On the Least Absolute Deviations Method for Ridge Estimation of SURE Models

Transcription

1 On the Least Absolute Devatons Method for Rdge Estmaton of SURE Models By Zangn Zeebar 1 & Ghaz Shukur 1, 1 Department of Economcs, Fnance and Statstcs, Jönköpng Internatonal Busness School, Sweden Department of Economcs and Statstcs, Lnnaeus Unversty, Sweden Abstract Ths paper examnes the applcaton of the Least Absolute Devatons (LAD) method for rdge-type parameter estmaton of Seemngly Unrelated Regresson Equatons (SURE) models. The methodology s amed to deal wth the SURE models wth non-gaussan error terms and hghly collnear predctors n each equaton. Some basng parameters used n the lterature are taken and the effcency of both Least Squares (LS) rdge estmaton and the LAD rdge estmaton of the SURE models, through the Mean Squared Error (MSE) of parameter estmators, s evaluated. Key-words: SURE Models, LAD Estmaton, Rdge Regresson, Effcency, Robustness JEL Classfcaton: C15, C30, C31. 1

2 1. Introducton The problems of Seemngly Unrelated Regresson Equatons (SURE) are extensvely dscussed n the lterature, after Zellner s (196) semnal paper on SURE models. The studes of Rao, (1975), Srvastava and Gles (1987), and Chb and Greenberg (1995) on the SURE models have been nfluental, too. The asymmetry of the error terms and the multcollnearty of the predctors can become severe problems to the SURE models. Each of these two problems has also been separately nvestgated by some researchers n the context of the SURE models. However, the problems of error asymmetry and multcollnearty are frst studed outsde the SURE context. The rdge estmaton method proposed by Hoerl and Kennard (1970a,b) s the most commonly used method to deal wth the multcollnearty of the predctors when the model under the study wth the selected predctors s mantaned. The Least Absolute Devatons (LAD) estmaton method s a robust method n the presence of outlers and asymmetrc error terms (Bassat and Koenker, 1978). In real data, t s the case to have estmates beset by the aggregaton of dfferent types of problems. On one hand, the problem of asymmetrc or heavy-taled error terms n the SURE context s, though relatvely less than and separately from the multcollnearty problem, dscussed by some researchers; Kuester (1987), Koenker and Portnoy (1990), and recently Shukur and Zeebar (011a,b), to menton some. On the other hand, the problem of multcollnearty of the predctors or sngularty of the covarance matrx of the cross-equaton errors, through rdge estmaton method, s more extensvely dscussed by many researchers, for nstance; Brown and Payne (1975), Brown and Zdek (1980), Srvastava and Gles (1987), Hatovsky (1987), Frnguett (1997), Alkhams and Shukur (008), Kbra, Shukur and Zeebar (011). A stuaton where both problems of the multcollnearty of the predctors and the asymmetry of the error terms can come together n a sngle equaton model s dealt wth by Zeebar (011a,b). Ths paper generalzes Zeebar s (011a,b) LAD rdge method n order to make t applcable to the SURE models. The methodology behnd the LAD method of the rdge-type SURE estmaton s dscussed n Secton. The desgn of the Monte Carlo smulaton study s explaned s Secton 3, and the smulaton results are nterpreted n Secton 4. Fnally, Secton 5 contans a bref summary of the conclusons. Extensve smulaton results are presented n the Appendx.

3 . Methodology Consder a general system of M regresson equatons Y Xβ ε, (.1) for 1,,, M, where Y s a T 1 vector of the dependent varable, ε s a T 1 vector of random errors wth var( ε ) I, X s a T T p matrx of observatons on ndependent varables ncludng a constant term and β s a p 1 vector of unknown coeffcents to be estmated. The M equatons n.1 can be expressed n Zellner s (196) compact form where X s a TM M 1 YXΒε, (.) p block dagonal matrx of the desgn matrces of the M equatons, wth p p, each of Y and ε are TM 1 vectors and cov( ε) Σ Ψ. The M M matrx of Σ s the nonsngular covarance matrx of the contemporaneous cross-equaton errors. T Zellner s transformaton of model., that takes the cross-equaton correlatons nto account, s the model wth the GLS estmator Ψ Y Ψ XΒ Ψ ε, (.3) bg ( X Ψ X) X Ψ Y, (.4) and the ordnary rdge verson of the GLS estmator b ( XΨ Xk I ) XΨ Y, (.5) GR wth the covarance matrx, p cov( b ) ( X Ψ XkI ) X Ψ X( X Ψ XkI ) GR p p. (.6) The same LAD rdge estmaton of Zeebar (011a,b) s adopted n the context of the SURE models. In other words, the specal mxed (augmented) model Ψ Y Ψ X Ψ ε β, (.7) 0p 1 k Ip ν p1 wth the assumpton that Ψ ε ν 0 s used and, nstead of applyng the OLS E( ) TM p estmaton on the model.3 whch gves us the rdge verson of the GLS estmator.4, the LAD estmaton method s used. It s worth mentonng that the cross-equaton covarance matrx does not generally exst and an estmaton of t s usually used nstead. 3

4 The only dfference of the LAD rdge estmaton procedure of Zeebar (011a,b) compared wth the LAD rdge estmaton for the SURE models s that the varables are not standardzed n the latter. The reason for avodng the standardzaton of the varables, n the context of the SURE models, s ts effect on the estmaton of the cross-equaton covarance matrx Σ ˆ S. The canoncal form of the transformed model (.3) s Ψ Y Ψ XΒ Ψ ε V Λ α Ψ ε, (.8) where the sngular value decomposton of the matrx 1 Ψ X s Ψ X V Λ U, wth V 1 1 a p TM orthogonal matrx, Λ a p p dagonal matrx of the egenvalues of X Ψ 1 X and 1 U a p p orthogonal matrx of the egenvectors of X Ψ X, and α Uβ. The GLS estmator of α n.8 (whch s the same as the OLS estmator) s 1 1 g Λ V Ψ Y. (.9) The LAD estmaton of α n.8, s the LAD parameter estmaton of the model, Ψ Y VΛ Ψ ε α. (.10) 0p 1 k Ip ν p1 3. Monte Carlo Experment In ths secton, 17 basng parameters presented n prevous works (see Table 3.1) and ther LAD versons are used for both the LS rdge and the LAD rdge estmaton of the SURE models. The lnear scale heteroscedastcty model of Koenker and Bassett (198), y x β ( x γ 1), (3.1) j j j j s employed to generate the jth value of the dependent varable n the th equaton, for 1,, M, j 1,,, T and d ε (,, ) St ξ, Ω, α, v, where St ξ, Ω, α, v j 1j Mj M s the skew t dstrbuton defned by Azzaln and Captano (003). The process of generatng the desred parameters of the skew t dstrbuton dscussed n Shukur and Zeebar (011a,b) s followed. The condtonal quantle functon of condtonal mean functon of Y s Y Q ( x ) x β γ Q ( ) Q ( ), whereas the Y becomes E( Y x ) x β γ E( ) E( ). Let us choose β (1,1,,1) and γ (0,1,,1) to get the populaton parameter for the condtonal M 4

5 quantle functon β ( ) (1 ( )). 1 at 0.5 and for the condtonal mean Q p Β(1 E( )). 1 p. Then, the generated data of the M equatons are put nto a compact form of Zellner s (196) SURE model,.e., the desgn matrx s the block dagonal matrx of the M equatons desgn matrces, Y ( Y1,, Y M ) and β ( β1,, β M ). Table 3.1: The basng parameters used n the smulaton experment k p gg, [15]*+ ˆ 0 k ˆ 1 max{ g }, [13]* k p gλg, [3]* ˆ k ˆ 4 medan{ g }, [16]* ˆ 6 max{ } k 3 5 p 1 p g 1 ˆ ( ), [17]* p 1 p ˆ, [18]* 1 k ( ( n p) g ) k g, [18]* k ˆ 7 mn{ g }, [18]* 8 p 1 1 p k g ˆ, [18]* k 9 p 1 1 p ˆ g, [18]* k g, [18]* k ˆ 11 medan{ g }, [18]* ˆ 10 medan{ } k ˆ max{ ( n p) max g }, [18]* p 1 p ˆ 14 ( ( ) max ), [18]* 1 k n p g k ˆ 13 max{ ( n p) max g }, [18]* p 1 p ˆ 15 ( ( ) max ), [18]* 1 k n p g k ˆ 16 medan{ ( n p) max g },[18]* * The [.] refers to the number of the reference from whch the basng parameter, s taken. + The LAD verson was already proposed by Pfaffenberger and Delman (1989). Samples of szes 30, 100 and 500 are generated as follows. Models wth 4 predctors are generated from multvarate normal dstrbutons wth mean 5 and dfferent varances, but such covarances that result n a correlaton matrx wth all mutual correlatons equal, and for each of the correlaton levels 0.75, 0.90 and 0.99 between the predctors. Whle keepng the generated desgn matrx fxed, d errors and consequently the dependent varable accordng to the (3.1), are generated for each of the M equatons. Then, the process of generatng the errors and the dependent varables wth the fxed predctors (for each sample sze) s repeated 000 tmes. Y, 5

6 Addtonally, n each replcaton after calculatng 17 dfferent basng parameters n Table 3.1 (based on both the OLS and the LAD ft), and ther resulted parameter estmates, the MSE of the regresson parameter estmators s calculated for all combnatons of the sample sze, the skewness level of error terms, the correlaton level of cross-equaton error terms, the correlaton (multcollnearty) level between predctors and the number of equatons nvolved n the SURE model. Table 3.: The factors varyng n the smulaton Factor Value Sample Sze 30, 100, 500 Number of Equatons 4* Skewness Level of Errors 0, 1.5 Correlaton Level Between Equatons 0, 0.5, 0.9 Predctors Correlaton (Multcollnearty) Level 0,9, 0,99 * The smulaton results of the -equaton models studed by the authors are consstent wth those of the 4-equaton models and excluded from beng presented. Dfferent basng parameters, used n the LS rdge estmaton, are shown n Table 3.1, where ˆ ( Ψ Y Ψ XbG)'( Ψ Y Ψ Xb G) ( TM p), b G s the GLS estmator of the 1 SURE parameter, g s as defned n.9, s the th egenvalue of X Ψ X, for 1,, p. For the LAD verson of these basng parameters, frst b G s replaced by β ˆ G whch s the Generalzed Least Absolute Devaton (GLAD) estmaton of β, proposed by Shukur and Zeebar (011a) and t s used to estmate ˆ. Then, g s replaced by ˆα, whch s the LAD estmator of the parameter n.8. Based on the smulaton results of ths study for each combnaton of the factor shown n Table 3., the basng parameters that performed relatvely poorly compared to the others are excluded from beng presented n the smulaton results n Appendx. However, the authors have extensve smulaton results for all the 17 basng parameters used n the study, and those results are avalable upon request. In each case (combnaton of factors n Table 3.), for the LAD rdge method, the MSE of the non-rdge,.e., the GLAD estmator, and the MSE of the SURE parameter LAD rdge estmators correspondng to each basng parameter are compared wth the MSE resulted from the k of Pfaffenberger and Delman (1989) and shaded f they are less than the latter. The least MSE s marked for each case. A smlar acton of comparson and markng the least MSE n the LS context s performed, n a way that each MSE s compared wth the one resulted from the basng parameter of Hoerl et al. (1975). 6

7 4. Smulaton Results Though the propertes of each k tself, lke; mean, medan, varance, skewnessurtoss and the range are extensvely studed, for some reasons, they are excluded from beng presented n the Appendx. Frstly, the propertes of the basng parameters can change from one case to another. Secondly, t s an attempt to reduce the sze of the paper, due to the fact that a presentaton of the propertes of those basng parameters s not the man am of ths study but ther nfluence on the LAD rdge and the OLS rdge estmaton n the SURE context. However, some propertes of those basng parameters are a matter of nterest and occasonally dscussed, f needed. Among the basng parameters shown n Table and especally k 1, from each of the LAD and the OLS ft, are generally very small wth very low varances. Consequently, they only slghtly shrnk the estmates. In some other stuatons where the other basng parameters take larger values3 4 shrnk the estmates to a larger extent. Generally, the value of k from the LAD ft s extremely large, and those of k 5 7 and occasonally k 6, from both the LAD and the OLS ft, are large wth huge varances of the basng parameter. Based on the results of ths study, the varance of k o s much less than those of unstable k, k 5 7. Therefore, for the cases n whch a relatvely large amount of shrnkage reduces the MSE, k o s unsurpassable among those over-shrnkage basng parameters. When the MSE s hghly senstve wth any ncrease n the amount of shrnkage 1 acts better than other under-shrnkage basng parameters lke k 3 4. For stuatons where a moderate amount of shrnkage mproves the MSE reducton, basng parameters other than those of the over-shrnkage ( k o 5 7 ) and under-shrnkage ( k ) perform better. If the over-shrnkage and the under-shrnkage basng parameters are not to be used n the LAD rdge estmaton and especally k show ther adequacy for a moderate shrnkage. In such stuatons, the basng parameters; k and especally k 13 moderately shrnk the OLS rdge estmates of the SURE models generated n the smulaton study. Wth each basng parameter, the MSE of the LAD rdge estmator of the SURE parameters decreases as the level of cross-equaton correlatons ncreases, regardless of the number of equatons, sample sze or the skewness level of the error terms. For such a reducton n MSE, 7

8 the LAD rdge estmaton of the SURE models seems to be consstent wth the GLAD (nonrdge) estmaton of the SURE models. A smlar argument holds for an ncrease n the sample sze, but the MSE reducton of the non-rdge estmaton s more rapd. Such a dfference n the MSE reducton of the rdge and the non-rdge LAD estmaton method wth an ncrease n the sample sze reduces the asymptotc effcency of the LAD rdge estmaton to the GLAD estmaton. An argument smlar to the above holds n the LS context, too. As the gap between the MSEs of the rdge LAD and the GALD estmaton methods asymptotcally vanshes, the under-shrnkage basng parameters, especally k 1, perform better than other basng parameters, f the LAD rdge estmaton s to be used. Among the basng parameters used n the study for the LAD rdge estmaton, that of Pfaffenberger and Delman (1989) results n the least MSE of the SURE parameter estmator, when the SURE model lacks both cross-equaton correlatons and the asymmetry of the error terms. However, n such cases there mght be no need of usng the LAD rdge estmaton. The relatve effcency of the rdge estmators, whether the LAD rdge or the LS rdge estmator, to the effcency of the non-rdge estmators reduces wth an ncrease n the skewness of the error terms. The effcency gan and/or the reducton n the relatve effcency, s more notceable n the LAD context than t s n the LS context. The reason s the fact that the LAD method s more robust than the LS methods n the presence of asymmetrc error terms. Ths means that any ncrease n the skewness of the error terms s less benefcal for the LAD rdge estmators than t s for the LAD estmators. As the multcollnearty level ncreases, the relatve effcency of the LAD rdge estmators to the GLAD estmators ncreases, as well. However, such an effcency gan s more notceable n the LS context, meanng that the LS estmators are beset by multcollnearty more than the GLAD estmators are. Larger MSEs are expected wth more equatons nvolved n the model, for the MSE, here, s meant by the total MSE. Nevertheless, based on the smulaton results, the relatve effcency of the LAD rdge estmators to the GLAD estmators does not change consderably wth any change n the sze of the system of equatons. Fnally, the cross-equaton correlaton reduces the MSE of the LAD, LAD rdge, GLS and the LS rdge estmators. However, the relatve effcency of the rdge methods to the not-rdge methods decreases slghtly, meanng that any ncrease n the cross-equaton correlaton s more benefcal for the non-rdge methods. 8

9 5. Conclusons The LAD rdge estmator of the SURE model parameters s more effcent than the GLAD estmator when the predctors are collnear. A strong cross-equaton correlaton mght slghtly affect the relatve effcency of the LAD rdge estmator to the GLAD estmator, whereas the multcollnearty level and the skewness of the error terms have a greater effect on the relatve effcency. The more collnear the predctors are, the hgher the relatve effcency of the LAD rdge estmator to the GLAD estmator s. However, the skewness of the error terms s more benefcal for the GLAD method, hence a reducton n the relatve effcency of the LAD rdge estmator to the GLAD estmator. But then agan, t does not mean that the MSE of the LAD rdge estmator decreases wth any ncrease n the level of cross-equatons correlaton. The basng parameter k 1 slghtly shrnks the LAD rdge estmates, due to ts small value and small varance, compared to other basng parameters shown n Table 3.1. Generally 5 7 extremely shrnk the LAD rdge estmates due to ther large values and large varances. The basng parameters k 3 4 cause just a lttle shrnkage lke k 1, but they can shrnk the parameter estmates more than other basng parameters when the resduals mean squared s relatvely large compared to magntude of the components of the parameter estmate. If the basng parameters are categorzed based on ther values as the over-shrnkage and under-shrnkage basng parameters, and based on ther varances as stable and unstable o behaves as a stable over-shrnkage basng parameter, whle k 5 7 are consdered as to be unstable over-shrnkage basng parameters. The basng parameters k are stable under-shrnkage basng parameters. The effect of the scale unt of measurement appears more n k An obvous reason s ther drect dependence on the largest egenvalue of the desgn matrx. Wth extremely large egenvalues 6 are almost equal. Wth large egenvalues, such an assocaton of tendency to each other s realzed between these pars of basng parameters; { k 7 13 }, { k 8 14 }, { k 9 15 } and { k }. If an over-shrnkage basng parameter s to be used o mght be a sutable canddate. For the cases when an under-shrnkage basng parameter s to be selected 1 could be an adequate one. Other basng parameters, lke k and especally k, perform better for the LAD rdge estmaton of the SURE models when a relatvely moderate level of shrnkage s needed. 9

10 References [1] Alkhams, M., Khalaf, G. and Shukur, G., (006), Some Modfcatons for Choosng Rdge Parameters, Communcatons n Statstcs- Theory and Methods, 35: [] Azzaln, A. and Captano, A. (003), Dstrbutons generated by Perturbaton of Symmetry wth Emphass on a Multvarate Skew t Dstrbuton, Royal Statstcal Socety, Seres B, 56,. [3] Azzaln, A. and Dalla Valle, A. (1996), The Multvarate Skew-Normal Dstrbuton, Bometrca, 83, 4: [4] Bassett, G., and Koenker, R. (1978), Asymptotc Theory of Least Absolute Error Regresson, Journal of the Amercan Statstcal Assocaton, Vol. 73, No. 363: [5] Brown, P. and Payne, C. (1975), Electon Nght Forecastng, Journal of the Royal Statstcal Socety. Seres A (General), Vol. 138, No. 4: [6] Brown, P. and Zdek, J. (1980), Adaptve Multvarate Rdge Regresson, The Annals of Statstcs, Vol. 8, No. 1: [7] Chb, S., and Greenberg, E. (1995). Herarchcal Analyss of SUR model wth Extensons to Correlated Seral Errors and Tme Varyng Parameter Models, Econometrcs, 68: [8] Ekberg, J., Hammarstedt, M. and Shukur, G. (009), SUR estmaton of earnngs dfferentals between three generatons of mmgrants and natves, Annals of Regonal Scences, 45, 3: [9] Frnguett, L. (1997), Rdge regresson n the context of a system of seemngly unrelated regresson equatons, Journal of Statstcal Computaton and Smulaton, 56,: [10] Gbbons, D.G., (1981), A Smulaton Study of Some Rdge Estmators, Journal of the Amercan Statstcal Assocaton, Vol. 76, No. 373: [11] Gruber, M.H.J., (010), Regresson Estmators: a Comparatve Study, nd edton, The Johns Hopkns Unversty Press, Baltmore. [] Hatovsky, Y. (1987), On multvarate rdge regresson, Bometrka, 74, 3: [13] Hoerl, A.E., and Kennard, R.W., (1970a), Rdge Regresson: Based Estmaton for Nonorthogonal Problems, Technometrcs, Vol., No. 1: [14] Hoerl, A.E., and Kennard, R.W., (1970b), Rdge Regresson: Applcaton to Nonorthogonal Problems, Technometrcs, Vol.. No. 1: [15] Hoerl, A.E., Kennard, R.W., and Baldwn, K.F., (1975), Rdge Regresson: Some Smulatons, Communcatons n Statstcs, 4: [16] Khalaf, G. and Shukur G., (005), Choosng Rdge Parameters for Regresson Problems, Communcatons n Statstcs- Theory and Methods, 34: [17] Kbra B.M.G., (003), Performance of Some New Rdge Regresson Estmators, Communcatons n Statstcs- Theory and Methods, 3: [18] Kbra, B.M.G., Månsson K., and Shukur, G., (010), Performance of Some Logstc Rdge Regresson Estmators, to appear n Computatonal Economcs. 10

11 [19] Kbra, B. M. G., Shukur, G. and Zeebar, Z. (011), Modfed Rdge Parameters for Seemngly Unrelated Regresson Model, to appear n Communcatons n Statstcs- Theory and Methods. [0] Koenker, R. and Bassett, G., (198), Robust Tests for Heteroscedastcty Based on Regresson Quantles, Econometrca, Vol. 50, No. 1: [1] Koenker, R. and Portnoy, S., (1990), M Estmaton of Multvarate Regressons, Journal of the Amercan Statstcal Assocaton, Vol. 85, No. 4: [] Kuester, K., (1987), Asymptotc Consstency and Normalty of Least Absolute Devatons Appled to Seemngly Unrelated Regresson Systems. Techncal report, Board of Governors of the Federal Reserve System. [3] Lawless, J.F., and Wang, P., (1976), A Smulaton Study of Rdge and Other Regresson Estmators, Communcatons n Statstcs, 5: [4] Marquardt, D.W., (1970), Generalzed Inverses, Rdge Regresson, Based Lnear Estmaton, and Nonlnear Estmaton, Technometrcs, Vol, No. 3: [5] McDonald G.C., Glarneau D.I., (1975), A Monte Carlo Evaluaton of Some Rdge-Type Estmators, Journal of the Amercan Statstcal Assocaton, Vol 70, No. 350: [6] Munz, G., and Kbra B.M.G., (009), On Some Rdge Regresson Estmators: An Emprcal Comparson Communcatons n Statstcs- Smulaton and Computaton, 38: 3, [7] Pfaffenberger, R.C., and Delman, T.E., (1989), A comparson of regresson estmators when both multcollnearty and outlers are present. In: Lawrence, K.D. and Arthur, J.L., edtors, Robust Regresson: Analyss and Applcatons, New York: Marcel Dekker, Inc. pp [8] Rao, C. R. (1975). Smultaneous Estmaton of Parameters n Dfferent Lnear Models and Applcatons to Bometrc Problems, Bometrcs, 3, : [9] Shukur, G., and Zeebar, Z. (011a), "On the medan regresson for SURE models wth applcatons to 3-generaton mmgrants data n Sweden", Economc Modellng, 8, 6: [30] Shukur, G., and Zeebar, Z. (011b), Medan Regresson for SUR Models wth the Same Explanatory Varables n Each Equaton, unpublshed (submtted to Journal of Appled Statstcs). [31] Srvastava, V., and Gles, D. (1987). Seemngly Unrelated Regresson Equatons Models, New York: Marcel Dekker. [3] Zeebar, Z. (011a), Developng Rdge Estmaton Method for Medan Regresson, submtted to Journal of Appled Statstcs. [33] Zeebar, Z. (011b), A Smulaton Study on the Least Absolute Devatons Method for Rdge Regresson, to appear n Communcatons n Statstcs- Theory and Methods. [34] Zellner, A. (196). An Effcent Method of Estmatng Seemngly Unrelated Regressons and Tests for Aggregaton Bas, Journal of the Amercan Statstcal Assocaton, 57, 98:

12 Cross-Equaton Correlaton = 0.9 Cross-Equaton Correlaton = 0.5 Cross-Equaton Correlaton = 0.0 Appendx Table A.1: The MSE of the LAD Rdge and the LS rdge Estmators of the 4-Equaton SURE Model Parameters, wth Sample sze=30 and Predctors Correlaton (multcollnearty) = 0.75 Skewness = 0 Skewness = 1.5 Skewness =.75 k LAD Rdge LS Rdge k LAD Rdge LS Rdge k LAD Rdge LS Rdge k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* * k=0 means non-rdge estmaton. Shaded MSE means less than that of k o. Flled cell marks the least MSE.

13 Cross-Equaton Correlaton = 0.9 Cross-Equaton Correlaton = 0.5 Cross-Equaton Correlaton = 0.0 Table A.1: The MSE of the LAD Rdge and the LS rdge Estmators of the 4-Equaton SURE Model Parameters, wth Sample sze=30 and Predctors Correlaton (multcollnearty) = 0.90 Skewness = 0 Skewness = 1.5 Skewness =.75 k LAD Rdge LS Rdge k LAD Rdge LS Rdge k LAD Rdge LS Rdge k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* * k=0 means non-rdge estmaton. Shaded MSE means less than that of k o. Flled cell marks the least MSE. 13

14 Cross-Equaton Correlaton = 0.9 Cross-Equaton Correlaton = 0.5 Cross-Equaton Correlaton = 0.0 Table A.1: The MSE of the LAD Rdge and the LS rdge Estmators of the 4-Equaton SURE Model Parameters, wth Sample sze=30 and Predctors Correlaton (multcollnearty) = 0.99 Skewness = 0 Skewness = 1.5 Skewness =.75 k LAD Rdge LS Rdge k LAD Rdge LS Rdge k LAD Rdge LS Rdge k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* * k=0 means non-rdge estmaton. Shaded MSE means less than that of k o. Flled cell marks the least MSE. 14

15 Cross-Equaton Correlaton = 0.9 Cross-Equaton Correlaton = 0.5 Cross-Equaton Correlaton = 0.0 Table A.1: The MSE of the LAD Rdge and the LS rdge Estmators of the 4-Equaton SURE Model Parameters, wth Sample sze=100 and Predctors Correlaton (multcollnearty) = 0.75 Skewness = 0 Skewness = 1.5 Skewness =.75 k LAD Rdge LS Rdge k LAD Rdge LS Rdge k LAD Rdge LS Rdge k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* * k=0 means non-rdge estmaton. Shaded MSE means less than that of k o. Flled cell marks the least MSE. 15

16 Cross-Equaton Correlaton = 0.9 Cross-Equaton Correlaton = 0.5 Cross-Equaton Correlaton = 0.0 Table A.1: The MSE of the LAD Rdge and the LS rdge Estmators of the 4-Equaton SURE Model Parameters, wth Sample sze=100 and Predctors Correlaton (multcollnearty) = 0.90 Skewness = 0 Skewness = 1.5 Skewness =.75 k LAD Rdge LS Rdge k LAD Rdge LS Rdge k LAD Rdge LS Rdge k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* * k=0 means non-rdge estmaton. Shaded MSE means less than that of k o. Flled cell marks the least MSE. 16

17 Cross-Equaton Correlaton = 0.9 Cross-Equaton Correlaton = 0.5 Cross-Equaton Correlaton = 0.0 Table A.1: The MSE of the LAD Rdge and the LS rdge Estmators of the 4-Equaton SURE Model Parameters, wth Sample sze=100 and Predctors Correlaton (multcollnearty) = 0.99 Skewness = 0 Skewness = 1.5 Skewness =.75 k LAD Rdge LS Rdge k LAD Rdge LS Rdge k LAD Rdge LS Rdge k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* * k=0 means non-rdge estmaton. Shaded MSE means less than that of k o. Flled cell marks the least MSE. 17

18 Cross-Equaton Correlaton = 0.9 Cross-Equaton Correlaton = 0.5 Cross-Equaton Correlaton = 0.0 Table A.1: The MSE of the LAD Rdge and the LS rdge Estmators of the 4-Equaton SURE Model Parameters, wth Sample sze=500 and Predctors Correlaton (multcollnearty) = 0.75 Skewness = 0 Skewness = 1.5 Skewness =.75 k LAD Rdge LS Rdge k LAD Rdge LS Rdge k LAD Rdge LS Rdge k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* * k=0 means non-rdge estmaton. Shaded MSE means less than that of k o. Flled cell marks the least MSE. 18

19 Cross-Equaton Correlaton = 0.9 Cross-Equaton Correlaton = 0.5 Cross-Equaton Correlaton = 0.0 Table A.1: The MSE of the LAD Rdge and the LS rdge Estmators of the 4-Equaton SURE Model Parameters, wth Sample sze=500 and Predctors Correlaton (multcollnearty) = 0.90 Skewness = 0 Skewness = 1.5 Skewness =.75 k LAD Rdge LS Rdge k LAD Rdge LS Rdge k LAD Rdge LS Rdge k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* * k=0 means non-rdge estmaton. Shaded MSE means less than that of k o. Flled cell marks the least MSE. 19

20 Cross-Equaton Correlaton = 0.9 Cross-Equaton Correlaton = 0.5 Cross-Equaton Correlaton = 0.0 Table A.1: The MSE of the LAD Rdge and the LS rdge Estmators of the 4-Equaton SURE Model Parameters, wth Sample sze=500 and Predctors Correlaton (multcollnearty) = 0.99 Skewness = 0 Skewness = 1.5 Skewness =.75 k LAD Rdge LS Rdge k LAD Rdge LS Rdge k LAD Rdge LS Rdge k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* k=0* * k=0 means non-rdge estmaton. Shaded MSE means less than that of k o. Flled cell marks the least MSE. 0