GENERALIZED METHOD OF MOMENTS CHARACTERISTICS AND ITS APPLICATION ON PANELDATA

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1 Sc.It.(Lahore),26(3), ,2014 ISSN ; CODEN: SINTE 8 GENERALIZED METHOD OF MOMENTS CHARACTERISTICS AND ITS APPLICATION ON PANELDATA Beradhta H. S. Utam 1, Warsoo 1, Da Kurasar 1, Mustofa Usma 1 ad Faz AM Elfak 2 1 Departmet of Mathematcs, Faculty of Mathematcs ad Sceces, Uversty of Lampug, Idoesa 2 Departmet of Sceces, Faculty of Egeerg, IIUM, P.O.Box 10, Kuala Lumpur, Malaysa ABSTRACT: Geeralzed Method of Momets () s a estmato procedure that allows ecoometrc models especally pael data to be specfed whle avodg ofte uwated or uecessary assumptos, such as specfyg a partcular dstrbuto for the errors. Pael data s combato of tme seres ad cross secto data that cota observatos o thousads of dvduals or famles, each observed at several pots tme. Furthermore, the Geeralzed Method of Momets estmator s obtaed by mmzg the crtero fucto by makg sample momet match the populato momet.the pot of ths research s to aalyze characterstcs estmator o pael data fxed effect models especally ubasedess, varace mmum, cosstecy, ad ormal asymptotc dstrbuted estmator propertes. Ths paper also provde the applcato of estmato o the area of Cost for Uted States Arles o Sx Frms from Keywords: Pael Data; Geeralzed Method of Momets; Ubasedess; Varace Mmum; Cosstecy; Normal Asymptotc Dstrbuted INTRODUCTION Ecoometrcs s the feld of ecoomcs that cocers tself wth the applcato of mathematcal statstcs ad the tools of statstcal ferece to the emprcal measuremet of relatoshps postulated by ecoomc theory (Greee, 2008)[1].The methodologes that combe mathematcal statstcs ad ecoomcs theory produce a ecoometrcs model. May recet studes ecoometrcs model have aalyzed pael or logtudal data sets that combe tme seres ad cross secto data sets (Johsto, 1984)[2]. Pael data sets aalyzed tme seres data o sets of frms, states, coutres, or dustres smultaeously so ts model lear may be wrtte as follows: y x z (1) there are K parameter slope t t t t x, wth = 1,2,, N show aalyss cross secto ad t = 1,2,,T show aalyss tme seres. The vector z s called dvdual effect wth z cotas a costat term ad a set of dvdual or group specfc varables (Greee, 2008)[3]. The varous cases of dvdual effect o pael data are pooled regresso, fxed effect, ad radom effect. Gujarat (2004)[4] wrote that usg pael data gvg more data ad formato so creasg degree of freedom, atcpatg heteroscedatcty problem ad provde better estmato ecoometrcs. O pael data aalyss, ofte produces over determed systems there are more momet equatos tha umber of parameters. Hase (1982)[5] troduced the estmato method to solve ths case s Geeralzed Method of Momets () by mmzg crtero weghted fucto. Geeralzed Method of Momets s coveet for estmatg terestg extesos of the basc uobserved effect model (Wooldrdge, 2001)[6]. The purpose of ths paper s to prove characterstcs of estmator o pael data especally ubasedess, varace mmum, cosstecy, ad ormal asymptotc dstrbuted propertes. To show all of the propertes, Secto 2 wll presets parameter estmato o pael data fxed effect model lear usg. Furthermore, Secto 3 wll show ubasedess estmator property, Secto 4 wll show varace mmum property, cosstecy property wll be show Secto 5 ad Secto 6 wll dscuss asymptotc ormal dstrbuted. Fally, Secto 7 wll preset estmato of estmator to estmate lear model pael data sets Cost for Uted States Arles o 6 Frms from PARAMETER LINEAR MODEL PANEL DATA ESTIMATION USING I exactly detfed cases, umber of equato momets equals to umber of parameters there wll be a sgle soluto by Method of Momets. But, whe the umber of momet codtos exceeds the umber of parameters, we caot hope to obta a estmator by settg the emprcal equvalet g ( ) of our momet codto equal to zero, (de Jog ad Ha, 2000)[7]. I other word, over determed system there s o uque soluto so t wll be ecessary to mmze crtero fucto as the crtero a weghted sum of squares q m( ) W m( ), ths estmato method s called Geeralzed Method of Momet (). The lear model pael data fxed effect s wrtte as: y t x t s K x 1 parameter vector ad t z, embodes all the observable effects ad specfes a estmable codtoal mea. Ths fxed effects approach takes to be a group-specfc costat term the regresso model (Greee, 2008). The precedg lear model Secto 1 help us to make sample momets equato as below: Usg, the crtero a weghted sum of squares s defed as the mmzg q as follows:

2 986 ISSN ; CODEN: SINTE 8 Sc.It.(Lahore),26(3), ,2014 m( ) 2 W m( ) 0. (2) From exactly detfed, we get the soluto of easly as: 1 1 ( ) Z y Z x m (3) The substtute (2) to (3) we wll get: Sce s ot radom varable ad E ( ) 0, we get Assocatve property o matrx algebra allows that: So we get [( Z ) W ( Z )] 1 ( Z ) W ( Z y ). I over detfed case (L > K), the weghted matrx W 1 ca be detty I or verse of covarace matrx V. Furthermore aalyss show that effcet ad cosstet estmator s obtaed by usg verse of asymptotc 1 covarace V, wth ad m l ( ) G. So, the estmator o pael data fxed effect model ca be wrtte as. 3. UNBIASEDNESS PROPERTY OF ESTIMATOR From the result of parameter estmato usg Secto 2, the the estmator ca be rewrtte as So, t s prove that s ubased estmator of. 4. VARIANCE MINIMUM PROPERTY OF ESTIMATOR Ecoometrcs model estmatos usg s oe of semparametrc estmato types that move away from parametrc assumptos, such as specfyg a partcular dstrbuto for the errors. Sometmes t makes some dffcultess to aalyze characterstc of a estmator. But, the semparametrc effcecy boud s assocated wth the mmum varace that plays the role of the Fsher Iformato boud a semparametrc settg as metoed by Nekpelov (2010)[8]. Sce estmator has ormal asymptotc ormal dstrbuted property ts probablty desty fucto s form of expoetal famly. Hogg ad Crag (1995)[9] defed that expoetal class oe parameter has probablty desty fucto of the cotuous type as follows: exp{p( )K(x) S(x) q( )}, a x b f (x; ) 0, otherwse : 1. Nether a or b depeds upo,, 2. p ( ) s a otrval cotuous fucto of,, 3. Each of K( x) 0 ad S(x) s a cotuous fucto of x, a < x < b Sce asymptotc property, t ca be assumed that dsturbaces have ormal multvarate dstrbuto wth mea ad matrx covarace V, ~ N(, V ) as Thus M [( Z ) V ( Z )] ( Z ) V Z. Thus

Sc.It.(Lahore),26(3),985-990,2014 ISSN 1013-5316; CODEN: SINTE 8 987 1 1 ( ) V ( ) l(2 V ) 2 2 The dervatve of l f ( ; ) wth respect to as follows: As we have defed earler secto 2, that ad m l ( ) G.

CONSISTENCY PROPERTY OF ESTIMATOR We have dscussed that estmator s obtaed by mmzg crtero fucto Ad the secod dervatve l f ( ; ) wth respect to s: We get Fsher Iformato as: Ad W s postve defte matrx as

3 Sc.It.(Lahore),26(3), ,2014 ISSN ; CODEN: SINTE ( ) V ( ) l(2 V ) 2 2 The dervatve of l f ( ; ) wth respect to as follows: As we have defed earler secto 2, that ad m l ( ) G. So we ca wrte varace of Rao-Cramer ad varace of estmator as relatoshp as follows: Sce varace of estmator less tha Rao-Cramer lower boud the t s prove that varace of estmator has varace mmum. 5. CONSISTENCY PROPERTY OF ESTIMATOR We have dscussed that estmator s obtaed by mmzg crtero fucto Ad the secod dervatve l f ( ; ) wth respect to s: We get Fsher Iformato as: Ad W s postve defte matrx as dscussed Newey (1985). It must frst be establshed that q ( ) coverges to a value q ( ), 0 So the Rao-Cramer lower boud s: So, q ( ) coverges to 0. For the proof of p lm reader see Greee (2008)[10]. So, estmator s cosstet estmator. 6. NORMAL ASYMPTOTICALLY DISTRIBUTED PROPERTY OF ESTIMATOR

4 988 ISSN ; CODEN: SINTE 8 Sc.It.(Lahore),26(3), ,2014 Asymptotc ormalty of estmators follows from takg a mea value expaso of the momet codtos aroud the true parameter, see (Che et al, 2002). To show ormal asymptotc dstrbuted property, the frst order codto for the estmator are: ow the quattes o the left- ad rght-had sdes have the same lmtg dstrbuto that s N[(, V ). Furthermore see Greee (2008), ad we have asymptotc ormal dstrbuto wth mea ad varace V, Let G ( ) ( m ). The G ( ) W m ( ) 0 (4) The orthogoalty equatos (4) are assumed that vector m to be cotuous at closure terval, ] [ 0 ad cotuously dfferetable at, ) so there ( 0 are ( 0, ), ad ths allows us to employ the Mea Value Theorem or t ca be wrtte as m ( ) ( ) ( )( m 0 G 0) s a pot betwee parameter 0. Substtute (4) to the (5) ad we get Usg left cacelato law by [ (5) ad the true 1 G ( ) WG ( )], obtaed ( 0) Ad multply by ( 0 ), produces 7. APPLICATION OF ESTIMATION ON PANEL DATA SETS I ths secto, we wll presets umerc aalyss o pael data sets of Cost for Uted States Arles o Sx Frms from (15 years) by has bee accessed o 20 th December Usg program R3.0.1, we get the pael data lear model about cost for Uted States arles o sx frms from wth s Y * * * 3 ad show the table as follows: T Table1: Parameter Estmato Usg Method Estmator Estmato Stadard Error Mea From the table, the estmatos of the dstrbuto varace of sample mea are , ad The measure of stadard error s flueced by stadard devato of populato ad umber of sample. Actually we should have expected the estmator to mprove the stadard errors. As a comparso, wll be presets parameter estmato usg Feasble Geeralzed Least Square (FGLS) method s preseted as below: Table2: Parameter Estmato Usg FGLS Stadard Method Estmator Estmato Error Mea FGLS So, we have pael data lear model about cost for Uted States arles o sx frms from usg FGLS s Y * * * 3. From the table we ca say that estmato for 1 s wth stadard error mea Whle the estmato for 2 s wth stadard error mea ad for 3 s wth stadard error mea I fact, the stadard error of mea of ad by usg s bgger tha usg FGLS but stadard error of mea of

Sc.It.(Lahore),26(3),985-990,2014 ISSN 1013-5316; CODEN: SINTE 8 989 by usg s smaller tha usg FGLS.

5 Sc.It.(Lahore),26(3), ,2014 ISSN ; CODEN: SINTE by usg s smaller tha usg FGLS. Ad the graph of Y ad usg s show as follows: Fgure 1: Plot Y ad Cost of Sx Arles Usg The fgure represet total cost of sx arles for fftee years so that we have 90 (ety) umber of cases. Blue le shows the real value of total costs o sx frms arles from ad red le states the estmato of total costs o sx frms arles from For example, the frst arle 1970, we have total cost of but usg the estmato we have From the fgure we ca state that estmato of total cost has closed value to real total cost. I the ceter of every hlls, the estmato s smlar wth real value. 8. CONCLUSION For the geeral case of the strumetal varable estmator, there are exactly as may momet equatos as there are parameters to be estmated. Thus, each of these are exactly detfed cases. There wll be a sgle soluto to the momet equatos, ths s called Method of Momet Estmato. But there are cases whch there are more momet equatos tha parameters, so the system s over determed. The Geeralzed Method of Momets techque s a exteso of the Method of Momets by mmzg crtero fucto as the crtero a weghted sum of squares. I fact, a large proporto of the recet emprcal work ecoometrcs, partcularly macroecoomcs ad face, has employed estmators. Based o the explaato the prevous chapters, we have that Geeralzed Method of Momets estmator o pael data lear model has characterstcs as ubasedess, varace mmum, cosstecy ad ormally asymptotc dstrbuted property. From umerc aalyss o pael data sets of Cost for Uted States Arles o Sx Frms from (15 years) usg we have model lear as Y * * * 3. The measure of stadard error s flueced by stadard Appedx 1. Y Value ad The Estmato Usg devato of populato ad umber of sample. Actually we should have expected the estmator to mprove the stadard errors. From the fgure plot Y ad, we ca state that estmato usg of total cost has closed value to real total cost, Although, we realze that estmato usg o real pael data sometmes wll be based. REFERENCES Che,., Lto, O. ad Kelegom. I,V Estmato of Semparametrc Models whe The Crtero Fucto s ot Smooth. CEMMAP workg paper. De Jog, R. ad Chrok, H The Propertes of L p - Estmators. Mchga State Uversty, USA. Greee, W.H Ecoometrc Aalyss.PretceHall, USA. Gujarat, D.N Basc Ecoometrcs. McGraw Hll, USA. Hase, P.L Large Sample Propertes of Geeralzed Method of Momets Estmators, Ecoometrca,Vol.20: Hogg, R.V. ad Crag,A.T Itroducto to Mathematcal Statstcs. Pretce-Hall, New Jersey. Johsto, J Ecoometrc Methods. McGraw Hll, USA. Nekpelov, D A Note o The Role of Regularty Codtos a Class of Models wth Iverse Weghtg. Uversty of Calfora, USA. Newey, W Geeralzed Method of Momets Specfcato Testg, Joural of Ecoometrcs Vol.29: Wooldrdge, J.M Applcato of Geeralzed Method of Momets Estmato, Joural of Ecoomc Perspectves:

6 990 ISSN ; CODEN: SINTE 8 Sc.It.(Lahore),26(3), ,2014 Arles Year Arles Year Y Y

Econometric Methods. Review of Estimation

Econometric Methods. Review of Estimation Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators