A TEST FOR SLOPE HETEROGENEITY IN FIXED EFFECTS MODELS

Transcription

1 A TEST FOR SLOPE HETEROGEEITY I FIXED EFFECTS MODELS TED JUHL AD OLEKSADR LUGOVSKYY Abstract. Typcal panel data models make use of the assumpton that the regresson parameters are the same for each ndvdual cross sectonal unt. We propose tests for slope heterogenety n panel data models. Our tests are based on the condtonal Gaussan lkelhood functon n order to avod the ncdental parameters problem nduced by the ncluson of ndvdual fxed effects for each cross sectonal unt. We derve the Condtonal Lagrange Multpler test that s vald n cases where and T s fxed. The test apples to both balanced and unbalanced panels. We expand the test to account for general heteroskedastcty where each cross sectonal unt has ts own form of heteroskedastcty. The modfcaton s possble f T s large enough to estmate regresson coeffcents for each cross sectonal unt by usng the MIQUE unbased estmator for regresson varances under heteroskedastcty. All versons of the test have a standard ormal dstrbuton under general assumptons on the error dstrbuton as. A Monte Carlo experment shows that the test has very good sze propertes under all specfcatons consdered, ncludng heteroskedastc errors. In addton, power of our test s very good relatve to exstng tests, partcularly when T s not large. The tests are appled to a model of trade and currency unons.. Introducton Panel data models are mportant for the analyss of data that s measured over tme for many cross-sectonal unts. One varant s the fxed effects model where t s assumed that each cross-sectonal unt has ts own ntercept term whch s mplctly estmated. Moreover, for panel data models to explot the large amount of data collected, t s often assumed that the slope coeffcents are the same for each cross-sectonal unt. In ths way, one can combne the data to get more effcent estmators of the common slope coeffcents relatve to estmatng a separate regresson for each cross-sectonal unt. If the cross-sectonal unts do not share the same regresson coeffcent vectors, t s natural to ask what a fxed effects estmator s estmatng. In general, fxed effects estmaton may not estmate any known parameters of nterest, ncludng the average of the crosssectonal regresson parameters when there are dfferent regresson parameters. Gven ths Verson: February 9, 200. We thank Bad Baltag, Roger Koenker, Walter Sosa-Escudero, Tm Vogelsang, and Zhje Xao for comments on earler versons of the paper. Ted Juhl, Department of Economcs, Unversty of Kansas, 45 Snow, Lawrence, KS 66045; Tel 785) ; juhl@ku.edu).

2 2 fact, a logcal queston to address s whether t s possble to test f the regresson slope coeffcents vary over the cross-sectonal unts. Baltag 2005) dscusses how to extend the Chow 960) test for testng slope heterogenety versus poolablty of the data. Moreover, Baltag, Grffn, and Xong 2000) explore the effects of estmatng models wth heterogenous slopes versus pooled models n an emprcal example of cgarette demand. Hsao 2003) suggests a modfcaton of the Breusch-Pagan 979) test wth the null hypothess of no slope heterogenety. The test s vald under the case where both tme T) and the number of cross sectonal unts ) tend to nfnty. Swamy 970) proposed a test where the null hypothess s constant slopes for each cross-sectonal unt but each unt s allowed to have a dfferent error varance. The asymptotc dstrbuton of the test s vald f T and s fxed. Pesaran and Yamagata 2008) recently proposed a re-scaled verson of the Swamy test that s vald as, but requres T. In Monte Carlo experments, they show that ther test performs very well relatve to exstng tests. In partcular, the Pesaran and Yamagata 2008) test has excellent sze and power and s the only test of those consdered that has good sze propertes over a varety of specfcatons of and T. In ths paper, we propose tests for slope heterogenety over cross-sectonal unts that are desgned to complement the tests of Pesaran and Yamagata 2008). For example, we wsh to formulate tests that are vald for very small values of T, such as cases where one could not even calculate any other tests. Moreover, t s well known that parameter heterogenety can be msnterpreted as heteroskedastcty, and we wsh to modfy our tests so that we can dsentangle the effects of heteroskedastcty. Our tests are based on the condtonal Gaussan lkelhood functon, yet are vald for general error dstrbutons. Each test s asymptotcally normal for cases where T s fxed and. Moreover, our tests can be modfed for varous forms of heteroskedastcty. In partcular, we propose versons of the test that are vald when there s group-wse heteroskedastcty as allowed n Swamy 970), and general heteroskedastcty where the form of heteroskedastcty can be dfferent for each cross-sectonal unt. The latter correcton s based on MIQUE estmaton as proposed n Rao 970). We evaluate the sze of the statstc usng a Monte Carlo experment based on data generated from a varety of dstrbutons. The sze s very close to the nomnal sze for all

3 dstrbutons consdered and for all value of and T explored n the experment. Power for our new tests s compettve wth exstng tests for large values of T. However, as expected from the asymptotc results whch do not depend on the sze of T, the proposed tests have excellent comparatve performance when T s small. Fnally, our experment shows that heteroskedastcty adversely affects sze of any tests that do not adjust for heteroskedastcty. However, our MIQUE based test retans excellent sze and power when there s heteroskedastcty n the model, yet does not sacrfce power f the errors are homoskedastc. The remander of the paper s structured as follows. In Secton 2, we llustrate the potental ptfalls of gnorng slope heterogenety. We revew the exstng tests n Secton 3. The tests and the resultng asymptotc theory are developed n Secton 4. Secton 5 contans the extensons to heteroskedastc errors. We provde a Monte Carlo experment n Secton 7 and an emprcal llustraton n Secton 8. Secton 9 concludes Model of Slope Heterogenety The basc model s wrtten as y t = α + x tβ + v t =,..., t =,..., T where β represents the regresson slope coeffcent for the th unt. To allow for potental slope heterogenety, we suppose that the k vector β s represented by β = β + η where Eη ) = 0. Let the covarance matrx of parameter heterogenety be gven as Eη η X ) = Σ η. Wrtten n matrx form, we have y = α ι T + X β + v,..., where y s T, X s T k, ι T s a T vector of ones, and v s T. Most panel data models assume that the slope s the same for each cross-sectonal unt. However, the α parameter s allowed to have some varaton over n typcal models. One such model s the fxed effects regresson where one estmates mplctly) each α parameter

4 4 n order to fnd a sutable estmator for β. The fxed effects model assumes that Σ η s zero so that β = β for each. Based on ths assumpton, fxed effects estmaton puts more weght on the cross sectonal unts wth more varaton n X. effects estmator can be wrtten as To be explct, the fxed ) ˆβ F E = X M 0 X X M 0 y where M 0 = I T ι T ι T T s the matrx that subtracts group means from each element of the group. It s convenent for our analyss to rewrte ˆβ F E as ˆβ F E = = = ) X M 0 X Xj M 0 X j j= Xj M 0 X j j= X M 0 y X M 0 X )X M 0 X ) X M 0 y X M 0 X ) ˆβ where ˆβ s the OLS estmator of regressng y on X and a constant. In ths way, we can see that ˆβ F E can be consdered as a weghted average of OLS for each cross sectonal unt. If β = β s the same for each cross-sectonal unt, ths s an optmal weghtng scheme because those groups wth the hghest ntra-group varaton X M 0X wll receve the most weght. However, f β = β + η s true, there are cases where ˆβ F E wll estmate no populaton parameter of nterest. For example, suppose that the ntra-group varaton s correlated wth β. That s, suppose that η s correlated wth X M 0X. For the moment, assume that each group has the same expectaton for X M 0X. Then we have the followng heurstc argument:

5 5 ) ˆβ F E = X M 0 X X M 0 X β + v ) ) = β + X M 0 X X M 0 X η + v ) p β + EX M 0 X )) EX M 0 X η ) β. From ths result, we see that the fxed effects estmator s not consstent for β, the populaton mean of β. Consder a refnement of our smple example where the number of cross sectonal unts s two. Suppose that for one unt, the slope s β = and for the second unt, the slope s β 2 = and we have the same number of tme seres observatons for each unt so that the average slope s zero. However, suppose that the ntra-group standard devaton s twce as large for the second group. Then, the formula for the fxed effects estmator smplfes to 2 2 ˆβ F E = Xj M 0 X j X M 0 X ) ˆβ j= 5 β β 2 = 3 5. We llustrate ths scenaro n Fgure. From the graph we see that the fxed effects estmator puts more weght on the unt wth larger ntra-group varaton, and n ths case, we wll not even be able to obtan a consstent estmate of the average of the two slopes, whch s zero for our example. The above dscusson hghlghts some of the mportant consequences of slope heterogenety. Frst, fxed effects estmaton has the potental to be very msleadng snce t s attemptng to combne parameters n a fashon that may render the estmator nconsstent for any populaton parameters. In the above case, the average of slopes s zero, yet we fnd a nonzero estmate based on fxed effects. However, even f we were to estmate the average slope of zero correctly), what does ths mply? Would we conclude that x does not affect y? Ths s certanly not true n our example snce x affects y for each cross sectonal unt.

6 6 Fgure Fxed Effects Estmaton Heterogeneous Slopes 3 β = y - β FE = -3/5-3 β 2 = x We can translate ths to an emprcal queston we may want to address. What good s an estmator for polcy purposes f t does not apply to any unts n the sample? The purpose of our test s to determne whether there s statstcally sgnfcant varaton n the slopes of our panel data model. 3. Exstng Tests There are several exstng tests for slope heterogenety. In theory, one could use a standard F test to test for slope heterogenety. In partcular, the F test n ths case s a form of Chow 960) test for dfferent regressons. Toyoda 974) showed analytcally and usng Monte Carlo that f there s group-wse heteroskedastcty, the Chow test may be severely overszed. The test proposed by Swamy 970) allows for group-wse heteroskedastcty so that each cross sectonal unt has ts own varance. The model has k varables n x t wth y t = α + x tβ + v t

7 where Ev v X ) = σ 2 I T groupwse heteroskedastcty). Gven the dfferent varances, Swamy proposes usng a more effcent estmator for β, namely ) X ˆβ W = M 0X X M 0y ˆσ 2 ˆσ 2, where ˆσ 2 s estmated for each cross sectonal unt. Frst, note that we must have T > k + to be able to estmate a regresson for each cross sectonal unt to obtan varance estmator. The test s based on squared dfferences between each estmated regresson coeffcent, ˆβ, and ˆβ W. S = ˆβ ˆβ W ) X M 0X ˆσ 2 ˆβ ˆβ W ). Under the null hypothess that each β s the same, S d χ 2 k ) f s fxed and T. Snce panel data models typcally employ the assumpton that, Pesaran and Yamagata 2008) recently proposed a novel centered verson of Swamy 970) whch employs estmates of σ 2 from fxed effects resduals. They defne S = ˆβ ˆβ W ) X M 0X σ 2 ˆβ ˆβ W ) where σ 2 s estmated usng the fxed effects estmator. The centered verson s P Y = ) S k. 2k 7 Under the null hypothess, P Y d 0, ) f, T, and /T Proposed Tests The test proposed by Pesaran and Yamagata 2008) performs very well n Monte Carlo experments. Our goal s to develop a test that complements ths test n the followng ways. Frst, we want to have a test that does not requre T. In fact, we propose one verson of the test for cases where T k + where t s mpossble to conduct any of the exstng tests. That s, all exstng tests of slope heterogenety requre an estmate of the slope β for every. If T s too small, we can t estmate a slope for each unt. One verson of our test s calculated under the null, and hence a fxed effects estmator s used, and the test s possble for T 3. Second, we want to address the possblty of heteroskedastcty n the errors that s more complcated than group-wse heteroskedastcty. For example, suppose

8 8 that each cross sectonal unt has ts own form of heteroskedastcty wthn that unt. We propose a test that s vald for those very general forms of heteroskedastcty. In order to test the null hypothess of no slope heterogenety, we wsh to test Eη η X ) = Σ η = 0. The lkelhood functon s gven as lα, β, σ 2 v, Σ η ) = 2 ln Σ ) 2 w Σ w w = y α ι T X β = X η + v Σ = Ew w X ) = X Σ η X + σ 2 vi T The lkelhood functon ncludes parameters assocated wth each of the ntercept terms α, yet we wsh to let. To avod the ncdental parameters problem assocated wth estmaton of α, we condton on the unbased suffcent statstc for α whch s gven by y Σ ι T /ι T Σ ι T, so that the condtonal lkelhood s gven by l C β, σv, 2 Σ η ) = 2 ln Σ ] 2 lnι T Σ ι T ) 2 w Σ Σ ι T Σ ) ] ι T ι T Σ w ι T We wsh to test the hypotheses H 0 : Σ η = 0 vs. H A : Σ η = γi k The alternatve hypothess may at frst appear restrctve snce t allows for only a very smple form of slope heterogenety. However, we wll show that the resultng test has power aganst a wde class of alternatves that are much more general than the smple form ndexed Smlar condtonng arguments are employed n Chamberlan 980), and, more recently, Inoue and Solon 2006).

9 by γ. One advantage of the current formulaton s that the resultng tests wll be one-sded and have a normal dstrbuton. To derve a test statstc, we examne the score evaluated at the true parameter vector under the null hypothess. 9 Theorem 4.. The dervatve of the condtonal lkelhood evaluated at θ 0 = β, σ 2 v, Σ η = 0) s gven by l C β, σ 2 v, Σ η ) γ = β,σ 2 v,0) )] v M 0 X X M 0 v σvtrace X 2 M 0 X. The above score functon suggests a statstc that s vald under group-wse heteroskedastcty. The ntuton of the test s as follows. Suppose that we could vew w = X η + v. Then, under the null hypothess, Σ η = 0 so that η = 0, and we have Ew M 0 X X M 0 w X ) = traceev v M 0 X X M 0 X ) = traceσ 2 M 0 X X M 0 ) = traceσ 2 X M 0 X ). Hence, the terms n the sum have mean zero under the null and we can apply a central lmt theorem. Under the alternatve, w = X η + v so that ] Ew M 0 X X M 0 w X ) = E η X + v )M 0 X X M 0 X η + v ) X = Ev M 0 X X M 0 v X ) + Eη X M 0 X X M 0 X η X ) = traceσ 2 X M 0 X ) + traceeη η X M 0 X X M 0 X X ) = traceσ 2 X M 0 X ) + traceσ η X M 0 X X M 0 X ) The dstrbuton of the statstc wll be pushed to the rght under the alternatve hypothess.

10 0 Based on ths dscusson, we propose two statstcs that are vald under group-wse heteroskedastscty. Frst, we have ṽ M 0 X X M 0ṽ σ 2trace X M )] 0X CLM s = M 0 ṽ = M 0 y X ˆβF E ) σ clms σ 2 = ṽ M 0ṽ T σ clms 2 = )] 2 ṽ M 0 X X M 0 ṽ σ 2 trace X M 0 X The notaton CLM s denotes a test that can be calculated f T s very small whch s due to the fact that every parameter s estmated usng the fxed effects model. We state the assumptons used to justfy the asymptotc dstrbuton of the CLM s statstc. Assumpton. v are ndependent across and Ev t X) = 0 where X = X..X ). Assumpton 2. The matrx X M 0 X s postve defnte and converges to a non-stochastc postve defnte matrx n the lmt. Assumpton 3. 0 < E v v ) 4 X ) ] = J < and E trace X M 0X ) 4]) = J 2 < for each. Assumpton requres strct exogenety. Assumpton 2 s used to guarantee root- consstency of ˆβ F E. Assumpton 3 ensures that there are no pathologcal cross sectonal unts where v moments of v. = 0 wth probablty one and also requres bounded eghth condtonal Theorem 4.2. Suppose that Assumptons -3 hold, Ev v X ) = σ 2 I T, and that. Then under the null hypothess of Σ η = 0, CLM s d 0, ). The CLM s test can be calculated when there are fewer tme seres observatons than varables, or T k +. However, f T > k +, we can estmate regresson parameters for

11 each cross sectonal unt, say ˆα, ˆβ. The new statstc s defned as ṽ M 0 X X M 0ṽ ˆσ 2trace X M )) 0X CLM g = ˆv = ˆv = y ˆα X ˆβ σ clmg ˆσ 2 = ˆv ˆv T k σ clmg = )] 2 ṽ M 0 X X M 0 ṽ ˆσ 2 trace X M 0 X The above statstc s desgned to have power n cases that CLM s may not. We wll dscuss the power dfferences n Secton 6. Assumpton 4. The matrx X M 0X s postve defnte for each. Theorem 4.3. Suppose that Assumptons -4 hold, Ev v X ) = σ 2 I T, and that. Then under the null hypothess of Σ η = 0, CLM g d 0, ). 5. Adjustments for General Heteroskedastcty Both versons of the CLM test derve ther asymptotc dstrbuton from the fact that we have subtracted the condtonal mean from each of the terms n the statstcs. That s, each term v M 0X X M 0v s centered by ts expectaton condtonal of X. If there s no general heteroskedastcty only group-wse heteroskedastcty), we appeal to Ev v X ) = σ 2I T. The centerng term wll change wth dfferent forms of heteroskedastcty. Suppose that we have a more general form of heteroskedastcty so that Ev v X ) = Ω wth σ σ Ω = σt 2 Ths means that each term n the statstc wll agan have a more complcated centerng term. In partcular, we have Ev M 0 X X M 0 v X ) = tracex M 0 Ω M 0 X )

12 2 whch requres an estmate of X M 0Ω M 0 X. Ths type of term s exactly of the form estmated usng Whte s 980) heteroskedastcty consstent covarance estmator. However, Whte s matrx n ths case would requre that T, a condton that we wsh to avod snce we have mantaned a fxed value of T. In addton, Whte s matrx s known to be a based estmator and we want a condtonally unbased estmator of the centerng terms. However, there s a condtonally unbased estmator for Ω suggested n Rao 970) whch s known as the MIQUE estmator. Denote M as the resdual maker for regressng on X and a constant for each, let e j be a T vector wth a one n the jth entry, and let W T = vece e ),, vece T e T )).2 Then the MIQUE estmator appled to each cross sectonal unt would be of the form ˆΩ = W T = W T I M M ) ˆv ˆv ) ]) ] )] I WT M M )W T W T M M )v v ) Takng condtonal expectatons and usng Ev v X ) = vecω ), we have EˆΩ X ) = WT = WT = Ω ] )] I WT M M )W T W T M M )vecω ) ] σ 2 I WT M M )W T W T M M )W T. σ 2 T An unbased estmator of Ev M 0X X M 0v X ) s then tracex M 0 ˆΩ M 0 X ) The test that allows general heteroskedastcty s gven by CLM h = ṽ M 0X X M 0ṽ trace X M 0 ˆΩ )] M 0 X σ lmh 2 See Magnus and eudecker 999) pg. 62 for the relatonshp between Hadamard multplcaton and the Kronecker product usng the matrx W T.

13 3 where ˆv = y ˆα X ˆβ ˆσ 2 = T k ˆv ˆv σ lmh 2 = )] 2 ṽ M 0 X X M 0 ṽ trace X M 0 ˆΩ M 0 X In order to calculate the new CLM h statstc, we need to be able to nvert M M for each. The followng assumpton s suffcent for the nverses to exst. Assumpton 5. The dagonal elements of X X X ) X are all less than 0.5 for each. Assumpton 3 s smlar to the usual Lndeberg condton for regresson models where t s assumed that each element on the dagonal of the matrx goes to zero wth ncreasng sample sze. Theorem 5.. Suppose that Assumptons -5 hold, Ev v X ) = Ω, and that. Then under the null hypothess of Σ η = 0, CLM h d 0, ). 6. Local Power In ths secton, we derve the local power expressons for each of the new tests. We make the followng assumpton concernng the nature of the local alternatve. Assumpton 6. Let η be ndependent of v and Eη η X ) = Σ η = Eη X ) = 0 Eη η η η X ) < J 3 / for some fnte J 3 and Ψ postve sem-defnte. Ψ /2 Ths assumpton lets the alternatve converge to zero at rate /2 and restrcts the parameter varaton to be uncorrelated wth X. 3 An dfferent way to vew the alternatve 3 We can obtan smlar but more complcated results for local power f η s correlated wth X but the results are less transparent and requre several new defntons wthout any addtonal nsghts to local power.

14 4 hypothess s that each slope parameter vares from the populaton mean by a factor proportonal to /4. Theorem 6.. Suppose that Assumptons -3 and 6 hold. Then ) d δs CLM s, σ clms where δ s = lm σclms 2 = lm E tracex M 0 X ΨX M 0 X ) ] T traceψx M 0 X )tracex M 0 X ) E v M 0 X X M 0 M ) ] 2 0 T tracex M 0 X ) v For the cases where T k +, we can estmate slope parameters for every cross sectonal unt. In these cases, gven Assumptons -6, local power takes a dfferent form. ) d δg CLM g, σ clmg δ g = lm σclmg 2 = lm δ h = lm σclmh 2 = lm ] E tracex M 0 X ΨX M 0 X ) E v ) ] M 0 X X M 2 M 0 T k tracex M 0 X ) v ) d δh CLM h, σ clmh ] E tracex M 0 X ΨX M 0 X ) ] 2 E v M 0 X X M 0 v tracex M 0 ˆΩ M 0 X ) The terms δ g and δ h are postve for nonzero p.s.d. Ψ. However, δ s could be zero or negatve for certan alternatves. Ths s the drawback of havng T < k Monte Carlo We compare the three versons of our test wth the test of Pesaran and Yamagata 2008) whch we denote PY n the tables and graphs. To begn, we examne the sze of our proposed tests. All of the statstcs have a lmtng normal dstrbuton based on the applcaton of

15 a central lmt theorem. To examne the accuracy of the lmtng normal dstrbuton, we nclude errors generated from a normal dstrbuton as well as errors from a t-dstrbuton wth fve degrees of freedom and a centered ch-square dstrbuton wth four degrees of freedom. The model s gven as y t = α + x tβ + v t where x t s a 5 vector, α s drawn from a 0, ) dstrbuton. In addton, we construct x t = α + ω t wth ω t 0, ) so that α and x t are correlated. The varances of the errors are gven by Ev 2 t ) = σ2 wth σ2 U, 2). Sample szes are T = 7, 0, 20, 30, 50, 00 and = 30, 50, 00, 200. For sze, we let β = β for all. For each experment, we perform 5000 replcatons. The sze of all tests are lsted n Tables -3. The frst thng to note s that sze s very good for all tests and all error dstrbutons. The CLM g verson of the test s slghtly overszed for T = 7 but dsappears very quckly as T ncreases. The CLM h test does not appear n tables when T = 7. Ths s due to the volaton of Assumpton 6 when T s small relatve to the dmenson of x t. For power comparsons, we let the β parameter vary across the dfferent cross-sectonal unts. In partcular, we have { β for =,..., /2 β = β + for = /2 +,..., c /4 T /2 so that the power s ndexed by the parameter c. The power graphs appear n Fgures 2-5. We note that when T = 20, the power of all three tests appears to be very smlar. As T decreases, the power for our CLM tests begns to domnate the PY test snce our tests do not depend on T. For T = 7, the CLM tests have much hgher power. The only test that can be calculated when T k + s the CLM s test. 4 5 The power appears n Fgure 5. otce that power s stll very good and mproves as T ncreases. 7.. Heteroskedastcty. The CLM h test that employs Rao s 970) MIQUE correcton s the only test consdered that allows for general heteroskedastcty beyond smple group-wse heteroskedastcty where σ 2 s dfferent for each. We examne the effects of heteroskedastcty on all of the tests by employng the same Monte Carlo desgn but wth 4 The sze of the test s below 5% for all cases. We do not report the sze table here snce there are no other tests to compare wth the CLM s.

16 6 Table. Sze wth ormal Errors P Y CLM s CLM g CLM h = A T = 7 = A = A = A = T = 0 = = = = T = 20 = = = = T = 30 = = = = T = 50 = = = = T = 00 = = = v t = σ ɛ t x t. The results appear n Table 4. otce that the CLM h test has good sze propertes for the values of T and consdered. However, all of the the other tests have very poor sze propertes under ths form of heteroskedastcty. In fact, the sze dstortons appear to get worse as we ncrease both and T. For example, when = 200 and T = 00 sze of the PY test s 98.06%.

17 7 Table 2. Sze wth t5 errors P Y CLM s CLM g CLM h = A T = 7 = A = A = A = T = 0 = = = = T = 20 = = = = T = 30 = = = = T = 50 = = = = T = 00 = = = We provde a graph of power for the CLM h test n Fgure 6. The power functon appears to drop off as T decreases but s stll very good. Although we do not provde a fgure here, the CLM h test performs very smlarly to the other tests when there s no heteroskedastcty and T s larger than 0. Ths suggests that the power costs of usng the CLM h test s mnor f there s not heteroskedastcty, but the benefts n terms of sze are very large f there s heteroskedastcty.

18 8 Table 3. Sze wth Ch-square errors P Y CLM s CLM g CLM h = A T = 7 = A = A = A = T = 0 = = = = T = 20 = = = = T = 30 = = = = T = 50 = = = = T = 00 = = = Emprcal Example Glck and Rose 2002) estmate a gravty model of nternatonal trade. Usng,78 country trade pars, they estmate the followng model: lny t ) = α + β CU t + β 2 lngdp prod t ) + β 3 lngdp P Cprod t ) + v t

19 9 Table 4. Sze wth Heteroskedastcty P Y CLM s CLM g CLM h = T = 0 = = = = T = 20 = = = = T = 30 = = = = T = 50 = = = = T = 00 = = = where s and ndex for each of the trade pars, Y t s value of blateral trade at tme t for par, GDP prod t s the product of real gdp for the country pars, and GDP P Cprod t s the per captal product. The varable of nterest s CU t whch ndcates f the par of countres s n the same currency unon at tme t. The data ncludes 27 countres and the tme seres observatons cover varous perods from 948 to 997. Glck and Rose 2002) estmate the model usng several technques, ncludng fxed effects estmaton. They obtan a fxed effects estmate of ˆβ = 0.65 wth a standard error of Such a fndng ndcates that jonng a currency unon has a large effect on the volume of trade between two countres.

20 20 We wsh to test whether the effect of a currency unon s the same for each country par. Frst apply the CLM s test to the entre data set. We fnd CLM s = so that the null hypothess of constant coeffcents over s rejected. ow we want to allow for group-wse heteroskedastcty as well as general heteroskedastcty. It turns out that t s not possble to calculate Pesaran and Yamagata s 2008) test or the CLM g and CLM h tests wth the full data set because most of the country pars have no varaton n the varable CU t. Ths means that we can t estmate a β coeffcent for every possble wth the full data. We then use only the data for each were T 20 and CU t has some varance n the sample. By reducng the data n ths manner, t s possble to estmate β for each. Ths substantally reduces the data to only 03 possble pars where β can be estmated. We examne the senstvty of the fxed effects estmator to the smaller sample. For fxed effects estmaton, we obtan ˆβ = 0.57 wth a standard error of usng the reduced sample. Evdently, the full sample result seems to be drven by ths reduced sample as the estmates are very smlar. Usng the test statstcs developed n ths paper gves CLM s = 4.03, CLM g = 4.09, and CLM h = 4.0 so that the null hypothess of constant coeffcents s rejected regardless of whch heteroskedastcty robust test we use. The estmate β are shown n Fgure 9. Some summary statstcs for the estmated values are a mean of , a medan of and a standard devaton of.86. Our CLM statstcs show that the model that mposes the same regresson coeffcents over each s rejected by the data. The fgure confrms that the estmated values of β vary wdely, and usng fxed effects estmaton as a summary measure of the effectveness of currency unons s a generalzaton that may be msleadng when tryng to predct the effect of a country jonng a currency unon. 9. Concluson Slope heterogenety creates three potental problems n panel data models. Frst, the popular fxed effects estmator may be nconsstent f there s slope heterogenety. The estmator wll converge to a weghted average of the actual slopes whch may not relate to the populaton average slope. Ths means that fxed effects estmaton may not be estmatng any meanngful quantty f there s slope heterogenety. Second, even f we do fnd an estmate of the average slope, s that a useful quantty to know? For example,

21 f we fnd that the average affect for a varable s zero, does that mean that the effect s zero? It could well be that the slope s postve for some cross sectonal unts and negatve for others. If so, we could fnd that a varable s not sgnfcant when we use fxed effects estmaton even though t s truly non-zero for every cross sectonal unt and we would have easly dscovered ths by gnorng the panel and estmatng a coeffcent for each unt. Fnally, n the one-way panel data model wth ndvdual effects ntercepts), the problem of gnorng ndvdual effects wll gve ncorrect standard errors. The same problem happens f one gnores heterogeneous coeffcents as the omtted varables become part of the error term. If one rejects usng a test for slope heterogenety, there are several possble remedes. Frst, one could smply model each slope as dfferent for each cross sectonal unt. Alternatvely, t s possble to model the heterogenety n the slope as a functon of other varables. These models are referred to as herarchcal models. Fnally, t s also possble to specfy partally heterogeneous models where some varables share a common slope and others are allowed to be heterogeneous. Examples of ths type of model were consdered n Polachek and Km 994). We have developed tests for slope heterogenety n fxed effects models based on a condtonal Lagrange multpler test. The advantage of our tests s that they do not depend on T. Moreover, we are able to adjust our tests for general forms of heteroskedastcty. A Monte Carlo experment confrms that our tests are partcularly effectve f T s small or f there s any heteroskedastcty. In an emprcal example for currency unons, we show that the fxed effects estmator wth the same coeffcents for each country par s rejected by the data. 2

22 22 Fgure 2 Power: T=20, = PY CLMs CLMg 60 Power c

23 23 Fgure 3 Power: T=0, = PY CLMs CLMg 60 Power c

24 24 Fgure 4 Power: T=7, =00 00 Power PY CLMs CLMg c

25 25 Fgure 5 Power for LM Test wth k+) Τ.0 Power T=6 T=5 T=4 T= c

26 26 Fgure 6 Power for LM h Test Heteroskedastc Errors T=50 T=30 T=20 T=2 0.6 Power c

27 27 Fgure 7 Densty of Estmated Currency Unon Coeffcents β

28 28 Proof of Theorem 4. l C β, σ 2 v, Σ η ) = Appendx A. Proofs = 2 2 ln Σ ] 2 lnι T Σ ι T ) w Σ w w ) ι T ι T Σ Σ ι T Σ ι T A + B + C + D ) We take dervatves wth respect to γ for each of the terms A, B, C, and D. A γ = ln Σ vecσ ) Σ η vecσ ) vecσ η ) γ = vecσ ) X X ) veci k ) w B γ = lnι T Σ ι T ) vecσ ) vecσ ) vecσ η ) vecσ ) vecσ ) vecσ η ) γ = ι ι T Σ T Σ ι T Σ )X X )veci k ) ι T ow for D, let C γ = w Σ w vecσ ) vecσ ) Σ η vecσ ) vecσ ) vecσ η ) γ = w w )Σ Σ )X X )veci k ) I T F γ) = ι T Σ ι T Gγ) = Σ ι T ι T Σ Usng Magnus and eudecker 999), we have vec F γ)gγ)) γ = Gγ) vecf γ)) I T ) + I T F γ)) vecgγ)) γ γ

29 29 In addton, we have and vecgγ)) γ vecf γ)) γ = veci T ) ι T Σ ι T ) vecσ = ι T Σ ι T ) 2 ι T Σ = vecσ ι T ι T Σ ) vecσ vecσ ) γ = ι T ι T Σ ) γ vecσ ) I T + I T Σ ι T ι T ) ι T Σ )X X )veci k ) ) Σ Σ )X X )veci k ) Usng these results, we have D γ = w + + Σ ι T ι T Σ w Σ ι T ι T Σ w Σ ι T Σ ι T w Σ ι T Σ ι T ) ι w T Σ w Σ ι T ι T Σ ι T Σ Combnng the terms gves us l C β, σv, 2 Σ η ) = vecσ ) ι T Σ A.) γ 2 A.2) A.3) A.4) A.5) w Σ w Σ ) ) ι T Σ ι T ) 2 ] X X )veci k ) X X )veci k ) ι T Σ ι T Σ ι T X X )veci k ) ) ) w Σ X X )veci k ) ) ι ι T ι T Σ w T Σ w Σ ι T ι T Σ w Σ ι T Σ ι T w Σ ι T Σ ι T w Σ ι T ι T Σ ι T Σ ) ) X X )veci k ) ι T Σ ι T ) 2 ] X X )veci k ) X X )veci k ) X X )veci k ) Evaluatng the score at β, σ 2 v, 0) and usng the Magnus and uedecker 999) result traceabcd) = vecd ) C A)vecB), A.) becomes 2σ 2 v tracex M 0 X ).

30 30 Smlarly, A.2) s then ext, A.3) s whch can be also wrtten as Fnally, A.4) and A.5) are and 2σ 4 v 2σ 4 v 2σ 4 v tracev X X v ). tracev M 0 v X M 0 X ) tracev M 0 X X M 0 v ). 2σ 4 v 2σ 4 v tracev X X M 0 v ). tracev M 0 X X v ). respectvely. The score s proportonal to ] v M 0 X X M 0 v σvtracex 2 M 0 X ) Proof of Theorem 4.2: We prove Theorem 4.2 here. Proofs of Theorems 4.3 and 5. are smlar and omtted. The fxed effects resduals wth group means subtracted are gven by So we can wrte A.6) A.7) A.8) A.9) = M 0 ṽ = M 0 y X ˆβF E ) = M 0 ι T α + X β + v X ˆβF E ) = M 0 v + M 0 X β ˆβ F E ) ] ṽ M 0 X X M 0 ṽ σ 2 tracex M 0 X ) + + 2β ˆβ F E ) X M 0 X X M 0 v β ˆβ F E ) X M 0 X X M 0 X β ˆβ F E ) ] v M 0 X X M 0 v σ 2 tracex M 0 X )

31 A.7) and A.8) are O p /2 ) by Assumptons and and the fact that β ˆβ F E ) = O p /2 ). Smlarly, we have σ 2 tracex M 0 X ) = + + = v M 0v T tracex M 0 X ) 2β ˆβ F E ) X M 0 v tracex M 0 X ) β ˆβ F E ) X M 0 X β ˆβ F E )tracex M 0 X ) v M 0v T tracex M 0 X ) + O p /2 ). Combnng these results and usng multvarate Lndeberg-Feller Central Lmt Theorem gves us ] ṽ M 0 X X M 0 ṽ σ 2 tracex M 0 X ) = v M 0 X X M 0 v v M ] 0v T tracex M 0 X ) + O p /2 ) = 0, σ 2 clms ) + O p /2 ) It remans to show that σ clms 2 p σclms 2. Usng smlar arguments, we have σ clms 2 = v M 0 X X M 0 v v M ) 0v ] 2 T trace X M 0 X + o p ) Let z = = Defne the quanttes v M 0 X X M 0 v v M 0v trace v v ) ] 2 T trace X M 0 X M 0 X X M 0 M ) ])] 2 0 T trace X M 0 X 3 Z = v v Z 2 = M 0 X X M 0 M 0 T trace X M 0 X )

32 32 so that Ez 2 ) E tracez = E E ] 2 ] ) 2 Z ) tracez2z 2 ) ] 2 ] ) ] 2 tracezz ) tracez2z 2 ) X Then J J 2 < 2 Ez 2 ) 0 and we can apply Chebychev s Weak Law of Large umbers to obtan σ 2 clms p lm = σ 2 clms E v M 0 X X M 0 M ) ] 2 0 T tracex M 0 X ) v Proof of Theorem 6.: We proove the theorem for CLMg and the other cases are smlar. Under the alternatve hypothess, the fxed effects resduals wth group means subtracted are gven by M 0 ṽ = M 0 y X ˆβF E ) In addton, under the alternatve we have ) ˆβ F E = X M 0 X = M 0 ι T α + X β + v X ˆβF E ) = M 0 v + M 0 X η + M 0 X β ˆβ F E ) ) = X M 0 X ) = β + X M 0 X = β + O p /2 ) X M 0 y X M 0 ι T α + X β + X η + v ) X M 0 X η + v ) where the last equalty comes from applyng the Central Lmt Theorem and the fact that Eη X ) = 0. Then

33 33 A.0) A.) A.2) A.3) A.4) A.5) Consder A.3). E = ṽ M 0 X X M 0 ṽ = E = 3/ v M 0 X X M 0 v η X M 0 X X M 0 v ) 2 = E trace η X M 0 X X M 0 X η η X M 0 X X M 0 X β ˆβ F E ) η X M 0 X X M 0 v v M 0 X X M 0 X β ˆβ F E ) β ˆβ F E ) X M 0 X X M 0 X β ˆβ F E ) ) 2 E η X M 0 X X M 0 v Σ η X M 0 X X M 0 v v M 0 X X M 0 X ) X ]) )] E trace ΨX M 0 X X M 0 Ω M 0 X X M 0 X so that A.3) s O p /4 ). The result for A.2) s smlar. The last two terms are O p /2 ). For A.), )] E trace η η X M 0 X X M 0 X = ote that )] E trace X M 0 X ΨX M 0 X )] 2 E trace η η X M 0 X X M 0 X ] ] < E traceη η η η X ) E tracex M 0 X X M 0 X X M 0 X X M 0 X ) < J 3 J 2

34 34 so that 2 )] 2 E trace η η X M 0 X X M 0 X 0 and Chebychev s Weak Law of Large umbers mples η X M 0 X X p )] M 0 X η lm E trace X M 0 X ΨX M 0 X = δ g. The term A.0) s centered approprately by ˆσ 2 tracex M 0M 0 X ) to gve asymptotc normalty. For the varance, σ clmg = )] 2 ṽ M 0 X X M 0 ṽ ˆσ 2 trace X M 0 X we use ṽ M 0 X X M 0 ṽ = v M 0 X X M 0 v + η X M 0 X X M 0 X η + β ˆβ F E ) X M 0 X X M 0 X β ˆβ F E ) + 2v M 0 X X M 0 X η + 2v M 0 X X M 0 X β ˆβ F E ) + 2η X M 0 X X M 0 X β ˆβ F E ) so that σ clmg = p σ 2 clmg )] 2 v M 0 X X M 0 v ˆσ 2 trace X M 0 X + op ) as before.

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