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1 Peranika J. Sci. & Technol. (1): (014) SCIENCE & TECHNOLOGY Journal homepage: hp:// The Performance of Robus Modificaion of Breusch-Godfrey Tes in he Presence of Ouliers Lim, H. A. 1, * and Midi, H. 1 1 Mahemaics Deparmen, Faculy of Science, Universii Pura Malaysia, Serdang, Selangor, Malaysia School of Applied Science and Foundaion Sudies, Kuala Lumpur Infrasrucure Universiy College, Jalan Ikram-Unien, Kajang, Malaysia ABSTRACT Auocorrelaion problem causes unduly effecs on he variance of Ordinary Leas Squares (OLS) esimaes. Hence, i is very essenial o deec he auocorrelaion problem so ha appropriae remedial measures can be aken. The Breusch-Godfrey (BG) es is he mos popular and commonly used es for he deecion of auocorrelaion. Since his es is based on he OLS esimaes, which are no robus, i is easily affeced by ouliers. In his paper, we propose a robus Breusch-Godfrey (MBG) es which is no easily affeced by ouliers. The resuls of he sudy indicae ha he MBG es is more powerful han he BG es in he deecion of auocorrelaion problem. Keywords: Auocorrelaion, ouliers, robus Breusch-Godfrey es INTRODUCTION Many saisicians employ he Ordinary Leas Squares (OLS) mehod o esimae he parameers of a linear model because of ease of compuaion. In many occasions, he assumpions of random and uncorrelaed errors are aken for graned by saisicians wihou any rigorous check. These assumpions Aricle hisory: Received: 30 May 011 Acceped: 0 January 01 addresses: lim_ha@yahoo.com (Lim, H. A.), habshahmidi@gmail.com (Midi, H.) *Corresponding Auhor may no be rue mos of he ime. The residuals may be correlaed wih he previous errors, which means ha E ( u i, u j ) 0 or cov( u i, u j ) 0 for i j. Many saisics praciioners are no aware of he consequences of he auocorrelaion problem. In specific, i ruins he imporan properies of OLS (Grassian & Boer, 1980; Whie & Brisbon, 1980). The OLS esimaors are no longer he Bes Linear Unbiased Esimaors (BLUE) in he sense ha he residual variance ˆ σ is likely o be underesimaed, he rue σ. Hence, less efficien esimaes are obained as a resul of employing an incorrec model based on he erroneous assumpion. ISSN: Universii Pura Malaysia Press.

2 Lim, H. A. and Midi, H. Addiionally, he usual and F ess of significance are no longer persuasive. These ess end o be saisically significan when in fac hey are no. The coefficien of deerminaion, R, becomes inflaed. As such, he esimaor will look more accurae as compared o is acual value. All hese problems conribue o he failure of he hypohesis esing. Hence, he auocorrelaion problem will mos likely give misleading conclusions abou he saisical significance of he esimaed regression coefficiens (Gujarai & Porer, 009). Therefore, i is very imporan o deec he presence of auocorrelaion. Many graphical mehods have been developed and hey are now available in he lieraure for deecing auocorrelaion (Davidson & MacKinnon, 1998; Gujarai & Porer, 009; Mirer, 1995; Murray, 006). However, due o he fac ha diagnosic plos can be very subjecive, i is necessary o have some saisical mehods o deec he problem of auocorrelaion. Rigorous procedures for esing he auocorrelaion of daa have also been suggesed in he lieraure (see Breusch, 1978; Durbin & Wason, 1951; Godfrey, 1978; Hosking, 1980; Hosking, 1981; Mirer, 1995; Murray, 006). Mos of hese echniques are based on he OLS esimaion. The Breusch-Godfrey (BG) es is he mos commonly used mehod o deec he presence of auocorrelaion. I was developed by Breusch (1978) and Godfrey (1978). This es has many pracical poins han oher exising ess of auocorrelaion such as Durbin-Wason Tes, Runs Tes, and Pormaneau Tes. Firs, i allows for nonsochasic regressors. Secondly, he regressors included in he regression may conain lagged values of he regressand Y, ha is Y 1, Y, ec. These lagged values may also appear as explanaory variables in he model. Thirdly, i allows he lagged values of he regressand o follow higher-order auoregressive scheme such as AR(1), AR(), ec. Oher exising ess are no applicable in hese circumsances (Breusch, 1978; Godfrey, 1978; Gujarai & Porer, 009; Mirer, 1995; Murray, 006). Suppose if he error erm Y = X β + u, (1) u u follows he ph-order auoregressive, AR(p), scheme = ρ 1 u 1 + ρ u ρ pu p + ε () where ε is a whie noise error erm ha saisfies all he classical assumpions. Then, he null hypohesis, H 0, o be esed is: H 0 : ρ1 = ρ =... = ρ p = 0 ha is, here is no serial correlaion beween u, u,..., 1 u p of any order. (3) The procedures of he BG es are as follows: Sep 1: Esimae he coefficiens of Eq. 1 by he OLS and obain he esimaed residual, Sep : Regress û on he original X and lagged values of he esimaed residuals in Sep (1). In summary, he following auxiliary regression is carried ou: û. 8 Peranika J. Sci. & Technol. (1): (014)

3 Robus Auocorrelaion Tesing in he Presence of Ouliers α ˆ ˆ ˆ + ε uˆ = X + ρ uˆ + ρ uˆ ρ puˆ 1 1 p ~ where α is he regression coefficiens of marix X. (4) Sep 3: Obain R = R from he above auxiliary regression. SSR SST R is given by:, (5) where SSR is he sum of he squared regression and SST is he sum of he squared oal of he auxiliary regression. When he sample size is large, he saisic ( n p) R is asympoically following he Chi-squared disribuion wih a degree of freedom of p, ha is ( n p) R ~ χ p. The null hypohesis is rejeced if he saisic ( n p) R exceeds he Chi-square value a he level of significan, which means a leas one ρ i in Eq. is saisically significanly differen from zero. In his aricle, a simple linear regression wih auocorrelaed errors are considered, as follows: Y µ = β 1 + β X +, (6) and he error erm is se o follow he firs-order auoregressive AR(1) scheme, u = ρ + ε, 1 < ρ 1. (7) u 1 < The auxiliary regression o be examined is herefore simplified o: uˆ α ˆ ˆ + ε (8) = X + ρu 1 ~ Since his es is based on he OLS esimaes, i is suspeced o be easily affeced by he ouliers. I is now eviden ha he oulier(s) have an unduly effec on he OLS esimaes (Midi, 1999; Habshah e al., 009; Rana e al., 008; Riazosham e al., 010). In his paper, an aemp was made o robusify he Breusch-Godfrey es by incorporaing he high efficien and high breakdown MM-esimaor (Yohai, 1987) in he formulaion of he new robus es for he idenificaion of auocorrelaion problem. We called his new es he Modified Breusch-Godfrey es (MBG). Real daa and simulaion experimens show ha he proposed MBG ouperforms he classical BG es in deecing auocorrelaion in he presence of ouliers. MATERIALS AND METHODS We have briefly discussed he Breusch-Godfrey (BG) es for auocorrelaion deecion. The BG es uses he OLS o esimae he regression coefficien, so we expec i o suffer a huge seback when ouliers are presen in he daa. Therefore, we propose a es which is robus agains ouliers. Here, we propose a new es which is a modificaion of Breusch-Godfrey es. We firs idenify he componens of he BG es ha are affeced by he ouliers and hen Peranika J. Sci. & Technol. (1): (014) 83

4 Lim, H. A. and Midi, H. replace hem wih robus alernaive. From he preceding procedures, we can see ha he BG es requires wo imes of minimizing he sum of squares residuals o ge he esimaed coefficiens. Firsly, we regress he original regression and hen regress he auxiliary regression. Edgeworh (1887) has proven ha squaring of he residual causes he leas square o become exremely vulnerable o he presence of ouliers. Therefore, he coefficiens obained are easily affeced by he oulier. The MM-esimaors inroduced by Yohai (1987), which combined high-breakdown poin and a high efficiency, are incorporaed ino he BG es. The robusified BG es is proven o minimize he impac of ouliers on he regression model. This es is called he Modified Breusch-Godfrey es, or in shor, MBG. The proposed MBG es is summarized as follows: Sep 1: Unlike he classical BG es, we esimae he coefficiens of he wo variables regression by MM-esimaor and ge he residuals, û. Sep : Regress X on he original X and u 1 or run he auxiliary regression saed in equaion (8) by he MM-esimaor. Sep 3: Find R from he auxiliary regression in Sep. R ˆ R for MBG es is defined as: SSR =, (9) ( SSE + SSR) where, SSE is he sum of squared errors and SSR is he sum of squared regression of he auxiliary regression. The null hypohesis of no serial correlaion beween µ and µ 1 will be rejeced if he saisic ( n 1) R exceeds he Chi-square value a 0.05 significan level. RESULTS AND DISCUSSION In his secion, a few real world examples and a simulaion sudy are presened o demonsrae he advanage of using he proposed Breusch-Godfrey es over he classical Breusch-Godfrey es in deecing serial auocorrelaion problems. Indexes of Real Compensaion and Produciviy Daa The firs example is he Indexes of Real Compensaion and Produciviy daa by Gujarai and Porer (009). The daa se conains 46 observaions ha give he Index of Oupu (X) and he Index of Real Compensaion per hour (Y) in U.S from 1960 o 005. The daa are shown in Table 1. In his sudy, he performances of he classical BG es and he MBG es in he original daa and conaminaed daa ses were examined. Three ypes of conaminaed daa ses were sudied. The firs ype of he conaminaed daa is he daa wih one oulier in he x direcion. An observaion in X is replaced wih an oulier; here will be a poin ha is in he far lower righ corner. The second ype of conaminaed daa is he daa wih one oulier in he y direcion. 84 Peranika J. Sci. & Technol. (1): (014)

5 Robus Auocorrelaion Tesing in he Presence of Ouliers One observaion in Y is replaced wih an oulier; here will be a poin ha i is in he far upper lef corner. The hird ype of he conaminaed daa is he daa wih a poin ha is in he far upper and far righ corner, and he oulier is in boh he x and y direcions. For his case, a good observaion is randomly replaced wih an oulier. There are many definiions of oulier. In his sudy, ouliers are considered as he values ha lay ouside he 3 deviaion scopes from is mean. Fig.1 shows a scaer plo of he original daa and he conaminaed daa. Fig. shows he scaer plo of he curren residuals (Res1) versus lagged residuals (Res(-1)) for he original daa. From he plo, i is clearly seen ha here is a posiive serial correlaion problem in he daa. Fig.1: Scaer plo for he original and conaminaed daa Fig.: Curren residuals (Res1) versus lagged residuals (Res(-1)) Peranika J. Sci. & Technol. (1): (014) 85

6 Lim, H. A. and Midi, H. TABLE 1 Original and modified Real Compensaion and Produciviy Daa, No X Y No X Y [171.7] (170) (00) {166} Noe: X = index of oupu Y = index of real compensaion per hour { } = oulier in X [ ] = oulier in Y ( ) = oulier in X and Y direcion The performances of he BG and MBG ess are evaluaed based on he p-values and he resuls are presened in Table. TABLE Auocorrelaion diagnosics for Real Compensaion and Produciviy Tes No Oulier One Oulier in X One Oulier in Y One Oulier in X and Y Direcion BG 7.667e e e e-01 MBG 5.703e e e e Peranika J. Sci. & Technol. (1): (014)

7 Robus Auocorrelaion Tesing in he Presence of Ouliers We observe from his able ha he classical BG es is able o deec auocorrelaion a 0.05 significance level if here is no oulier in he daa. However, i fails o deec he problem of auocorrelaion when he oulier occurs in he daa se. We now observe he resuls of he MBG es on he original and modified Indexes of Real Compensaion and Produciviy daa. Unlike he BG es, he MBG es can successfully deec he auocorrelaion in he presence of an oulier yielding a highly significan p-value. Economic Repor of he Presiden 198 Daa Our nex example is he economic repor of he presiden daa given by Mirer (1995). These daa conain 5 observaions ha show he relaionship beween personal consumpion expendiures (CON) and disposable personal income (DPI). We deliberaely replace a good observaion wih an oulier ino he daa se in order o ge he modified daa in verical direcion, horizonal direcion, as well as boh verical and horizonal direcions. This daa se, ogeher wih he conaminaed daa, is presened in Table 3. TABLE 3 Original and modified Economic Repor of he Presiden 198 daa No DPI(X) CON(Y) No DPI(X) CON(Y) (1570.0) 461.4(1461.4) {1400.0} [1557.5] Noe: { } = oulier in X [ ] = oulier in Y ( ) = oulier in X and Y direcions Fig.3 shows he scaer plo of he original and modified economic repors of he presiden 198 daa, while Fig.4 illusraes he scaer plo of he curren residuals (Res1) versus lagged residuals (Res(-1)) for he original daa. Mos of he residuals are bunched, in he firs and he hird quadrans, suggesing a posiive correlaion in he daa. Peranika J. Sci. & Technol. (1): (014) 87

8 Lim, H. A. and Midi, H. Fig.3: Scaer plos for he original and conaminaed daa for he Economic Repor of he Presiden daa Fig.4: Curren residuals (Res1) versus lagged residuals (Res(-1)) for he Economic Repor of he Presiden daa The resuls of he newly proposed MBG es and he classical BG es in deecing auocorrelaion for he Economic Repor of Presiden daa are presened in Table 4. Table 4 signifies ha he classical BG es can only correcly idenify he auocorrelaion problem a 0.05 significance level, i.e. when he daa are free from conaminaion alhough hey give a false deecion in he presence of ouliers. The MBG es sill successfully deecs he presence of auocorrelaion problem wih and wihou he presence of ouliers. 88 Peranika J. Sci. & Technol. (1): (014)

9 Robus Auocorrelaion Tesing in he Presence of Ouliers TABLE 4 Auocorrelaion diagnosics for he real compensaion and produciviy daa Tes No Oulier One Oulier in X One Oulier in Y One Oulier in X and Y Direcion 8.75e-0 BG 1.17e e e-01 MBG 8.765e e e e-03 Invenories and Sales in U.S. Manufacuring, daa For he las numerical example, we consider invenories and sales aken from Gujarai and Porer (009). Once again, we randomly replace a good observaion in he sales and invenories wih he ouliers and replace a coordinae paired wih a conaminaed pair in he sales and invenories direcion. The original and conaminaed daa are shown in Table 5, and he scaer po of each daa se is shown in Fig.5. I can be seen by looking a he residual plo in Figure 6 ha he daa have posiive auocorrelaion problem. TABLE 5 Original and conaminaed Invenories and Sales daa No Sales(X) Invenories(Y) No Sales(X) Invenories(Y) [80567] {579000} (547551) 31185( ) Noe: { } =oulier in X [ ] =oulier in Y ( ) =oulier in X and Y direcions Peranika J. Sci. & Technol. (1): (014) 89

10 Lim, H. A. and Midi, H. Fig.5: Scaer plo for he original and conaminaed daa for he Invenories and Sales daa Fig.6: Curren residuals (Res1) versus lagged residuals (Res(-1)) for he Invenories and Sales daa We employ he classical BG and MBG ess o he sales and invenories daa. The es resuls are exhibied in Table 6. Similar resuls are obained as in he previous examples. The power of deecion of he classical BG es becomes poor when he ouliers are presen in he daa. The MBG es is reliable in deecing he serial correlaion irrespecive of he presence of ouliers a 0.05 significance level. 90 Peranika J. Sci. & Technol. (1): (014)

11 Robus Auocorrelaion Tesing in he Presence of Ouliers TABLE 6 Auocorrelaion diagnosics for he real compensaion and produciviy daa Tes BG No Oulier.789e-09 One Oulier in X 5.043e-0 One Oulier in Y 4.937e-01 One Oulier in X and Y Direcion 5.047e-0 MBG 3.097e e e e-04 Simulaion Sudy We have seen he performance of he MBG es in he real world daa. Now, we wan o verify he resuls by checking a Mone Carlo simulaion experimen. In his sudy, we considered hree differen samples sizes, n = 0, 60 and 100, o represen he small, medium and large samples. For each sample, n good daa are generaed according o he following relaion: Y = + 4X + u (10) where, all he values of X are generaed from Uniform Disribuion, U(0,10). The error erm u is generaed by he firs-order auoregressive scheme, as follows: u =.9u ε (11) wih an iniial value of u 1 equals o. The whie noise, ε is generaed from he Normal disribuion, wih mean 0 and sandard deviaion 0.1. This auoregressive scheme is repeaed for every 10 observaions. Based on our experiences, he value of 0.9 is chosen in Eq. 11 o ensure he exisence of a high auocorrelaion problem. We would like o compare he performance of he BG and MBG ess wih 5% and 10% oulier in x, y and boh x and y direcions. For each sample size, ouliers are generaed by deleing he good observaions and subsiuing hem wih bad daa poins. The ouliers in x are represened by a uniform disribued variae x i from Uniform Disribuion U(15,0), wih y i being randomly seleced Y values which are less han 15. Similarly, he ouliers in he y direcion are represened by generaing he y i variae from a Uniform Disribuion U(50,60), wih x i being randomly chosen X values which are less han 4. Finally, he daa ses wih he ouliers in boh x and y direcions are creaed by randomly replacing good observaions wih x i from U(15,0) and yi from U(50,60). In his sudy, we se he significance level o 0.05 and in each simulaion run, here are 10,000 simulaions. Table 7 exhibis he classical BG and MBG ess. The classical BG es performs very poorly in he simulaion. Throughou he simulaion, he classical BG ess show inconsisency in deecing auocorrelaion. In fac, he BG ess fail when here are ouliers in he daa se for all he hree sample sizes. Noneheless, he MBG es performs superbly hroughou. This es is robus when he daa are conaminaed wih he ouliers. The MBG es also has higher power of deecion wih he increase of sample sizes. Thus, he MBG es ouperforms he classical BG es in every respec of conaminaion. Peranika J. Sci. & Technol. (1): (014) 91

12 Lim, H. A. and Midi, H. TABLE 7 Simulaion resuls of auocorrelaion Sample sizes Tess No Oulier X Y 5% of Ouliers 10% of Ouliers Boh X and Y X Y Boh X and Y n = 0 BG 1.643e e e e e e e-01 MBG 4.14e e e e e e e-0 n = 60 BG 5.906e e e e e e e-01 MBG 5.099e e e e e e e-05 n = 100 BG 1.759e e e e e e e-01 MBG 6.958e e e e e e e-07 CONCLUSION In his research, he commonly used es for deecing auocorrelaion has been shown o fail when ouliers are presen in any respec of he daa. Hence, we formulae a simple bu robus modificaion of he Breusch-Godfrey es o overcome he problem. Meanwhile, he comparison using he real daa and Mone Carlo simulaion experimens proved ha he proposed Breusch- Godfrey es is consisen and reliable in offering subsanial improvemens over he classical Breusch-Godfrey es and also performs excellenly in he deecion of auocorrelaion in he presence of ouliers. REFERENCES Breusch, T. S. (1978). Tesing for auocorrelaion in dynamic model. Ausralian Economic Papers, 17, Davidson, R. and MacKinnon, J. G. (1998). Graphical mehods for invesigaing he size and power of es saisics. The Mancheser School, 66, 1-6. Durbin. J., & Wason G. S. (1951). Tesing for serial correlaion in leas squares regression II. Biomerika, 38, Edgeworh, F. Y. (1887). On Observaions Relaing o Several Quaniies. Hermahena, 6, Grassian, A., & Boer, E. S. (1980). Some mehods of growh curve fiing. Mah. Scienis, 5, Godfrey, L. G. (1978). Tesing for higher order serial correlaion in regression equaions when he regressors include lagged dependen variables. Economerica, 46, Gujarai, D. N., & Porer, D. C. (009). Auocorrelaion: Wha Happens If he Error Terms Are Correlaed? In M. A. Grove, M. Io, H. Kim, P. V. Wunnava, & A. Paizis (Eds), Basic Economerics (pp ). New York: McGraw-Hill. Hampel, F. R., Ronchei, E. M., Rousseeuw, P J., & Sahel, W. A. (1986). Robus Saisics: The Approach Based on Influence Funcions. New York: John Wiley & Son, Inc. Habshah, M., Norazan, M. R., & Imon, A. H. M. R. (009). The performance of diagnosic-robus generalized poenials for he idenificaion of muliple high leverage poins in linear regression. Journal of Applied Saisics, 36(5), Peranika J. Sci. & Technol. (1): (014)

13 Robus Auocorrelaion Tesing in he Presence of Ouliers Hosking, J. R. M. (1980). The mulivariae pormaneau saisics. Journal of American Saisical Associaion, 75, Hosking, J. R. M. (1981). Lagrange-muliplier ess of mulivariae ime series model. Journal of he Royal Saisical Sociey, B43, Midi, H. (1999). Preliminary esimaors for robus non-linear regression esimaion. Journal of Applied Saisics, 6(5), Mirer, T. W. (1995). Heeroscedasiciy and Auocorrelaion. Economic Saisics and Economerics. In P. V. Wunnava, K. Lahiri, B. Bechdol, & G. Chowdhury-Bose (Eds). Unied Saes of America: Prenice-Hall. Murray, M. (006). Auoregressive Disurbances. Economerics A Modern Inroducion (pp ). D. Clinon (Eds). Unied Saes of America: Pearson Addison-Wesley. Rana, M. S., Midi, H., & Imon, A. A. H. M. R. (008). A robus odificaion of he Goldfeld-Quand es for he deecion of heeroscedasiciy in he presence of ouliers. Journal of Mahemaics and Saisics, 4(4), Riazoshams, H., Midi, H., & Sharipov, O. (010). The performance of robus wo-sage esimaor in nonlinear regression wih auocorrelaed error. Communicaions in Saisics. Simulaion and Compuaion, 39(6), Yohai, V. (1987). High Breakdown-poin and High Efficiency Esimaes for Regression. The Annals of Saisics, 15, Whie, G. C., & Brisbon, I. L. (1980). Esimaion and comparison of parameers in sochasic growh model for barn owls. Growh, 44, Peranika J. Sci. & Technol. (1): (014) 93

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