Fractional Response Models - A Replication Exercise of Papke and Wooldridge (1996)

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1 56 Vol. 6 Issue Harald Oberhofer and Mchael Pfaffermayr Prmary submsson: Fnal acceptance: Fractonal Response Models - A Replcaton Exercse of Papke and Wooldrdge (1996 Harald Oberhofer 1 and Mchael Pfaffermayr,3 ABSTRACT Key words: JEL Classfcaton: Ths paper replcates the estmates of a fractonal response model for share data reported n the semnal paper of Lesle E. Papke and Jeffrey M. Wooldrdge publshed n the Journal of Appled Econometrcs 11(6, 1996, pp We have been able to replcate all of the reported estmaton results concernng the determnants of employee partcpaton rates n 401(k penson plans usng the standard routnes provded n Stata. As an alternatve, we estmate a two-part model that s capable of copng wth the excessve number of boundary values equallng one n the data. The estmated margnal effects are smlar to those derved n the paper. A small-scale Monte Carlo smulaton exercse suggests that the RESET tests proposed by Papke and Wooldrdge n ther robust form are useful for detectng neglected non-lneartes n small samples. replcaton exercse, fractonal response models, two-part models, Monte Carlo smulaton C15, C1 1 Unversty of Salzburg, Austra Unversty of Innsbruck, Austra 3 Austran Insttute of Economc Research, Austra Introducton In many applcatons, the stuaton n whch share data are confned to the [0,1] nterval must be addressed, and n addton, the data may nclude a sgnfcant amount of observatons of the dependent varable takng on values at the boundares of 0 or 1. Whle share data can be handled usng log-odds transformed varables, the combnaton of these two ssues s complex. In ther semnal paper, Lesle E. Papke and Jeffrey M. Wooldrdge (1996 propose a fractonal response model that extends the generalsed lnear model (GLM lterature from statstcs. In a recent paper, Papke and Wooldrdge (008 ntroduce fractonal response models for panel data. The authors ntroduce a quas-maxmum lkelhood estmator (QLME to obtan a robust method for estmatng Corespondence concernng to ths artcle should be addressed to: harald.oberhofer@sbg.ac.at fractonal response models wthout an ad hoc transformaton of the boundary values. The paper shows that the proposed QLME s consstent gven that the condtonal mean functon s correctly specfed (see ther equaton 4. In addton, the authors ntroduce robust Ramsey RESET tests for the correct specfcaton of the mean functon. Fnally, the paper provdes an applcaton of ths estmaton procedure: estmatng a model of employee partcpaton rates n 401(k pensons plans. Ramalho, Ramalho and Murtera (011 provde a comprehensve up-to-date overvew on the econometrcs of fractonal response models. Papke and Wooldrdge (1996 consder the followng model for the condtonal expectaton of the fractonal response varable: E( y x = x β, = 1,..., N, (1 where 0 y 1 denotes the dependent varable and (the 1 k vector x refers to the explanatory varables of observaton. Typcally, G (. s a ds- CONTEMPORARY ECONOMICS DOI: /ce

2 Fractonal Response Models - A Replcaton Exercse of Papke and Wooldrdge ( trbuton functon smlar to the logstc functon G ( z = exp( z/(1+ exp( z, whch maps z to the (0,1 nterval. The authors follow the methods of McCullagh and Nelder (1991 and suggest maxmsng the Bernoull log lkelhood wth the ndvdual contrbuton gven by the followng (Papke & Wooldrdge, 1993 also consder the case n whch the group sze s known and s gven by n. They show that n ths case, the condtonal lkelhood for observaton s the same as that n (, but t s weghted by n : l ( β = y log[ x β] + (1 y log[1 x β]. ( In ths formulaton of the lkelhood functon, the number of draws (here, the number of elgble employees of each frm drops out be cause t does not depend on the parameters. Rather, the share of successes,.e., the partcpaton rate, enters the lkelhood drectly (see McCullagh & Nelder, 1991, p The consstency of the QLME follows from the study by Goureroux, Monfort & Trognon (1984 because the densty upon whch the lkelhood functon s based s a member of the lnear exponental famly, and the assumpton that the condtonal expectaton of y s correctly specfed valdates ths fndng. In fact, the QLME s N -asymptotcally normal regardless of the dstrbuton of y condtonal on x. Papke and Wooldrdge (1996 provde vald (robust estmators of the asymptotc varance of β based on the well-known sandwch formula (see Cameron & Trved, 005 and the non-lnear condtonal mean G (.. Papke and Wooldrdge (1996 ntroduce and apply extended Ramsey RESET tests for H = 0, = 0 0: 1 3 1( xβ ( xβ n the augmented model G ( x β + +. Ther frst RESET test s non-robust because t mantans the GLM varance assumpton: Va r ( y x = x β[1 x β]. The robust RESET test only requres the correct specfcaton of the condtonal mean. Detals on calculatng the RESET test are provded on pages n ther paper. In many applcatons, ncludng that presented n ths paper, there s a sgnfcant share of boundary values. Consderng the data-generatng process n the paper by Papke and Wooldrdge (1996 lterally, one would use the number of elgble employees as the number of Bernoull draws. However, n the full sample, the mean frm sze s 461 and the medan frm sze s 68. Basng the Bernoull draws on these numbers makes a boundary value of 1 n PRATE a rare event. Thus, n the case where 4.7 per cent of the boundary values n the data are equal to 1, t appears plausble to assume that frms that exhbt 100 per cent partcpaton rates n ther penson plans behave dfferently and are not well descrbed by the Bernoull model. Accordng to problem 19.8 n Wooldrdge (00, Ramalho and Vdgal da Slva (009 and Ramalho, Ramalho and Murtera (011, we can alternatvely consder a two-part model that accounts for an excessve number of boundary values that are equal to one (refer to Pohlmeer and Ulrch (1995 for an early applcaton of a twopart model for count data. We defne: 0 f y [0,1 y = (3 1 f y = 1 and assume for the frst part of the model that P( y = 1 x = P( y = 1 x = x γ, where x γ denotes the cumulatve logstc dstrbuton functon. The second part of the model s the fractonal response model that refers to observatons y ε [0,1. Then, the condtonal mean of the two-part model s specfed as the followng: E[ y x ] = P( y = 0 x E[ y x, y = 0 ] + P( y = 1 x = (1 x γ x β + x γ. (4 The margnal effects of the explanatory varables can be derved as follows: E[ y x ] P( y = 1 x = (1 E[ y x, y = 0] x x j + (1 P( y j E[ y x, y = 0 ] = 1 x. (5 x Ths model allows the explanatory varables to affect the outcome ( y = 1 and sze of y at y ε [0,1 n a dfferent way. More mportantly, the explanatory varables n the frst and second parts of the model are not requred to be the same. Under ths specfcaton (quas maxmum lkelhood estmaton s straghtforward because t can be separated nto the estmaton of the logt model explanng P = 1 x usng all of the observatons ( y and the estmaton of the parameters of the condtonal densty f ( y x, y = 0 based only on the observaton where y < 1. Essentally, the condtonal dstrbuton j Vzja Press&IT

3 58 Vol. 6 Issue Harald Oberhofer and Mchael Pfaffermayr of y x, y = 0 s derved from the uncondtonal b- nomal dstrbuton through dvson by 1 G ( x β so that n n ( 1 1 (, = 0 = y n y n f y x y xβ (1 xβ (1 x β, n y where n s the number of elgble employees (see also Papke & Wooldrdge, In the case where n s large, the last term wll be equal to approxmately 1. In the followng dervaton, we neglect ths term. In fact, the second part of the model s defned as the fractonal response model, as ntroduced above. Agan, the crtcal assumpton that s necessary to obtan consstent parameters s the correct specfcaton of the condtonal mean, whch now requres the correct specfca- tons of P = 1 x and E[ y x, y = 0 ]. ( y The replcaton exercse In ther applcaton, Papke and Wooldrdge (1996 are nterested n an econometrc model of the partcpaton rates n 401(k penson plans. These plans are employersponsored penson plans n whch employees are permtted to make pre-tax contrbutons and the employer may match part of the contrbuton. The dependent varable (PRATE s defned as the number of actve penson accounts dvded by the number of employees elgble to partcpate for a sample of US manufacturng frms. The explanatory varables of ther model nclude the plan match rate of the employer (MRATE, the log sze of the frm measured n terms of employment and the square of ths value, the plan s age and the square of ths value and a dummy varable called SOLE that ndcates whether the 401(k penson plan s the only plan of ts type offered by the frm. In sum, the followng specfcaton s estmated n Tables II and III of ths paper: E( PRATE x= β + EMP 1 + βmrate + β3 log( EMP β4log( + β 5AGE+ β 6 AGE + β7sole. (6 The lnear specfcaton assumes G ( z = z, whle n the non-lnear fractonal response regresson,. s specfed as a logstc functon,.e., G ( z = exp( z/(1+ exp( z. In a second specfcaton, the authors addtonally nclude MRATE as an explanatory varable. Tables II and III n the paper report smple OLS estmates and the QMLE of the fractonal response model. The estmates n Table II use only the observatons where n MRATE < 1, whle the estmaton results n Table III are based on all of the observatons. There are no zeros n the dependent varable, but 4.7 per cent of the sample represents the frms n whch all of the employees partcpate n 401(k penson plans so that PRATE = 1. In Table II of ther paper, the authors report that the frm s matchng rate has a sgnfcant postve mpact. The log frm sze and the age of the plan enter nonlnearly. The mpact of the log frm sze s sgnfcantly negatve, but ncreases for large frms. AGE s sgnfcantly postve but also has a decreasng effect. Last, the varable SOLE s nsgnfcant. In Table II of the paper, the OLS estmates are rejected by both the non-robust and robust RESET tests, suggestng that the lnear model neglects mportant non-lneartes. However, the sgns of the estmated parameters are the same for the OLS and the QLME estmates for all varables. There s an mportant dfference between the OLS and QMLE estmates because the RESET tests (both n ther robust and non-robust versons do not reject the fractonal response model. Furthermore, the R of the fractonal response model s 6 percentage ponts hgher compared to the lnear model. From an economc pont of vew, the dfference between the two models s mportant because the fractonal response model mples that there s a decreasng margnal effect of MRATE. The authors also conclude that smply addng ( MRATE to the lnear model s not suffcent to capture ths non-lnearty. The results n Table III of ther paper show that the basc results do not change f the models are estmated over the entre sample. The only clear dfferences are that the quadratc term n MRATE s now sgnfcant and the RESET test does not reject the fractonal response model that ncludes rejects the baselne specfcaton. MRATE, but t The authors estmated and tested the fractonal response model usng GAUSS-code. We were able to easly replcate and verfy ther estmated results usng the now avalable standard Stata code and specfcally, the Stata procedure glm wth optons fam(bn, lnk(logt and scale(x for non-robust standard errors and optons fam(bn, lnk(logt and rob for robust standard errors. In ths way, we have been able to replcate each entry n Tables II and III. Therefore, the fractonal response model s attractve because t can be easly estmated usng standard econometrc software. The Stata code s avalable upon request from the authors. CONTEMPORARY ECONOMICS DOI: /ce

4 Fractonal Response Models - A Replcaton Exercse of Papke and Wooldrdge ( We also estmated the two-part model usng the basc specfcaton proposed by Papke and Wooldrdge (1996, whch s reported n the frst two columns of Table II n ther paper. As noted above, these estmates exclude observatons where MRATE > 1. For comparson, we reproduced the correspondng estmates n Table 1. In the logt model of the two-part model, the same varables that enter the fractonal response model determne whether all of the employees partcpate n the 401(k penson plans. Approxmately all of the explanatory varables are sgnfcant, and for MRATE, log( EMP and log (EMP, we obtan the same sgns as those n the fractonal response model. In contrast to the results of the fractonal response model, AGE turns out to be nsgnfcant, whle AGE s postve at a p-value slghtly hgher than The varable SOLE s sgnfcantly postve, whch s also n contrast to the estmate n the fractonal response model. Table 1. Results for the Restrcted Sample (1 ( (3 (4 Varable OLS QMLE Two-Part Model MRATE Logt QMLE (0.01 (0.100 (0.160 (0.089 [0.011] [0.107] [0.166] [0.097] log(emp (0.014 (0.111 (0.00 (0.09 [0.013] [0.110] [0.197] [0.094] log(emp (0.001 (0.007 (0.013 (0.006 [0.001] [0.007] [0.013] [0.006] AGE (0.001 (0.009 (0.014 (0.007 [0.001] [0.009] [0.016] [0.006] AGE (0.000 (0.000 (0.000 (0.000 [0.000] [0.000] [0.000] [0.000] SOLE (0.006 (0.047 (0.078 (0.039 [0.006] [0.050] [0.078] [0.040] ONE (0.051 (0.47 (0.740 (0.354 [0.048] [0.41] [0.741] [0.35] Observatons 3,784 3,784 3,784,489 SSR SER R RESET (0.000 ( (0.000 Robust-RESET (0.000 ( (0.000 Notes: See Table II n Papke and Wooldrdge (1996. In the logt model, the value of the dependent varable s one f all employees partcpate n the 401(k penson plan and s zero otherwse. The QMLE of the two-part model s estmated only for PRATE<1. Vzja Press&IT

5 60 Vol. 6 Issue Harald Oberhofer and Mchael Pfaffermayr The second part of the fractonal response model uses observatons where PRATE < 1. Wth the excepton of the sgnfcant negatve mpact of SOLE, we obtan qualtatvely smlar results as those of Model n Table II of Papke and Wooldrdge (1996. However, n quanttatve terms, the parameter estmates are qute dfferent. The ft of the two-part model s comparable to the orgnal estmates wth to the value of R amountng to Smlar R for the non-lnear fractonal response model n Papke and Wooldrdge (1996, the value of R for the two-part model s based on the predcted values of all of the observatons, ncludng the boundary values. However, both the robust and non-robust RESET tests are rejected, ndcatng a possble msspecfcaton of the second part of the fractonal response model. The man advantage of both the fractonal response model and the two-part model s ther ablty to capture non-lneartes, partcularly n the decreasng effect of the matchng rate. Table reproduces the margnal model results n relatvely small margnal effects at low values of MRATE, but n a less pronounced decrease n the margnal effects as MRATE ncreases. When observng the n-sample predctons of the estmated models, we found two puzzlng results. Frst, t can easly be observed from the specfcaton of the condtonal mean under the logstc lnk assumpton,.e., G ( z = exp( z/(1+ exp( z, that both of the consdered models rule out values of 1 n the dependent varable. Put dfferently, the models by defnton always predct a value that s lower than one for those observatons of PRATE that fall on the boundary 1. Table 3 presents the calculatons of the mean of the resduals resultng from the estmates n Table II n the paper as well as for the two-part model wthn each quntle of PRATE and, separately, for the values on the boundary cases where PRATE = 1. As expected, the resduals are postve for the values of PRATE = 1 Table. The Margnal Effects from the QMLE and the Two-Part Model EMP=00 EMP=4,60 EMP=100,000 MRATE QMLE Two-Part QMLE Two-Part QMLE Two-Part effects of the matchng rate of the estmated model, whch are presented n columns 1 and n Table II of the paper. To obtan ths result, SOLE s set equal to 0, AGE = 13 and EMP = 00;4,60; 100,000. The partal effects are computed at the matchng rate values of 0, 0.5 an 1. Whle the margnal effect under the lnear model amounts to 0.156, t dmnshes for both the fractonal response model and the two-part model. The fractonal response model mples an ncrease n PRATE by.9 percentage ponts as a response to an ncrease n MRATE from 0 to 0.1. Under the two-part model, the effect s smaller and amounts to.1 percentage ponts. Conversely, at MRATE = 1, the margnal effect of the two-part model s 1.3 per cent compared to an effect of 1. per cent, whch was mpled by the fractonal response model. Generally, the two-part for both the OLS estmaton and the QMLE. Addtonally, there s vrtually no dfference between the consdered models. Second, we observed systematc effects n the resduals of both the lnear and non-lnear models. For the observatons where PRATE < 1, all of the consdered models overpredct for the lower three quntles of PRATE and underpredct for the two upper quntles. The same pattern s found for the resduals of the two-part model. In fact, the resduals of the four estmated models n Table II of Papke and Wooldrdge (1996 and those of the two-part model are hghly correlated, wth correlatons as hgh as As n many applcatons, there s only a mnor dfference between the lnear and non-lnear models n terms of the root mean squared predcton error, and usng a logstc lnk functon leads to only small mprovements. CONTEMPORARY ECONOMICS DOI: /ce

6 Fractonal Response Models - A Replcaton Exercse of Papke and Wooldrdge ( Table 3. Resduals from the OLS, the QMLE and the Two-Part Model Prate OLS QMLE Two-Part 1 st Quntle nd Quntle rd Quntle th Quntle th Quntle PRATE= Total Note: The fgures are based on the means wthn the respectve quntle. A small scale Monte Carlo exercse on the performance of the proposed RESET tests To nvestgate the performance of the proposed RESET test, we establshed a small Monte-Carlo smulaton exercse. Bernoull random varables were generated usng the predcted partcpaton rates of column 4 of Table II n the paper, assumng that the reported parameters are the true values (see Equatons and 3. Because the Bernoull random varable measures the number of successes n n trals, we set n = 10 n the frst experment to generate a large share of ones (approxmately 0 per cent. To obtan the share varables, we dvded the resultng Bernoull random number by n (and smlarly n the other experments. The drawback of ths desgn s that we obtaned only nne dfferent realsatons of the generated random varable. In Experment, we set n = 1000, whle n Experment 3, we allowed n to vary and assumed that n = EMP. The latter experment ntroduces addtonal heterogenety and volates the nomnal varance assumpton because the log of the number of employees and ts square are used as regressors (see equaton 6 n the paper and the dscusson below. Experments 4 and 5 are dentcal to Experments and 3, but assume that the estmated logt model s the true data generatng process for the boundary values. We generated a unformly dstrbuted random varable and set the smulated value of PRATE to 1 f ths random varable s lower than the predcted probablty, as mpled by the logt model. Then, we appled the two-part model and estmate a fractonal response model usng only the non-boundary values. We calculated the bas and root mean squared error (RMSE of the estmated parameters resultng from 10,000 replcatons of Monte Carlo experments. Followng the methods of Kelejan and Prucha (1999, we defne the bas as med ( θ θ and RMSE as ( Bas + ( IQ 0.5 /1.35, where IQ s the nterquantle range. In all of the experments, the estmated parameters are vrtually unbased. Wth the excepton of Experment 1, the RMSEs are relatvely small. In partcular, the RMSEs are consderably smaller than the standard errors reported n the paper, whch orgnate from estmated models that have a sgnfcant share of boundary values. To obtan the power curves of the RESET tests, we assume that γ 1 takes on values n the range of values ncludng{ 0.05, 0.015, 0.005,0,0.005,0.015, 0.05} and γ s 1/5 th of γ 1. Because the power had a sgnfcantly low value n Experment 1, we scaled the γ - values for ths experment by a factor of 10. In each 3 experment, we added γ ( x β + γ ( x β to the lnear 1 γ 1 = γ = predctor. Therefore, at 0, the share of rejectons n the respectve experment s an estmate of the sze of the RESET tests, and at γ 0 or γ 0, the 1 value for the power of the test s obtaned. In Tables 4 and 5, the smulated sze (n bold fgures and power of the RESET tests are dsplayed for a nomnal sze of For each value of g 1, the frst lnes n the tables refer to the non-robust RESET test and the second lnes refer to the robust verson. Whle the RESET tests are properly szed under Experment 1 and, we fnd the correct sze for only the robust RE- SET test under Experment 3, as expected. Although the constructon of the share varable often remans unobserved emprcally, ts calculaton s mportant for estmatng and testng the fractonal response models, as argued by Papke and Wooldrdge (1996. In Vzja Press&IT

7 6 Vol. 6 Issue Harald Oberhofer and Mchael Pfaffermayr Table 4. Power and sze of the RESET tests under the Fractonal Response Model for Experments 1, and 3 Experment g g g g Notes: The DGP s assumed to be Model 4 reported n Table II of Papke and Wooldrdge (1996. Bold fgures refer to the sze of the test; the remanng fgures refer to the power. For each value of g1, the frst lne n the table refers to the non-robust verson of the RESET test and the second lne refers to the robust verson. Experment 1: Bernoull random varable scaled by 10. Experment : Bernoull random varable scaled by Experment 3: Bernoull random varable scaled by employment. CONTEMPORARY ECONOMICS DOI: /ce

8 Fractonal Response Models - A Replcaton Exercse of Papke and Wooldrdge ( Table 5. Power and sze of the RESET tests under the Two-Part Model for Experments 4 Experment g g Notes: The DGP s assumed to be Model 4 reported n Table II of Papke and Wooldrdge (1996. Bold fgures refer to the sze of the test; the remanng fgures refer to the power. For each value of g1, the frst lne n the table refers to the non-robust verson of the RESET test and the second lne refers to the robust verson. Experment 4: Bernoull random varable scaled by Logt model of Table 1 s assumed to be the DGP. Experment 5: Bernoull random varable scaled by employment. Logt model of Table s assumed to be the DGP. ths respect, our fndngs confrm the dscusson of the RESET tests n the paper. The results of Experments 4 and 5 refer to the two-part model and confrm the fndngs of Experments and 3. Alternatvely, we also nvestgated the case n whch a fractonal response model usng all of the observatons s estmated n the case of a large share of boundary values. The results, whch are avalable upon request from the authors, ndcate that n ths case, the RESET tests are overszed and ther power tends to be consderably lower, even when takng nto account that the tests are overszed. However, ths fndng has to be expected because ths setup volates the condtonal mean assumpton. Generally, the RESET tests exhbt suffcent power to detect neglected non-lneartes. Only at small n values, as n Experment 1, the power s not satsfactory. For ths experment, we obtaned power fgures comparable to the other experments when scalng γ 1 and γ by a factor of 10. The hghest power of the RESET test s observed when ether γ 1 or γ s zero and the correspondng non-zero value has a hgh absolute value. However, at large absolute values of γ 1 and γ wth dfferent sgns, the power of the RESET test results n a very low value. Ths result holds for the robust and non-robust versons of the RESET test. Vzja Press&IT

9 64 Vol. 6 Issue Harald Oberhofer and Mchael Pfaffermayr Conclusons Ths paper replcated the results of the semnal paper of Lesle E. Papke and Jeffrey M. Wooldrdge (1996 concernng a fractonal response model for employee partcpaton rates n 401(k penson plans n US manufacturng frms. Usng the now avalable standard Stata code, we have been able to replcate each estmaton result n the paper. An mportant feature of ther dependent varable s that more than 40 per cent of these data are equal to one, ndcatng full employee partcpaton. To cope wth the excessve number of boundary values, we addtonally estmated a two-part model. The frst part of ths model estmates the probablty of a boundary observaton by a smple logt model. The second part of the model refers to non-boundary values and s estmated by the same fractonal response model. The estmaton of the second part of the model yelds slghtly dfferent results. However, the margnal effects of the matchng rate that take both parts nto account are comparable n sze. The effects are slghtly smaller, and the dmnshng mpact of the matchng rate s less pronounced. Therefore, n the presence of a hgh share of boundary values, the two-part model s a useful alternatve to the fractonal response model. Moreover, t s as easy to perform the calculatons usng ths model wth the avalable standard software. Lookng at the n-sample predctons of the estmated model reveals some complextes. Frst, for all of the observatons wth a boundary value of one n the dependent varable, the correspondng predctons by defnton are less than one. Second, n all of the estmated models, there are systematc dfferences n the remanng resduals, dependng on the sze of the partcpaton rate. A small-scale Monte Carlo smulaton exercse confrms that the proposed RESET tests are useful for detectng neglected non-lneartes n small samples. In ther robust form, the RESET tests are always properly szed and equpped wth power n approxmately all of the consdered cases. McCullagh, P., & Nelder, J. A. (1991. Generalzed Lnear Models. (nd ed., London, UK: Chapman and Hall. Kelejan, H. H., & Prucha, I. R. (1999. A generalzed moments estmator for the autoregressve parameter n a spatal model. Internatonal Economc Revew, 40(, Papke, L. E., & Wooldrdge, J. M. (1993. Econometrc methods for fractonal response varables wth an applcaton to 401(k plan partcpaton rates (Techncal Workng Paper No Natonal Bureau of Economc Research. Papke, L. E, & Wooldrdge, J. M. (1996. Econometrc methods for fractonal response varables wth an applcaton to 401(k plan partcpaton rates. Journal of Appled Econometrcs, 11(6, Papke, L. E., & Wooldrdge, J. M. (008. Panel data methods for fractonal response varables wth an applcaton to test pass rates. Journal of Econometrcs, 145(1-, Pohlmeer, W., & Ulrch, V. (1995. An econometrc model of the two-part decsonmakng process n the demand for health care. Journal of Human Resources, 30(, Ramalho, J. J. S., & Vdgal da Slva, J. (009. A twopart fractonal regresson model for the fnancal leverage decsons of mcro, small, medum and large frms. Quanttatve Fnance, 9(5, Ramalho, E. A., Ramalho, J. J. S., & Murtera, J. M. R. (011. Alternatve estmatng and testng emprcal strateges for fractonal regresson models. Journal of Economc Surveys, 5(1, Wooldrdge, J. M. (00. Econometrc Analyss of Cross Secton and Panel Data. Cambrdge, MA: MIT Press. References Cameron, C. A., & Trved, P. K. (005. Mcroeconometrcs: Methods and Applcatons. Cambrdge: Cambrdge Unversty Press. Goureroux, C., Monfort, A, & Trognon, A. (1984. Pseudo-maxmum lkelhood methods: Theory. Econometrca, 5(3, CONTEMPORARY ECONOMICS DOI: /ce

10 65 Vzja Press&IT