Nonparametric Identification of the Production Functions

Transcription

1 Proceedgs of he World Cogress o Egeerg 2011 Vol I, July 6-8, 2011, Lodo, U.. Noparamerc Idefcao of he Produco Fucos Geady oshk, ad Aa ayeva Absrac A class of sem-recursve kerel plug- esmaes of fucos depedg o mulvarae desy fucoals ad her dervaves s cosdered. The approach eables o esmae he produco fuco, margal producvy ad margal rae of echcal subsuo of pus. The pecewse smoohed appromaos of hese esmaes are proposed. The ma pars of he asympoc mea square errors AMSE of he esmaes are foud. The resuls are geeralzed o he produco fucos wh he lagged values of he oupu. Ide Terms Almos surely covergece, kerel recursve esmaor, mea square covergece, pecewse smooh appromao. I. INTRODUCTION NUMEROUS sascal problems such as defcao, classfcao, flerg, predco, ec. are coeced o esmao of cera characerscs of he followg epressos: J = H {a }, {a 1j }, = 1, s, j = 1, m = = H a, a 1j. 1 Here R m, H : R m+1s R 1 s a gve fuco, a 0j = a = a 1,..., a s, a 1j = a 1j 1,..., a 1j s, a = g yf, ydy, = 1, s, a 1j = a, = 1, s, j = 1, m, j g 1,..., g s are he kow Borel fucos,, R 1 f, s a ukow probably desy fuco p.d.f. for he observed radom vecor Z = X, Y R m+1. If g y 1, he a = f, ydy = p, p s he margal p.d.f. of he radom varable X, ad fy = f, y/p s he codoal p.d.f. Here are he well kow eamples of such fucos: Mauscrp receved March 6, Ths work was suppored by he Russa Foudao for Basc Research projec o ad FP Scefc ad scefc-pedagogcal cadres of ovave Russa, projec o A. ayeva s wh Ieraoal Maageme Isue, Tomsk Polyechc Uversy, Russa, emal: olz@omske.ru., G. oshk s wh Tomsk Sae Uversy, Deparme of Appled Mahemacs ad Cyberecs, 36, Le sr., Tomsk, Russa ad wh Deparme of Iformazao Problem, Tomsk Scefc Ceer SB RAS, emal: kgm@mal.su.ru. he codoal al momes µ m = y m fy dy, Ha 1, a 2 = a 1 /a 2, m 1, g 1 y = y m, g 2 y = 1; µ 1 = r s he regresso le; he codoal ceral momes V m = y r m fy dy, g 1 y = y, g 2 y = y 2,..., g m y = y m, g m+1 y = 1; V 2 = D s he codoal varace; he codoal coeffce of skewess β 1 = EY r 3 [DY ] 3/2, b = a /a 1, g y = y 1, Ha 1, a 2, a 3, a 4 = b 4 3b 3 b 2 + 2b 3 2/b 3 b 2 2 3/2 ; he sesvy fucos T j = r j, g 1 y = 1, g 2 y = y, H a 1, a 2, a 1j 1, a 1j 2 = a1j 1 a 2 a 1a 1j 2 a 2 2 II. PROBLEM STATEMENT = b 1j 2. Take he followg epresso as a esmae of he fucoal a = a 0j r = 0 ad s dervaves a 1j r = 1 a a po : a rj = 1 gy h m+r rj X. 2 Here Z = X, Y, = 1,, s he m + 1-dmesoal radom sample from p.d.f. f,, s a sequece of posve badwdhs edg o 0 as, 0j u = u = m u s a m-dmesoal mulplcave fuco whch does o eed o possess he properes of p.d.f., 1j u = u u j, gy = g 1 y,..., g s y, a rj = a rj 1,..., arj s. Noe ha 2 ca be compued recursvely by 1 [ a rj a rj = a rj 1 1 gy h m+r rj X h ]. 3 Ths propery s parcularly useful whe he sample sze s large sce 3 ca be easly updaed wh each addoal observao. ISSN: Pr; ISSN: Ole

2 Proceedgs of he World Cogress o Egeerg 2011 Vol I, July 6-8, 2011, Lodo, U.. The recursve kerel esmae of p he case whe m = 1, s = 1, gy = 1, Ha 1 = a 1 was roduced by Wolvero ad Wager [1] ad apparely depedely by Yamao [2], ad has bee horoughly eamed [3]. The sem-recursve kerel esmaes of codoal fucoals b = b 1,..., b s 1, b = a /p = g yfy dy a a po are desged as g s = 1 gy X h m b = = a 1 X p = a0j a 0j s. h m Such esmaes are called sem-recursve because hey ca be updaed sequeally by addg era erms o boh he umeraor ad deomaor whe ew observaos became avalable. If g 1 y = y s = 2, we oba sem-recursve kerel esmaes of he regresso le [4] [6]. Weak ad srog uversal cossecy of such esmaes was vesgaed [7] [11]. For esmao of 1 we are gog o use he followg sascs { J = H a rj }, j = 1, m, r = 0, 1. 4 Plug- esmaes 4 are ofe used for he esmao of raos. There are problems coeced wh uboudedess of he esmaes a some pos see [12] for deals. Ths problems ca be solved by make usg of he pecewse smooh appromao [13] J = J 1 + δ J τ ρ, 5 τ > 0, ρ > 0, ρτ 1, δ 0 as. Deoe: III. MEAN SQUARE ERRORS sup = sup, 1 u = du R m du, T j = u j udu, j = 1, 2,.... Defo 1. A fuco H : R s R 1 belogs o he class N ν H N ν f s couously dffereable up o he order ν a he po R s. A fuco H N ν R f s couously dffereable up o he order ν for ay z R s. Defo 2. A Borel fuco A r, A 0 = A f r u du <, ad u du = 1. Defo 3. A Borel fuco A r ν, A 0 ν = A ν f A r, T j = 0, j = 1,..., ν 1, T ν 0, u ν u du <, ad u = u. Defo 4. A sequece h Hm + r + q f h + 1/h m+r+q 1 0, h λ = S λ h λ + oh λ, ISSN: Pr; ISSN: Ole λ s a real umber, S λ s a cosa depede o ; r, q = 0, 1. Defo 5. Le, X 1,..., X are vecors, ad = X 1,..., X. A sequece of fucos {H } belogs o he class Mγ f for ay possble values X 1,..., X he sequece { H } s domaed by a sequece of umbers C 0 d γ, d as, 0 γ <, C 0 s a cosa. Pu for r, q = 0, 1;, p = 1, s; j = 1, m : { } A = A = a rj ; H jr = HA/ a rj { H he se a rj } = HA ; a s+ = a, p = g yg p y f, y dy; a 1+, p = g yg p y f, y dy; L r, q = r u q u du; r, q B, p = L r, q L 0, 0 m 1 a, p ; ω rj ν = T ν ν! m l=1 ν a rj ν l ; g s y f, ydy; ; {0} f j r = 0; Q = {1} f j r = 1; {0, 1} f j r = 0 r = 1. Theorem 1 he AMSE of he esmae J. If for, p = 1, s, j = 1, m, r Q : 1 he fucos a, p N 0 R, sup a 1+, p <, sup a 1+ <, sup a 4+ < ;, sup r, f Q = {0, 1} he r N 0 R, f 1 Q he 2 he kerel fuco A mar ν lm u sup u = 0; 3 a rj N ν R, sup ν a rj <, l,,..., q = 1, m; l... q 4 he sequece h Hm + 2 mar; 5 H N 2 A; 6 {HA } Mγ, 0 γ 1/4. The AMSE of he esmae J as [ u 2 J = S m+2 mar, q B +O s m, p=1 j, k=1 r, q Q r, q, p h m+r+q < a rj <, H jr H pkq + Sν 2 ω rj ν ω p qk ν h 2ν [ ] h 2ν 2 m+2 mar. h I s mpora ha we do o eed codo 6 of Theorem 1 whe pecewse smooh appromao 5 s used. ] +

3 Proceedgs of he World Cogress o Egeerg 2011 Vol I, July 6-8, 2011, Lodo, U.. Theorem 2 he AMSE of he pecewse smooh appromao J, ν. Suppose ha codos 1 5 of Theorem 1 hold ad resrco 6 s replaced by 6 J = HA 0 or τ 4, τ s a posve eger. The as u 2 J u 2 J. The proofs are gve [14]. IV. NONPARAMETRIC SEMI-RECURSIVE IDENTIFICATION OF THE PRODUCTION FUNCTION AND ITS CHARACTERISTICS Apply he resuls o esmae he produco fuco ad s characerscs. A. Esmao of he produco fuco Le r, = 1, 2, 3 R 3 be he regresso model of he hree-facor produco fuco, a = a 1, a 2, a 1 = yf, ydy, a 2 = f, ydy = p. Here 1 > 0 s he capal pu, 2 > 0 s he labor pu, 3 > 0 s he aure pu, y > 0 s a produc, ad f, y > 0 oly f 1 > 0, 2 > 0, 3 > 0, y > 0. The Y X J = r = = a0j 1 a 0j 2 = 1 X = a 1 p. 6 Le u = u 1 u 2 u 3, A ν, sup u <, ad h H3. To fd he AMSE of he esmae r, we use Theorem 1. I vew of 1 4 codos of he heorem fucos a z, = 1, 2, ad her dervaves are couously dffereable up o he order ν for ay z R 3, ad he fuco y 4 f, ydy s bouded o R 3. If p > 0, he codo 5 s fulflled. I seems mpossble o fd a majorzg sequece d codo 6 of Theorem 1, sce he deomaor 6 may be equal o zero. I some cases we ca fd a majorzg sequece accordg o Defo 5 wh γ = 0 uder ν = 2 f, for eample, 0, ad Y < [15]. For ν > 2 we ca use he pecewse smooh appromao r : r = r 1 + δ, ν r τ ρ, τ > 0, ρ > 0, ρτ 1, δ, ν = O h 2ν + 1/h 3, δ, ν 0 as. I vew of codo 6 of Theorem 2 s eough o ake eve τ 4, ad as u 2 r = = 2, p =1 ω 1ν = T ν ν! H H p S 3 CB, p +O h 3 [ 1 h 3 ν a 1 ν 1 + S 2 ν ω ν ω pν h 2ν ] 3/2 + h 2ν, + ν a 1 ν ν a 1 ν, 3 ω 2ν = T ν ν p ν! ν + ν p 1 ν + ν p 2 ν, 3 H 1 = 1 p, H 2 = r p 2 ; B 1,1 = B 1, 2 = B 2,1 = yf, ydy, B 2, 2 = p ; C = 2 udu. y 2 f, ydy, B. Esmao of he margal producvy fuco I he case of he margal producvy fuco T 1 = r a doma sequece fdg dffcules force us o 1 use he pecewse smooh appromao T 1 : T 1 = T 1 = Y T δ T 1 τ ρ, Y h 4 X [ 11 X 1 X Y h 4 1 X 11 X ] 2, 7 11 u = 1 u 1 u 2 u 3. The kerel has o sasfy he addoal codos: sup 1 u <, lm u = 0, u α N 0 R, α = 1, 2; fucos a 1, a 2 ad her dervaves up o he order ν + 1 eed o be couous ad bouded o R 3 ; he sequece h H4. C. Esmao of he margal rae of echcal subsuo Le T j = r/ j ad MRT S 12, = T 1 /T 2 be he esmae of he margal rae of echcal subsuo of a pu 2 wh a pu 1, he deomaor T 2 s gve by 7, 11 u s replaced by 12 u = u 1 1 u 2 u 3. The pecewse smooh appromao of he esmae MRT S 12, ca be wre easly. I vew of codo 5 of Theorem 1 he codo r a / 1 p has o 2 2 hold addo o he prevous resrcos. ISSN: Pr; ISSN: Ole

4 Proceedgs of he World Cogress o Egeerg 2011 Vol I, July 6-8, 2011, Lodo, U.. V. NONPARAMETRIC SEMI-RECURSIVE IDENTIFICATION OF THE DYNAMIC PRODUCTION FUNCTION Noe ha above resuls are gve for depede observaos radom samples. The resuls ca be geeralzed o me seres. I [16] a auoregressve heeroscedasc model sasfyg geomerc ergodcy codos s cosdered. The approach allows us o esmae a dyamc produco fucos wh lagged values of he oupu. Suppose ha a sequece Y =..., 1,0,1,2,... s geeraed by a olear homoscedasc ARX process of order m, s Y = ΨY 1,..., Y m, X + ξ = ΨU + ξ, 8 X = X 1,..., X s are eogeous varables, U = Y 1,..., Y m, X, 1 1 < 2 <... < m s he kow subsequece of aural umbers, ξ s a sequece of depede decally dsrbued wh desy posve o R 1 radom varables wh zero mea, fe varace, zero hrd, ad fe fourh momes, Ψ s a ukow operodc fuco bouded o compacs. Assume ha he process s srcly saoary. Crera for geomerc ergodcy of a olear heeroscedasc auoregresso ad ARX models whc ur mply α-mg have bee gve by may auhors see for eample [17] [21]. Le Y 1,..., Y be observaos geeraed by he process 8. The codoal epecao Ψ, z = Ψu = EY U = u = EY u,, z = u R m+s we esmae by he sasc, whcs a sem-recursve couerpar of he Nadaraya Waso esmae [22], [23] smlarly o 6: u U Ψ, m+s u = Y h m+s 1 h m+s h. 9 u U Ths quay may be erpreed as he predced value based o he pas formao. Sce he observaos are depede, vesgao of he esmaes properes becomes much harder. For eample, he ma par of he Nadaraya-Waso esmae s AMSE for srogly mg s. m. sequeces was foud oly 1999 [24]; he auhors also proved ha hs esmae coverges wh probably oe. We proved [16] ha f he observed sequece sasfes he s. m. codo wh a s. m. coeffce ατ such ha 0 h τ 2 [ατ] δ 2+δ dτ < 10 for some 0 < δ < 2, he Theorems 1 3 hold. Noe ha a s. m. coeffce wh he geomerc rae sasfes codo 10. We apply 9 uder U = Y 1, X 1 1, X 2 1, X 3 1 o vesgae he depedece of Russa Federao s Idusral Produco Ide Y o he dollar echage rae X 1, drec vesme X 2, ad epor X 3 for he perod from Sepember 1994 ll March The daa are avalable from: hp:// ad hp://sophs.hse.ru/. The esmae Ψ, 4 Y 1, X 1 1, X 2 1, X 3 1 = H = = 1 1 H Y H 4 h j, = j=1 Y 1 Y 1 h 1, Y 1 Y 1 j=1 h 1 3 Xj 1 X j 1. h j+1 To fd he AMSE of he esmae Ψ, 4 u we use Theorem 2 [16]. Le f, be he saoary dsrbuo of he vecor U, Y. Suppose ha A ν, u = 4 u, sup u <, he sequece h H4, ad λ = 4. Le fucos a u, = 0, 1, ad her dervaves up o ad cludg he order ν be couous ad bouded o R 4 ; fucos y 2 fu, y dy ad y 4 fu, y dy be bouded o R 4 ; ad, moreover, y 2 fu, y dy ad y 2+δ fu, y dy be couous a he po u. The codos 1 5 of Theorem 2 [16] hold; we also suppose ha codo 6 Theorem 2 [16] holds. If p u > 0, he codo 7 Theorem 2 [16] holds oo. If he radom varables Y are uformly bouded, ad we selec a oegave kerel, he s easy o show ha Ψ, 4 u are bouded for ν = 2. By codo 8 Theorem 2 [16], hs s equvale o he esece of a majorzg sequece wh γ = 0. For ν > 2 he pecewse smooh appromao solves he problem see he prevous seco. I Table 1 he relave errors of he forecas REF obaed wh Ψ, 4 for each year from 1995 ll 2004 are gve. Toal REF s 6.68%. TABLE 1 Errors of Forecass The resul of 1998 ca be eplaed by 1998 Russa facal crss Ruble crss Augus The kerel used s he Gaussa kerel ad he badwdhs h j = 0.17ˆσ j 1/8, ˆσ j, j = 1, 2, 3, 4 are he correspodg sample mea square devaos, he cosa 0.17 s chose subjecvely. The margal producvy fuco ad margal rae of echcal subsuo are esmaed he same way o he base of 7. VI. CONCLUSION Ths work preses a ufyg approach o esmag he dyamc produco fuco ad s characerscs he margal producvy fuco, margal rae of echcal subsuo. The approacs based o plug- esmag of fucos depedg o fucoals of he jo saoary dsrbuo of he vecor of eplaaory varables ISSN: Pr; ISSN: Ole

5 Proceedgs of he World Cogress o Egeerg 2011 Vol I, July 6-8, 2011, Lodo, U.. U = Y, Y 1,..., Y m, X 1,..., X s, X = X 1,..., X s are eogeous varables, Y s a oupu produc, 2 <... < m s he kow subsequece of aural umbers. Noe ha m may be large, whle m s small. We assume ha he process Y s a olear homoscedasc srcly saoary ARX process, whch sasfes o he s. m. codo wh he geomerc rae. The plug- esmaes are semrecursve,.e., we recursvely compue oly he kerel esmaes of fucoals 3. By usg he pecewse smooh appromaos of he esmaes, we have maaged o avod he problems cocerg o he majorzg sequece s esece eeded for obag of he ma pars of he esmae s AMSE. [22] E. A. Nadaraya, O Regresso Esmaes, Theor. Prob. App., vol. 19, o. 1, pp , [23] G. S. Waso, Smooh regresso aalyss, Sakhya. Ida J. Sas., vol. A26, pp , [24] D. Bosq, N. Cheze-Payaud, Opmal Asympoc Quadrac Error of Noparamerc Regresso Fuco Esmaes for a Couous-Tme Process from Sampled-Daa, Sascs, vol. 32, pp , REFERENCES [1] C. T. Wolvero ad T. J. Wager, Recursve esmaes of probably deses, IEEE Tras. Sys. Sc. ad Cybere., vol. 5, o. 3, pp , [2] H. Yamao, Sequeal esmao of a couous probably desy fuco ad mode, Bulle of Mahemacal Sascs, vol. 14, pp. 1 12, [3] E. J. Wegma ad H. I. Daves, Remarks o some recursve esmaes of a probably desy fuco, A. Sas., vol. 7, o. 2, pp , [4] J. A. Ahmad ad P. E. L, Noparamerc sequeal esmao of a mulple regresso fuco, Bull. Mah. Sas., vol. 17,o. 1 2, pp , [5] Buldakov, V.M. ad G.M. oshk, O he recursve esmaes of a probably ad a regresso le, Problems Iform. Tras., vol. 13, pp , [6] L. Devroye ad T. J. Wager, O he L 1 covergece of kerel esmaes of regresso fucos wh applcaos dscrmao, Z. Wahrsch. Verw. Gebee, vol. 51, pp , [7] A. rzyźak ad M. Pawlak, Almos every covergece of a recursve regresso fuco esmae ad classfcao, IEEE Tras. Iform. Theory, vol. IT 30, pp , [8] W. Greblck ad M. Pawlak, Necessary ad suffce cossecy codos for recursve kerel regresso esmae, J. Mulvarae Aal., vol. 23, pp , [9] A. rzyźak, Global covergece of he recursve kerel esmaes wh applcaos classfcao ad olear sysem esmao, IEEE Tras. Iform. Theory, vol. IT 38, pp , [10] L. Györf, M. ohler ad H. Walk, Weak ad srog uversal cossecy of sem-recursve kerel ad parog regresso esmaes, Sas. Decsos, vol. 16, pp. 1 18, [11] H. Walk, Srog uversal powse cossecy of recursve kerel regresso esmaes, A. Is. Sas. Mah., vol. 53, o. 4, pp , [12] H. Cramér, Mahemacal Mehods of Sascs. Prceo, [13] G. M. oshk, Devao momes of he subsuo esmae ad s pecewse smooh appromaos, Sbera Mah. J., vol. 40, o. 3, pp , [14] A. V. aeva ad G. M. oshk, Recurre Noparamerc Esmao of Fucos from Fucoals of Muldmesoal Desy ad Ther Dervaves, Auom. Remoe Corol, o. 3, pp , [15] G. Collomb, Esmao o paramerque de la regresso par la mehode du oyau, Thèse Doceur Igéeur, Uv. Paul-Sabaer: Toulouse, [16] A. V. aeva ad G. M. oshk, Sem-recursve oparamerc defcao he geeral sese of a olear heeroscedasc auoregresso, Auom. Remoe Corol, vol. 71, o. 2, pp , [17] E. Masry ad D. Tjøshem, Noparamerc esmao ad defcao of olear ARCH me seres, Ecoom. Theory, vol. 11, pp , [18] Z. D. Lu, O he geomerc ergodcy of a o-lear auoregressve model wh a auoregressve codoal heeroscedasc erm, Sas. Sca, vol. 8, pp , [19] Z. D. Lu ad Z. Jag, L1 geomerc ergodcy of a mulvarae olear AR model wh a ARCH erm, Sas. Probab. Le., vol. 51, pp , [20] P. Doukha ad A. Tsybakov, Noparamerc robus esmao olear ARX models, Problems of Iformao Trasmsso, vol. 29, o. 4, pp , [21] P. Doukha, Mg: properes ad eamples. Lecure Noes Sascs, vol. 85, Sprger-Verlag, ISSN: Pr; ISSN: Ole