ESTIMATES OF TECHNICAL INEFFICIENCY IN STOCHASTIC FRONTIER MODELS WITH PANEL DATA: GENERALIZED PANEL JACKKNIFE ESTIMATION

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1 ESIMAES OF ECHNICAL INEFFICIENCY IN SOCHASIC FRONIER MODELS WIH PANEL DAA: GENERALIZED PANEL JACKKNIFE ESIMAION Panutat Satchacha Mchgan State Unversty Peter Schmdt Mchgan State Unversty Yonse Unversty Estmates of techncal neffcency based on fxed effects estmaton of the stochastc fronter model wth panel data are based upward. Prevous work has attempted to correct ths bas usng the bootstrap, but n smulatons the bootstrap corrects only part of the bas. he usual panel jackknfe s based on the assumpton that the bas s of order and s smlar to the bootstrap. / We show that when there s a te or a near te for the best frm, the bas s of order, not, and ths calls for a dfferent form of the jackknfe. he generalzed panel jackknfe s qute successful n removng the bas. However, the resultng estmates have a large varance. July 6, 009

2 . INRODUCION In ths paper we consder the stochastc fronter model wth tme-nvarant techncal neffcency n a panel data settng. hs model was frst consdered by Ptt and Lee 98), who estmated the model by MLE gven a dstrbutonal assumpton for techncal neffcency. Wthout such a dstrbutonal assumpton, Schmdt and Sckles 984) proposed fxed effects estmaton. In ths approach, the fronter ntercept s estmated as the maxmum of the estmated frm-specfc ntercepts, and a frm s level of neffcency s measured by the dfference between the fronter ntercept and the frm s ntercept. It s well understood that the max operaton causes the estmated fronter ntercept, and therefore the estmated neffcency levels, to be based upward. Schmdt and Sckles 984), Park and Smar 994) and Km, Km and Schmdt 007) dscuss ths problem. Hall, Härdle and Smar 995) show that the bootstrap s asymptotcally as wth N fxed) vald n ths settng, provded that there s a unque best frm no te for the largest populaton ntercept), and Km, Km and Schmdt 007) use the bootstrap to construct a bas-corrected estmate of the fronter ntercept and therefore of neffcency levels). he bootstrap s used to estmate the bas, whch s then subtracted from the orgnal estmate. In ther smulatons, Km, Km and Schmdt 007) found that the bas correcton was partally successful. It removed some but not all of the bas. Often t seemed to remove about half of the bas. Why t removed half of the bas, as opposed to some other fracton, s an nterestng puzzle. In ths paper we consder bas correctons based on the jackknfe. here are two motvatons for dong so. Our frst motvaton s that the jackknfe s thought to be smlar to the bootstrap, but t s analytcally smpler. herefore we use the jackknfe to explan why t s that under certan crcumstances we remove half of the bas, and so we at least partally resolve the

3 puzzle of the prevous paragraph. he second motvaton s to nvestgate whether the jackknfe s practcally useful as a bas-reducton technque n ths model. Here we are less successful, because the jackknfe does effectvely remove the bas of the estmate, but the varance and MSE of the jackknfe estmate are unfortunately rather large. he approprate form the jackknfe depends on the order of the leadng term n an expanson of the bas of the estmate. If the bas of the estmate s of order, the usual delete-one panel jackknfe estmator as n Hahn and Newey 004)) should remove the bas. However, ntutvely we would expect the jackknfe bas correcton to be smlar to the bootstrap bas correcton, whch was only partally successful. hus t would seem that the fnte-sample relevance of the bas beng of order may be questonable. In ths paper we analyze the case of an exact te for the best frm. In ths case the bootstrap s not asymptotcally vald. Furthermore, we show that the bas of the fxed effects estmate of the fronter ntercept s of order /, not. In ths case the usual delete-one panel jackknfe does not properly remove the bas. Indeed, we show that t removes approxmately) half of the bas. A dfferent form of the jackknfe, whch we call the generalzed panel jackknfe, does remove the bas n ths case. In the smulatons of Km, Km and Schmdt 007) there was not an exact te, and an exact te may also be unlkely n actual data. However, f there s nearly a te, n the sense that there s substantal uncertanty ex post about whch s the best frm, t s not clear whether asymptotcs that assume no te are more relevant than asymptotcs that assume an exact te. In order to further analyze a near te, we gve a specfc defnton nvolvng a local parameterzaton) of a near te, and we show that the bas s agan of order to successfully remove the bas. /, so that the generalzed panel jackknfe s needed 3

4 We then perform smulatons to assess the fnte-sample relevance of these results. he plan of the paper s as follows. In Secton, we defne some notaton and gve a bref revew of fxed effects estmaton of the stochastc fronter model wth panel data. In Secton 3 we show that the bas s of order / for the case of an exact te or a near te. Secton 4 descrbes the generalzed panel jackknfe that s approprate n ths crcumstance. In Secton 5 we explan the desgn of our Monte Carlo experments, and Secton 6 gves ts results. Fnally, Secton 7 contans our concludng remarks.. FIXED EFFECS ESIMAION OF HE MODEL Consder a sngle-output producton functon wth tme-nvarant techncal neffcency u 0. here are N frms, ndexed by,..., N, over tme perods, ndexed by t,...,. We consder the lnear regresson model of Schmdt and Sckles 984): y α + x β + v u,,..., N; t,...,, ) t t t where y t s the logarthm of output for frm at tme t ; x t s a vector of K nputs e.g., n logarthms for a Cobb-Douglas producton functon); β s a K vector of coeffcents; and v t s an..d. dosyncratc error wth mean zero and fnte varance. he v t represent uncontrollable shocks that affect the level of output, e.g., luck, weather, or machne performance. he tme-nvarant techncal neffcency u satsfes u 0 for all and u > 0 for some. here s no dstrbutonal assumpton on u except that t s one-sded. Defnng α α u, we can wrte ) as a standard panel data model: y α + x β + v. ) t t t 4

5 Obvously, α α snce u 0. When α and u ) s treatng as fxed, ) leads to a fxed effects estmaton problem n whch nether a dstrbuton for techncal neffcency nor the ndependence between techncal neffcency and x t or v t or both) s needed. We assume strct exogenety of the regressors x t n the sense that x,..., x ) s ndependent of v,..., v ). here s no restrcton on the dstrbuton of v t other than zero mean and fnte varance. o estmate β, we use the fxed effects estmate βˆ, whch can be estmated as least squares wth dummy varables, by regressng y t on x t and a set of N dummy varables, or as the wthn estmator, by regressng y y ) on x x ). Gven the estmate βˆ, the estmates t t can be recovered as the averages of the frm-specfc resduals,.e., ˆ α y x ˆ β where y t y t and x t x t, or equvalently as the coeffcents of the frm-specfc dummy varables. he wthn estmator βˆ s consstent as N or, and the frm-specfc ntercepts are consstent as. o estmate α and u, Schmdt and Sckles 984) suggested the followng estmators: ˆ α max ˆ α uˆ j, j,..., N ˆ α ˆ α,,..., N. 3) Park and Smar 994) show that these estmates are consstent as N,, and / ln N) 0. In ths paper, to mantan the connecton to the earler lterature on bootstrappng of ths model, and also the lterature on the jackknfe, we wll consder asymptotc arguments as wth N fxed. In ths case we can only measure neffcency relatve to the best of the N frms. 5

6 For ease of presentaton, we follow Km, Km and Schmdt 007) and rank the ntercepts α such that α α )... α N ), so that N) ndexes the frm wth the largest value of α among the N frms, whch we wll call the best frm. Smlarly, we rank the levels of techncal neffcency u n the opposte order such that )... N ) u u u. Obvously, α α u ) for all and specfcally α N ) α u N ). Now we defne the relatve neffcency measures u u u α α. 4) N ) N ) hese are the focus of ths paper snce, as wth N fxed, s a consstent estmate of α N ), not α, and u ˆ s a consstent estmate of u, not u. Although s consstent for α N ) as wth N fxed), t s based upward for fnte. hs s true because ˆ α ˆ α N ) and E ˆ α N ) ) α N ). hat s, the max operator n 3) nduces an upward bas: the largest s more lkely to contan postve estmaton error than negatve error. he upward bas n the estmate nduces an upward bas n the estmates of relatve techncal neffcency. hat s, E ˆ) α α N ) E u ) u ˆ. herefore we wll smply evaluate the bas of as an estmate of α N ) ; there s no need to separately evaluate the bas of the estmates of relatve techncal neffcency. he bas of as an estmate of α N ) corresponds to what Km, Km and Schmdt 007) call the frst-level bas. o correct ths frst-level bas, Km, Km and Schmdt 007) consder a bootstrap bas correcton for the fxed effects estmate. hey evaluate the second-level bas, boot E ˆ α ) ˆ α, and use t to correct the frst-level bas. hat s, f the second-level bas equals the frst-level bas, we would want to evaluate 6

7 boot boot ˆ α [ E ˆ α ) ˆ] α ˆ α E ˆ α ). 5) he feasble verson of ths s B b boot b) ˆ α ˆ ˆ BC α B α, 6) where b represents a sngle bootstrap replcaton and B s the total number of bootstrap replcatons. In ther smulatons see ther able 4), ths estmate removes some but not all of the bas n. Often t seems to remove about half of the bas. As noted n the ntroducton, the fact that half of the bas s removed s the puzzle that at least partally motvated our nterest n the jackknfe. 3. DERIVING HE ORDER IN PROBABILIY OF HE BIAS In ths secton, we show that the bas of s of order f there s no te for the best frm; that s, f α N ) s strctly larger than all of the other α. However, f there s a te for the best frm, or f there s a near te n a sense defned precsely below), the bas s of order For smplcty, we wll dscuss the smple case of no regressors: y /. α + v,,..., N; t,...,, 7) t t where v t are..d. wth mean zero and varance σ. hus y. he varous are ndependent and ˆ α α ) N0, σ ). However, the ncluson of regressors would not alter our results snce the wthn estmator of β s unbased, and our results really only requre that the vector whose th element s α ) ˆ α s normal wth mean zero and fnte varance matrx. See Hall, Härdle and Smar 995), Appendx ), equaton A.) for ths condton, whch would stll hold wth regressors. 7

8 3. he Case of No e that α α Suppose frst that there s no te for the best frm. hat s, there s a unque frm such N ). Hall, Härdle and Smar 995) show the equvalence of ) there s no te for the best frm, and ) the asymptotc dstrbuton of s normal. More precsely, they show that f there s no te, P ˆ α ˆ α ) as, so that the asymptotc dstrbuton of s the same as the N ) ˆ N asymptotc dstrbuton of α, the estmate of α N ) that would be used f the dentty of the best frm were known. Snce N ) s unbased, t follows that tmes the bas of must go to zero as. hus we conclude that the bas of s of an order smaller than /. We presume that t s of order. 3. he Case of an Exact e Suppose now that there s a te for the best frm the largest α ). Specfcally suppose that the frst k frms are ted, so that α N ) α α... α k for k N. Agan the dscusson n Hall, Härdle and Smar 995, Appendx )) apples. Wth a probablty that approaches one as, wll equal for some wth k, that s, the estmated best frm wll be one of the k truly best frms. herefore wth a probablty that approaches one, ˆ α α N ) ) max ˆ α α max{ ˆ α α N ) N ), ˆ α α ), N ) ˆ α α,..., ˆ α α k N ) ),..., N ) ) ˆ α k α N ) )} 8) 8

9 and therefore ˆ α N ) Z where Z s the maxmum of a set of k normals wth zero mean. α For k >, Z s not normal, and E Z) > 0. he bas of s therefore, for large, / E Z), whch s of order /. We can gve an explct expresson for the case of N k and the smple model above wth no regressors). We frst state the followng Lemma. Lemma Suppose X and X are..d. N μ,σ ),.e., X X ~ μ σ N, μ 0 0, σ then E [max X, X ) μ ] π ) σ. 9) Proof. Let Y Z X X X ~ μ σ N, 0 σ σ. σ So, ρ σ σ σ and E X X > X ) E Y Z > 0) μ + ) σλ0), where λ ) s the normal hazard functon μ + ) σ π ), snce λ0) φ0) Φ0)) π μ + π ) σ. Hence, E X X > X ) μ π ) σ and 9

10 E[max X, X )] E X X > X ) + E X X > X ), by symmetry E X X > X ), snce X and X are..d. herefore, bas E[max X, X )] μ π ) σ. In the present settng, X and X are ˆα and ˆα, μ α α, the varance s σ, and the bas of α max ˆ α, ˆ α ) equals π ) / σ. Clearly, ths s proportonal to /. ˆ σ 3.3 he Case of a Near e In the prevous sectons we saw that the bas of s of order best frm, whle t s of order f there s no te for the / f there s an exact te. It s not clear how relevant ether set of results wll be n fnte samples f there s n some sense) nearly a te. Intutvely that wll depend on how close we are to a te, whch depends not only on how close the α are to each other, but also on / σ, whch s the standard devaton of the. One way to model ths s by a local to te parameterzaton. So, to keep thngs smple, let N, α > α, and α / α c for c > 0, where c does not depend on. hen n our smple no regressors) model, ˆ α α ) N0, ). Also ˆ α α ) N0, ) and so σ / ˆ α α + c) N0, σ ), or ˆ α α) N c, σ ). hen σ [max ˆ α ˆ ) ] max[ ˆ ), ˆ, α α α α α α)] Z 0) 0

11 where Z s the max of a N 0, σ ) random varable and a N c, σ ) random varable. Clearly E Z) E N0, σ )) 0 and the bas of s agan for large ) / E Z), whch s of order /. A smlar analyss apples f α α γ c where c > 0 and γ. he value of c matters as above) when γ but t does not affect the lmt dstrbuton f γ >. So the asymptotcs for the case of a near te are very smlar to those for an exact te f a te s near enough. Once agan we can gve an explct expresson for the case of N k and the smple model no regressors). We state wthout proof the followng Lemma. Lemma Let X and X be ndependent normals, where X ~ N0, σ ) and X ~ N μ, σ ). hen where μ μ σ. E [max X, X )] [ Φ μ ) μ + φ μ )] σ, ) o apply ths to our model, X and X are ˆα and ˆα, σ s σ, μ / c, and / μ c / σ c. σ So the bas s whch s ndeed proportonal to / bas [ Φ c σ ) c σ ) + φ c σ )] σ, ) /. 4. CORRECING BIAS WIH HE PANEL JACKKNIFE AND HE GENERALIZED PANEL JACKKNIFE

12 4. he Panel Jackknfe Jackknfe estmaton s an automatc bas reducton tool under the assumpton of the exstence of a seres expanson for the bas of an estmator. Quenoulle 956) and ukey 958) show that usng the jackknfe estmates based on removng data and then recalculatng the estmator removes the frst order bas from an ntal estmator. For a basc background dscusson of jackknfe estmaton, see Mller 974). o descrbe the jackknfe n a general settng, let the data be ndexed by t,,...,. Let ˆ be the estmator based on all observatons, and let be the delete-observaton-t estmator that omts observaton t and uses the other observatons. hen the jackknfe estmator s hs estmator s sad to remove the bas of order hen So f the bas s of order. ˆ t ˆ J ˆ) ) ˆ. 3) t t ), n the followng sense. Suppose that 3 E ˆ) + B + D + O ). 4) E[ J ˆ)] + D + O ) + O ). 5), n the sense that 4) holds, the jackknfe leaves only the bas of order Hahn and Kuerstener 004), Hahn and Newey 004), and Fernández-Val and Vella 007) apply the jackknfe to nonlnear panel data models and dynamc panel data models. In the panel data settng, even though there are really N observatons, we treat the number of observatons n 3) as, and to calculate we delete the ˆ t th t perod observaton for each

13 cross-sectonal unt. hs s done because, n the models they consder, the bas s of order We refer to ths procedure as the panel jackknfe. Other smlar versons of the jackknfe can remove bas of order. For example, Dhaene, Jochmans and huysbaert 006) propose the half-panel jackknfe estmator:.) ˆ panel J half ) ˆ ˆ ˆ + ), 6) where ˆ s the fxed effects estmator based on the full sample; ˆ and ˆ are based on the frst- and second- halves of the panel sample, where each half-panel conssts of consecutve observatons over tme for all cross-sectonal unts. hey show that the half-panel jackknfe estmator also removes the bas of order that the bas s of order from the orgnal estmator. However, for the case, we wll consder only the standard panel jackknfe as descrbed above. It s obvous that when there s no te, the panel jackknfe wll remove the frst-level bas of the estmate of α N ) hence, the bas of the estmates of relatve techncal neffcency the bas s of order estmator that can handle bas of order generalzed jackknfe. u ) snce. For the cases of an exact te and a near te, however, we need a jackknfe /. he dfference n the order of the bas leads us to the 4. he Generalzed Jackknfe Schucany, Gray and Owen 97) were the frst to propose a jackknfe estmator that can handle a more general form of bas. It was not untl later that Gray and Schucany 97) gave t the name generalzed jackknfe. Gray and Schucany 97) defne the generalzed jackknfe as the followng. 3

14 Defnton Gray and Schucany 97) s Defnton.. Let ˆ and ˆ be two estmators for. hen, for any real number R, the generalzed jackknfe estmator G ˆ, ˆ ) s defned as ˆ ˆ ˆ ˆ R G, ). 7) R he usual Quenoulle) jackknfe corresponds to ˆ ˆ, ˆ ˆ t t), and R ). If we can express the bas of the estmators n terms of the sample sze and the true parameter, we can choose R so that the generalzed jackknfe s unbased. heorem Gray and Schucany 97) s heorem.. If the bas of the estmators ˆ and ˆ can be expressed as E ˆ ) + b k k, ), k, ; b, ) 0; and b, ) R b, ), then E G ˆ, ˆ )] [. 4

15 5 Proof. )., ), snce, ), ), )], [ )], [ )] ˆ, ˆ [ b Rb R Rb b R b R b G E In general, we do not have a bas expresson of the form of the prevous theorem, but we have a seres expanson of the bas wth a leadng term of known order. hen the generalzed jackknfe removes the leadng term of the seres expanson of the bas. heorem Gray and Schucany 97) s heorem.. If the bas of the estmators ˆ and ˆ can be expanded as an nfnte seres:, ),, ) ˆ + k b E k k and, ), ), b b R then. ), ), )] ˆ, ˆ [ R b R b G E + Proof. Smlar to the proof of heorem.

16 / 4.3 he Generalzed Panel Jackknfe When the Bas Is of Order We are specfcally nterested n the case that the bas of ˆ s of order the followng expanson holds: As before, we let ˆ ˆ observatons) and ˆ ˆ s equal to and the generalzed jackknfe s /. Suppose that / 3 / E ˆ) + B + D + O ). 8) t t). hen the weght R n heorem R B ) B ) 9). 0) G ˆ) ˆ ˆ t t) It s then easy to verfy that the bas of G ˆ ) s of order ˆ has been removed. ; that s, the / term n the bas of In the panel data case, once agan we treat the number of observatons as, and s calculated by deletng the ths the generalzed panel jackknfe. th t tme perod observaton for each cross-sectonal unt. We wll call he generalzed jackknfe removes bas more aggressvely than the usual jackknfe, n the ˆ t sense that the weghts attached to ˆ and to ˆ t t) are larger. For example, for 0 we have J ˆ) 0 ˆ 9 G ˆ) 9.5 ˆ 8.5 ˆ t t) ) ˆ t t) ). 6

17 Smlarly for 50 we have J ˆ) 50 ˆ 49 G ˆ) 99.5 ˆ 98.5 A detal that we do not pursue n ths paper s that t s possble to consder a second level ˆ t t) ) ˆ t t) ). of the jackknfe. If the bas of the orgnal estmate s of order /, the bas of ) G ˆ s of order. he usual panel jackknfe appled to the estmator ) G ˆ would remove the bas of order. he resultng estmator would be a lnear combnaton of the orgnal estmate, the varous drop one observaton estmates, and the varous drop two observatons estmates. 4.4 What If he Wrong Jackknfe Is Used? We have seen that the usual panel jackknfe s approprate when the bas s of order whereas the generalzed panel jackknfe s approprate when the bas s of order the queston of what happens f the wrong verson of the jackknfe s used., /. hs rases heorem 3 If the bas of ˆ s of order half of the bas. /, the usual panel jackknfe corrects approxmately Proof. We have E ˆ) + / B + hgher order terms. So, droppng the hgher order terms, we calculate B E[ J ˆ)] + B B +. ) + Comparng the bas n ths expresson to the orgnal bas of / B, we have removed about half of the frst-order bas term. 7

18 8 heorem 3 s our explanaton of the puzzle that the bootstrap and the jackknfe often correct half of the bas. Cases where ths occurs correspond to more or less an exact te. heorem 4 If the bas of ˆ s of order, the bas of the generalzed panel jackknfe s approxmately the negatve of the bas of the orgnal estmate. Proof. Suppose. ˆ) hgher order terms B E + + So, agan droppng the hgher order terms,. ) ] ) [ ] ) [ ] [ ] [ ˆ)] [ / / / / B B B B B B B G E t ) So the bas of ) ˆ G, B ), s approxmately the negatve of the orgnal bas, B. 5. DESIGN OF HE MONE CARLO EXPERIMENS In ths secton, we conduct Monte Carlo smulatons to nvestgate the fnte sample performance of the followng estmators of ) N α : ), the maxmum of the fxed effects

19 estmates; ) J ), the panel jackknfe estmate; ) G ) estmate; and v) boot BC, the bas-corrected bootstrap pont estmate., the generalzed panel jackknfe We are prmarly nterested n the bas of these estmators. However, we wll also report ther varance and mean square error. hese measures are defned precsely later n ths secton. he model s the smple panel data model wth no regressors, as gven n 7). hus, the data generatng process s y t α + v u α + v,,..., N; t,...,, t t 3) where α α u ; the u are..d. half-normal: u U where U ~ N0, σ ) ; and the v t are normal wth mean zero and varance σ v. hese dstrbutonal assumptons are not used n estmaton. hey just characterze the process that generates the data. he set of parameters s { α, σ v, σ u, N, } but ths can be reduced somewhat. All of the results bas, varance, and MSE) are nvarant wth respect to α, so we set t equal to one, wthout loss of generalty. Also, only ratos of varances matter. If we multply both σ u and σ v by a u constant q, the bases of the estmates change by q and the MSE s change by q. So we really only need to consder three parameters: N,, and a relatve varance parameter. Km, Km and Schmdt 007) used the relatve varance parameter γ σ ) [ σ + σ ) ], where u v u σ u ) var u) π ) π ) σ u. We wll use nstead the parameter μ defned by σ ). 4) u μ / σ v hs s not a matter of substance. We use μ because we fnd t easer to nterpret. It measures the 9

20 standard devaton of the α n unts of the standard devaton of the. Also, for reasons gven below, wth ths parameterzaton t turns out that does not matter very much. Only μ and N turn out to be mportant. So, n the end, our parameter space s { μ, N, }. We set scale by settng σ 0., whch for a gven determnes σ v. hen, for a gven μ, σ u ) s determned. We consder / 0,0,,0 / μ, and 0. Wth σ v 0., for a gven value of μ, the values of σ u ) and σ u are as follows: ) 0 μ 0. : σ ) 0. 00; σ ; u ) 0 / μ : σ ) 0. 0; σ ; u 3) μ : σ ) 0. u ; σ u ; 4) 0 / μ 3. 63; σ ) ; σ. 759 ; u 5) μ 0 : σ ) 0 u ; σ u We consder sample szes N,0, 0,50, and 00, and we set 0 u u u v. We also consdered 5,0,50, and 00, and the results for these values of are avalable n a supplementary set of tables, avalable from the authors on request. he basc outcomes that we would expect n the smulatons are as follows. Frst, bas wll be larger when N s larger, but the effect of N on the relatve performance of the varous bas-corrected methods s not obvous. Second, bas wll be larger when μ s smaller, snce then the varablty of the α s smaller relatve to the samplng varablty of the. We mght expect the ordnary panel jackknfe or the bootstrap to be better than the generalzed jackknfe when μ 0

21 s large we are farther from a te), and vce-versa. hrd, condtonal on μ, we do not expect to be very mportant. When we change n our experment, holdng constant μ and σ v, t means that σ v ncreases proportonally to, and σ u ) s unchanged. herefore nether the varablty of the α nor the samplng varablty of the changes. he only reason that should matter s that the jackknfe s weghts on ˆ and ˆ t t) depend on. We consder three dfferent varatons of the setup we have just descrbed. Experment I No e). he setup of ths experment s exactly as just descrbed. here are no restrctons on the α. hey just follow from the draws of the half-normal u. hs setup s very smlar to that of Km, Km and Schmdt 007). Experment II Exact e). We generate data as descrbed above. Now we the data generator) know whch frm s the best and the value α N ) of ts ntercept. We randomly select one of the other N ) frms and set ts ntercept also equal to α N ). herefore we have created an exact two-way te for the best frm. Experment III Near e). We start as n Experment II. However, once we have observed the best frm and α N ), we randomly select one of the other N ) frms and set ts ntercept equal to / α N α N α N ]. 5) [ So, for example, f 0, we have now created a new second-best frm whose ntercept s tmes closer to α N ) than the prevously second-best frm s ntercept. For each confguraton of { μ, N, }, we perform,000 replcatons. Wthn each replcaton, the bas-corrected bootstrap estmate s based on,000 bootstrap replcatons.

22 boot For each of the estmators α, J ˆ), α G ˆ), α ˆ α ) ˆ we calculate bas, varance and mean square error. he parameter beng estmated, α N ), vares across replcatons because of the random draws of the half-normal u that determne α α u. herefore we wll explctly state BC our defnton of bas, varance and MSE. Frst defne: ) NREP number of replcatons; ) r ndex of replcaton, r,..., NREP ; ) r value of α N ) n replcaton r ; v) ˆ r value of ˆ n replcaton r for any of the four estmators lsted above); and v) NREP r r ˆ and NREP ˆ. r r he defnton of bas s straghtforward: bas ˆ) NREP ˆ ) ˆ. r r 6) r hen we defne the mean squared error as MSE ˆ) NREP NREP r r ˆ ) r [ ˆ ) bas ˆ)] r r r + bas ˆ) 7) and the varance as var ˆ) MSE ˆ) bas ˆ) NREP r [ ˆ ) bas ˆ)] r r. 8) 6. RESULS OF HE MONE CARLO EXPERIMENS ables,, and 3 gve the results of Experment I n whch there s no te. All of these results are for 0. able gves the bas of the estmates, whle able gves varance and able 3 gves MSE. In all three tables, column ) gves results for ; column ) gves results for

23 the panel jackknfe J ) ; column 3) gves results for the generalzed panel jackknfe G ) ; and column 4) gves results for the bas-corrected bootstrap pont estmate boot BC. Consder frst able, whch gves the bas of the varous estmates as an estmate of α N ). hs s equvalent to the bas of estmated relatve techncal neffcency u ˆ as an estmate of As expected, the bas of s larger when N s larger the max s taken over more frms) and when μ s smaller we are closer to a te). he panel jackknfe and the bas-corrected bootstrap are less based than the fxed effects estmate. However, they only correct part of the bas. In most cases the jackknfe corrects more of the bas than the bas-corrected bootstrap. he generalzed panel jackknfe overcorrects so the orgnal upward bas now becomes a downward bas). When μ s very small, so that the varablty of the α s very small relatve to the samplng varablty of the, we are n a sense close to a te. In these cases the exact te asymptotcs appear to be relevant: the generalzed panel jackknfe s nearly unbased, and the panel jackknfe and also the bas-corrected bootstrap) corrects about half of the bas, as predcted by heorem 3. Conversely, when μ s large we are far from a te, the panel jackknfe and the bas-corrected bootstrap are nearly unbased, and the downward bas of the generalzed panel jackknfe s almost as large as the upward bas of, as predcted by heorem 4. able gves the varance of the varous estmates. hey are easy to summarze. he varance of the s less than the varance of the bas-corrected bootstrap pont estmate, whch s less than the varance of the panel jackknfe, whch s less than the varance of the generalzed panel jackknfe. he varance of the generalzed panel jackknfe s consderably larger than the varance of the other estmators. o properly nterpret these varances, remember that we are u. 3

24 ultmately nterested n estmatng the relatve sze of the u, whose varance s σ u ), and that n our setup σ ) 0.00, 0.0, 0.,, and 0 for μ 0,0 /,,0 /, and 0, respectvely. u So the varance of these estmators s large enough to be an ssue, except perhaps for the larger values of μ. able 3 gves the MSE of the estmates. In terms of MSE, the two varetes of the jackknfe are domnated by the bas-corrected bootstrap. he bas-corrected bootstrap s also generally better than the fxed effects estmate, except n those cases where the bas of s small.e., when N s small and μ s large). 6. Now we turn to Experment II, the case of an exact te. hese results are n able 4, 5, and In terms of bas, we see n able 4 that the generalzed panel jackknfe s clearly the best. It overcorrects the bas, but not by as much as the panel jackknfe and the bas-corrected bootstrap undercorrect. As expected from heorem 3, the panel jackknfe corrects about half of the bas. he bas-corrected bootstrap, whch s not vald asymptotcally n the case of an exact te, also appears to correct about half of the bas. In able 5, the varances of the estmates are rather smlar to the varances for the case of no te able ). he man dfference s that now the varance does not depend as strongly on μ, presumably because, once we have forced a te, the smlarty of the other α s not of as much mportance. he rankng of the estmators, n order of ncreasng varance, s stll the same as n able, bas-corrected bootstrap, panel jackknfe and generalzed panel jackknfe). In terms of MSE, we see n able 6 that the bas-corrected bootstrap stll domnates both varetes of the jackknfe. It s also generally better than the fxed effects estmate. hs 4

25 favorable performance of the bas-corrected bootstrap s perhaps surprsng, gven that t s not asymptotcally vald n the case of an exact te. Our last experment s Experment III, the case of a near te. he results for ths experment are gven n able 7, 8, and 9. As a general statement, the results are between those of Experment I and Experment II, whch s not surprsng. For small values of μ nearer te), the bas results n able 7 are qute smlar to those of able 4 for an exact te. In these cases the generalzed panel jackknfe has lttle bas, whle the panel jackknfe and the bas-corrected bootstrap correct about half of the bas. For large values of μ less near te), the panel jackknfe and the bas-corrected bootstrap stll correct only some of the bas, but the generalzed panel jackknfe overcorrects. Stll, t s generally true n able 7 that the generalzed panel jackknfe has the smallest bas. In terms of varance able 8) and MSE able 9), the results are farly smlar to those for both the case of no te and the case of an exact te. Once agan the bas-corrected bootstrap s generally the best, and the generalzed panel jackknfe s the worst. he last ssue we consder s the effect of changng. We consder the same three knds of experments as just descrbed, wth 5,0,50, and 00 n addton to 0, whch we have just dscussed). hese results are gven n a set of 36 Supplemental ables, avalable on request from the authors. In ths paper we wll dsplay the results only for μ and N 0. ables 0,, and gve the bas, varance, and MSE of the varous estmates. As dscussed n Secton 5, we do not expect changes n to be very mportant, because we are holdng constant N, μ, and σ v, or equvalently we are holdng constant N, σ u ), and σ v. Indeed, the motvaton for adoptng ths parameterzaton was that we expected t to 5

26 make one of the parameters unmportant. We expect changng to be more mportant for the jackknfe estmates than for the other two estmates, because the value of affects the weghts that the jackknfe puts on the orgnal estmate versus the average of the delete-one-observaton estmates. What we see n ables 0 s not surprsng. In able 0, the effect of changng on the bas of the estmates s very mnor. In able, changng does not affect the varance of the fxed effects estmate or the bas-corrected bootstrap pont estmate very much, but the varance of the jackknfe estmates ncreases notceably as ncreases. Correspondngly, n able the MSE of the jackknfe estmates ncreases as ncreases. However, t remans true that the value of s much less mportant than the values of N and μ n determnng the relatve performance of the varous estmates. 7. CONCLUDING REMARKS In the stochastc fronter model wth panel data, the fxed effects estmate of the fronter ntercept s based upward. Prevous work found that the bas-corrected bootstrap corrected only part of ths bas. hs paper has tred to explan that fndng and to see whether we can more successfully remove the bas usng the jackknfe. he bootstrap s known to be asymptotcally as wth N fxed) vald f there s no te for the best frm, and not vald f there s an exact te. So whether there s a te, and how close we are to havng a te f there s not an exact te, s a reasonable ssue to focus on. When there s an exact te, we show that the bas of the fxed effects estmate s of order / rather than. Not only s the bootstrap not vald, but the usual panel jackknfe, whch s based on the assumpton that the bas s of order, also does not work correctly. More 6

27 specfcally, we show that t removes approxmately) half of the bas. A dfferent form of the jackknfe, whch we call the generalzed panel jackknfe, s needed to remove the bas of order /. If there s no te, the bootstrap s vald and the panel jackknfe should also be effectve n removng bas, snce now the bas s of order. In ths case the generalzed panel jackknfe wll not work correctly, and ndeed we show that ts bas s the negatve of the bas of the fxed effects estmate; t reverses the bas. We also consder the case of a near te, whch we defne as the case that the dfference between the fronter ntercept and the ntercept of the second-best frm s O / ). In ths case the bas s agan of order / and so the generalzed panel jackknfe should remove t. Our smulatons support the fnte-sample relevance of these arguments. When there s a te or a near te, the generalzed panel jackknfe removes the bas effectvely, whereas the panel jackknfe and the bas-corrected bootstrap remove about half of the bas. When there s not a te, the generalzed panel jackknfe overcorrects the bas, and the panel jackknfe and the bas-corrected bootstrap are much better at removng the bas. he major drawback of the jackknfe s that ts varance s large. hs s true for both versons of the jackknfe but the varance s the largest for the generalzed panel jackknfe. here does not seem to be any good reason to prefer the panel jackknfe to the bas-corrected bootstrap, snce t has a larger varance and does not do a better job of correctng bas. However, whle the generalzed panel jackknfe s clearly domnated by the bas-corrected bootstrap n terms of MSE, t does do a very good job of removng bas when there s an exact te or a near te. Emprcally, presumably that corresponds to cases where the dentty of the best frm s n substantal doubt. he nablty of the generalzed panel jackknfe to beat the bas-corrected bootstrap n 7

28 terms of MSE when there s an exact or a near te s perhaps surprsng, snce the bootstrap s not vald f there s a te. However, not vald here has a specfc meanng, namely that we cannot clam that the dstrbuton of the bootstrap estmate around the orgnal estmate matches the dstrbuton of the orgnal estmate around the true parameter. Apparently the bas-corrected bootstrap s nevertheless a useful pont estmate. 8

29 REFERENCES Dhaene, G., K. Jochmans, and B. huysbaert, 006, Splt-Panel Jackknfe Estmaton of Fxed Effects Models Prevous tle: Jackknfe Bas Reducton for Nonlnear Dynamc Panel Data Models wth Fxed Effects), Workng Paper. Fernández-Val, I. and F. Vella, 007, Bas Correctons for wo-step Fxed Effects Panel Data Estmator, IZA Dscusson Paper 690, Insttute for the Study of Labor IZA) Gray, H.L., and W.R. Schucany, 97, he Generalzed Jackknfe Statstc, Marcel Dekker, Inc., New York. Hahn, J. and G. Kuerstener, 004, Bas Reducton for Dynamc Nonlnear Panel Models wth Fxed Effects, Unpublshed Manuscrpt. Hahn, J., and W. Newey, 004, Jackknfe and Analytcal Bas Reducton for Nonlnear Panel Models, Econometrca, 7, Hall, P., W.Härdle, and L. Smar, 995, Iterated Bootstrap wth Applcatons to Fronter Models, Journal of Productvty Analyss, 6, Km, M., Y. Km, and P. Schmdt, 007, On the Accuracy of Bootstrap Confdence Intervals for Effcency Levels n Stochastc Fronter Models wth Panel Data, Journal of Productvty Analyss, 8, Mller, R.G., 974, he Jackknfe A Revew, Bometrka, 6, -5. Park, B.U., and L. Smar, 994, Effcent Semparametrc Estmaton n a Stochastc Fronter Model, Journal of the Amercan Statstcs Assocaton, 89, Ptt, M.M., and L.F. Lee, 98, he Measurement and Sources of echncal Ineffcency n the Indonesan Weavng Industry, Journal of Development Economcs, 9, Quenoulle, M.H., 956, Note on Bas n Estmaton, Bometrka, 6, Schmdt, P., and R. Sckles, 984, Producton Fronters and Panel Data, Journal of Busness and Economc Statstcs,, Schucany, W.R., H.L. Gray, and D.B. Owen, 97, On Bas Reducton n Estmaton, Journal of the Amercan Statstcal Assocaton, 66, ukey, J.W., 958, Bas and Confdence n Not Qute Large Samples, Abstract), Annals of Mathematcal Statstcs, 8, 64. 9

30 ABLE EXPERIMEN I: NO IE 0 BIAS OF HE ESIMAES μ N ) E ˆ α α N ) ) E J ˆ α α N ) 3) E G ˆ α α N ) 4) boot E ˆ α α / / / / / / / / / / BC N ) 30

31 ABLE EXPERIMEN I: NO IE 0 VARIANCE OF HE ESIMAES μ N ) ) J 3) G 4) / / / / / / / / / / boot BC 3

32 ABLE 3 EXPERIMEN I: NO IE 0 MSE OF HE ESIMAES μ N ) ) J 3) G 4) / / / / / / / / / / boot BC 3

33 ABLE 4 EXPERIMEN II: EXAC IE 0 BIAS OF HE ESIMAES μ N ) E ˆ α α N ) ) E J ˆ α α N ) 3) E G ˆ α α N ) 4) boot E ˆ α α / / / / / / / / BC N ) Note: value of μ s rrelevant when N and there s an exact te. 33

34 ABLE 5 EXPERIMEN II: EXAC IE 0 VARIANCE OF HE ESIMAES μ N ) ) J 3) G 4) / / / / / / / / boot BC Note: value of μ s rrelevant when N and there s an exact te. 34

35 ABLE 6 EXPERIMEN II: EXAC IE 0 MSE OF HE ESIMAES μ N ) ) J 3) G 4) / / / / / / / / boot BC Note: value of μ s rrelevant when N and there s an exact te. 35

36 ABLE 7 EXPERIMEN III: NEAR IE 0 BIAS OF HE ESIMAES μ N ) E ˆ α α N ) ) E J ˆ α α N ) 3) E G ˆ α α N ) 4) boot E ˆ α α / / / / / / / / / / BC N ) 36

37 ABLE 8 EXPERIMEN III: NEAR IE 0 VARIANCE OF HE ESIMAES μ N ) ) J 3) G 4) / / / / / / / / / / boot BC 37

38 ABLE 9 EXPERIMEN III: NEAR IE 0 MSE OF HE ESIMAES μ N ) ) J 3) G 4) / / / / / / / / / / boot BC 38

39 ABLE 0 EFFEC OF CHANGING μ, N 0 BIAS OF HE ESIMAES Experment I NO IE II EXAC IE III NEAR IE ) E ˆ α α N ) ) E J ˆ α α N ) 3) E G ˆ α α N ) 4) boot E ˆ α α BC N ) 39

40 ABLE EFFEC OF CHANGING μ, N 0 VARIANCE OF HE ESIMAES Experment ) I NO IE II EXAC IE III NEAR IE ) J 3) G 4) boot BC 40

41 ABLE EFFEC OF CHANGING μ, N 0 MSE OF HE ESIMAES Experment ) I NO IE II EXAC IE III NEAR IE ) J 3) G 4) boot BC 4