Supplementary Material to A General Method for Third-Order Bias and Variance Corrections on a Nonlinear Estimator

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1 Supplemetary Material to A Geeral Method for Third-Order Bias ad Variace Correctios o a Noliear Estimator Zheli Yag School of Ecoomics, Sigapore Maagemet Uiversity, 90 Stamford Road, Sigapore s: zlyag@smu.edu.sg March, 2012 Summary This supplemetary material cotais exteded Mote Carlo results icludig those reported i a paper etitled A Geeral Method for Third-Order Bias ad Variace Correctios o a Noliear Estimator. More detailed aalyses relatig to the geeral observatios reported i the paper are also give. 1 Itroductio Extesive Mote Carlo experimets are carried out to ivestigate (i) the fiite sample performace of the QMLE ad the bias-corrected QMLEs bc2 ad bc3 of the spatial lag parameter λ, (ii) the fiite sample performace of the corrected stadard errors (se) of the origial QMLE as well the bias-corrected QMLE, ad (iii) the impact of the bias ad se correctios o the subsequet ifereces for λ. Also, through the process of our ivestigatio o the higher-order properties of the SAR model, we foud some aomalies i the documeted Mote Carlo results i Bao ad Ullah (2007a) ad i Lee (2004a). Their Mote Carlo experimets are reru ad the ameded results reported. A compariso is thus made with the aalytical approach of Bao ad Ullah (2007a). 2 Mote Carlo Experimet I To demostrate the effectiveess of the proposed bias-correctio procedure, ad to ivestigate the impact of bias ad se correctios o the subsequet iterfereces, a comprehesive set of Mote Carlo experimets is carried out based o the followig geeral SAR model: Y = λw Y + β X 1 β 1 + X 2 β 2 + u, where 1 is a -vector of oes. For all the Mote Carlo experimets, β = {β 0, β 1, β 2 } is set at {5, 1, 1} or {.5,.1,.1}, σ at 1 or 2, λ takes values {.5,.25,0,.25,.5}, ad takes values

2 {50, 100, 200,500}. 1 Several ways of geeratig W, (X 1, X 2 ), ad u are cosidered. First, the values {x 1i } or {x 1,ir } of X 1, ad the values {x 2i } or {x 2,ir } of X 2 are, MRSAR-A: {x 1i } iid N(0, 1)/ 2, ad {x 2i } iid N(0, 1)/ 2, or MRSAR-B: {x 1,ir } = (2z r + z ir )/ 7, ad {x 2,ir } = (v r + v ir )/ 7, where i MRSAR-B, {z r, z ir, v r, v ir } iid N(0, 1), across all i ad r. Apparetly, MRSAR-A gives iid X values, ad MRSAR-B gives o-iid X values, or differet group meas uder group iteractio, see Lee (2004a) ad below for details. Both schemes give sigal-to-oise ratios 1 or 1/2 whe σ = 1 or 2. Spatial layouts. Three geeral spatial layouts are cosidered i the Mote Carlo experimets. The first is based o Rook cotiguity, the secod is based o Quee cotiguity ad the third is based o the otio of group iteractios. The methods used i geeratig these three spatial layouts are similar to those used i Yag (2010). The detail for geeratig the W matrix uder rook cotiguity is as follows: (i) idex the spatial uits by {1, 2,, }, radomly permute these idices ad the allocate them ito a lattice of k m( ) squares, (ii) let W ij = 1 if the idex j is i a square which is o immediate left, or right, or above, or below the square which cotais the idex i, otherwise W ij = 0, i, j = 1,,, to form a matrix, ad (iii) divide each elemet of this matrix by its row sum to give W. Similarly, oe geerates the W matrix uder Quee cotiguity with additioal eighbors sharig a commo vertex with the uit of iterest. To geerate the W matrix accordig to the group iteractio scheme, suppose we have k groups of sizes m 1, m 2,, m k. Defie W = diag{w j /(m j 1), j = 1,, k}, a matrix formed by placig the submatrices W j alog the diagoal directio, where W j is a m j m j matrix with oes o the off-diagoal positios ad zeros o the diagoal positios. The group sizes {m j } ca be the same or differet, ad idepedet or depedet o, allowig for a full rage of spatial scearios cosidered i Lee (2004a). The details are as follows: (i) calculate the umber of groups accordig to k = K(), ad the approximate average group size m = /k, (ii) geerate the group sizes (m 1, m 2,, m k ) accordig to a discrete distributio cetered at m, ad (iii) adjust the group sizes so that k j=1 m j =. 2 I our Mote Carlo experimets, we use K() = Roud( ǫ ) with ǫ = 0.35, 0.50, ad 0.75, 1 As i Lee (2007a), the maximizatio of l c (λ) is performed globally without imposig a restricted lower boud o λ. This is importat whe the true λ value is egative ad big, because QMLE is dowward biased ad a restricted lower boud, say, would result i the searchig process to hit the lower boud quite ofte, thus failig to reach the true maximum poit. This would i tur give a wrog impressio that the QMLE ca be upward biased ad the bias-correctio may ot work i certai cases. This is believed to be the reaso for the icoheret Mote Carlo results of Bao ad Ullah (2007a). See Aseli (1988, p ) for a theoretical discussio o the parameter space of λ i relatio to the eigevalues of W. 2 Clearly, this desig covers the sceario cosidered i Case (1991). Typical forms of K() iclude K() = /m where m is a prespecified costat idepedet of ad K() = Roud( ǫ ). Lee (2007b) shows that the group size variatio plays a importat role i the idetificatio ad estimatio of ecoometric 2

3 represetig respectively the situatios where (a) there are few groups of may spatial uits i each, (b) the umber of groups ad the sizes of the groups are of the same magitude, ad (c) there are may groups of few elemets i each. Clearly, h = O( 1 ǫ ). The group sizes are draw from a discrete uiform distributio from 0.5m to 1.5m. Error distributios. To geerate u = σe, three distributios are cosidered: dgp1: the elemets {e i } of e are iid stadard ormal, dgp2: {e i } are iid stadardized ormal mixture, ad dgp3: {e i } are iid stadardized log-ormal. Specifically, for dgp2, e i = ((1 ξ i )Z i + ξ i τz i )/(1 p + p σ 2 ) 0.5, where ξ Beroulli(p), ad Z i N(0, 1) idepedet of ξ. The parameter p represets the proportio of mixig the two ormal populatios. I our experimets, we choose p = 0.1, meaig that 90% of the radom variates are from stadard ormal ad the remaiig 10% are from aother ormal populatio with stadard deviatio τ. We choose τ = 4 to simulate the situatio where there are gross errors i the data. For dgp3, e i = [exp(z i ) exp(0.5)]/[exp(2) exp(1)] 0.5, which gives a error distributio that is both skewed ad leptokurtic. The ormal mixture gives a error distributio that is still symmetric like ormal but leptokurtic. Other oormal distributios, such as ormal-gamma mixture ad chi-square, are also cosidered ad the results (available from the author upo request) exhibit a similar patter. Fiite sample performace of the spatial estimators. We report the Mote Carlo mea, rmse ad sd of, bc2 ad bc3 uder various combiatios of the values for (, λ, σ), the error distributios, ad the spatial layouts. We also report the averages (over Mote Carlo samples) of the 1st-, 2d- ad 3rd-order ses: V 1 ( ) 1 2, V 2 ( ) 1 2, V 3 ( ) 1 2 ad V 3 ( bc3 ) 1 2, defied by Corollaries 3.3 ad 3.4 ad calculated based o the proposed bootstrap method. Each set of results is based o 10,000 Mote Carlo samples, ad B = floor( 0.75 ) bootstrap samples for each Mote Carlo sample. Table 1 summarizes the results (reported i the paper) with β = {5, 1, 1} ad σ = 1, where the regressors values are geerated accordig to MRSAR-A for Quee cotiguity spatial layout, ad MRSAR-B for group iteractio spatial layout. Table 1-1 cotais additioal results uder the same setup as the group iteractio scheme but with differet group size cofiguratios. Table 1-2 replicates the results of Table 1 uder β = {.5,.1,.1}, ad Table 1-3 replicates the results of Table 1 uder σ = 2. For the results i both Tables 1b ad 1c, 1,000 Mote Carlo rus ad floor( 0.5 ) bootstrap samples for each Mote Carlo ru are used. Some geeral observatios are i order: models with group iteractios, cotextual factors ad fixed effects. Yag (2010) shows that it also plays a importat role i the robustess of the LM test of spatial error compoets. 3

4 (i) the bias-corrected QMLEs bc2 ad outperform the origial QMLE ; bc3 are i geeral early ubiased ad clearly (ii) is always dowward biased ad the biasess ca be very serious depedig o the (iii) spatial layout, the sample size ad the error stadard deviatio; bc3 bc2 bc2 improves over, but usig seems to be sufficiet uder most of the situatios as far as bias-correctio is cocered; (iv) spatial layouts ca have a huge impact o the fiite sample performace of the stroger the spatial depedece the worse performs; (v) the values of σ ad the slope parameters also have a big impact the bigger the σ is, or the smaller the β 1 ad β 2 are, the bigger are the biases, rmses ad sds of ; (vi) the value of λ ad the way the regressors beig geerated affect the fiite sample performace of as λ decreases, the bias of decreases uder iid regressors but icreases uder o-iid regressors, whereas the [rmse](se) of always icreases as λ decreases, with a sharper amout for the case of o-iid regressors; (vii) the error distributio does ot affect much o the geeral performace of the three estimators, showig the robustess of the proposed approach. (viii) The empirical sd of ca be slightly differet from that of small, suggestig that the variaces of ad bc3 bc3 whe sample size is may differ o higher-order term (see paels (a)-(c), Table 1). The results i the last four colums of Table 1 show that V 3 ( bc3 ) provides the best approximatio to the variace of. The empirical sds of ad bc2 bc3 bc3 ad agree closely, suggestig that the fiite sample variaces of bc3 are about the same. These are cosistet with the result of Corollary 3.4. I summary, the proposed bias-correctio procedure works excelletly i geeral, it is simple ad widely applicable, ad thus should be recommeded for the practitioers. The performace of t-ratios. The fiite sample behavior of the t-ratios t ij for testig H 0 : λ = 0, defied i (23) are ivestigated. Partial Mote Carlo results i terms of meas, sds, ad tail probabilities are reported i Table 2. From the results, the followig coclusios ca be draw: (i) the asymptotic t-ratio t 11 ca perform quite badly with severe distortios o mea ad sizes; (ii) use of secod-order bias-corrected estimator oly (t 21 ) immediately improves; (iii) use of both secod-order bias-corrected estimator ad secod-order variace gives further improvemets; ad (iv) use of the third-order bias-corrected estimator ad the third-order variace estimate gives the best results. bc2 4

5 3 Mote Carlo Experimet II The Mote Carlo experimet coducted by Bao ad Ullah (2007a) is replicated ad exteded for two purposes: (i) to compare our bootstrap-based bias correctio with their aalytical bias correctio, ad (ii) to ivestigate the cause of icoheret Mote Carlo results of Bao ad Ullah (2007a) for the case of λ = 0.9 ad J = 10, where J is the umber of eighbors each spatial uit has whe the circular-world desig is used. The empirical meas ad rmses for the four estimators: the MLE, the secod-order bias-corrected MLE bc2, ad the aalytically bias-corrected MLE BU of Bao ad Ullah (2007a), are summarized i Table 3. The results show that the secod-order bias correctio works excelletly i geeral: (i) i case of ormal errors, the bootstrap method ad aalytical method produce almost idetical results, cofirmig the validity of the proposed bootstrap method, ad (ii) i case of o-ormal errors, bootstrap method produces slightly better results, showig that the proposed method is more robust agaist the error distributio tha the aalytical method that is based o the ormality assumptio. The results also show a very coheret behavior of the MLE ad the bias-corrected MLEs i that the MLE is almost ubiased whe J is small ad is dowward biased whe J is big. I cases where the MLE is biased, the bias-corrected estimators always be able to correct the bias. However, the Mote Carlo results of Bao ad Ullah show a differet picture for the cases where J is large ad λ is large ad egative: the MLE is upward biased ad the bias-correctio does ot work. A possible explaatio for this ca be foud i Footote 9. 4 Mote Carlo Experimet III The Mote Carlo experimets of Lee (2004) are reru ad exteded agai for two purposes: i) to further compare MLEs (estimators uder ormal errors) ad secod-order bias-corrected MLEs uder a model without itercept, ad ii) to amed the aomalies i his Mote Carlo results. Uder the idetical set-up but with more Mote Carlo replicatios (10,000 vs 400) for each case, we foud that with the umber of districts R = (30, 60, 120) ad the umber of members i each district m = (2, 3, 5,1020,50,100),the MLEs all perform very well i all cases, showig a big cotrast to the results of Lee (2004a). However, whe the values for R ad m are switched, the results obtaied are more i lie with those of Lee (2004a) i terms of bias but ot i terms of sd. Table 4a correspods to Tables I & II i Lee (2004a). Tables 4b summarize the results of the Mote Carlo experimets whe the values for R ad m are switched, i.e., R = (3, 5, 10,20,50, 100) ad m = (30, 60, 120). The results i Tables 4b are ow more i lie with Lee s results i terms of bias but ot i terms of sd of. The results further show that for a give R value, icreasig the m value does ot ecessarily reduce the bias of. 5

6 However, for a give m, icreasig R clearly improves the performace of the estimators with both bias ad sd sigificatly reduced. From the theoretical poit of view, our results make more sese as icreasig m elarges the degree of spatial depedece, makig the estimatio of the spatial parameter harder (or at least ot easier). I cotrast, icreasig R clearly reduces the degree of spatial depedece, makig the estimatio of the spatial parameter much easier. 6

7 Table 1. Empirical Mea[rmse](sd) of Estimators of λ, ad Averaged Bootstrap SEs λ bc2 bc3 se 1 se 2 se 3 se c 3 (a) Quee Cotiguity, Normal Errors, MRSAR-A [.195](.174).492 [.175](.175).497 [.175](.175) [.123](.116).498 [.117](.117).500 [.117](.117) [.078](.076).499 [.075](.075).499 [.075](.075) [.049](.048).501 [.048](.048).501 [.048](.048) [.222](.204).242 [.209](.209).246 [.210](.210) [.146](.140).248 [.142](.142).250 [.143](.143) [.094](.092).250 [.093](.093).250 [.093](.093) [.060](.060).250 [.060](.060).250 [.060](.060) [.229](.216) [.224](.224) [.226](.226) [.157](.153) [.156](.156) [.157](.157) [.106](.104) [.105](.105).000 [.105](.105) [.068](.067) [.068](.068) [.068](.068) [.233](.223) [.236](.236) [.237](.237) [.164](.161) [.166](.166) [.166](.166) [.112](.111) [.112](.112) [.112](.112) [.073](.072) [.073](.073) [.073](.073) [.228](.222) [.236](.236) [.237](.237) [.162](.161) [.166](.166) [.166](.166) [.113](.113) [.114](.114) [.114](.114) [.073](.073) [.073](.073) [.073](.073) (b) Quee Cotiguity, Normal Mixture Errors, MRSAR-A [.182](.164).494 [.165](.165).498 [.165](.165) [.120](.114).499 [.114](.114).500 [.114](.114) [.076](.074).500 [.074](.074).500 [.074](.074) [.049](.048).500 [.048](.048).500 [.048](.048) [.207](.190).241 [.195](.195).244 [.195](.195) [.140](.135).248 [.136](.136).249 [.137](.137) [.092](.090).249 [.090](.090).249 [.090](.090) [.060](.060).250 [.060](.060).250 [.060](.060) [.217](.206) [.213](.213) [.214](.214) [.150](.147) [.150](.150) [.150](.150) [.104](.103) [.103](.103) [.103](.103) [.068](.067) [.067](.067) [.067](.067) [.223](.213) [.224](.224) [.225](.225) [.155](.153) [.157](.157) [.157](.157) [.111](.110) [.112](.112) [.112](.112) [.072](.072) [.072](.072) [.072](.072) [.218](.212) [.224](.224) [.225](.225) [.155](.153) [.158](.158) [.158](.158) [.112](.111) [.113](.113) [.113](.113) [.074](.073) [.074](.074) [.074](.074) Note: se 1 = mea( b V 1( ) 1 2 ), se2 = mea( b V 2( ) 1 2 ), se3 = mea( b V 3( ) 1 2 ) ad se c 3 = mea( b V 3( bc3 ) 1 2 ). 7

8 Table 1 (cot d). Empirical Mea[rmse](sd) of Estimators of λ, ad Averaged Bootstrap SEs λ bc2 bc3 se 1 se 2 se 3 se c 3 (c) Quee Cotiguity, Logormal Errors, MRSAR-A [.163](.146).491 [.146](.146).493 [.146](.146) [.110](.105).498 [.105](.105).498 [.105](.105) [.072](.069).499 [.069](.069).499 [.069](.069) [.047](.046).499 [.046](.046).499 [.046](.046) [.185](.171).241 [.174](.174).244 [.174](.174) [.128](.124).247 [.126](.125).248 [.126](.125) [.087](.085).249 [.085](.085).249 [.085](.085) [.058](.057).249 [.057](.057).249 [.057](.057) [.198](.186) [.192](.191) [.192](.192) [.139](.136) [.138](.138) [.138](.138) [.099](.097) [.098](.098) [.098](.098) [.065](.064) [.065](.065).000 [.065](.065) [.199](.191) [.198](.198) [.199](.199) [.142](.140) [.144](.144) [.144](.144) [.105](.104) [.105](.105) [.105](.105) [.070](.070) [.070](.070) [.070](.070) [.196](.190) [.200](.199) [.200](.200) [.145](.144) [.148](.148) [.148](.148) [.106](.106) [.107](.107) [.107](.107) [.070](.070) [.070](.070) [.070](.070) (d) Group Iteractio with k = 0.5, Normal Errors, MRSAR-B [.145](.124).495 [.122](.122).499 [.122](.122) [.112](.099).498 [.097](.097).500 [.097](.097) [.073](.068).499 [.067](.067).500 [.067](.067) [.042](.041).500 [.041](.041).500 [.041](.041) [.209](.179).239 [.178](.178).244 [.179](.178) [.160](.143).248 [.141](.141).250 [.141](.141) [.108](.102).249 [.101](.101).249 [.101](.101) [.062](.060).250 [.061](.061).250 [.061](.061) [.261](.224) [.226](.226) [.227](.227) [.206](.183) [.182](.182) [.182](.182) [.137](.128).001 [.128](.128).001 [.128](.128) [.081](.079) [.079](.079) [.079](.079) [.303](.260) [.266](.266) [.267](.267) [.243](.218) [.219](.219) [.219](.219) [.170](.160) [.160](.160) [.160](.160) [.101](.098) [.098](.098) [.098](.098) [.337](.291) [.302](.301) [.303](.302) [.281](.254) [.257](.257) [.257](.257) [.198](.186) [.187](.187) [.187](.187) [.118](.115) [.116](.116) [.116](.116) Note: se 1 = mea( b V 1( ) 1 2 ), se2 = mea( b V 2( ) 1 2 ), se3 = mea( b V 3( ) 1 2 ) ad se c 3 = mea( b V 3( bc3 ) 1 2 ). 8

9 Table 1 (cot d). Empirical Mea[rmse](sd) of Estimators of λ, ad Averaged Bootstrap SEs λ bc2 bc3 se 1 se 2 se 3 se c 3 (e) Group Iteractio with k = 0.5, Normal Mixture Errors, MRSAR-B [.144](.124).489 [.121](.120).493 [.120](.120) [.111](.098).495 [.096](.096).497 [.096](.096) [.073](.068).498 [.068](.068).499 [.068](.068) [.042](.041).500 [.041](.041).500 [.041](.041) [.203](.175).234 [.172](.171).239 [.172](.171) [.157](.140).245 [.138](.137).248 [.138](.138) [.104](.097).249 [.097](.097).249 [.097](.097) [.062](.060).250 [.060](.060).250 [.060](.060) [.250](.215) [.214](.213) [.214](.214) [.203](.182) [.180](.180) [.181](.181) [.137](.129) [.129](.128) [.129](.129) [.082](.080) [.080](.080) [.080](.080) [.294](.256) [.259](.259) [.260](.259) [.241](.216) [.216](.216) [.216](.216) [.165](.156) [.155](.155) [.155](.155) [.101](.098) [.099](.099) [.099](.099) [.330](.289) [.298](.297) [.298](.298) [.270](.243) [.245](.245) [.245](.245) [.197](.185) [.186](.186) [.186](.186) [.120](.117) [.117](.117) [.117](.117) (f) Group Iteractio with k = 0.5, Logormal Errors, MRSAR-B [.127](.110).488 [.107](.106).492 [.106](.106) [.104](.092).492 [.090](.090).494 [.090](.090) [.071](.067).497 [.066](.066).497 [.066](.066) [.041](.039).499 [.040](.040).499 [.040](.040) [.180](.158).236 [.156](.155).241 [.155](.155) [.146](.131).242 [.128](.128).244 [.128](.128) [.101](.094).247 [.094](.093).247 [.093](.093) [.060](.059).248 [.059](.059).248 [.059](.059) [.231](.203) [.202](.201) [.201](.201) [.179](.160) [.159](.159) [.159](.158) [.131](.123) [.122](.122) [.122](.122) [.078](.076) [.077](.077) [.077](.076) [.274](.243) [.245](.244) [.244](.244) [.216](.193) [.194](.194) [.194](.193) [.159](.148) [.148](.148) [.148](.148) [.099](.096) [.097](.097) [.097](.097) [.312](.279) [.286](.285) [.285](.285) [.245](.219) [.222](.221) [.222](.221) [.186](.174) [.174](.174) [.174](.174) [.117](.114) [.114](.114) [.114](.114) Note: se 1 = mea( b V 1( ) 1 2 ), se2 = mea( b V 2( ) 1 2 ), se3 = mea( b V 3( ) 1 2 ) ad se c 3 = mea( b V 3( bc3 ) 1 2 ). 9

10 Table 1-1. Additioal Results i Coectio to Table 1 λ bc2 bc3 se 1 se 2 se 3 se c 3 (g) Group Iteractio with k = 0.35, Normal Errors, MRSAR-B [.149](.126).491 [.129](.128).494 [.129](.129) [.127](.109).496 [.112](.112).498 [.113](.113) [.086](.080).499 [.080](.080).500 [.080](.080) [.051](.049).500 [.050](.050).500 [.050](.050) [.213](.180).237 [.185](.185).241 [.186](.186) [.189](.161).241 [.166](.166).243 [.166](.166) [.130](.120).247 [.120](.120).248 [.120](.120) [.076](.073).249 [.074](.074).249 [.074](.074) [.272](.232) [.241](.241) [.241](.241) [.246](.212) [.219](.219) [.219](.219) [.170](.156) [.157](.157) [.157](.157) [.100](.097).001 [.098](.098).001 [.098](.098) [.328](.279) [.291](.290) [.291](.291) [.301](.256) [.266](.266) [.267](.267) [.211](.195) [.195](.195) [.195](.195) [.125](.120) [.122](.122) [.122](.122) [.384](.325) [.343](.342) [.343](.343) [.360](.307) [.320](.320) [.321](.320) [.247](.229) [.230](.230) [.230](.230) [.150](.145) [.146](.146) [.146](.146) (h) Group Iteractio with k = 0.35, Normal Mixture, MRSAR-B [.149](.127).485 [.127](.126).490 [.127](.126) [.128](.110).491 [.112](.112).493 [.112](.112) [.086](.080).498 [.079](.079).498 [.079](.079) [.050](.048).500 [.048](.048).500 [.048](.048) [.214](.183).229 [.185](.183).235 [.184](.183) [.187](.161).238 [.165](.165).241 [.165](.165) [.128](.118).246 [.117](.117).247 [.117](.117) [.076](.074).248 [.074](.074).248 [.074](.074) [.277](.238) [.242](.240) [.241](.240) [.243](.210) [.215](.214) [.214](.214) [.169](.156) [.156](.156) [.156](.156) [.101](.097) [.098](.098) [.098](.098) [.324](.278) [.284](.283) [.283](.282) [.297](.254) [.262](.262) [.262](.261) [.211](.196) [.196](.196) [.196](.196) [.124](.119) [.121](.121) [.121](.121) [.384](.329) [.339](.336) [.338](.336) [.353](.304) [.314](.313) [.313](.313) [.248](.230) [.231](.231) [.231](.231) [.148](.143) [.145](.145) [.145](.145) Note: se 1 = mea( b V 1( ) 1 2 ), se2 = mea( b V 2( ) 1 2 ), se3 = mea( b V 3( ) 1 2 ) ad se c 3 = mea( b V 3( bc3 ) 1 2 ). 10

11 Table 1-1 (cot d). Empirical Mea[rmse](sd), ad Averaged Bootstrap SEs λ bc2 bc3 se 1 se 2 se 3 se c 3 (i) Group Iteractio with k = 0.35, Logormal Errors, MRSAR-B [.137](.120).484 [.118](.117).489 [.117](.116) [.124](.108).488 [.108](.107).492 [.107](.107) [.085](.080).496 [.079](.079).497 [.079](.078) [.051](.050).499 [.050](.050).499 [.050](.050) [.199](.174).225 [.172](.170).232 [.170](.169) [.177](.155).234 [.156](.155).239 [.154](.154) [.126](.117).245 [.116](.116).246 [.116](.116) [.077](.074).247 [.075](.075).247 [.075](.075) [.252](.223) [.221](.220) [.220](.219) [.231](.200) [.201](.200) [.199](.198) [.171](.159) [.157](.156) [.156](.156) [.102](.098) [.099](.099) [.099](.099) [.319](.282) [.282](.279) [.280](.278) [.289](.252) [.255](.254) [.252](.251) [.208](.194) [.192](.192) [.192](.191) [.126](.121) [.122](.122) [.122](.122) [.376](.333) [.334](.330) [.330](.328) [.333](.288) [.294](.292) [.291](.290) [.243](.226) [.224](.224) [.224](.224) [.152](.147) [.149](.149) [.149](.149) (j) Group Iteractio with k = 0.65, Normal Error, MRSAR-B [.099](.091).498 [.089](.089).500 [.089](.089) [.078](.074).501 [.072](.072).502 [.072](.072) [.058](.055).499 [.054](.054).499 [.054](.054) [.035](.035).500 [.035](.035).500 [.035](.035) [.134](.123).249 [.123](.123).251 [.124](.124) [.109](.102).250 [.101](.101).252 [.101](.101) [.080](.077).250 [.077](.077).251 [.077](.077) [.052](.051).250 [.051](.051).250 [.051](.051) [.159](.146) [.149](.149) [.149](.149) [.133](.125).001 [.125](.125).002 [.125](.125) [.102](.098).000 [.098](.098).001 [.098](.098) [.066](.065) [.065](.065) [.065](.065) [.178](.166) [.171](.171) [.172](.172) [.150](.141) [.142](.142) [.143](.143) [.120](.116) [.116](.116) [.116](.116) [.078](.077) [.077](.077) [.077](.077) [.182](.171) [.179](.179) [.180](.180) [.159](.151) [.155](.155) [.155](.155) [.131](.127) [.128](.128) [.128](.128) [.091](.090) [.090](.090) [.090](.090) Note: se 1 = mea( b V 1( ) 1 2 ), se2 = mea( b V 2( ) 1 2 ), se3 = mea( b V 3( ) 1 2 ) ad se c 3 = mea( b V 3( bc3 ) 1 2 ). 11

12 Table 1-1 (cot d). Empirical Mea[rmse](sd), ad Averaged Bootstrap SEs λ bc2 bc3 se 1 se 2 se 3 se c 3 (k) Group Iteractio with k = 0.65, Normal Mixture, MRSAR-B [.094](.086).498 [.084](.084).500 [.084](.084) [.077](.072).499 [.070](.070).500 [.070](.070) [.057](.054).499 [.054](.054).500 [.054](.054) [.035](.035).500 [.035](.035).500 [.035](.035) [.128](.118).246 [.118](.118).247 [.118](.118) [.106](.099).249 [.098](.098).250 [.098](.098) [.080](.076).249 [.076](.076).249 [.076](.076) [.050](.049).250 [.049](.049).250 [.049](.049) [.152](.141) [.143](.143) [.143](.143) [.128](.120) [.120](.120) [.120](.120) [.099](.095).001 [.095](.095).002 [.095](.095) [.066](.065) [.065](.065) [.065](.065) [.168](.157) [.161](.161) [.161](.161) [.146](.138) [.140](.140) [.140](.140) [.117](.113) [.114](.114) [.114](.114) [.079](.078) [.078](.078) [.078](.078) [.173](.163) [.170](.170) [.171](.170) [.155](.147) [.152](.152) [.152](.152) [.129](.125) [.127](.127) [.127](.127) [.091](.089) [.090](.090) [.090](.090) (l) Group Iteractio with k = 0.65, Logormal Errors, MRSAR-B [.085](.078).496 [.076](.076).498 [.076](.076) [.072](.067).497 [.065](.065).498 [.065](.065) [.053](.050).499 [.050](.050).500 [.050](.050) [.035](.034).500 [.034](.034).500 [.034](.034) [.111](.104).247 [.103](.103).249 [.103](.103) [.097](.091).247 [.090](.090).248 [.090](.090) [.073](.070).250 [.070](.070).250 [.070](.070) [.049](.049).250 [.048](.048).250 [.048](.048) [.135](.125) [.126](.126) [.126](.126) [.117](.109) [.109](.109) [.109](.109) [.092](.088) [.088](.088) [.088](.088) [.063](.062) [.062](.062) [.062](.062) [.150](.141) [.144](.144) [.144](.144) [.133](.125) [.127](.127) [.127](.127) [.109](.106) [.106](.106) [.106](.106) [.076](.075) [.075](.075) [.075](.075) [.161](.153) [.159](.158) [.159](.159) [.145](.138) [.143](.143) [.143](.143) [.124](.121) [.123](.123) [.123](.123) [.089](.088) [.088](.088) [.088](.088) Note: se 1 = mea( b V 1( ) 1 2 ), se2 = mea( b V 2( ) 1 2 ), se3 = mea( b V 3( ) 1 2 ) ad se c 3 = mea( b V 3( bc3 ) 1 2 ). 12

13 Table 1-2. Replicatio of Table 1, uder β = {.5,.1,.1} λ bc2 bc3 se 1 se 2 se 3 se c 3 (a) Quee, Normal Error, MRSAR-A [.252](.215).486 [.215](.215).493 [.216](.216) [.150](.140).500 [.139](.139).501 [.139](.139) [.098](.093).497 [.093](.093).497 [.093](.093) [.059](.058).500 [.058](.058).500 [.058](.058) [.277](.246).236 [.253](.253).241 [.255](.255) [.181](.171).246 [.173](.173).247 [.174](.174) [.115](.111).247 [.112](.112).247 [.112](.112) [.073](.071).249 [.071](.071).249 [.071](.071) [.296](.264) [.282](.281) [.284](.283) [.198](.188) [.193](.193) [.194](.193) [.135](.131) [.133](.133) [.133](.133) [.080](.080).001 [.080](.080).001 [.080](.080) [.284](.269) [.290](.290) [.292](.292) [.197](.195) [.203](.202) [.203](.203) [.135](.133) [.136](.136) [.136](.136) [.088](.087) [.088](.088) [.088](.088) [.272](.259) [.284](.284) [.286](.286) [.190](.188) [.197](.197) [.198](.198) [.138](.137) [.141](.140) [.141](.141) [.086](.086) [.087](.087) [.087](.087) (b) Quee, Normal Mixture, MRSAR-A [.234](.203).497 [.203](.203).503 [.204](.204) [.150](.138).496 [.138](.138).497 [.138](.138) [.094](.090).502 [.089](.089).502 [.089](.089) [.059](.058).501 [.058](.058).501 [.058](.058) [.259](.229).243 [.235](.235).248 [.237](.237) [.166](.158).252 [.160](.160).253 [.161](.160) [.113](.111).257 [.112](.112).258 [.112](.112) [.071](.070).252 [.071](.071).252 [.071](.071) [.268](.251).009 [.264](.264).012 [.267](.266) [.181](.175).001 [.180](.180).002 [.180](.180) [.128](.127).005 [.128](.128).005 [.128](.128) [.079](.078).001 [.079](.079).001 [.079](.079) [.266](.246) [.265](.265) [.267](.267) [.185](.180) [.186](.186) [.187](.187) [.139](.137) [.139](.139) [.139](.139) [.087](.087) [.087](.087) [.087](.087) [.266](.255) [.278](.277) [.279](.279) [.197](.195) [.204](.204) [.204](.204) [.136](.135) [.138](.138) [.138](.138) [.087](.086) [.087](.087) [.087](.087) Note: se 1 = mea( bv 1( ) 1 2 ), se2 = mea( bv 2( ) 1 2 ), se3 = mea( bv 3( ) 1 2 ) ad se c 3 = mea( bv 3( bc3 ) 1 2 ). 13

14 Table 1-2 (Cot d). Empirical Mea[rmse](sd), ad Averaged Bootstrap SEs λ bc2 bc3 se 1 se 2 se 3 se c 3 (c) Quee, logormal, MRSAR-A [.214](.179).493 [.178](.177).499 [.178](.178) [.135](.125).499 [.124](.124).500 [.124](.124) [.089](.086).502 [.085](.085).502 [.085](.085) [.055](.054).499 [.054](.054).499 [.054](.054) [.234](.205).247 [.210](.210).251 [.211](.211) [.159](.150).248 [.151](.151).249 [.151](.151) [.111](.108).248 [.108](.108).249 [.108](.108) [.068](.067).250 [.067](.067).250 [.067](.067) [.263](.233) [.246](.245) [.247](.246) [.166](.161).007 [.165](.165).008 [.165](.165) [.127](.123) [.125](.125) [.125](.125) [.080](.079).001 [.080](.080).001 [.080](.080) [.262](.242) [.260](.259) [.261](.261) [.177](.170) [.176](.176) [.176](.176) [.130](.128) [.130](.130) [.130](.130) [.081](.081) [.081](.081) [.081](.081) [.254](.245) [.266](.266) [.268](.268) [.174](.171) [.178](.178) [.178](.178) [.129](.128) [.131](.131) [.131](.131) [.084](.083) [.084](.084) [.084](.084) (d) Group Iteractio with k = 0.5, Normal Errors, MRSAR-B [.302](.238).488 [.198](.198).502 [.198](.198) [.195](.161).499 [.138](.138).504 [.137](.137) [.141](.125).502 [.112](.112).504 [.112](.112) [.100](.089).498 [.082](.082).498 [.082](.082) [.381](.297).247 [.257](.257).262 [.260](.260) [.285](.235).242 [.208](.208).248 [.208](.208) [.209](.186).255 [.168](.168).257 [.168](.168) [.151](.136).249 [.127](.127).250 [.127](.127) [.464](.365) [.333](.333).010 [.338](.338) [.364](.304) [.277](.277) [.278](.278) [.263](.230).001 [.209](.209).003 [.209](.209) [.188](.167) [.156](.156) [.156](.155) [.564](.441) [.428](.427) [.435](.434) [.396](.330) [.306](.306) [.309](.309) [.330](.296) [.271](.271) [.272](.272) [.241](.216) [.204](.204) [.204](.204) [.568](.436) [.444](.442) [.450](.449) [.465](.387) [.375](.374) [.378](.377) [.374](.321) [.297](.297) [.298](.298) [.260](.237) [.223](.223) [.224](.223) Note: se 1 = mea( b V 1( ) 1 2 ), se2 = mea( b V 2( ) 1 2 ), se3 = mea( b V 3( ) 1 2 ) ad se c 3 = mea( b V 3( bc3 ) 1 2 ). 14

15 Table 1-2 (Cot d). Empirical Mea[rmse](sd), ad Averaged Bootstrap SEs λ bc2 bc3 se 1 se 2 se 3 se c 3 (e) Group Iteractio with k = 0.5, Normal Mixture, MRSAR-B [.284](.224).493 [.184](.184).507 [.184](.184) [.189](.158).502 [.137](.136).507 [.136](.136) [.135](.117).502 [.105](.105).504 [.105](.105) [.099](.090).501 [.083](.083).502 [.083](.083) [.362](.290).258 [.255](.255).272 [.258](.257) [.272](.223).243 [.198](.198).249 [.198](.198) [.200](.174).250 [.157](.157).252 [.157](.157) [.145](.131).252 [.123](.123).253 [.123](.123) [.451](.358) [.328](.328).011 [.334](.334) [.355](.294) [.267](.267) [.269](.269) [.263](.226) [.206](.205) [.206](.206) [.196](.174) [.163](.162) [.162](.162) [.503](.399) [.385](.385) [.393](.393) [.399](.329) [.307](.307) [.310](.310) [.302](.262) [.241](.241) [.242](.242) [.231](.206) [.194](.194) [.194](.194) [.576](.461) [.477](.476) [.484](.484) [.427](.351) [.341](.341) [.346](.346) [.351](.301) [.279](.279) [.280](.280) [.265](.239) [.224](.224) [.224](.224) (f) Group Iteractio with k = 0.5, logormal, MRSAR-B [.235](.184).511 [.155](.154).523 [.156](.154) [.183](.147).495 [.127](.127).499 [.127](.127) [.123](.108).506 [.097](.097).506 [.097](.097) [.095](.083).497 [.077](.077).497 [.077](.077) [.336](.257).251 [.228](.228).263 [.230](.230) [.246](.202).255 [.179](.179).259 [.180](.179) [.179](.155).256 [.140](.140).257 [.140](.140) [.140](.125).250 [.116](.116).249 [.116](.116) [.417](.319) [.291](.291).007 [.295](.295) [.309](.252) [.229](.229).003 [.231](.231) [.243](.207) [.189](.189) [.189](.189) [.178](.161).005 [.151](.151).004 [.151](.151) [.475](.368) [.359](.359) [.365](.365) [.343](.282) [.264](.264) [.267](.266) [.300](.257) [.237](.237) [.237](.237) [.225](.200) [.187](.187) [.187](.187) [.510](.401) [.407](.407) [.414](.414) [.385](.320) [.312](.312) [.316](.316) [.323](.279) [.260](.260) [.261](.261) [.247](.222) [.209](.209) [.210](.210) Note: se 1 = mea( b V 1( ) 1 2 ), se2 = mea( b V 2( ) 1 2 ), se3 = mea( b V 3( ) 1 2 ) ad se c 3 = mea( b V 3( bc3 ) 1 2 ). 15

16 λ bc2 Table 1-3. Replicatio of Table 1, uder σ = 2 bc3 se 1 se 2 se 3 se c 3 (a) Quee, Normal Error, MRSAR-A [.234](.203).487 [.203](.203).493 [.204](.203) [.143](.134).500 [.134](.134).501 [.133](.133) [.089](.085).497 [.085](.084).498 [.085](.084) [.054](.053).500 [.053](.053).500 [.053](.053) [.256](.229).238 [.235](.235).243 [.236](.236) [.173](.164).245 [.166](.166).246 [.166](.166) [.109](.105).245 [.105](.105).246 [.105](.105) [.067](.066).250 [.066](.066).250 [.066](.066) [.275](.249) [.264](.263) [.266](.266) [.188](.179) [.184](.184) [.184](.184) [.125](.122) [.124](.123) [.124](.124) [.074](.074).001 [.075](.075).001 [.075](.075) [.268](.255) [.273](.273) [.275](.275) [.187](.185) [.191](.191) [.192](.192) [.129](.127) [.130](.130) [.130](.130) [.082](.082) [.082](.082) [.082](.082) [.260](.249) [.270](.270) [.272](.272) [.182](.180) [.188](.188) [.188](.188) [.130](.130) [.132](.132) [.133](.132) [.080](.080) [.081](.081) [.081](.081) (b) Quee, Normal Mixture, MRSAR-A [.215](.188).497 [.189](.189).502 [.190](.190) [.141](.131).497 [.131](.131).498 [.131](.131) [.088](.085).501 [.084](.084).502 [.084](.084) [.054](.053).501 [.053](.053).501 [.053](.053) [.243](.219).244 [.225](.225).248 [.226](.226) [.156](.149).252 [.151](.151).254 [.151](.151) [.108](.106).256 [.107](.107).256 [.107](.107) [.065](.064).251 [.064](.064).251 [.064](.064) [.249](.235).008 [.246](.246).010 [.248](.247) [.170](.165) [.169](.169).001 [.169](.169) [.120](.118).003 [.119](.119).003 [.120](.120) [.073](.073).001 [.073](.073).001 [.073](.073) [.253](.236) [.251](.251) [.253](.253) [.175](.171) [.176](.176) [.176](.176) [.129](.128) [.129](.129) [.130](.130) [.081](.080) [.081](.081) [.081](.081) [.251](.242) [.261](.260) [.262](.262) [.188](.186) [.194](.194) [.194](.194) [.129](.128) [.131](.131) [.131](.131) [.081](.081) [.082](.082) [.082](.082) Note: se 1 = mea( bv 1( ) 1 2 ), se2 = mea( bv 2( ) 1 2 ), se3 = mea( bv 3( ) 1 2 ) ad se c 3 = mea( bv 3( bc3 ) 1 2 ). 16

17 Table 1-3 (cot d). Empirical Mea[rmse](sd), ad Averaged Bootstrap SEs λ bc2 bc3 se 1 se 2 se 3 se c 3 (c) Quee, Logormal Error, MRSAR-A [.194](.166).491 [.166](.165).496 [.166](.166) [.129](.119).497 [.119](.119).498 [.119](.119) [.084](.082).502 [.081](.081).502 [.081](.081) [.050](.048).498 [.048](.048).498 [.048](.048) [.223](.199).242 [.203](.203).246 [.204](.204) [.150](.143).250 [.145](.145).251 [.145](.145) [.105](.102).248 [.103](.103).249 [.103](.103) [.062](.061).250 [.062](.062).250 [.062](.062) [.242](.218) [.228](.227) [.229](.228) [.161](.156).003 [.160](.160).004 [.160](.160) [.120](.116) [.118](.118) [.118](.118) [.074](.073).000 [.073](.073).000 [.073](.073) [.244](.225) [.240](.239) [.241](.241) [.169](.163) [.168](.167) [.168](.168) [.121](.120) [.121](.121) [.121](.121) [.076](.075) [.076](.076) [.076](.076) [.236](.230) [.246](.246) [.247](.247) [.166](.164) [.170](.170) [.170](.170) [.121](.120) [.123](.123) [.123](.123) [.078](.078) [.079](.079) [.079](.079) (d) Group Iteractio with k = 0.5, Normal Error, MRSAR-B [.233](.188).486 [.171](.170).495 [.171](.171) [.145](.122).498 [.115](.115).501 [.115](.115) [.114](.102).501 [.096](.096).502 [.096](.096) [.076](.070).497 [.068](.068).497 [.068](.068) [.314](.254).238 [.237](.237).248 [.238](.238) [.214](.181).241 [.173](.173).244 [.174](.173) [.169](.154).254 [.145](.145).255 [.145](.145) [.112](.103).247 [.100](.100).248 [.100](.100) [.366](.297) [.284](.284).006 [.287](.287) [.275](.237) [.230](.230) [.231](.231) [.218](.195) [.185](.185).001 [.185](.185) [.145](.134) [.130](.130) [.130](.130) [.455](.366) [.363](.362) [.366](.365) [.305](.263) [.257](.257) [.258](.258) [.277](.253) [.241](.241) [.241](.241) [.179](.167) [.163](.163) [.163](.163) [.467](.371) [.382](.381) [.385](.385) [.368](.320) [.320](.320) [.322](.322) [.302](.266) [.256](.256) [.256](.256) [.204](.191) [.187](.187) [.187](.187) Note: se 1 = mea( b V 1( ) 1 2 ), se2 = mea( b V 2( ) 1 2 ), se3 = mea( b V 3( ) 1 2 ) ad se c 3 = mea( b V 3( bc3 ) 1 2 ). 17