables, the tme-nvarant varables, the rarely changng varables and the unexplaned part of the fxed effects vector. Ths

4 4 stage s requred to control for multcollnearty and to adjust the degrees of freedom n estmatng the standard errors of the coeffcents. 1 Based on Monte Carlo smulatons we demonstrate that the vector decomposton model has better fnte sample propertes n estmatng models that nclude ether tme-nvarant or almost tme-nvarant varables correlated wth unt effects than competng estmators. In the analyses dealng wth the estmaton of tme-nvarant varables, we compare the vector decomposton model to the fxed effects model, the random effects model, pooled OLS and the Hausman- Taylor model. We fnd that whle the fxed effects model does not compute coeffcents for the tme-nvarant varables, the vector decomposton model performs far better than pooled OLS, random effects and the Hausman-Taylor procedure f both tme-nvarant and tme-varyng varables are correlated wth the unt effects. The analyss of the rarely changng varables takes these results one step further. Agan based on Monte Carlo smulatons, we show that the vector decomposton method s more effcent than the fxed effects model 2 and thus gves more relable estmates than the fxed effects model under a wde varety of constellatons. Specfcally, we fnd that the vector decomposton model s superor to the fxed effects model when the rato between the between varance and the wthn varance s large, when the overall R² s low, and when the correlaton between the rarely changng / tme-nvarant varable and the unt effects (.e. the hgher the effectvely used between varance) s low. These advantages of the fevd model equally apply to both cross-sectonal and tmeseres domnant panel data. What matters for the estmaton problem provded by tme-nvarant and rarely changng varables s not so much whether the data set at hand ncludes more cases or perods, but whether the between varaton exceeds the wthn varaton by a certan threshold. 1 The procedure we suggest s superfcally smlar to that suggested by Hsao (2003: 52). However, Hsao only clams that hs estmate for tme-nvarant varables (γ ) s consstent as N approaches nfnty. We are nterested n the small sample propertes of our estmator and thus explore tme-seres cross sectonal (TSCS) data. Hsao (correctly) notes that hs γ s nconsstent for TSCS. Moreover, he does not provde standard errors for hs estmate of γ, nor does he compare hs estmator to others. Snce we fully develop our estmator, we do not further consder Hsao's bref dscusson. 2 We also ran all smulatons on rarely changng varables for the random effects model and pooled OLS. Unless the tme-varyng varables are uncorrelated wth the unt effects, the vector decomposton model performs strctly better than both compettors. For the sake of clarty and smplcty, we do not report smulaton output for pooled OLS and the RE model n the secton dealng wth rarely changng varables.

5 5 In a substantve perspectve, ths artcle contrbutes to an ongong debate about the pros and cons of fxed effects models (Green et al. 2001, Beck/ Katz 2001; Plümper et al. 2005; Wlson/ Butler 2003; Beck 2001). Whle the varous partes n the debate put forward many reasons for and aganst fxed effects models, ths paper analyzes the condtons under whch the fxed effects model s nferor to alternatve estmaton procedures. Most mportantly, t suggests a superor alternatve for the cases n whch the FE model s neffcency mpedes relable pont estmates. We proceed as follows: In secton 2 we llustrate the estmaton problem and dscuss how appled researchers dealt wth t. In secton 3, we descrbe the econometrcs of the fxed effects vector decomposton procedure n detal. Secton 4 explans the setup of the Monte Carlo experments. Secton 5 analyzes the fnte sample propertes of the proposed fevd procedure relatve to the fxed effects and the random effects model, the pooled OLS estmator, and the Hausman-Taylor procedure n estmatng tme-nvarant varables and secton 6 presents MC analyses for rarely changng varables n whch we wthout loss of generalty compare only the fxed effects model to the vector decomposton model. Secton 7 concludes. 2. Estmaton of Tme-Invarant and Rarely Changng Varables Tme-nvarant varables can be subdvded nto two broadly defned categores. The frst category subsumes varables that are tme-nvarant by defnton. Often, these varables measure geography or nhertance. Swtzerland and Hungary are both landlocked countres, they are both located n Central Europe, and there s lttle nature and (hopefully) poltcs wll do about t for the foreseeable future. Along smlar lnes, a country may or may not have a colonal hertage or a clmate prone to tropcal dseases. The second category covers varables that are tme-nvarant for the perod under analyss or because of researchers selecton of cases. For nstance, consttutons n postwar OECD countres have proven to be hghly durable. Swtzerland s a democracy snce 1291 and the US mantaned a presdental system snce Independence Day. Yet, by ncreasng the number of perods and/or the number of cases t would be possble to render these varables tmevarant. A small change n the sample can turn tme-nvarant varables of the second category nto a varable wth very low wthn varaton an almost tme-

6 6 nvarant or rarely changng varable. The level of democracy, the status of the presdent, electoral rules, central bank autonomy, or federalsm to menton just a few do not change often even n relatvely long pooled tme-seres datasets. Other poltcally relevant varables, such as the sze of the mnmum wnnng coalton, and the number of veto-players change more frequently, but the wthn varance, the varance over tme, typcally falls short of the between varance, the varance across unts. The same may hold true for some macroeconomc aggregates. Indeed, government spendng, socal welfare, tax rates, polluton levels, or per capta ncome change from year to year, but panels of these varables can stll be domnantly cross-sectonal. Unfortunately, the problem of rarely changng varables n panel data wth unt effects remaned by-and-large unobserved. 3 Snce the fxed effects model can compute a coeffcent f regressors are almost tme-nvarant, t seems far to say that most appled researchers have accepted the resultng neffcency of the estmate wthout payng too much attenton. Yet, as Nathanel Beck has unmstakably formulated: ( ) although we can estmate ( ) wth slowly changng ndependent varables, the fxed effect wll soak up most of the explanatory power of these slowly changng varables. Thus, f a varable ( ) changes over tme, but slowly, the fxed effects wll make t hard for such varables to appear ether substantvely or statstcally sgnfcant. (Beck 2001: 285) Perhaps even more mportantly, neffcency does not just mply low levels of sgnfcance; pont estmates are also unrelable snce the nfluence of the error on the estmated coeffcents becomes larger as the neffcency of the estmator ncreases. In comparson, by far more attenton was devoted to the problem of tmenvarant varables. Wth the fxed effects model not computng coeffcents for tme-nvarant varables, most appled researchers apparently estmated emprcal models that nclude tme-nvarant varables by random effects models or by pooled-ols (see for example Elbadaw/ Sambans 2002; Acemoglu et al. 2002; Knack 1993; Huber/ Stephens 2001). Daron Acemoglu et al. (2002) justfy not controllng for unt effects by statng the followng: Recall that our nterest s n the hstorcally-determned component of nsttutons (that s more clearly exogenous), hence not n the varatons n nsttutons from year-to-year. As a 3 None of the three man textbooks on panel data analyss (Baltag 2001, Hsao 2003, Wooldrdge 2002) refers explctly to the neffcency of estmatng rarely changng varables n a fxed effects approach.

7 7 result, ths regresson does not (cannot) control for a full set of country dummes. (Acemoglu et al. 2002: 27) Clearly, both the random effects model and pooled-ols are nconsstent and based when regressors are correlated wth the unt effects. Employng these models trades the ablty to compute estmates of tme-nvarant varables for the unbased estmaton of tme-varyng varables. Thus, they may be a secondbest soluton f researchers are solely nterested n the coeffcents of the tmenvarant varables. In contrast, econometrc textbooks typcally recommend the Hausman-Taylor procedure for panel data wth tme-nvarant varables and correlated unt effects (Hausman/ Taylor 1981; see Wooldrdge 2002: ; Hsao 2003: 53). The dea of the estmator s to overcome the bas of the random effects model n the presence of correlated unt effects and the soluton s standard: If a varable s endogenous use approprate nstruments. In bref, ths procedure estmates a random effects model and uses exogenous tme-varyng varables as nstruments for the endogenous tme-varyng varables and exogenous tme-nvarant varables plus the unt means of the exogenous tme varyng varables as nstruments for the endogenous tme-nvarant varables (textbook characterzatons of the Hausman-Taylor model can be found n Wooldrdge 2002, pp and Hsao 2003, pp. 53ff). In an econometrc perspectve, the procedure s a consstent soluton to the potentally severe problem of correlaton between unt effects and tme-nvarant varables. Unfortunately, the procedure can only work well f the nstruments are uncorrelated wth the errors and the unt effects and hghly correlated wth the endogenous regressors. Identfyng those nstruments s a formdable task especally snce the unt effects are unobserved (and often unobservable). Nevertheless, the Hausman-Taylor estmator has recently ganed n popularty at least among economsts (Egger/ Pfaffermayr 2004). 3. Fxed Effects Vector Decomposton Recall the data-generatng process of a fxed effects model wth tme nvarant varables: K y = α + β x + γ z + u + ε t k k t m m t k= 1 m= 1 M. (1)

8 8 where the x-varables are tme-varyng and the z-varables are assumed to be tme-nvarant. 4 u denotes the unt specfc effects (fxed effects) of the data generatng process and ε t s the d error term, β and γ are the parameters to estmate. In the frst stage, the fxed effects vector decomposton procedure estmates a standard fxed effects model. The fxed effects transformaton can be obtaned by frst averagng equaton (1) over T: K y = β x + γ z + e + u M (2) k k m m k= 1 m= 1 where y T 1 = y, x T t = 1 t T 1 = x, e T t = 1 t T 1 = e T t = 1 t and e stands for the resdual of the estmated model. Then equaton 2 s subtracted from equaton 1. As s well known, ths transformaton removes the ndvdual effects u and the tme-nvarant varables z. We get K M ( ) ( ) ( ) ( ) y y = β x x +γ z z + e e + u u t k k t k m m m t k= 1 m= 1 K y = β x + e t k k t t k= 1 (3), x k t = xk t xk, and e t = e t e denotng the demeaned wth y t = y t y varables of the fxed effects transformaton. We run ths fxed effects model wth the sole ntenton to obtan estmates of the unt effects û. At ths pont, t s mportant to note that the estmated unt effects û do not equal the unt effects u n the data generatng process snce estmated unt effects nclude all tme nvarant varables the overall constant term and the mean effects of the tme-varyng varables x. Equaton 4 explans how the unt effects are computed and what explanatory varables account for these unt effects K FE = βk k k= 1 û y x e (4) where FE β k s the pooled OLS estmate of the demeaned model n equaton 3. The û nclude the unobserved unt specfc effects as well as the observed unt specfc effects z, the unt means of the resduals e and the tme-varyng varables x k, whereas u only account for unobservable unt specfc effects. In 4 In secton 5 we assume that one z-varable s rarely changng and thus only almost tmenvarant.

9 9 stage 2 we regress the unt effects û from stage 1 on the observed tmenvarant and rarely changng varables the z-varables (see equaton 5) to obtan the unexplaned part h (whch s the resdual from regresson the unt specfc effect on the z-varables). In other words, we decompose the estmated unt effects nto two parts, an explaned and an unexplaned part that we dub h : M û = γ z + h, (5) m m m= 1 The unexplaned part h s obtaned by predctng the resduals form equaton 5: M h = uˆ - γ z. (6) m m m= 1 As we sad above, ths crucal stage decomposes the unt effects nto an unexplaned part and a part explaned by the tme-nvarant varables. We are solely nterested n the unexplaned part h. In stage 3 we re-run the full model wthout the unt effects but ncludng the unexplaned part h of the decomposed unt fxed effect vector obtaned n stage 2. Ths stage s estmated by pooled OLS. K y = α + β x + γ z + δ h + ε t k k t m m t k= 1 m= 1 M. (7) By constructon, h s no longer correlated wth the vector of the z-varables. The estmaton of stage 3 s necessary for varous correctons. Perhaps most mportantly, we use correct degrees of freedom n calculatng the standard errors of the coeffcentsβ and γ. Even though the thrd stage s estmated as a pooled OLS model the procedure s based on a fxed effects setup that has to be mrrored by the computaton of the standard errors for both the tme-varyng and tme-nvarant varables. Correct n contrast to the second stage n the Hsao procedure here means therefore that we use a fxed effects demeaned varance-covarance matrx for the estmaton of the standard errors of βk and we also employ the rght number of degrees of freedom for the computaton of all standard errors (the number of coeffcents to be estmated plus the number of unt specfc effects). The fevd estmator thus gves standard errors whch devate from the pooled OLS standard errors snce we reduce the OLS degrees of freedom by the number

10 10 of unts (N-1) to account for the number of estmated unt effects n stage 1. The devaton of fevd standard errors from pooled OLS standard errors of the same model ncreases n N and decreases n T. Not correctng the degrees of freedom leads to a potentally serous underestmaton of standard errors and overconfdence n the results. In adjustng the standard errors we explctly control for the specfc characterstcs of the three step approach. Estmatng the model requres that heteroscedastcty and seral correlaton must be elmnated. If the structure of the data at hand s as such, we suggest runnng a robust Sandwch-estmator or a model wth panel corrected standard errors (n stage 3) and ncluson of the lagged dependent varable (Beck and Katz 1995) or/and model the dynamcs by an MA1 process (Pras-Wnsten transformaton of the orgnal data n stage 1 and 3). 5 The coeffcents of the tme-nvarant varables are estmated n a procedure smlar to cross-sectonal OLS. Accordngly, the estmaton of tme nvarant varables shares the pooled OLS propertes. However, the estmaton deals wth an omtted varable (the unobserved unt effects), the estmator remans nconsstent even f N approaches nfnty. A potental soluton s to use nstruments for the tmenvarant and rarely changng varables correlated wth the unt effects. However, such nstruments are notorously dffcult to fnd, especally snce unt effects are unobservable. In the absence of approprate nstruments, all exstng estmators gve based results. In the case of the fxed effects vector decomposton model, ths s the case because n order to compute coeffcents for the tme nvarant varables, we need to make a stark assumpton: All varance s attrbuted to the rarely changng or tme-nvarant varables and the covarance between the z-varables and the fxed effects s assumed to be zero. The bas of γ m estmated by OLS n the second stage depends on the covarance between the z-varables and the fxed effects and the cross-sectonal varance of the z-varables. The bas of γ m s postve, the coeffcent of the z-varables tends to be larger than the true value, f the rarely changng and tme-nvarant 5 Snce pcse and robust optons only manpulate the VC matrx and therefore the standard errors of the coeffcents t s sensble to do these correctons only n stage 3 because stage 1 s solely used to receve the fxed effects (whch are not altered by ether pcse or robust VC matrx). A correcton for seral correlaton by a Pras-Wnsten transformaton also affects the estmates and therefore the estmated fxed effects n stage 1 and s accordngly mplemented n both stage 1 and stage 3 of the procedure.

11 11 varables co-vary postvely wth the unt fxed effects and vce versa. The larger the between varance of the z-varables the smaller the actual bas of γ m. 4. Desgn of the Monte Carlo Smulatons To compare the fnte sample propertes 6 of our estmator to competng procedures, we conduct a seres of Monte Carlo analyses evaluatng the competng estmators by ther root mean squared errors (RMSE). In partcular, we are nterested n the fnte sample propertes of competng estmators n relaton to estmatng the coeffcents of tme-nvarant (secton 4) and rarely changng (secton 5) varables. The RMSE thus provdes a unfed vew of the two man sources of wrong pont estmates: bas and neffcency. Kng, Keohane and Verba (1994: 74) hghlght the fundamental trade-off between bas and effcency: We would ( ) be wllng to sacrfce unbasedness ( ) n order to obtan a sgnfcantly more effcent estmator. ( ) The dea s to choose the estmator wth the mnmum mean squared error snce t shows precsely how an estmator wth some bas can be preferred f t has a smaller varance. Ths potental trade-off between effcency and unbasedness mples that the choce of the best estmator typcally depends on the sample sze. If researchers always went for the estmator wth the best asymptotc propertes (as typcally recommended n econometrc textbooks) they would always choose the best estmator for very large samples. Unfortunately, ths estmator could perform poorly n estmatng the fnte sample at hand. All experments use smulated data, whch are generated to dscrmnate between the varous estmators, whle at the same tme mmc some propertes of panel data. Specfcally, the data generatng process underlyng our smulatons s as follows: y t = α+ β1x1 t + β2x2 t + β3x3 t + β4z1 + β5z2 + β6z3 +u + ε t, where the x-varables are tme varyng and the z-varables are tme-nvarant, both groups are drawn from a normal dstrbuton. u denotes the unt specfc unobserved effects and also follows normal dstrbuton. The dosyncratc error 6 For two reasons we are not nterested n analyzng the nfnte sample propertes: Frst, econometrc textbook wsdom suggests that pooled OLS s the best estmator f N equals nfnty whle the FE model has the best propertes for the problem at hand f N s fnte and T nfnte. And second, data sets used by appled researchers have typcally farly lmted szes. Adolph, Butler, and Wlson (2005, pp 4-5) show that most data sets analyzed by poltcal scentsts consst of between 20 and 100 cases typcally observed over between 20 and 50 perods. Unfortunately, an estmator wth optmal asymptotc propertes does not need to perform best wth fnte samples.

12 12 ε t s whte nose and s for each run repeatedly drawn from a standard normal dstrbuton. The R² s fxed at 50 percent for all smulatons. Whle x3 s a tme varyng varable correlated wth the unt effects u, z3 s tme-nvarant n secton 4 and rarely changng n secton 5. In both cases, z3 s correlated wth u. We hold the coeffcents of the true model constant throughout all experments at the followng values: α =1, β1 = 0.5, β2 =2, β3 = -1.5, β4 = -2.5, β5 =1.8, β 6 = 3. Among these sx varables, only varables x3 and z3 are of analytcal nterest snce only these two varables are correlated wth the unt specfc effects u. Varables x1, x2, z1 and z2 do not co-vary wth u. However, we nclude these addtonal tme-varant and tme-nvarant varables nto the data generatng process, because we want to ensure that the Hausman-Taylor nstrumental estmaton s at least just dentfed or even overdentfed (Hausman/ Taylor 1981). We hold ths outlne of the smulatons constant n secton 5, where we analyze the propertes of the FE model and the vector decomposton technque n the presence of rarely changng varables correlated wth the unt effects. Whle the ncluson of the uncorrelated varables x1, x2, z1 and z2 s not necessary n secton 5, these varables do not adversely affect the smulatons and we keep them to mantan comparablty across all experments. All tmevaryng x varables and tme-nvarant z varables as well as the unt specfc effects u follow a standard normal dstrbuton. 7 In the experments, we vared the number of unts (N=15, 30, 50, 70, 100), the number of tme perods (T=20, 40, 70, 100), the correlaton between x3 and the unt effects (x3,u )={0.0, 0.1, 0.2,.., 0.9, 0.99} and the correlaton between z3 and the unt effects (z3,u )= {0.0, 0.1, 0.2,.., 0.9, 0.99}. The number of possble permutatons of these settngs s 2000, whch would have led to 2000 tmes the aggregated number of estmators used n both experments tmes 1000 sngle estmatons n the Monte Carlo analyses. In total, ths would have gven 18 mllon regressons. However, wthout loss of generalty, we smplfed the Monte Carlos and estmated only 980,000 sngle regresson models. We report only representatve examples of these Monte Carlo analyses here, but the output of the smulatons s avalable upon request. 7 N~(0,1); z3 n chapter 5 s rarely changng, the between and wthn standard devaton for ths varable are changed accordng to the specfcatons n fgures 5-7.

13 13 5. The Estmaton of Tme-Invarant Varables We report the RMSE and the bas of the fve estmators, averaged over 10 experments wth varyng correlaton between z3 and u. The Monte Carlo analyss underlyng Table 1 holds the sample sze and the correlaton between x3 and u constant. In other words, we vary only the correlaton between the correlated tme-nvarant varable z3 and the unt effects corr(u,z3). Table 1 about here Observe frst, that (n ths and all followng tables) we hghlght all estmaton results, n whch the estmator performs best or wthn a narrow range of ±10% (of the RMSE) to the best estmator. Table 1 reveals that estmators vary wdely n respect to the correlated explanatory varables x3 and z3. Whle the vector decomposton model, Hausman-Taylor, and the fxed effects model estmate the coeffcent of the correlated tme-varyng varable (x3) wth almost dentcal accuracy, pooled OLS, the vector decomposton model and the random effects model perform more or less equally well n estmatng the effects of the correlated tme-nvarant varable (z3). In other words, only the fxed effects vector decomposton model performs best wth respect to both varables correlated wth the unt effects, x3 and z3. The poor performance of Hausman-Taylor results from the neffcency of nstrumental varable models. Whle t holds true that one can reduce the neffcency of the Hausman-Taylor procedure by mprovng the qualty of the nstruments (Breusch/ Mzon/ Schmdt 1989; Amemya/ MaCurdy 1986, Baltag/ Khant-Akom 1990, Baltag / Bresson/ Protte 2003, Oaxaca/ Gesler 2003), all carefully selected nstruments have to satsfy two condtons smultaneously: they have to be uncorrelated wth the unt effects and correlated wth the endogenous varables. Needless to say that fndng nstruments whch smultaneously satsfy these two condtons s a dffcult task especally snce the unt effects cannot be observed but only estmated. Pooled OLS and the random effects model fal to adequately account for the correlaton between the unt effects and both the tme-nvarant and the tmevaryng varables. Hence, parameter estmates for all varables correlated wth the unt effects are based. When appled researchers are theoretcally nterested

14 14 n both tme-varyng and tme-nvarant varables, the fxed effect vector decomposton technque s superor to ts alternatves. Fgures 1a-d allow an equally easy comparson of the fve competng estmators. Note that n the smulatons underlyng these fgures, we held all parameters constant and vared only the correlaton between the tme-nvarant varable z3 and u (Fgures 1a and 1b) and the tme-varyng varable x3 and u (Fgures 1c and 1d), respectvely. Fgures 1a and 1c dsplay the effect of ths varaton on the RMSE of the estmates for the tme-varyng varable x3, Fgures 1b and 1d the effect on the coeffcent of the tme-nvarant varable z3. corr(z3, u ) affects corr(x3, u ) affects the RMSE RSME (x3) random effects fxed effects pooled OLS hausman-taylor 0.05 of x corr (z3, u) xtfevd Fgure 1a: corr(z3, u ) on RSME(x3) Parameter settngs: N=30, T=20, rho( u,x3)=0.3 RMSE (x3) pooled OLS random effects hausman-taylor, xtfevd, fxed effects corr (x3, u) Fgure 1c: corr(x3, u ) on RSME(x3) Parameter settngs: N=30, T=20, rho( u,z3)=0.3 the RMSE RMSE (z3) xtfevd hausman-taylor random effects, pooled OLS 0.0 of z corr (z3, u) RMSE (z3) hausman-taylor pooled OLS, random effects xtfevd corr (x3, u) Fgure 1b: corr(z3, u ) on RSME(z3) Parameter settngs: N=30, T=20, rho( u,x3)=0.3 Fgure 1d: corr(x3, u ) on RSME(z3) Parameter settngs: N=30, T=20, rho( u,z3)=0.3 Fgures 1 a-d: Change n the RMSE over varaton n the correlaton between the unt effects and z3, x3, respectvely

15 15 Fgures 1a to 1d re-establsh the results of Table 1. We fnd that fevd, random effects and pooled OLS perform equally well n estmatng the coeffcent of the correlated tme-nvarant varable z3, whle fxed effects, Hausman-Taylor and fevd are superor n estmatng the coeffcent of tme-varyng varable x3. We fnd that the advantages of the vector decomposton procedure over ts alternatves do not depend on the sze of the correlaton between the regressors and the unt effects but rather hold over the entre bandwdth of correlatons. The fxed effects vector decomposton model s the sole model whch gves relable fnte sample estmates f the dataset to be estmated ncludes tmevaryng and tme-nvarant varables correlated wth the unt effects. 8 Ths seems to suggest that there s no reason to use the fevd estmator n the absence of tme-nvarant varables. In the followng secton, we demonstrate that ths concluson s not correct. The fevd estmator also gves more relable estmates of the coeffcents of varables whch vary over tme but whch are almost tmenvarant. We call those varable rarely changng varables. 6. Rarely Changng Varables One advantage of the fxed effects vector decomposton procedure over the Hausman-Taylor procedure and the Hsao suggestons s that t extends ncely to almost tme-nvarant varables. Estmaton of these varables by fxed effects gves a coeffcent, but the estmaton s extremely neffcent and hence the estmated coeffcents are unrelable (Green et al Beck/ Katz 2001). However, f we do not estmate the model by fxed effects, than estmated coeffcents are based f the regressor s correlated wth the unt effects. Snce t seems not unreasonable to assume that the unt effects are made up prmarly of geographcal and varous nsttutonal varables, t s not unreasonable to perform an orthogonal decomposton of the explaned part and an unexplaned part as descrbed above. Clearly, the orthogonalty assumpton s often ncorrect and ths wll nevtably bas the estmated coeffcents of the almost tme-nvarant varables. As we wll demonstrate n ths secton, ths bas s under dentfable condtons less harmful than the neffcency caused by fxed effects estmaton. At the same tme, our procedure s also superor to 8 Appendx A demonstrates that ths result also holds true when we vary the sample sze. Even wth a comparably large T and N the fxed effects vector decomposton model performs best.

16 16 estmaton by random effects or pooled OLS as we leave the tme-varyng varables unbased whereas the latter two procedures do not. Obvously the performance of fevd wll depend on what exactly the data generatng process s. In our smulatons we show that unless the DGP s hghly unfavorable for fevd, our procedure performs reasonably well and s generally better than ts alternatves. Before we report the results of the Monte Carlo smulatons, let us brefly explan why the estmaton of almost tme-nvarant varables by the standard fxed effects model s problematc due to neffcency and what that neffcency does to the estmate. The neffcency of the FE model results from the fact that t dsregards the between varaton. Thus, the FE model does not take all the avalable nformaton nto account. In techncal terms, the estmaton problem stems from the asymptotc varance of the fxed effects estmator that s shown n equaton 8: ( ˆ FE β ) 1 N 2 Avar ˆ = ˆ σ u X ' X. (8) = 1 When the wthn transformaton of the FE model s performed on a varable wth lttle wthn varance, the varance of the estmates can approach nfnty. Thus, f the wthn varaton becomes very small, the pont estmates of the fxed effects estmator become unrelable. When the wthn varance s small, the FE model does not only compute large standard errors, but n addton the samplng varance gets large and therefore the relablty of pont predctons s low and the probablty that the estmated coeffcent devates largely from the true coeffcent ncreases. Our Monte Carlo smulatons seek to dentfy the condtons under whch the fxed effects vector decomposton model computes more relable coeffcents than the fxed effects model. Table 2 reports the output of a typcal smulaton analogous to Table 1: 9 Table 2 about here 9 We have also compared the vector decomposton and the fxed effects model to pooled OLS and the random effects model. Snce all fndngs for tme-nvarant varables carry over to rarely changng varables, ndcatng that the vector decomposton model domnates pooled OLS and random effects models, we report the results of the RE and pooled OLS Monte Carlos only n the onlne appendx.

17 17 Results dsplayed n Table 2 mrror those reported n Table 1. As before, we fnd that only the fevd procedure gves suffcently relable estmates for both the correlated tme-varyng x3 and the rarely changng varable z3. As expected, the fxed effects model provdes far less relable estmates of the coeffcents of rarely changng varables. There can thus be no doubt that the fxed effects vector decomposton model can mprove the relablty of the estmaton n the presence of varables wth low wthn and relatvely hgh between varance. We also fnd that pooled OLS and the RE model estmate rarely changng varables wth more or less the same degree of relablty as the fevd model but are far worse n estmatng the coeffcents of tme-varyng varables. Note that these results are robust regardless of sample sze. 10 Snce any further dscusson of these ssues would be redundant, we do not further consder the RE and the pooled OLS model n ths secton. Rather, ths secton provdes answers to two nterrelated questons: Frst, can the vector decomposton model gve more relable estmates (a lower RMSE) than the FE model? And second, n case we can answer the frst queston postvely, what are the condtons that determne the relatve performance of both estmators? To answer these questons, we assess the fnte sample propertes of the competng models n estmatng rarely changng varables by a second seres of Monte Carlo experments. Wth one notable excepton, the data generatng process n ths secton s dentcal to the one used n secton 5. The excepton s that now z3 s not tme-nvarant but a rarely changng varable wth a low wthn varaton and a defned rato of between to wthn varance. The easest way to explore the relatve performance of the fxed effects model and the vector decomposton model s to change the rato between the between varance and the wthn varance across experments. We call ths rato the b/wrato and compute t by dvdng the between standard devaton by the wthn standard devaton of a varable. There are two ways to vary ths rato systematcally: we can hold the between varaton constant and vary the wthn varaton or we can hold the wthn varaton constant and vary the between varaton. We use both technques. In Fgure 2, we hold the between standard 10 We re-ran all Monte Carlo experments on rarely changng varables for dfferent sample szes. Specfcally, we analyzed all permutatons of N={15, 30, 50, 70, 100} and T={20, 40, 70, 100}. The results are shown n Table A2 of Appendx A (see the Poltcal Analyss webpage). All fndngs for rarely changng varables reman vald for larger and smaller samples, as well as for N exceedng T and T exceedng N.

18 18 devaton constant at 1.2 and change the wthn standard devaton successvely from 0.15 to 1.73, so that the rato of between to wthn varaton vares between 8 and 0.7. In Fgure 3, we hold the wthn varance constant and change the between varance Parameter settngs: N=30 T=20 RMSE (z3) fxed effects fevd rho(u,x3)=0; rho(u,z3)=0.3 between SD (z3): 1.2 wthn SD (z3) rato between/wthn standard devaton (z3) Fgure 2: The rato of between to wthn SD (z3) on RSME (z3) Recall that the estmator wth the lower RMSE gves more relable estmates. Hence, Fgure 2 shows that when the wthn varance ncreases relatve to the between varance, the fxed effects model becomes ncreasngly relable. Snce the relablty of the vector decomposton model does not change, we fnd that the choce of an estmator s contngent. Below a between to wthn standard devaton of approxmately 1.7, the fxed effects estmator performs better than the vector decomposton model. Above ths threshold, t s better to trade unbasedness for the effcency of the vector decomposton model. Accordngly, the b/w-rato should clearly nform the choce of estmators. Ths fndng depends on the correlaton of the rarely changng varable z3 wth the unt fxed effects of 0.3. Furthermore, the threshold rato s dependent on the relaton of between varance to the varance of the overall error term. We obtan smlar results when we change the wthn varaton and keep the between varaton constant. Fgure 3 shows smultaneously the results of two slghtly dfferent experments. In the one experment (dotted lne), we vared the wthn varaton and kept the between varaton and the error constant. In the other experment, we kept the between varaton constant but vared the wthn varaton and the error varance n a way that the fevd-r² remaned constant.

19 19 RMSE (z3) fxed effects Parameter settngs: N=30 T=20, rho(u,x3)=0 rho(u,z3)=0.3 between SD (z3): wthn SD (z3): fevd rato between/wthn standard devaton (z3) Fgure 3: The rato of between to wthn SD (z3) on RSME (z3) In both experments, we fnd the threshold to be at approxmately 1.7 for a correlaton of z3 and u of 0.3. We can conclude that the result s not merely the result of the way n whch we computed varaton n the b/w-rato, snce the threshold level remaned constant over the two experments. Unfortunately, the relatve performance of the fxed effects model and the vector decomposton model does not solely depend on the b/w-rato. Rather, we also expected and found a strong nfluence of the correlaton between the rarely changng varable and the unt effects. The nfluence of the correlaton between the unt effects and the rarely changng varable obvously results from the fact that t affects the bas of the vector decomposton model but does not nfluence the neffcency of the fxed effects model. Thus, a larger correlaton between the unt effects and the rarely changng varable renders the vector decomposton model worse relatve to the fxed effects model. We llustrate the strength of ths effect by relatng t to the level of the b/w-rato, at whch the FE model and the fevd model gve dentcal RMSE. Accordngly, Fgure 4 dsplays the dependence of the threshold level of the b/w-rato on the correlaton between the rarely changng varable and the unt effects.

20 Parameter settngs: N= T=20, b/w-rato fevd gves lower RMSE FE gves lower RMSE R² = 0.5 rho(u,x3)=0, rho(u,z3)= {0, 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 0.9} between SD (z3): wthn SD (z3): 1 correlaton of z3 and u Fgure 4: The correlaton between z3 and u and the mnmum rato between the between and wthn standard devaton that renders fevd superor to the fxed effects model Note that, as expected, the threshold b/w-rato s strctly ncreasng n the correlaton between the rarely changng varable and the unobserved unt effects. In the case where the rarely changng varable s uncorrelated wth u, the threshold of b/w-rato s as small as 0.2. At a correlaton of 0.3, fevd s superor to the FE model f the b/w-rato s larger than approxmately 1.7; at a correlaton of 0.5 the threshold ncreases to about 2.8 and at a correlaton of 0.8 the threshold gets close to 3.8. Therefore, we cannot offer a smple rule of thumb whch nforms appled researchers of when a partcular varable s better estmated as nvarant varable by fevd or as tme-varyng varable. Even worse, the correlaton between the unt effects and the rarely changng varable s not drectly observable, because the unt effects are unobservable. However, the odds are that at a b/w-raton of at least 2.8, the varable s better ncluded nto the stage 2 estmaton of fevd than estmated by a standard FE model. Appled researchers can mprove estmates created by the vector decomposton model by reducng the potental for correlaton. To do so, stage 2 of the fevd model needs to be studed carefully. We can reduce the potental for bas of the estmaton by ncludng addtonal tme-nvarant or rarely-changng varables nto stage 2. Ths may reduce bas but s lkely to also reduce effcency. Alternatvely, appled researchers can use varables whch are uncorrelated wth the unt effects as nstruments for potentally correlated tme-nvarant or rarely changng varables a strategy whch resembles the Hausman-Taylor model.

21 21 Yet, as we have repeatedly ponted out: t s mpossble to tell good from bad nstruments snce the unt effects can not be observed. The decson whether to treat a varable as tme nvarant or varyng depends on the rato of between to wthn varaton of ths varable and on the correlaton between the unt effects and the rarely changng varables. In ths respect, the estmaton of tme-nvarant varables s just a specal case of the estmaton of rarely changng varables a specal case n whch the between-towthn varance rato equals nfnty and fevd s consequently better. These fndngs suggest that strctly speakng the level of wthn varaton does not nfluence the relatve performance of fevd and FE models. However, wth a relatvely large wthn varance, the problem of neffcency does not matter much the RMSE of the FE estmator wll be low. Stll, f the wthn varance s large but the between varance s much larger, the vector decomposton model wll perform better on average. Wth a large wthn varance, the actual absolute advantage n relablty of the fevd estmator wll be tny. From a more general perspectve, the man result of ths secton s that the choce between the fxed effects model and the fevd estmator depends on the relatve effcency of the estmators and on the bas. As Kng, Keohane and Verba have argued (1994: p. 74), appled researchers are not well advsed f they base ther choce of the estmator solely on unbasedness. At tmes, pont predctons become more relable (the RMSE s smaller) when researchers use the more effcent estmator. The fxed effects vector decomposton model s more effcent than the fxed effects model snce t uses more nformaton. Rather than just relyng on the wthn varance, our estmator also uses the between varance to compute coeffcents. 6. Concluson Under dentfable condtons, the vector decomposton model produces more relable estmates for tme-nvarant and rarely changng varables n panel data wth unt effects than any alternatve estmator of whch we are aware. The case for the vector decomposton model s clear when researchers are nterested n tme-nvarant varables. Whle the fxed effects model does not compute coeffcents of tme-nvarant varables, the vector decomposton model performs better than the Hausman-Taylor model, pooled OLS and the random effects model.

22 22 The case for the vector decomposton model s less straghtforward, when at least one regressor s not strctly tme-nvarant but shows some varaton across tme. Nevertheless, under many condtons the vector decomposton technque produces more relable estmates. These condtons are: frst and most mportantly, the between varaton needs to be larger than the wthn varaton; and second, the hgher the correlaton between the rarely changng varable and the unt effects, the worse the vector decomposton model performs relatve to the fxed effects model and the hgher the b/w-rato needs to be to render fevd more relable. From our Monte Carlo results, we can derve the followng rules that may nform the appled researcher s selecton of an estmator on a more general level: Estmaton by Pooled-OLS or random effects models s only approprate f unt effects do not exst or f the Hausman-test suggests that exstng unt effects are uncorrelated wth the regressors. If ether of these condtons s not satsfed, the fxed effects model and the vector decomposton model compute more relable estmates for tme-varyng varables. Among these models, the fxed effects model performs best f the wthn varance of all regressors of nterest s suffcently large n comparson to ther between varance. We suggest estmatng a fxed effects model, unless the rato of the between-to-wthn varance exceeds 2.8 for at least one varable of nterest. Otherwse, the effcency of the fxed effects vector decomposton model becomes more mportant than the unbasedness of the fxed effects model. Therefore, the vector decomposton procedure s the model of choce f at least one regressor s tme-nvarant or f the between varaton of at least one regressor exceeds t's wthn varaton by at least a factor of 2.8 and f the Hausman-test suggests that regressors are correlated wth the unt effects.

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