SOME NEW ADJUSTED RIDGE ESTIMATORS OF LINEAR REGRESSION MODEL

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1 Internatonal Journal of Cvl Engneerng and Technology (IJCIET) Volume 9, Issue 11, November 18, pp , Artcle ID: IJCIET_9_11_84 Avalable onlne at ISSN Prnt: and ISSN Onlne: IAEME Publcaton Scopus Indexed SOME NEW ADJUSTED RIDGE ESTIMATORS OF LINEAR REGRESSION MODEL Kayode Aynde Department of Statstcs, Federal Unversty of Technology, Akure, Ondo State, Ngera. Adewale F. Lukman Department of Mathematcs, Landmark Unversty, Omu-Aran, Kwara State, Ngera. Samuel O. Olarenwaju and Omokova M. Attah Department of Statstcs, Unversty of Abuja, Abuja, Ngera. ABSTRACT The rdge estmator for handlng multcollnearty problem n lnear regresson model requres the use the basng parameter. In ths paper, some new adjusted rdge parameters whch do not requre the basng parameter are proposed. The performances of the proposed Adjusted Rdge Estmators are compared wth a recently proposed Adjusted Rdge Estmator, Generalzed Rdge Regresson Estmator (E), Ordnary Rdge Regresson Estmator (E) and Ordnary Least Square estmator (E) va Monte Carlo study by countng the number of tmes each estmator has smallest Mean Square Error (MSE) n ten thousand (1,) replcatons. The proposed Adjusted Rdge Estmator s most effcent especally when multcollnearty s severe and the error varance s hgh. Keywords: Adjusted Rdge estmator, Generalzed Rdge Estmator, Ordnary Rdge Estmator Cte ths Artcle: Kayode Aynde, Adewale F. Lukman, Samuel O. Olarenwaju and Omokova M. Attah, Some New Adjusted Rdge Estmators of Lnear Regresson Model, Internatonal Journal of Cvl Engneerng and Technology, 9(11), 18, pp INTRODUCTION In lnear regresson model, one of the crucal condtons for the applcaton of Ordnary Least Squares () estmator to estmate the model parameters s that the explanatory varables are not strongly or perfectly correlated. Wth strong correlatons or lnear relatonshp among the explanatory varables, multcollnearty problem arses. In practce, ths problem s nherent n edtor@aeme.com

2 Kayode Aynde, Adewale F. Lukman, Samuel O. Olarenwaju and Omokova M. Attah most economc functons due to the nature of economc varables movng together over tme (Kuotsoyanns, 3). It s therefore common n tme seres data, even though, t mght occur n cross-sectonal data as well. For example n tme seres data, exchange rate and nflaton rate tends to ncrease together as tme ncreases. There s no conclusve evdence as regards the degree of multcollnearty that wll affect the parameter estmates when collnearty s present (Kuotsoyanns, 3). The serousness of ths problem s a functon of the degree of ntercorrelaton. The use of estmator n ths stuaton produces unstable and mprecse parameter estmates, questonable predctons and nvald statstcal nferences about the model parameters (Kuotsoyanns, 3). Varous correctve measures have been dscussed n lterature to handle the problem of multcollnaerty. Brown et al. (1973) suggested that the data dsaggregaton wll reduce the level of multcollnearty. Gujarat (1995) suggested that ncreasng the sample szes wll reduce the degree of multcollnearty. However, n practce, these are not often feasble but the possblty cannot be overlooked. A smple way to handle ths problem s to drop one of the collnear varables. Johnston (197) stated that deletng a relevant varable may lead to specfcaton bas. Based estmaton technques such as prncpal component regresson (Massey, 1965), Sten estmator (Sten, 1956), rdge regresson (Hoerl and Kennard, 197), Lu estmator (Lu, 1993) had been suggested n lterature to handle ths problem. In ths study, rdge regresson estmator proposed by Hoerl and Kennard (197) s consdered whch requres the addton of a postve constant, k, s added to the dagonal elements of the matrx to llcondton the matrx so as to reduce multcollnearty whch n turn makes the mean squared error (MSE) for rdge regresson to be smaller than the MSE of. Dfferent estmators for k have been suggested by dfferent authors at dfferent perod of tmes. These nclude Hoerl and Kennard (197), Hoerl et al. (1975), McDonald and Galarneau (1975), Lawless and Wang (1976), Hockng et al. (1976), Dempster et al. (1977), Wchern and Churchll (1978), Gbbons (1981), Nordberg (198), Saleh and Kbra (1993), Haq and Kbra (1996), Sngh and Tracy (1999), Kbra (3), Khalaf and Shukur (5), Alkhamset al. (6), Alkhams and Shukur (8), Munz and Kbra (9), Dorugade and Kashd (1), Mansson et al. (1), Khalaf (13), Ghadhan and Mohamed (14), Dorugade (14), Lukman and Aynde (16), Adnan et al. (16) and Dorugade (16). Dorugade (16) modfed the Ordnary Rdge Regresson estmator and provded an estmator whch avods computng the rdge parameter, k. The modfed estmator s referred to as adjusted rdge parameter. It s most effcent when multcollnearty s severe. Ths study also provdes some adjusted estmators followng the dea of Dorugade (16). The performances of the proposed estmators wth some exstng ones were compared. Ths artcle s organzed as follows. Secton dscusses the lnear regresson model and ther estmators ncludng the proposed ones. Smulaton study s provded n Secton 3 and ts result dscussed n Secton 4. Fnally, a concludng remark s made n secton 5... MODEL AND ESTIMATORS Consder the multple lnear regresson model: y=xβ+ u (1) Where y s an n 1 vector of response varable, X s an n p full rank matrx of known regressors varables augmented wth a column of ones. β s p 1 vector of the unknown regresson coeffcents and u s the nx1 vector of error terms such that u~(,σ I ) and I s an nxn dentty matrx. The estmator of β s defned as: edtor@aeme.com

3 Some New Adjusted Rdge Estmators of Lnear Regresson Model β =(X X) X y () Assume that the response varable y s centered and the regressors X s are standardzed. Let and T be the matrces of egen values and egen vectors of X X respectvely such that X XT = = dagonal ( λ1, λ, λp), where λ1 represents the th egenvalue of X X and T = =Ip. The equvalent model for equaton (1) s y = Zα + u (3) Where Z = XT such that = and α =T β. The estmator of α s defned as: α = ( Z Z) -1 Z Y = -1 Z Y (4) The relatonshp between the estmator of β and α s gven as: β= Tα. (5) To crcumvent the problem of multcollnearty and mprove the estmator, Hoerl and Kennard (197) suggested the rdge estmator as an alternatve method of parameter estmaton by addng a basng parameter, k, to the dagonal elements of the X X matrx n equaton (). The Generalzed rdge regresson estmator (E) was suggested by them requre varyng rdge parameter to each regressors n the dagonal elements of X X matrx. Stephen and Chrstopher (11), among many others, clamed that the estmator yelds a smaller MSE when compared wth n the presence of multcollnearty. The E of α s defned by Dorugade (16) as: 1 α = ( I K( Λ + K) ) αˆ (6) Where K = dag(k1, k kp),k, = 1,,,p. Hence, E for s ˆ β = ˆ GR Tα GR The mean square error of E s λ k ˆ α σ (8) ) p p = 1 ( λ + k ) = 1 ( λ + k MSE ( ) = ˆ + E reduces to Ordnary Rdge Regresson Estmator (E) by addng a fxed rdge parameter to the dagonal elements of the X X matrx. That s when k1= k =kp = k and k. The mean square error of E becomes λ ˆ α σ (9) k) p p = 1 ( λ + k) = 1 ( λ + MSE (! ) = ˆ + k The MSE of becomes MSE of when k=. The MSE of s gven as: p 1 MSE ( "# ) = ˆ σ (1) λ = 1 Some of the well-known methods of estmatng the rdge parameter are presented as follows. Hoerl and Kennard (197) for the E proposed $ %&' = () * ' ) (11) They also suggested estmatng rdge parameter k by takng the maxmum (Fxed Maxmum) of +. Ths s provded as: (7) 84 edtor@aeme.com

4 Kayode Aynde, Adewale F. Lukman, Samuel O. Olarenwaju and Omokova M. Attah $, -. %& = () ) (1)./(* ' ) The non-lnear functon of the rdge parameter k s a major drawback of the rdge regresson estmator (Lu, 1993; Dorugade, 16). Ths makes t very dffcult to choose a value for k even though dfferent authors have suggested dfferent technques of estmaton. Dorugade (16) suggested a modfcaton of the Ordnary Rdge Regresson estmator where calculatng the rdge parameter k can be avoded. The proposed estmator was obtaned by addng the dagonal elements of the correlaton coeffcent between X and Y. The adjusted rdge regresson estmator (ARRE) suggested by Dorugade (16) s gven as: 1 α 1 = ( I C( Λ + K) ) αˆ (13) Or 1 α 1 = ( Λ + C) Z Y (14) ˆ Tα AR 1 ( ). 5 dag Z Where C= ' Y The ARRE of s ˆ β = The MSE of ARRE s gven as: MSE( ˆ α ) = p ˆ σ λ + ( ˆ α c ) ( λ c ) = 1 + Whereαˆ, =1,, p s the th element of estmator of α and σ establshed that the MSE (α ) ˆ ) ( ˆ (15) (16) Y Y ˆ α Z Y =. It was n p MSE α f and only f ( λ + ) k ( + k) c.1. Proposed Adjusted Rdge Regresson Estmators Followng Dorugade (16), the followng estmators are proposed: λ. c α 13 = Λ + ) 1 ( C ) Z Y (17) Where C= ' 1/ k dag ( Z Y ) for =1,,3 8. The value of k + s defned as follows: k =, k =p, k 5 =n, k 6 =np, k 7 =p p, k 8 =n p, k 9 =(np) p, k : =(np) p+5 It should be noted that wth k =, the estmator becomes the adjusted rdge estmator proposed by Dorugade (16) and that the k ncreases as ncreases. Thus, C I n as k ; Ths s often the case for = 4, 5, 6, 7 and 8. More so, the addton of 5 to p n k : s a further attempt to force C8 to an dentty matrx. 3.. SIMULATION STUDY The performances of the proposed adjusted rdge estmators are compared wth adjusted rdge estmator proposed by Dorugade (16), E, E by Hoerland Kennard (197) and estmator va Monte Carlo study. Tme seres processor (TSP) software was used to wrte the programme for the smulatons study. The mean square error of the estmators was compared edtor@aeme.com

5 Some New Adjusted Rdge Estmators of Lnear Regresson Model at varyng degree of multcollnearty, sample szes and error varances as recently done by Dorugade (16). To generate the varyng degree of multcollnearty among the regresors, the procedure adopted by McDonald and Galarneau (1975), Wchern and Churchll (1978), Gbbons (1981) and Dorugade (16) was used. X ;3 =(1 ρ )? )Z ;3 +ρz ;A, t=1,, 3 n. =1,, p. (18) Where Z ;3 are ndependent standard pseudo-random numbers, ρ represents the correlaton between any two regressors taken as,,, 5, 9, 99 and 999; p s the number of regressors taken to be ether three (3) or seven (7). The dependent varable for the n observatons are determned usng the model Y + =β B +β X 3 +β X 3 + +β A X 3A +U + t=1,,,n ; (19) WhereU 3 ~N(,σ ). The true values of the parameter when p=3 are β B =14, β =5, β = and β 5 =6. When p=7, the values of β were β =.4, β =.1, β 5 =, β 6 =., β 7 =.5, β 8 =.3, β 9 =.53. Sample szes were vared between 1,, 3, 4, 5 and 1. Ten thousand smulatons are run for dfferent values of σ =.5, 1, 9, 5 and 1. The mean square error of the estmators at each replcaton was computed usng the followng equaton: p ˆ ( β ) MSE( ˆ) β = β = 1 Where the estmator βˆ provdes the th estmate of β; and β s the true value of the parameter prevously mentoned. The number of tmes at whch each estmator has the smallest mean square error, MSE (βˆ ), n ten thousand replcaton s counted. A sample of ths s provded n Table 1 and when n=1 and respectvely. Furthermore, for ease of results presentaton on the performance of the estmators, the followng classfcatons were done. The sample sze (n) was classfed nto small (1 and ), moderate (3, 4 and 5) and hgh (1). The multcollnearty level were classfed nto moderate (N= and ), hgh (N= and 5) and severe (N=9, 99 and 999). Also, the error varances were classfed nto low (O =.5 and 1), moderate (O =9 and 5) and hgh (O =1). The number of tmes each estmator has the hghest number of smallest MSE on the bass of the classfcatons are counted. Thus, the hgher the number the better the estmator. Moreover, each combnaton of classfcaton has ts expected number of counts. For nstance, the expected number of counts wth small sample (1 and ), moderate multcollnearty (N= and ) and low level of error varance (O =.5 and 1) s eght ( =8). 4.. RESULTS AND DISCUSSION The results from the smulaton study are presented pctorally n Fgure 1 to 1. From Fgure 1 to 1, t was observed that as the sample szes, level of autocorrelaton and standard error ncreases the performances of some estmators are affected. Moreover, AR8 and perform well and occasonally. The results of the hghest number of smallest MSE on the bass of the dfferent classfcaton s provded n Table 3. From Table 3, the generalzed rdge regresson () estmator performs consstently well when the multcollnearty level s moderate at the dfferent level of error varance. The proposed estmator, AR8, performs especally when the multcollnearty level s moderate and severe coupled wth both moderate and hgh level of varance. Irrespectve of the sample szes. The performance s not satsfactory when the error varance s small but the adjusted rdge estmator,, proposed by Dorugade () 84 edtor@aeme.com

6 Kayode Aynde, Adewale F. Lukman, Samuel O. Olarenwaju and Omokova M. Attah (16) perform consstently well under ths condton. Ocassonally, the proposed estmator, AR, competes favourably wth especally when the number of regressors ncreased. and AR8 perform equally especally when the sample sze s low and there s hgh varance. It was also observed that AR7 and AR8 perform equally especally when the sample szes and the number of regressors ncreases. It was observed that ncreasng the multcollnearty level and error varances mproves the performance of adjusted rdge regresson estmator over the exstng ones. Table 1: Number of tmes each estmator has smallest MSE n ten thousand replcatons when n=1 P p=3 p=7 Estmator Q R AR AR AR AR AR AR AR AR AR AR AR edtor@aeme.com

7 Some New Adjusted Rdge Estmators of Lnear Regresson Model AR AR AR AR AR AR AR AR AR AR AR6 4 1 AR AR Table : Number of tmes each estmator has smallest MSE n ten thousand replcatons when n= P p=3 p=7 Estmator Q R AR edtor@aeme.com

8 Kayode Aynde, Adewale F. Lukman, Samuel O. Olarenwaju and Omokova M. Attah AR6 7 7 AR AR AR AR AR AR AR AR AR AR AR AR AR AR AR edtor@aeme.com

9 Some New Adjusted Rdge Estmators of Lnear Regresson Model AR AR AR AR AR AR AR Table 3: Estmator and the number of tmes each has hghest number of smallest MSE based on the each classfcaton Q R Small Moderate Hgh Small Small sample sze Moderate Sample sze Hgh Sample sze p=3 Multcollnearty level Moderate Hgh Severe Moderate Hgh Severe Moderate Hgh Severe (4) (4) (3) (4) AR8(1) () AR8() (3) (1) () AR8() (1) (7) AR8(8) (1) AR8(3) () AR(4) AR8() (1) (1) (1) () AR8(1) (6) (6) (1) (11) AR8(6) (6) (7) AR(3) AR8() p=7 (6) (4) AR8() Moderate (4) AR8(4) AR8(8) AR8(1) (1) Hgh () AR8() AR8(4) AR8(6) (6) (1) () AR8 (1) (5) AR8 (1) (4) AR(3) AR8(5) (7) AR8(5) (5) AR8(1) (3) (13) AR(1) AR8(1) (4) () () AR8 (18) (4) (4) AR8(9) () () (6) AR(9) AR8(3) AR8(18) NOTE: Estmator wth the hghest number of counts s bolded () (1) AR8(1) (3) AR8(1) (1) () AR8(1) (4) AR8(9) () () () (4) AR8 (6) AR8 (3) () AR(4) AR6() AR8(4) AR8() AR6(1) edtor@aeme.com

10 Kayode Aynde, Adewale F. Lukman, Samuel O. Olarenwaju and Omokova M. Attah Number of tmes each esmators has smallest MSE Sample Szes and Level of Autocorrelaton AR AR6 Fgure 1: Number of tmes each estmator has smallest MSE n ten thousand replcatons when p=3 and Q=S.U Number of tmes each estmators has the smallest MSE Sample szes and Level of Autocorrelaton AR AR6 AR7 AR8 Fgure : Number of tmes each estmator has smallest MSE n ten thousand replcatons when p=3 and Q=V edtor@aeme.com

11 Fgure 4: Number of tmes each estmator has smallest MSE n ten thousand replcatons when p=3 Some New Adjusted Rdge Estmators of Lnear Regresson Model Number of tmes each estmators has smallest MSE Sample szes and Level of Autocorrelaton AR AR6 AR7 AR8 Fgure 3: Number of tmes each estmator has smallest MSE n ten thousand replcatons when p=3 and Q=W Number of tmes each estmators has smallest MSE Samples Szes and Level of Autocorrelaton AR AR6 AR7

12 Kayode Aynde, Adewale F. Lukman, Samuel O. Olarenwaju and Omokova M. Attah Number of tmes each estmators has smallest MSE Sample szes and Level of Autocorrelaton AR Fgure 5: Number of tmes each estmator has smallest MSE n ten thousand replcatons when p=3 and Q=VS Number of tmes each estmator has smallest MSE Sample szes and Level of Autocorrelaton AR AR6 Fgure 6: Number of tmes each estmator has smallest MSE n ten thousand replcatons when p=7 and Q=S.U Number of tmes each estmators has smallest MSE Sample szes and Level of Autocorrelaton AR Fgure 7: Number of tmes each estmator has smallest MSE n ten thousand replcatons when p=7 and Q=V edtor@aeme.com

13 Some New Adjusted Rdge Estmators of Lnear Regresson Model Sample szes and Level of Autocorrelaton AR Fgure 8: Number of tmes each estmator has smallest MSE n ten thousand replcatons when p=7 and Q=W Number of tmes each estmators has smallest MSE Sample szes and Level of Autocorrelaton AR AR6 Fgure 9: Number of tmes each estmator has smallest MSE n ten thousand replcatons when p=7 and Q=U Axs Number of tmes each estmator has smallest MSE Sample szes and level of Autocorrelaton AR Fgure 1: Number of tmes each estmator has smallest MSE n ten thousand replcatons when p=7 and Q=VS 85 edtor@aeme.com

14 Kayode Aynde, Adewale F. Lukman, Samuel O. Olarenwaju and Omokova M. Attah 5.. CONCLUSION The proposed adjusted rdge estmator s superor to other exstng estmators especally when the multcollnearty level s severe and error varance s moderate or hgh provded the sample sze s small, moderate or hgh. Generalzed rdge regresson estmator performs well when there s moderate multcollnearty, moderate and hgh varances provded the sample sze s moderate or hgh. Ocassonally, the proposed compete favourably wth t. Adjusted rdge estmator by Dorugade (16) performs consstently well when the multcollnearty level s severe and the error varance s small provded the sample sze s small. Fnally, under the condton that the multcollnearty level s severe, the rdge regresson estmator that nvolves estmatng the rdge parameter, k, can be avoded and replaced wth the adjusted rdge regresson estmator. REFERENCES [1] Adnan, K., Yasn, A. and Asr, G. (16). Some new modfcatons of Kbra s and Dorugade s methods: An applcaton to Turksh GDP data. Journal of the Assocaton of Arab Unverstes for Basc and Appled Scences,, [] Alkhams, M., Khalaf, G. and Shukur, G. (6). Some modfcatons for choosng rdge parameters. Communcatons n Statstcs - Theory and Methods, 35(11), 5-. [3] Alkhams, M. and Shukur, G. (8). Developng rdge parameters for SUR model.communcatons n Statstcs - Theory and Methods, 37(4), [4] Brown, W.G., Fard, N. and Joe, B. S. (1973). The Oregon Bg Game Resources: An Economc Evaluaton. Oregon Agrcultural Experment Staton Specal Report #379, Corvalls. [5] Dempster, A.P., Schatzoff, M. and Wermuth, N. (1977). A smulaton study of alternatves to Ordnary Least Squares. Journal of the Amercan Statstcal Assocaton, 7, [6] Dorugade, A. V. (14). New rdge parameters for rdge regresson. Journal of the Assocaton of Arab Unverstes for Basc and Appled Scences, 15, [7] Dorugade, A. V. (16). Adjusted rdge estmator and comparson wth Kbra s method n lnear Regresson. Journal of the Assocaton of Arab Unverstes for Basc and Appled Scences, 1, [8] Dorugade, A. V. and Kashd, D. N. (1). Alternatve method for choosng rdge parameter for regresson. Internatonal Journal of Appled Mathematcal Scences, 4(9), [9] Gbbons, D. G. (1981). A smulaton study of some rdge estmators. Journal of the Amercan Statstcal Assocaton, 76, [1] Ghadban, K. and Mohamed, I. (14). Rdge Regresson and Ill-Condtonng. Journal of Modern Appled Statstcal Methods, 13() [11] Gujarat, D. N. ( 1995). Basc Econometrcs, McGraw-Hll, New York. [1] Haq, M. S. and Kbra, B. M. G. (1996). A shrnkage estmator for the restrcted lnear regresson model: Rdge Regresson Approach. Journal of Appled Statstcal Scence,3(4), [13] Hockng, R., Speed, F. M. and Lynn, M. J. (1976). A class of based estmators n lnear regresson. Technometrcs, 18 (4), [14] Hoerl, A.E. and Kennard, R.W. (197). Rdge regresson: based estmaton for nonorthogonal problems. Technometrcs, 1, [15] Hoerl, A. E., Kennard, R. W. and Baldwn, K. F. (1975). Rdge regresson: Some smulaton. Communcatons n Statstcs, 4 (), edtor@aeme.com

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