Corrected Maximum Likelihood Estimators in Linear Heteroskedastic Regression Models *

Transcription

1 Corrected Maxmum Lkelhood Estmators n Lnear Heteroskedastc Regresson Models * Gauss M. Cordero ** Abstract The lnear heteroskedastc regresson model, for whch the varance of the response s gven by a sutable functon of a set of lnear exogenous varables, s very useful n econometrc applcatons. We derve a smple matrx formula for the n 1 bases of the maxmum lkelhood estmators of the parameters n the varance of the response, where n s the sample sze. These bases are easly obtaned as a vector of regresson coeffcents n a smple weghted least squares regresson. We use smulaton to compare the uncorrected estmators wth the bas-corrected ones to conclude the superorty of the corrected estmators over the uncorrected ones wth regard to the normal approxmaton. The practcal use of such bases s llustrated n two applcatons to real data sets. Keywords: Bas Correcton, Fsher Scorng Method, Heteroskedastc Model, Lnear Regresson, Maxmum Lkelhood, Weghted Least Squares. JEL Codes: C4, C5. * Submtted n October Revsed n March The author s very grateful to two referees and to the Edtor for helpful comments that consderably mproved the paper. ** Departamento de Estatístca e Informátca, Unversdade Federal Rural de Pernambuco (UFRPE). Rua Dom Manoel de Mederos, s/n, Recfe, PE, Brazl. E-mal: gauss@denfo.ufrpe.br Brazlan Revew of Econometrcs v. 28, n o 1, pp May 2008

2 Gauss M. Cordero 1. Introducton When economc data are used, heteroskedastcty can arse from a varety of sources, and the soluton depends on the nature of the problem dentfed. Heteroskedastc regresson models have recently ganed popularty n applcatons for analyzng unreplcated experments, experments for robust desgn and the analyss of process data. Many authors have also consdered varance modelng to obtan corrected standard errors and confdence ntervals for mean parameters n regresson analyss. Judge et al. (1985) proposed a varety of solutons when the heteroskedastcty s thought to be related to exogenous varables. They bascally dscussed the estmaton of parameters n the followng cases when the varance σ 2 of the response y s a functon of exogenous varables: () σ 2 = (z Tγ)2 ; () σ 2 = z Tγ; () σ2 = (x T β)p ; (v) σ 2 = exp(z T γ). In case (v), the log of the varance s a lnear functon of exogenous varables whch leads to a multplcatve heteroskedastc model. Clearly, there are other ways to model heteroskedastcty. If we are only nterested n parameters of the mean n the presence of heteroskedastcty, then robust nference does not nvolve modelng the varance. An mportant ssue n asymptotc lkelhood theory s the calculaton of the second-order bases of the maxmum lkelhood estmators (MLEs). These estmators typcally have bases of order O ( n 1) for large sample sze n, whch are commonly gnored n practce, the justfcaton beng that they are small when compared to the standard errors of the parameter estmators of order O ( n 1/2). For small sample szes, however, the second-order bases can be apprecable and of the same magntude as the correspondng standard errors. In such cases the bases should not be neglected, and ncludng already second-order bas correctons may substantally mprove the MLEs, whereas hgher-order correctons are usually much more dffcult to evaluate (Cordero and Barroso, 2007) and yeld lesser mprovement n comparson wth second-order bas correctons. In order to make the estmaton of second-order bas correctons feasble n practcal applcatons, correspondng formulas for ther calculaton need to be establshed for a wde range of regresson models. Unfortunately, t s common n econometrc applcatons to consder that the MLEs are unbased and the man textbooks n econometrcs usually do not gve expressons or references for secondorder bases of these estmators for the most mportant regresson models. Practtoners usually gnore these bases based on the frst-order asymptotc theory. To mprove the accuracy of the MLEs, substantal effort has been placed upon computng the cumulants of log-lkelhood dervatves whch are, however, notorously cumbersome; see, e.g. Cordero and McCullagh (1991) and the references theren. Ths paper shows how to easly compute the second-order bases of the MLEs n lnear heteroskedastc regresson models (wthout computng cumulants) from a vector of regresson coeffcents n a sutable weghted lnear regresson. A general formula for the O ( n 1) multparameter bases of the MLEs for a model wth p parameters was gven by Cox and Snell (1968) and Cordero and 2 Brazlan Revew of Econometrcs 28(1) May 2008

3 Corrected Maxmum Lkelhood Estmators n Lnear Heteroskedastc Regresson Models McCullagh (1991). Some closed-form expressons for second-order bases of the MLEs n regresson models are avalable n the lterature whch do not nvolve cumulants of log-lkelhood dervatves. In ths paper we obtan bas-corrected estmators for lnear heteroskedastc regresson models for whch the varance of the response takes the form σ 2 = h(z T γ), where h( ) s a known functon of a lnear combnaton of exogenous varables τ = z T γ for = 1,...,n. There are many possble ways to choose h( ) such as those cases (), () and (v) mentoned before. We thus generalze the results by Cordero (1993), whch hold only for the multplcatve heteroskedastc model (case (v)). We propose bas-corrected estmators that have good performance n small samples as llustrated n some Monte Carlo smulaton studes. We consder the lnear heteroskedastc regresson models E(y) = µ = Xβ, Cov(y) = Λ Λ = dag{σ 2 1,..., σ2 n } σ 2 = h(z T γ) (1) where X s a specfed n p matrx of full rank p < n, β = (β 1,..., β p ) T s a set of unknown parameters, h( ) s any contnuous twce dfferentable functon, γ s a q 1 vector of unknown parameters and z T s a 1 q vector of known exogenous varables. The varables z T s may be functons of the regresson varables n X. Let Z T = (z 1,..., z n ) be a q n matrx obtaned from the z s assumed to be of full rank q < n, and τ = Zγ. Further, we consder that the responses y 1,..., y n are normally dstrbuted y N(µ, σ 2 ) for = 1,...,n. Throughout the paper, we assume that X and z are non-stochastc and the parameters β and γ functonally ndependent and then there are p + q parameters to be estmated. The paper s organzed as follows. In Secton 2, we show that any software wth a weghted lnear regresson routne can be used to teratvely calculate the MLEs ˆβ and ˆγ of the parameters β and γ. In Secton 3, we show that the second-order bas of ˆβ s zero and provde a smple formula for the n 1 bas vector of ˆγ. Some specal cases are also dscussed. In Secton 4, we present some smulaton results for model (1) to show that the bas-corrected estmators produce smaller bases than the usual MLEs wthout ncreasng the mean squared errors. Moreover, we use the smulaton results to assess the adequacy of the normal approxmaton for the emprcal dstrbutons of the uncorrected and corrected estmators. Fnally, Secton 5 gves two applcatons of bas correcton to real data sets. 2. Maxmum Lkelhood Estmator For the lnear heteroskedastc regresson model (1), the total log-lkelhood for θ = (β T, γ T ) T gven y = (y 1,...,y n ) T can be wrtten apart from a constant Brazlan Revew of Econometrcs 28(1) May

4 Gauss M. Cordero l(θ) = 1 2 n log h 1 2 =1 n (y µ ) 2 =1 h (2) where h = h(z T γ). We assume that the functon l = l(θ) s regular wth respect to all β and γ dervatves up to and ncludng those of fourth order. We now ntroduce the followng total log-lkelhood dervatves n whch we reserve lower-case subscrpts r, s, t,... to denote components of the β vector and upper-case subscrpts R, S, T,... for components of the γ vector: U r = l/ β r, U rs = 2 l/ β r γ S, U rst = 3 l/ β r γ S γ T, and so on. The standard notaton wll be adopted for the cumulants of the log-lkelhood dervatves κ rs = E(U rs ), κ r,s = E(U r U s ), κ rst = E(U rst ), etc., where all κ s refer to a total over the sample and are, n general, of order n. We also defne the dervatves of the cumulants by κ (t) rs = κ rs / β t, κ (T) rs = κ rs / γ T, etc. We can easly obtan the followng second-order dervatves of the log-lkelhood from (1) and U RS = 1 2 U r = U R = 1 2 (y µ )x r h, U rs = ( ) h z Rz S h x r x s h h z R + 1 (y µ ) 2 h z R h 2 h 2 (y µ ) 2 ( h h 2 ) z Rz S U rs = (y µ )h x rz S h 2 where the dashes here and henceforth denote dervatves wth respect to τ = z T γ. The score functons for β and γ are then gven by and U β = X T Λ 1 (y µ) (3) U γ = 1 2 ZT Λ 1 F ZT Λ 2 F v (4) where Λ = dag{h } and F = dag{h } are dagonal matrces of order n, 1 s an n 1 vector of ones and v = (v 1,..., v n ) T has the th component gven by v = (y µ ) 2. Let κ r,s = κ rs and κ R,S = κ RS be typcal elements of the nformaton matrces K β and K γ for β and γ, respectvely. We have 4 Brazlan Revew of Econometrcs 28(1) May 2008

5 Corrected Maxmum Lkelhood Estmators n Lnear Heteroskedastc Regresson Models K β = X T Λ 1 X and K γ = Z T V Z, where V = dag{ h 2 } s a dagonal matrx of order n. 2h 2 The parameters β and γ are globally orthogonal snce κ rs = 0 and then the estmates ˆβ and ˆγ are asymptotcally ndependent due to ther normalty and to the block dagonal structure of the nformaton matrx. We obtan ˆβ and ˆγ teratvely from Fsher scorng method ( β (m+1) γ (m+1) ) ( β (m) = γ (m) ) + ( K (m) 1 β 0 0 K (m) 1 γ )( U (m) β U (m) γ where U β and U γ are gven by (3) and (4), respectvely. From (5) we can wrte ) (5) and where β (m+1) = (X T Λ (m) 1 X) 1 X T Λ (m) 1 y (6) γ (m+1) = (X T V (m) Z) 1 Z T V (m) (τ (m) + δ (m) ) (7) δ = 1 2 V 1 Λ 1 (Λ 1 Fv F1) s an n 1 vector. Each loop through the teratve scheme gven by equatons (6) and (7) can be solved by teratvely weghted least squares and any software wth a weghted lnear regresson routne can be used to calculate ˆβ and ˆγ teratvely. Intal approxmatons β (1) and γ (1) for the teratve algorthm are used to evaluate Λ (1), V (1), τ (1), F (1), v (1) and δ (1) from whch equatons (6) and (7) can be used to obtan the next estmates β (2) and γ (2). These new values can update Λ, V, τ, F, v and δ and so the teratons contnue untl convergence s observed. The asymptotc covarance matrces of the estmators ˆβ and ˆγ are just gven by K 1 β = (X T Λ 1 X) 1 and K 1 γ 3. Bases of the Estmators of β and γ = (Z T V Z) 1, respectvely. In recent years there has been consderable nterest n fndng formulas for the n 1 bases of the MLEs n some classes of regresson models. The methodology has been appled to normal nonlnear models (Cook et al., 1986), generalzed lnear models (Cordero and McCullagh, 1991), multplcatve regresson models (Cordero, 1993), ARMA models (Cordero and Klen, 1994), generalzed lnear models wth dsperson covarates (Botter and Cordero, 1998) and overdspersed generalzed lnear models (Cordero and Botter, 2001). All these papers gve closed-form expressons for the bases of the MLEs wthout nvolvng cumulants of log-lkelhood dervatves. The bas approxmaton may be used to produce a bas-reduced estmator by subtractng the bas approxmaton from the MLE. Brazlan Revew of Econometrcs 28(1) May

6 Gauss M. Cordero Alternatvely, an examnaton of the form of the bas may suggest a reparameterzaton of the model that results s less based estmates. In regular parametrc problems, there s an alternatve bas correcton scheme proposed by Frth (1993) who shows how to remove the frst-order term from the asymptotc bas of these estmates by a sutable modfcaton of the score functon. In exponental famles wth canoncal parameterzaton hs correcton scheme conssts n penalzng the lkelhood by the Jeffreys nvarant pror. Ths s a preventve approach to bas adjustment whch has ts merts, but the connectons between our results and hs work are not pursued n ths paper snce they could be developed n future research. We should also stress that t s possble to avod cumbersome and tedous algebra on cumulant calculatons by usng Efron s bootstrap; see Efron and Tbshran (1993). We use the analytcal approach here snce ths leads to a nce formula. Moreover, the applcaton of the analytcal bas approxmaton seems to generally be the most feasble procedure to use and t contnues to receve attenton n the lterature. A smple matrx verson for bas correcton s developed to provde a bass for mprovng nference on the parameters modelng the varance. We denote by κ r,s = κ rs and κ R,S = κ RS the correspondng elements of the nverses K 1 β and Kγ 1 of the nformaton matrces for β and γ, respectvely. These nverse matrces are of order O(n 1 ). We obtan the followng cumulants needed for the second-order bases of ˆβ and ˆγ. κ rst = κ (t) rs = κ RSt = 0, κ Rst = ( h h 2 ) x s x t z R κ RST = 1 (4h 3 3h h h 2 h 3 ) z R z S z T and κ (T) RS = ( ) h 2 2h 2 z Rz Sz T Let B(ˆβ r ) and B(ˆγ R ) be the second-order bases of the MLEs ˆβ r and ˆγ R, respectvely, for r = 1,...,p and R = 1,...,q. From the general formulas for the n 1 bases of the MLEs n regular problems gven by Cox and Snell (1968) and Cordero and McCullagh (1991), B(ˆβ a ) follows due to the orthogonalty between β and γ as ( B(ˆβ a ) = κ ar κ st κ (t) rs 1 ) 2 κ rst 1 2 κar κ ST κ rst where Ensten summaton conventon s adopted wth the ndces varyng over the correspondng parameters. All thrd-order cumulants n the above expresson vansh and then B(ˆβ a ) = 0,.e., the estmator ˆβ has no bas to order O(n 1 ). Most obvously, ths s to be expected for the lnear homoskedastc model. However, t s not evdent that ths happens for any lnear heteroskedastc model (1) snce the MLE ˆβ s obtaned from the nonlnear equaton (5). Ths surprsng result could 6 Brazlan Revew of Econometrcs 28(1) May 2008

7 Corrected Maxmum Lkelhood Estmators n Lnear Heteroskedastc Regresson Models be further nvestgated by computng the bas of order n 2 of ˆβ followng the same lnes descrbed by Cordero and Barroso (2007) n the context of the generalzed lnear models to check whether t also vanshes. However, ths wll be done n a future research because of the complcated algebra nvolvng jont cumulants of log-lkelhood dervatves of fourth and ffth orders. See, formulas (6) (8) of ther paper, where there are 52 terms to be computed nvolvng these cumulants. Usng the orthogonalty between β and γ, the n 1 bas of ˆγ A s gven by B(ˆγ A ) = 1 ( 2 κar κ st κ Rst + κ AR κ ST κ (T) RS 1 ) 2 κ RST where, agan, we use Ensten summaton conventon over the parameters. We have κ (T) RS 1 2 κ RST = 1 4 h h h 2 z R z S z T Hence, we can wrte the n 1 bas of ˆγ A as ( ) B(ˆγ A ) 1 κ AR h z R 2 h 2 κ st x s x t,r s,t 1 κ AR h z h R 4 h 2 κ ST z S z T,R S,T where runs over the observatons and the other ndces run over the parameters. We now defne the n n matrces A = ZKγ 1ZT = Z(Z T V Z) 1 Z T and B = XK 1 β XT = X(X T Λ 1 X) 1 X T. Let A d = dag{a } and B d = dag{b } be the dagonal matrces wth the correspondng elements of A and B, respectvely. Note that A and B are the asymptotc covarance matrces of Zˆγ and X ˆβ, respectvely. We then have: B(ˆγ A ) = 1 2,R κ AR h h 2 z R b Fnally, after some algebra, we obtan the n 1 bas of ˆγ,R κ AR h h h 2 z R a B(ˆγ) = (Z T V Z) 1 Z T V ξ (8) where ξ = V 1 (B d H 1 +A d H 2 )1 s an n 1 vector and H 1 = dag{ h } and H 2h 2 2 = dag{ h h } are dagonal matrces of order n. The bas vector B(ˆγ) depends only 4h 2 on the matrx Z and the dagonal matrces V, H 1, H 2, A d and B d. These matrces are functons of the heteroskedastc functon h and of ts frst two dervatves. The Brazlan Revew of Econometrcs 28(1) May

8 Gauss M. Cordero bas vector B(ˆγ) s smply the set of coeffcents from the weghted least squares regresson of ξ n the columns of Z wth V as a weghtng matrx. Equaton (8) represents the man result of ths paper. For homoskedastc regresson models, q = 1, Z = 1, γ = σ 2 s a scalar parameter, h s the dentty functon whch mples H 2 = 0, Λ = σ 2 I, V = H 1 = (2σ 2 ) 1 I, A = 2σ4 n 11T, where I s the dentty matrx, B = σ 2 X(X T X) 1 X T and t s easy to show from (8) that B(ˆγ) = pσ 2 /n as expected. We now gve expressons of V, H 1 and H 2 for some heteroskedastc functons. For the heteroskedastc power functon h(τ ) = τ θ wth τ = z T γ, we obtan V = θ dag{τ }, H 1 = θ θ 1 2dag{τ }, and H 2 = θ2 (θ 1) 4 dag{τ 3 } and, therefore, the weghts n the regresson (8) ncrease wth θ. When the varance s the square of a lnear functon of exogenous varables, we have V = 2Λ 1 and H 1 = H 2 = Λ 3/2. When the varance s a lnear functon of exogenous varables, V = H 1 = 1 2 Λ 2 and H 2 vanshes. For the multplcatve heteroskedastc model for whch h = exp(z Tγ), we have V = 1 2 I, H 1 = I and H 2 = 1 2I. In ths case, equaton (8) reduces to Cordero s (1993) expresson (7). The bases of the MLEs of the varances follow by expandng ˆσ 2 = h(ˆτ ) n Taylor seres and usng B(ˆτ ) = z TB(ˆγ) and V ar(ˆτ ) = a B(ˆσ 2 ) = h (τ ) z T B(ˆγ) h (τ ) a We can defne the bas-corrected estmator (BCE) γ = ˆγ B(ˆγ) (9) where B(ˆγ) s the value of (8) n the MLE (ˆβ T, ˆγ T ) T. The BCE defned n (8) removes the term of order n 1 from the asymptotc bas of ˆγ and s, qute generally, second-order effcent. Fnally, we do not know of any formula for the O ( n 2) covarance of the BCE for a multparameter model, although ts O ( n 1) covarance s dentcal to the asymptotc covarance of the MLE gven by the nverse of the nformaton matrx. For one-parameter models, Ferrar et al. (1996) compared the BCEs and the MLEs n terms of ther mean squared errors showng that the bas correcton could ncrease or decrease the varance, but no results are avalable n multparameter models. 4. Smulaton Results Some Monte Carlo smulatons have been developed to compare the performance of the MLE ˆγ and the bas-corrected counterpart γ n lnear models wth heteroskedastc functon h(z T γ) = (zt γ)θ for θ = 1 and 2 and h(z T γ) = exp(zt γ). The systematc component for the models s defned by 8 Brazlan Revew of Econometrcs 28(1) May 2008

9 Corrected Maxmum Lkelhood Estmators n Lnear Heteroskedastc Regresson Models µ = β 1 + β 2 x 2 + β 3 x 3 and τ = γ 1 + γ 2 z 2 + γ 3 z 3 and the true values of the parameters for the smulatons were taken as β 1 = 1, β 2 = 1, β 3 = 2, γ 1 = 2, γ 2 = 1 and γ 3 = 1. All ndependent explanatory varables n X and Z were obtaned as random draws from a unform dstrbuton U(0, 1) and ther values were held fxed throughout the smulatons wth equal sample szes. The number of observatons was set at n = 20, 30 and 40 and the smulatons run-sze was 10, 000 n each case. The calculatons were performed usng the statstcal package SAS. In each of the 10, 000 replcatons, we ftted the model and computed the uncorrected estmates ˆβ and ˆγ through the procedures NLP and NLMIXED of SAS. Then, we obtaned B(ˆγ) from (8) wth all quanttes evaluated at (ˆβ T, ˆγ T ) T and γ from (9). For each n, we computed the sample means and standard errors of the uncorrected ˆγ and corrected estmates γ based on ther values from the 10, 000 trals. Tables 1 3 gve the sample means of both estmates wth ther respectve standard errors n parentheses. We can see that the bas-corrected estmators γ are closer to the true parameter values than the uncorrected estmators ˆγ, thus correctly sgnalzng the drecton of the bas correctons. They also show that the bases of the estmators ˆγ are negatve for the three heteroskedastc functons consdered. The bases are greater n magntude for the multplcatve model and smaller for the square heteroskedastc functon. Some smulatons wth other models, not gven here, suggest that the heteroskedastc functon h(z T γ) = (zt γ)1/2 reduces the magntude of the bases of the corrected estmators ˆγ. The bas correcton has less mpact as n ncreases and the estmators γ s tend to have slghtly smaller standard errors than ˆγ s for small-szed samples. It s common belef that bas correcton entals some varance nflaton. However, these smulaton results show otherwse. In these cases, the bas correcton can lead to substantal mprovement n terms of bas and mean squared error. The smulatons suggest that bas correcton n lnear heteroskedastc regresson models can then be used to obtan mproved estmators wth more relable fnte sample behavor. Brazlan Revew of Econometrcs 28(1) May

10 Gauss M. Cordero Table 1 Uncorrected and corrected estmators of γ parameters for the heteroskedastc functon σ 2 = (z T γ) 2 n Estmate γ 1 = 2 γ 2 = 1 γ 3 = 1 20 MLE (2.71) (1.83) (1.79) BCE (2.40) (1.80) (1.71) 30 MLE (2.29) (1.72) (1.73) BCE (2.05) (1.75) (1.70) 40 MLE (1.91) (1.68) (1.65) BCE (1.88) (1.63) (1.62) Table 2 Uncorrected and corrected estmators of γ parameters for the heteroskedastc functon σ 2 = z T γ n Estmate γ 1 = 2 γ 2 = 1 γ 3 = 1 20 MLE (2.68) (1.87) (1.80) BCE (2.39) (1.83) (1.68) 30 MLE (2.27) (1.76) (1.71) BCE (2.08) (1.72) (1.69) 40 MLE (1.90) (1.65) (1.67) BCE (1.87) (1.60) (1.58) Even though econometrcans need to compute pont estmates and ther standard errors, the nterest typcally les n performng hypothess testng nference on the regresson parameters. We now show a smulaton study to evaluate nterval estmaton based on the corrected estmators n (9). We consder only the model wth heteroskedastcty functon σ 2 = exp(zt γ) and n = 30, although the results are smlar under the other two heteroskedastc models consdered here and for other sample szes. 10 Brazlan Revew of Econometrcs 28(1) May 2008

11 Corrected Maxmum Lkelhood Estmators n Lnear Heteroskedastc Regresson Models Table 3 Uncorrected and corrected estmators of γ parameters for the heteroskedastc functon σ 2 = exp(z T γ) n Estmate γ 1 = 2 γ 2 = 1 γ 3 = 1 20 MLE (2.40) (1.99) (1.84) BCE (2.19) (1.87) (1.80) 30 MLE (2.20) (1.90) (1.69) BCE (2.06) (1.92) (1.70) 40 MLE (1.96) (1.88) (1.56) BCE (1.85) (1.80) (1.54) Frst, n Table 4, we gve the sample mean, varance, skewness and kurtoss of the 10, 000 smulated values for both (uncorrected and bas-corrected) standardzed estmates (estmate true value)/standarderror), namely USEs and CSEs, respectvely, where the common standarderror s for both estmates are equal to the square roots of the dagonal elements of the nverse nformaton matrx K 1 γ = (Z T V Z) 1. Good agreement wth the normal dstrbuton happens when these fgures are, on average, close to 0, 1, 0 and 3, respectvely. The four cumulants of the CSEs are generally closer to ther lmtng values than the USEs, the most sgnfcant beng the mean and varance, as expected. Second, the hstograms of the 10, 000 uncorrected and corrected values for both standardzed estmates are gven n Fgure 1 together wth the graph of the probablty densty functon of a standard normal dstrbuton. We can see that the emprcal dstrbutons of the CSEs tend to be closer to the standard normal dstrbuton than the emprcal dstrbutons of the USEs. For example, n terms of shape, the emprcal dstrbuton of ˆγ 1 has a more acute peak around the mean than the dstrbuton of γ 1,.e. a hgher probablty than a normally dstrbuted varable for values near the mean. Table 4 Comparson of the uncorrected and corrected standardzed estmators for the heteroskedastc functon σ 2 = exp(z T γ) and n = 30 mean varance skewness kurtoss USE CSE USE CSE USE CSE USE CSE Further, we constructed asymptotc 95% confdence ntervals gven by estmate 1.96 standard error and noted that the ntervals from CSEs seem to be more precse than those from the MLEs. In fact, the observed frequences that the true parameters γ 1 = 2, γ 2 = 1 and γ 3 = 1 are wthn the ntervals based on the Brazlan Revew of Econometrcs 28(1) May

12 Gauss M. Cordero USEs were 9647, 9416 and 9428, whereas those frequences based on the CSEs were 9514, 9532 and 9522, respectvely. To verfy numercally that the dstrbuton of the corrected estmates s generally closer to a standard normal dstrbuton than the dstrbuton of the uncorrected estmates, we perform a one-sample Kolmogorov-Smrnov (K-S) test on both sets of 10, 000 USEs and CSEs. In Table 5, we gve the values of the K-S statstc (whch s a metrc on the set of dstrbuton functons), and the correspondng p values by applyng the K-S test for normalty on each set of 10, 000 USEs and each set of 10, 000 CSEs. From Table 5, we conclude that the K-S statstcs are smaller for the CSEs than for the correspondng USEs. Then, the emprcal dstrbutons of the corrected estmates are closer to the normal dstrbuton than the emprcal dstrbutons of the uncorrected estmates. The arguments above reveal the superorty of the BCEs γ s over the MLEs ˆγ wth regard to the normal approxmaton. Further, we should not expect a great mprovement n regard to the normal approxmaton snce we are adjustng only the frst moment, yet, here, we obtaned some mprovement. These results show that a smple adjustment of the frst moment of the MLE to order n 1 may yeld some mprovement n the normal approxmaton for ts dstrbuton. Table 5 K-S statstcs for comparng the USEs and CSEs for the heteroskedastc functon σ 2 = exp(z T γ) and n = 30 K-S stat. for USE p-value for USE K-S stat. for CSE p-value for CSE < < Econometrcans tend to avod estmaton by maxmum lkelhood when they use the lnear regresson model because maxmum lkelhood requres an addtonal assumpton about the dstrbuton of the data. For that reason, least squares estmaton s typcally used. We now show some smulatons (wth the heteroskedastc functon σ 2 = exp(zt γ) and n = 30) consderng that the normalty assumpton does not hold. In order to do ths, we use formula (9) for the BCE but generate the data from three other symmetrc dstrbutons, namely Student t 4 and logstc (types I and II) dstrbutons, to nvestgate how the bases of ˆγ and γ were senstve to msspecfcaton of the dstrbuton. 12 Brazlan Revew of Econometrcs 28(1) May 2008

13 Corrected Maxmum Lkelhood Estmators n Lnear Heteroskedastc Regresson Models Uncorrected estmate (true γ=2) Corrected estmate (true γ=2) Uncorrected estmate (true γ=1) Corrected estmate (true γ = 1) Uncorrected estmate (true γ = 1) Corrected estmate (true γ= 1) Fgure 1 Emprcal dstrbutons of the USEs and CSEs Brazlan Revew of Econometrcs 28(1) May

14 Gauss M. Cordero The results are n Table 6. The fgures n ths table show that the BCEs are more relable than the MLEs even under msspecfcaton n the sense that they always dsplayed smaller bases. The same does not happen n relaton to the mean squared errors. Although, these results may look unfamlar, we note that the normal dstrbuton s n fact the unnormalzed saddlepont approxmaton for all symmetrc dstrbutons and, therefore, asymptotc results under the normal model could n prncple be appled approxmately for other models provded the saddlepont approxmaton holds. Table 6 MLEs and BCEs under msspecfcaton Dstrbuton Estmate γ 1 = 2 γ 2 = 1 γ 3 = 1 t 4 MLE (2.31) (1.60) (2.41) BCE (2.05) (1.71) (2.17) logstc I MLE (2.20) (1.90) (1.69) BCE (1.93) (1.92) (1.70) logstc II MLE (2.08) (1.95) (1.78) BCE (2.13) (2.01) (1.51) 5. Applcatons In ths secton, we provde two applcatons to llustrate the bas-corrected estmators. Frst, we consder a data set n portfolo market (Smonoff and Tsa, 1994) that represents the monthly excess returns over the rskless rate for the Acme Cleveland Corporaton (Y ) and the return for the market (x) for the perod January 1986 December Qute often, the response varable that represents the return of a securty dsplays heteroskedastcty. In the orgnal analyss of these data, they suggested a lnear heteroskedastc model to predct Y n terms of a lnear functon of the return of the market x by consderng two systematc components µ = β 1 + β 2 x and σ 2 = exp(γ 1 + γ 2 x ) for = 1,..., 59. We are nterested n testng the null hypothess of homoskedastcty H : γ 2 = 0. The uncorrected estmates (wth ther correspondng standard errors n parentheses) are: ˆβ1 = (0.5782), ˆβ2 = (0.0021), ˆγ 1 = (2.3781) and ˆγ 2 = (5.3127). The bas-corrected estmates are γ 1 = (2.3781) and γ 2 = (5.3127), yeldng a corrected t rato of for testng H : γ 2 = 0 that s sgnfcant at 5%, whereas the usual t test s not. 14 Brazlan Revew of Econometrcs 28(1) May 2008

15 Corrected Maxmum Lkelhood Estmators n Lnear Heteroskedastc Regresson Models Second, we now reanalyze the example dscussed by (Ramanathan, 1993, Table 10.1) n a study n whch seven varables were observed n 40 metropoltan areas. The response varable s the number (n thousands) of cable TV subscrbers (Y ) and the explanatory varables are the number (n thousands) of homes n the area (home), the per capta ncome for each televson market wth cable (ncome), the nstallaton fee (f ee), the monthly servce charge (charge), the number of televson sgnals carred by each cable system (sgnals) and the number of televson sgnals receved wth good qualty wthout cable (wcable). Because Y corresponds to count data, Cysneros and Paula (2005) used a square root transformaton n order to try to obtan a common varance. Alternatvely, we ftted a normal lnear model wth exponental heteroskedastcty and estmated the vectors β and γ usng SAS. The systematc components of the model were taken as µ = β 1 + β 2 home + β 3 charge and σ 2 = exp(γ 1 + γ 2 charge ) snce the coeffcents of the other explanatory varables were not sgnfcant n wder ftted models. The excluded varables do not contan much nformaton about the response varable n the analyss. The MLEs are: ˆβ1 = ( ), ˆβ 2 = ( ), ˆβ 3 = ( ), ˆγ 1 = ( ) and ˆγ 2 = ( ). The corrected estmates are γ 1 = ( ) and ˆγ 2 = ( ), changng the sgnfcance (at 5%) of γ 1 and mprovng the sgnfcance of γ 2. References Botter, D. A. & Cordero, G. M. (1998). Improved estmators for generalzed lnear models wth dsperson covarates. Journal of Statstcal Computaton and Smulaton, 62: Cook, D. R., Tsa, C. L., & We, B. C. (1986). Bas n nonlnear regresson. Bometrka, 73: Cordero, G. M. (1993). Barlett correctons and bas correcton for two heteroskedastc regresson models. Communcatons n Statstcs Theory and Methods, 22: Cordero, G. M. & Barroso, L. P. (2007). A thrd-order bas-corrected estmate n generalzed lnear models. Test, 16: Cordero, G. M. & Botter, D. A. (2001). Second-order bases of maxmum lkelhood estmates n overdspersed generalzed lnear models. Statstcs and Probablty Letters, 55: Cordero, G. M. & Klen, R. (1994). Bas correcton n ARMA models. Statstcs and Probablty Letters, 19: Brazlan Revew of Econometrcs 28(1) May

16 Gauss M. Cordero Cordero, G. M. & McCullagh, P. (1991). Bas correcton n generalzed lnear models. Journal Royal Statstcal Socety B, 53: Cox, D. R. & Snell, E. J. (1968). A general defnton of resduals (wth dscusson). Journal Royal Statstcal Socety B, 30: Cysneros, F. J. A. & Paula, G. A. (2005). Restrcted methods n symmetrcal lnear regresson models. Computatonal Statstcs & Data Analyss, 49: Efron, B. & Tbshran, R. J. (1993). An Introducton to the Bootstrap. Chapman and Hall, New York. Ferrar, S. L. P., Botter, D. A., Cordero, G. M., & Crbar-Neto, F. (1996). Secondand thrd-order bas reducton for one-parameter famly models. Statstcs and Probablty Letters, 30: Frth, D. (1993). Bas reducton of maxmum lkelhood estmates. Bometrka, 80: Judge, G. G., Grffths, W. E., Hll, R. C., Lütkepohl, H., & Lee, T.-C. (1985). The Theory and Practce of Econometrcs. John Wley, New York. Ramanathan, R. (1993). Statstcal Methods n Econometrcs. John Wley, New York. Smonoff, J. S. & Tsa, C.-H. (1994). Use of modfed profle lkelhood for mproved tests of constancy of varance n regresson. Appled Statstcs, 43: Brazlan Revew of Econometrcs 28(1) May 2008