h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa, K. a a Graduae School of Engineering, Universiy of Tokyo, Bunkyo-ku, Tokyo 3-8656, Japan Email: nawaa@mi..u-okyo.ac.jp Absrac: The Box-Cox (964) ransformaion (hereafer called he BC ) is widely used in various fields of economerics and saisics. However, since he error erms canno be normal excep in cases in which he ransformaion parameer is zero, he likelihood funcion under he normaliy assumpion (hereafer he BC likelihood funcion) is misspecified and he maximum likelihood esimaor (hereafer he BC MLE) canno be consisen. Alernaive disribuions of he error erms and ransformaions for he BC have been proposed by various auhors. However, hese alernaive esimaors are no inconsisen if he disribuions of he error erms are misspecified. Foser, Tain, and Wei () and Nawaa and Kawabuchi (3) proposed semiparameric esimaors. However, heir esimaors are no consisen under heeroscedasiciy. Powell (996) proposed a semiparameric esimaor based on he momen resricion. Alhough Powell s esimaor is consisen under heeroscedasiciy, he problems of he esimaor are: (i) o idenify he ransformaion parameer,, we need o inroduce one or more insrumenal variables, w, which saisfy E(w u ) = and are no included in x, and he resul of he esimaion changes depending on he selecion of insrumenal variables, (ii) as poined ou by Khazzoom (989), when all observaions are y <, he objecive funcion is always minimized a = (or a = if y > for all observaions), so ha a raher arbirary rescaling of y is necessary, and (iii) is finie-sample properies are no good and i ofen performs poorly, as shown in he Mone Carlo experimens. Here I propose a new robus esimaor of he power ransformaion (he Box-Cox ransformaion excluding he cases in which he ransformaion parameer is zero) given by z = x ' β + u, z = y, y, =,,..., T. The esimaor is based on only he firs- and hird-momen resricions of he error erms and does no require he assumpion of a specific disribuion. The esimaor is a roo of he equaions; ( z x β ) 3 =, and x ( z x ' β ) =. The esimaor is consisen if he firs- and hird-momens of he error erms are zero; ha is, i is consisen even under heeroscedasiciy. Moreover, i can be easily calculaed by he leas-squares and scanning mehods. The resuls of he Mone Carlo experimens show he superioriy of he proposed esimaor over he BC MLE and Powell s esimaor. Keywords: Box-Cox ransformaion, power ransformaion, heeroscedasiciy, robus esimaor, momen resricion 84
Nawaa, Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion. INTRODUCTION The Box-Cox (964) ransformaion (hereafer called BC ) is widely used in various fields of economerics and saisics. However, since he error erms canno be normal excep in cases in which he ransformaion parameer is zero, he likelihood funcion under he normaliy assumpion (hereafer he BC likelihood funcion) is misspecified and he maximum likelihood esimaor (hereafer he BC MLE) canno be consisen. Alernaive disribuions of he error erms and ransformaions for he BC have been proposed by various auhors. However, hese alernaive esimaors are no inconsisen if he disribuions of he error erms are misspecified. Foser, Tain, and Wei () and Nawaa and Kawabuchi (3) proposed semiparameric esimaors. However, hese alernaive esimaors are no consisen under heeroscedasiciy. Powell (996) proposed a semiparameric esimaor based on he momen resricion. Alhough Powell s esimaor is consisen under heeroscedasiciy, he problems of he esimaor are: (i) o idenify he ransformaion parameer,, we need o inroduce one or more insrumenal variables, w, which saisfy E ( w u ) = and are no included in x, and he resul of he esimaion changes depending on he selecion of insrumenal variables, (ii) as poined ou by Khazzoom (989), when all observaions are y <, he objecive funcion is always minimized a = (or a = if y > for all observaions), so ha a raher arbirary rescaling of y is necessary, and (iii) is finie-sample properies are no good and i ofen performs poorly as shown in he Mone Carlo experimens. Here I propose a new robus esimaor of he power ransformaion : he Box-Cox ransformaion excluding he cases in which he ransformaion parameer is zero. The esimaor is based on only he firs- and hird-momen resricions of he error erms and does no require he assumpion of a specific disribuion. The esimaor is consisen even under heeroscedasiciy. Is asympoic disribuion is obained, and he resuls of Mone Carlo experimens are also presened. MODEL We consider he simple power ransformaion z = x ' β + u, z =, y, =,,..., T, () y where x and β are k-h dimensional vecors of explanaory variables and he coefficiens, respecively, and * is he ransformaion parameer. Le ( y ) / = x * ' β + v and v = u /, in which case we obain he BC. However, o ensure he asympoic disribuion of he esimaor, we only considered he case and did no consider he = case. Therefore, we call his a power ransformaion raher han a BC. { x } and { u } do no have o be independen and idenically disribued (i.i.d.) random variables, and heeroscedasiciy can be assumed. The following assumpions are made: Assumpion. {( x, u )} are independen bu no necessary idenically disribued. The disribuion of u may depend on x. Assumpion. u follows disribuions whereby he suppors are bounded from below (i.e., f ( u) = if u a for some a > where f (u) is he probabiliy (densiy) funcion.) For any, he following momen condiions are saisfied: (i) E ( u x ) =, (ii) ( 3 6 E u x ) =, and (iii) δ < E( u x ) < δ for some < δ < δ <. Assumpion 3. { x } are independen and is fourh momens are finie. The disribuions of { x } and he parameer space of β are resriced so ha inf( x ' β ) > a and inf( x ' β ) > c for some c > in he neighborhood of β where β is he rue parameer value of β. Here, insead of he BC likelihood funcion, we use he hird-momen resricion and he roos of he equaions; 3 GT ( θ ) = ( z xβ ) g ( θ ) =, and x ( z x ' β ) =, () where θ ' = (, β), are considered. Noe ha he second equaion of () gives he leas-squares esimaor when he value of is given. Le θ ' = (, β) be he rue parameer value of θ. Since E [ G( θ )] =, we obain he following proposiion: 85
Nawaa, Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Proposiion Among he roos of (), here exiss a consisen roo. Le ˆ θ ' = ( ˆ, ˆ' β ) be he consisen roo. The asympoic disribuion of θˆ is obained by he following proposiion. Proposiion ( θ ) Le ξ ( θ ) = x ( z x ' β ) and ( θ )' = [ g ( θ ), ξ ( θ )']. Suppose ha θ converges o a T θ ' nonsingular marix A in probabiliy and ha E[ ( θ) ( θo )'] converges o a nonsingular marix B. Then T he asympoic disribuion of θˆ is given by T ( ˆ θ θ ) N[, A B( A') ( θ ) where A = p lim θ, and B = lim E[ ( ) ( )']. θ θo T θ ' T T [Proof] Le ], (3) Then GT ( θ ) ( ) ( ) ( ). θ = θ = ξ θ (4) ˆ T ( θ θ ) = [ * ] ( θ), (5) θ T θ ' T where * θ is some value beween θˆ and θ. Here, 3 u ( θ) = x. u (6) σ Therefore, E[ ( θ )] =. Since he variables { ( θ )} are independen and he Lindberg condiion is saisfied under Assumpions and 3, we obain ( θ) N(, B), (7) T from Theorem 3..6 in Amemiya (985, p. 9). 3 ( z x ' β ) z log( y ) 3 ( z x ' β ) x Since / θ =, zx ' β log( y ) x x ' ( θ ) T θ ' P * A θ, from Theorem 4..4 in Amemiya (985, pp.-3). From Theorem 4..3 in Amemiya (985, p.), he asympoic disribuion of θˆ is given by Equaion (3). (8) 86
Nawaa, Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion 3. MONTE CARLO STUDY In his secion some Mone Carlo resuls are presened for he BC MLE, he newly proposed esimaor, and Powell s esimaor. The behaviors of he esimaors under boh homoscedasiciy and heeroscedasiciy are sudied. The basic is z β + x + u, z =, y, =,,..., T. (9) = β y Noe ha when is given, β and β are obained by he leas-squares mehod. The BC MLE and he proposed esimaor are calculaed by he following scanning mehod (Nawaa 994; Nawaa and Nagase 996). i) Choose < < 3 <... < n from. o. wih an inerval of.. ii) Calculae ˆ β ( ) and ˆ β ( ) for each by he leas-squares mehod. iii) For BC MLE, choose ˆBC, which maximizes he BC likelihood funcion. For he proposed esimaor, choose ˆ N, which saisfies GT ( θ i ) GT ( θi+ ) < where θ ˆ ( ), ˆ i ' = ( i, β i β( i )). iv) Choose i in he neighborhood of ˆBC and ˆN wih an inerval of., and repea seps (ii) and (iii). v) Deermine he final esimaor. For he proposed esimaor, here are wo possible problems. They are: i) Equaion () has muliple soluions, and Equaion () does no have a soluion. However, all rials have jus one soluion and he above problems do no occur in he Mone Carlo sudy. Since Powell (996) suggesed a funcion of x as he insrumen variable w, we use w = x and consider he momen resricions, E ( x u ) = and ( ) =. Since heeroscedasiciy is also considered, he generalized mehod of momen (GMM) ype esimaor is no used and Powell s esimaor is obained by minimizing. S = { x ( z β βx )} + { x ( z β βx )}. () Powell s esimaor is also calculaed by he scanning mehod over [,.]. As he proposed esimaor, here are wo possible problems for Powell s esimaor. They are: (i) S is no minimized in (.,.) and S is minimized on he boundary (i.e., =. or =.), and (ii) S becomes by muliple values of θ. Unlike he proposed esimaor, hese problems happen in many rials. Since we canno ge accurae values of he esimaor in hese rials, he resuls of Powell s esimaor are calculaed for rials wihou hese problems. 3.. Under homoscedasiciy In his secion, he behavior of he esimaors under homoscedasiciy is analyzed. { x } are i.i.d. random variables disribued uniformly on (, ). { u } are i.i.d. random variables disribued uniformly on (-5, 5). The rue parameer values are =.4, β = 5. and β =.. Values of 5, and are considered for he sample size T. The number of rials was, for all cases. The resuls are presened in Table. Noe ha he following noaion is used in he ables: STD, sandard deviaion; Ql, firs quarile; and Q3, hird quarile. For Powell s esimaor, he following noaion is also used: N, number of rials where S is minimized a =. ; N, number of rials where S is minimized a =. ; and N3, number of rials where S = becomes a muliple values of θ. The BC MLE underesimaes and has a fairly large bias for all cases. Alhough he sandard deviaions of he proposed esimaor are abou.7 imes larger han hose of he BC MLE, he biases of he proposed esimaor are much smaller. The bias almos disappears, even when T = 5. In erms of he mean squared error (MSE), he proposed esimaor is beer han he BC MLE if T. (When T= 5, he MSEs of he wo esimaors are similar values.) The BC MLE also underesimaes β and β. Alhough he sandard deviaions of he proposed esimaor for β and β are abou.5 and. imes larger han hose of he BC MLE, hey are mainly caused by he scaling effec of he ransformaion. Since he BC MLE underesimaes, he equaion z = y holds and is variaion become smaller han he rue value. This effec makes he sandard deviaions smaller. Therefore, he smaller sandard deviaion does no direcly indicae he superioriy of he 87
Nawaa, Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion esimaors for β and β. Powell s esimaor performs poorly. In many rials, we canno ge accurae values of he esimaor because of he problems menioned earlier. Moreover, alhough he biases are smaller han hose of he BC MLE, he sandard deviaions are much larger han hose of he newly proposed esimaor even for rials wihou he problems. 3.. Under heeroscedasiciy In his secion, he effec of heeroscedasiciy is analyzed. The values of x are chosen in he same way as in he previous secion. The rue parameer values are =.4, β =. 5 and β =.5. The error erms are given by u = ( + x /) ε () where { ε } represen i.i.d. random variables disribued uniformly on (-.5,.5). As before, Values of 5, and are considered for he sample size T., are considered in he Mone Carlo sudy. The resuls are presened in Tables. The BC MLE underesimaes and he biases of he BC MLE are larger han hose under homoscedasiciy for all cases. This coincides wih a previous repor (Showaler, 994) in which large biases of he BC MLE under heeroscedasiciy were described. The sandard deviaions of he proposed esimaor are abou.5 imes larger han hose of he BC MLE. However, he biases of he proposed esimaor are much smaller han hose of he BC MLE. As a resul, in erms of he MSE, he proposed esimaor is beer han he BC MLE in all cases. As before, Powell s esimaor performs poorly. In many rials, we canno ge accurae values of he esimaor. The sandard deviaions are much larger han hose of he newly proposed esimaor even for he rials wihou he problems. 4. CONCLUSION Alhough he BC is widely used in various fields, he BC MLE is no consisen. In his paper, a new robus esimaor of he power ransformaion is proposed. The esimaor is based on he firs- and hird-momen resricions of he error erms. The esimaor is consisen even under heeroscedasiciy and is asympoic disribuion is also obained. Moreover, he esimaor is easily calculaed by he leas-squares and scanning mehods. The resuls of he Mone Carlo experimens show he superioriy of he proposed esimaor over he BC MLE and Powell s esimaor. However, he performance of he esimaors may depend on he ; he findings may differ in oher s. Furher invesigaion is hus necessary o deermine he condiions under which he proposed esimaor shows superioriy REFERENCES Amemiya, T. (985). Advanced Economerics. Harvard Universiy Press, Cambridge, MA. Foser, A. M., L. Tain, and L. J. Wei (). Esimaion for he Box-Cox Transformaion Model wihou Assuming Parameric Error Disribuion. Journal of he American Saisical Associaion, 96,:97-. Khazzoom, D. J. (989). A Noe on he Applicaion of he Non-linear wo-sage leas squares esimaor o a Box-Cox Transformed Model. Journal of Economerics, 4: 37-379. Nawaa, K. (994). Esimaion of sample selecion bias s by he maximum likelihood esimaor and Heckman's wo-sep esimaor. Economic Leers, 45: 33-4. Nawaa K. and K. Kawabuchi (3). Evaluaion of he 6 revision of he medical paymen sysem in Japan by a new esimaor of he power ransformaion -An analysis of he lengh of he hospial say for caarac operaions-. mimeo, Graduae School of Engineering, Universiy of Tokyo. Nawaa, K., and Nagase, N. (996), Esimaion of sample selecion biases s. Economeric Reviews, 5: 387-4. Powell, J. L. (996). Rescaled Mehod-of-Momens Esimaion for he Box Cox Regression Model. Economics Leers, 5: 59 65. Showaler, M. H. (994). A Mone Carlo Invesigaion of he Box-Cox Model and a Nonlinear Leas Squares Alernaive. Review of Economics and Saisics, 76: 56-57. 88
Nawaa, Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Table. BC MLE, proposed esimaor, and Powell s esimaor under homoscedasiciy ( =.4) BC MLE Mean STD Q Median Q3 T = 5.956.689.49.96.3397 β 3.499.443.4383 3.77 4.47 β.534.868..464.935 T =.97.457.66.956.369 β 3.93.853.6735 3.57 3.7539 β.58.568.74.485.836 T =.96.35.74.96.36 β 3.358.5995.84 3.439 3.569 β.5.377.6.478.78 Proposed Esimaor T = 5.45..367.3985.4863 β 6.74 6.49 3.3644 4.9468 7.658 β.375.3435.35.87.59 T =.47.776.356.3994.454 β 5.6734.6783 3.968 5.3 6.645 β.64.599.34.899.84 T =.44.556.3663.48.4379 β 5.3683.7 4.65 5.67 6.4 β.45.96.488.939.5 Powell's Esimaor T = 5.55.3834.754.3985.83 β 89.886 889.88.84 4.87 33.638 β.844 3.746.3.45.669 N=56, N=, N3=4 T =.4933.359.89.379.776 β 4.97 44.587.9585 4.3445 7.7987 β.856 3.677.8.538.74 N=3, N=, N3=9 T =.473.357.879.359.79 β 7.58 69.878.9 4.75.4976 β.3.438.9.68.483 N=3, N=, N3=65 89
Nawaa, Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Table. BC MLE, proposed esimaor, and Powell s esimaor under heeroscedasiciy ( =.4) BC MLE Mean STD Q Median Q3 T = 5.56.46.49.544.848 β.763.3765.4876.6988.9863 β.7.645.594.936.39 T =.569.3.347.553.777 β.7486.77.5564.743.8933 β.994.46.7.95.56 T =.564.7.48.56.75 β.737.83.66.73.848 β.975.87.778.957.63 Proposed Esimaor T = 5.434.93.3356.453.4836 β.658.459.966.45 3.6 β.5666.78.48.483.543 T =.435.776.3598.466.457 β.658.7699.544.4843.969 β.334.783.67.64.48 T =.466.56.378.4.439 β.557.4757.45.4935.85 β.896.597.843.54.35 Powell's Esimaor T = 5.453.336.433.3869.75 β 3.34 4.93.8.783 3.66 β.8866 3.677.7.89.68 N=79, N=, N3=7 T =.485.374.36.987.684 β.938 3.84.667.79 3.33 β.4593.78.86..863 N=34, N=, N3=64 T =.48.99.744.49.37 β.9569.46.333.7966 3.787 β.648.4834.343.59.68 N=5, N=, N3=6 9