Heteroscedasticityinstochastic frontier models: A Monte Carlo Analysis

Heterosceasticityinstochastic frontier moels: A Monte Carlo Analysis by C. Guermat University of Exeter, Exeter EX4 4RJ, UK an K. Hari City University, Northampton Square, Lonon EC1V 0HB, UK First version: March 1999 Abstract: This paper uses Monte Carlo experimentation to investigate the nite sample properties of the maximum likelihoo (ML) estimators of the halfnormal stochastic frontier prouction functions in the presence of heterosceasticity. It is foun that when heterosceasticity exists correcting for it leas not only to a substantial improvement of the statistical properties of estimators but also to improve e ciency an ranking measures. On the other han correcting for heterosceasticity when there is none has serious averse results. Hence, there is a nee for testing for heterosceasticity an if there is any the appropriate correction shoul be mae. JEL classi cation: C15; C21; C24; D24; Q12. Keywors: Stochastic Frontier Prouction; Heterosceasticity; Technical E ciency; Monte Carlo; Maximum likelihoo Estimation; Tests. Corresponing author: Kaour Hari Department of Economics City University Lonon, EC1 0HB, UK E-mail: K.Hari@City.ac.uk

1. Introuction The original speci cation of stochastic frontier prouction for cross-section ata was inepenently propose by Aigner, Lovell an Schmit (1977), Battese an Corra (1977) an Meeusen an van en Broeck (1977). There has been consierable research to exten an apply the initial moel. A recent survey of this research is provie by Green (1993). Because these stuies are base on ata from rms varying enormously in size, the presence of heterosceasticity in these moels is very likely. Economists have for a long time associate the presence ofheterosceasticity incross-sectional ata with certain size-relate characteristics of the rms observe. In their seminal paper, Prais an Houthakker (1955) n expenitures for househol with high incomes to be more sprea than expenitures for lower incomes househols. This is expecte because househol with high incomes have more freeom to behave i erently. It is well known that the consequences of heterosceasticity for least squares estimation is quite serious. Estimators remain unbiase, but are no longer e cient. But more importantly, the stanar errors usually compute for the least squares estimators are no longer appropriate, an hence con ence intervals an hypothesis tests that use these stanar errors are invali. It is only recently that some of the e ects of heterosceasticity in stochastic frontier moels have been investigate. Cauill an For (1993), using a limite Monte Carlo experiment, showe that heterosceasticity in the one-sie error in Cobb-Douglas stochastic frontier prouction function leas to overestimation of the intercept an unerestimation of the slope coe cients. Cauill, For an Gropper (1995) estimate a stochastic frontier bank costs an bank-speci c ine ciency measures using maximum likelihoo metho an accounting for heterosceasticity only in the one-sie error term. They rightly pointe out that the measures of ine ciency use in the previous stuies are base on resiuals erive from the estimation of a frontier. They observe that resiuals are sensitive to speci cation errors, particularly in frontier moels, an that this sensitivity is passe on to the ine ciency measures. Hari (1999) using the same ata as the one use by Cauill et al. (1995) accounte for heterosceasticity in both ranom terms. The new speci cation was strongly supporte by the ata. Moreover, he foun that rm-speci c ine ciency measures are very sensitive to the propose correction. The rm rankings are also a ecte. In Hari, Guermat an Whittaker (1999) the same technique is extene to stochastic prouction frontier functions an to a set of panel ata on 102 farms for the years 1982 to 1987. 2

The small sample properties of these corrections for heterosceasticity have not been investigate yet. This paper proposes to ll this gap. The paper is organize as follows. In section 2 we iscuss the moels an notation while in section 3 we escribe the Monte Carlo experiment. Next, in section 4, the results are presente an iscusse. The nal section contains concluing remarks. 2. Moels an Notation The basic moel use in the literature to escribe a frontier prouction function can be writtenas follows: y i =X i +w i v i ; (1) wherey i enotes the logarithm of the prouction for the ith sample farm (i = 1;:::;N) ;X i is a (1 k) vector of the logarithm of the inputs associate with theith sample farm (the rst element woul be one when an intercept term is inclue); is a (k 1) vector of unknown parameters to be estimate;w i is a twosie error term withe[w i ]=0,E[w i w j ]=0 for alli anj,i6=j; var(w i )=¾ 2 w ; v i is a non-negative one-sie error term withe[v i ]>0,E[v i v j ]=0 for alli anj, i 6=j; an var(v i )=¾ 2 v : Furthermore, it is assume thatw anv are uncorrelate. The one-sieisturbance v re ects the fact that each rm s prouction must lie on or below its frontier. Such a term represents factors uner the rm s control. The two-sie error term represents factors outsie the rm s control. If we assume thatv i is half-normal anw i is normal, then the ensity function of their sum, erive by Weinstein (1964), takes the form: f(² i )=(2=¾)f (² i =¾)(1 F ( ² i =¾)); 1<² i <+1; (2) where² i = w i +v i ;¾ 2 = ¾ 2 w +¾2 v ; =¾ v=¾ w anf (:) anf (:) are, respectively, the stanar normal ensity an istribution functions. The avantage of stochastic frontier estimation is that it permits the estimation of rm-speci c ine ciency. The most wiely use measure of rm-speci c ine ciency suggeste by Jonrow, Lovell, Materov an Schmit (1982) base on the conitional expecte value ofv i given² i is given by where¾ =¾ v ¾ w =¾: E[v i j² i ]=¾ [ ² i =¾+f (² i =¾)=F ( ² i =¾)]; (3) 3

In what follows, we erive the log-likelihoo functions for the three possible cases: heterosceasticity in the one-sie, two-sie an both error terms. These erivations are use for estimation an to evaluate log-likelihoo ratios for testing purposes. Following Cauill et al. (1995) an Hari (1999) we assume the following multiplicative heterosceasticity for the one-sie error term ¾ vi =exp(z i ); (4) wherez i is a vector of nonstochastic explanatory variables relate to characteristics of rm management an is a vector of unknown parameters. Z i is assume to inclue an intercept term. The stanar eviation of the two-sie error term is also written in exponential form so that ¾ w = exp(µ): The ensity function corresponing to the moel where only the one-sie error term is assume heterosceastic is givenby: f i (² i )=(2=¾ i )f (² i =¾ i )(1 F ( i² i =¾ i )); 1<² i <+1; (5) where¾ 2 i =¾ 2 w+¾ 2 vi; i=¾ vi =¾ w anf (:) anf (:) are as e ne previously. The log-likelihoo functionis logl( ; ;µ)= NX i=1 log(f i (² i )): (6) In the cross-section imension the two-sie error is likely to be a ecte by size-relateheterosceasticity. The misspeci cation resulting from not incorporating heterosceasticity in the ML estimation of our frontier can cause parameters estimators to be inconsistent as well as invaliating stanar techniques of inference, see White (1982). In orer to incorporate heterosceasticity in the two-sie error term we write¾ wi =exp(w i µ);wherew i is a vector of nonstochastic explanatory variables relate generally to characteristics of rm size an µ is a vector of unknown parameters. W i is assume to inclue an intercept term. The stanar eviation of the one-sie error term, assume here to be homosceastic, is now ¾ v =exp( ): The ensity function is still as in (5) but now we have¾ 2 i =¾2 wi +¾2 v ; an i=¾ v =¾ wi. Last but not least, the most likely correct speci cation is the one where the two error terms are assumeto be concurrently heterosceastic. (5) is still appropriate but now we have¾ 2 i = ¾ 2 wi +¾2 vi ; an i =¾ vi =¾ wi where ¾ wi = exp(w i µ) an ¾ vi =exp(z i ): 4

3. The Monte Carlo Design In orer to investigate the nite sample properties of the maximum likelihoo (ML) estimators of the half-normal stochastic frontier prouction functions in presence of heterosceasticity we use a Monte Carlo experiment. We simulate a frontier moel using a simple Cobb-Douglass function lnq i = 0+ 1lnL i + 2lnK i +W i V i ; where W an V are a normal error variable an a half normal error respectively. We generate ata using the following proceure: fl i ;K j g are pairs (i;j), i = 1,:::; p N, an j = 1;:::; p N recursively. The variablez i was generate accoring to fz i g= p i,i=1;:::;n: The two error terms were generate asw»n(0;¾ 2 w ) anv» jn(0;¾2 v )j. Heterosceasticity was moele as follows: ¾ v =exp( 0 +± 1 lnl i +± 2 lnk i ) ¾ w =exp( 0 +± 1 lnz i ) The parameters were set at: 0= 0 = 0 = 1 = 2 = 1 =1 1= 2=0:5 (constant return to scale) The parameter ± measures the egree of heterosceasticity. We use several egrees of heterosceasticity by letting ± to increase from 0 to 0.5 by increments of 0.05. When ± = 0 we obtain the homosceastic case. We consiere also i erent sample sizes: 50, 100, 200, 300 an 400 observations. To analyze the e ect of heterosceasticity we estimate four moels: ignoring heterosceasticity (H0), allowing for heterosceasticity in W (HW), allowing for heterosceasticity in V (HV), an nally allowing for heterosceasticity in V an W (HVW). In each case we use 1000 replications. This setting allows us to n the consequences of ignoring heterosceasticity when it is present as well as the consequences of imposing heterosceasticity when there is none. To n out if the number of replication of 1000 (see Hari an Garry (1998) on the importance of the number of replications) was su cient, we carrie out some experiments with 20000 replications. We foun that the results were not signi cantly i erent. The main results of the Monte Carlo experiment are presente in Table 1, 2, 3 an 4 corresponing to the case of ignoring heterosceasticity (H0), allowing heterosceasticity in the two sie error term (HW), allowing heterosceasticity 5

in the e ciency term (HV), an allowing for heterosceasticity in both terms (HVW) respectively. Each table shows the biases for the ve sample sizes use ranging from 50 to 400. For each sample size, results are given for 11 i erent levels of heterosceasticity (±), varying from 0 (homosceastic case) to 0.5 (highly heterosceastic). The mean bias of parameters, the stanar eviations, an the mean square errors () are shown. Since we are intereste in the e ect on the value an the ranking of technical e ciency measures, we also isplay the bias, the for estimate e ciency, an the mean of rank correlation coe cients between the true rank anthe estimateone. 4. The Results Table 1 gives the results from the estimation of moelh0; namely not accounting for heterosceasticity in both isturbance terms. It shows that the bias for the intercept term ( 0) increases algebraically with the increase of the egree of heterosceasticity. The bias also increases algebraically with the size of the sample. The bias of 1 is negative an increases in absolute with the increase in the egree of heterosceasticity. The bias grows in absolute values with the size ofthe sample inicating inconsistency when omitting to correct for heterosceasticity when it is present. The bias of 2 is generally negative an increases in absolute value with the increase of the egree of heterosceasticity an with the size of sample con- rming the inconsistency. Strangely enough the bias of the measure of e ciency, generally, ecreases with the augmentation of the egree of heterosceasticity an the increase of sample size. Finally, the rank correlation between the estimate measure of e ciency an the true one iminishes with the size of the egree of heterosceasticity. Table 2 shows the outcome of estimating the moel HW in which the heterosceasticity is accounte for only in the symmetrical error term. The results follow similar patterns as those obtaine using H0: There is a signi cant ecrease in the bias of the intercept, but a slight increase in the bias of the slope parameters. The bias of the measure of e ciency increases nonmonotically with the egree of heterosceasticity but ecreases with the size of samples. The rank correlations followthe same pattern as above. Table 3 reports the estimates obtaine from moelhv: The bias of the intercept is negative but relatively small. The bias of the rst slope parameter is positive an increases with ±; the egree of heterosceasticity, but remains all but small. The bias of the secon slope parameter is negative an increases in 6

absolute value but still remaining relatively small. The fourth table summarizes the results obtaine by using the right speci - cation namely HV W: As expecte the results show negligeable biases even when there is no heterosceasticity. Moreover, the bias ecreases with increasing level of heterosceasticity. For su ciently large samples the rank correlation is very close to one, while the e ciency bias becomes negligible as± ann increase. The e ects of accounting for heterosceasticity when none is present is reporte in Table 5 for sample size N = 200. This misspeci cation oes not a ect the bias of the intercept but causes a slight increase in the bias of 2. However, is seems that the speci cation HVW has the ege in terms of overall parameter bias although marginally. Misspeci cation oes not appear to have any e ect on e ciency bias either. However, misspeci cation has a sizeable negative e ect on rank correlation. For example, the mean rank correlation rops from 0.944 when we use the right speci cation (H0) to 0.59 when we use HVW. Similar results are obtaine for other sample sizes. Tables 6 an 7 gather outcomes from estimating moels correcting for heterosceasticity when it is present for two egrees of heterosceasticity namely 0.25 an 0.5 respectively. As expecte using the right speci cation leas to a signi cant reuction in bias for all three prouction function parameters. As ± increases, biases in H0 an HW become more pronounce. The bias in e ciency shows a slightly i erent picture. There is little i erence in e ciency bias between H0, HV an HVW where the bias tens to zero as±increases. The rank correlation results clearly inicate that using the right speci cation is important. For instance, the mean rank correlation using the right speci cation HVW leas to an increase of more than 27% of the rank correlation atn=200. Finally, Figures 1 an 2 give a pictorial an clear view of the parameters biases uner the four speci cations for samples of size 50 an 400 respectively. Figures 3 an 4 show the biases of the measure of e ciency an the rank correlations respectively. 5. Conclusion The results of the present Monte Carlo experiment show that ignoring heterosceasticity whenit is present leas to substantial biases an even inconsistent estimates of the prouction function parameters. When we ignore heterosceasticity the measures ofe ciency are aversely a ecteleaing to incorrect ranking of rms. On the other han, imposing correction for heterosceasticity when none exists 7

also has serious negative consequences on the ranking of the e ciency measure. This paper shows that there is a nee for testing for heterosceasticity an if there is any the appropriate correction shoulbe mae. References Aigner, D.J., Lovell, C.A.K., an Schmit, P. (1977), Formulation an Estimation ofstochastic Frontier ProuctionFunctionMoels, Journal of Econometrics, 6, 21-37. Battese, G.E., an Corra, G.S. (1977), Estimation of a Prouction Frontier Moel: With Application to the Pastoral Zone of Eastern Australia, Australian Journal of Agricultural Economics, 21, 169-179. Cauill, S.B. an For, J.M. (1993) Biases in frontier estimation ue to heterosceasticity, Economics Letters, 41, 17-20. Cauill, S.B, For, J.M., an Gropper, D.M. (1995), Frontier Estimation an Firm- Speci c Ine ciency Measures in the Presence of Heterosceasticity, Journal of Business & Economic Statistics, 13, 105-111. Green, W.H. (1993), The econometric Approach to E ciency Analysis, in The measurement of Prouctive E ciency, es. H.O. Frie, C.A.K. Lovell, an S.S. Schmit, New York: Oxfor University Press, pp. 68-119. Hari, K., an J. Whittaker. (1995) e ciency, environmental contaminants an farm size: testing for links using stochastic prouction frontiers iscussion paper in economics 95/05, Exeter University. Hari, K. (1997). A frontier Approachto Disequilibrium Moels, Applie Economic Letters, No. 4, pp. 699-701. Hari, K., C. Guermat, an J. Whittaker. (1999) Doubly heterosceastic stochastic prouction frontiers with application to English cereals farms Discussion paper in economics, Exeter University. Hari, K. an G.D.A. Phillips. (1999) The accuracy of the higher orer bias approximation for the 2SLS estimator, Economics Letters, 62, 167-174. Hari, K. (1999) Estimation of a oubly heterosceastic stochastic frontier cost function, forthcoming in the Journal of Business & Economic Statistic. Jonraw, J., Lovell, C.A.K., Materov, I., anschmit, P. (1982), On the Estimation of Technical Ine ciency in Stochastic Prouction Function Moel, Journal of Econometrics, 19, 233-238. Meeusen, W., an van en Broeck, J. (1977), E ciency Estimation from Cobb- Douglas Prouction Function with Compose Error, International Economic Review, 18, 435-444. 8

Prais, S.J. an H.S. Houthaker. (1955), The analysis of family bugets, Cambrige University Press, Cambrige. Weistein, M.A. (1964), The Sum of Values From a Normal ana Truncate Normal Distribution, Technometrics, 6, 104-105 (with some aitional material, 469-470). White, H. (1982), Maximum Likelihoo Estimation ofmisspeci e Moels, Econometrica, 50, 1-25. 9

Table 1. Estimation Results Using H0 Moel. N δ Bias SD Bias SD Bias SD Bias Eff Eff Mean Rank 50 0.00-0.330 1.839 3.491-0.013 0.685 0.469 0.028 0.643 0.414 0.290 0.239 0.883 0.05-0.086 1.914 3.671-0.053 0.772 0.598-0.101 0.721 0.530 0.268 0.221 0.860 0.10-0.167 2.009 4.064-0.165 0.812 0.687-0.204 0.772 0.637 0.310 0.251 0.817 0.15 0.182 2.193 4.843-0.320 0.885 0.886-0.364 0.850 0.856 0.293 0.239 0.757 0.20 0.610 2.332 5.808-0.468 1.025 1.269-0.530 0.898 1.087 0.271 0.227 0.701 0.25 0.924 2.559 7.402-0.657 1.055 1.546-0.750 1.036 1.637 0.264 0.220 0.651 0.30 1.485 2.667 9.316-0.770 1.173 1.967-0.953 1.122 2.166 0.224 0.189 0.615 0.35 1.965 2.989 12.795-1.035 1.363 2.929-1.307 1.277 3.339 0.220 0.186 0.575 0.40 2.706 3.105 16.962-1.345 1.433 3.864-1.599 1.392 4.495 0.179 0.153 0.545 0.45 3.441 3.275 22.567-1.553 1.709 5.334-1.909 1.534 5.998 0.163 0.139 0.525 0.50 4.467 3.341 31.111-1.856 1.809 6.717-2.329 1.733 8.430 0.125 0.111 0.509 100 0.00-0.434 1.552 2.596-0.009 0.462 0.213-0.012 0.438 0.192 0.261 0.203 0.917 0.05-0.073 1.657 2.750-0.104 0.491 0.252-0.123 0.490 0.255 0.222 0.175 0.890 0.10 0.234 1.798 3.287-0.188 0.527 0.313-0.230 0.517 0.320 0.202 0.161 0.842 0.15 0.642 1.954 4.229-0.378 0.591 0.493-0.410 0.557 0.479 0.184 0.146 0.783 0.20 1.199 2.164 6.118-0.579 0.669 0.784-0.621 0.637 0.792 0.178 0.143 0.731 0.25 1.873 2.227 8.466-0.745 0.763 1.136-0.870 0.755 1.326 0.134 0.109 0.691 0.30 2.883 2.413 14.134-1.036 0.868 1.826-1.175 0.825 2.061 0.100 0.085 0.656 0.35 3.882 2.581 21.733-1.295 1.037 2.753-1.567 0.914 3.291 0.083 0.071 0.628 0.40 5.223 2.742 34.797-1.724 1.106 4.194-2.010 1.026 5.093 0.062 0.055 0.605 0.45 6.856 2.879 55.297-2.082 1.289 5.994-2.532 1.173 7.787 0.034 0.034 0.590 0.50 8.740 3.027 85.550-2.444 1.520 8.285-3.190 1.376 12.06 0.019 0.022 0.579 200 0.00-0.364 1.257 1.713 0.008 0.292 0.085-0.008 0.279 0.078 0.186 0.138 0.944 0.05-0.036 1.426 2.035-0.091 0.314 0.107-0.110 0.303 0.104 0.171 0.128 0.908 0.10 0.501 1.518 2.556-0.238 0.370 0.194-0.268 0.374 0.212 0.136 0.102 0.841 0.15 1.256 1.585 4.088-0.422 0.402 0.340-0.474 0.401 0.386 0.094 0.072 0.781 0.20 2.074 1.712 7.233-0.594 0.452 0.557-0.730 0.470 0.753 0.067 0.052 0.735 0.25 3.006 1.802 12.280-0.878 0.527 1.049-0.991 0.516 1.247 0.048 0.039 0.697 0.30 4.423 1.878 23.093-1.173 0.631 1.774-1.394 0.616 2.322 0.021 0.021 0.666 0.35 6.237 1.975 42.795-1.522 0.700 2.808-1.862 0.712 3.973 0.005 0.010 0.645 0.40 8.257 1.951 71.990-1.944 0.855 4.510-2.388 0.805 6.350-0.006 0.003 0.630 0.45 10.757 2.209 120.59-2.533 0.962 7.343-3.094 0.942 10.46-0.007 0.002 0.616 0.50 13.607 2.542 191.61-3.112 1.161 11.03-3.893 1.150 16.47-0.007 0.001 0.610 300 0.00-0.299 1.159 1.433 0.008 0.229 0.053 0.005 0.234 0.055 0.137 0.099 0.955 0.05 0.118 1.242 1.557-0.102 0.258 0.077-0.107 0.252 0.075 0.116 0.083 0.913 0.10 0.711 1.351 2.331-0.250 0.284 0.143-0.274 0.301 0.166 0.091 0.065 0.847 0.15 1.626 1.391 4.580-0.429 0.332 0.294-0.508 0.341 0.374 0.047 0.035 0.791 0.20 2.658 1.417 9.071-0.665 0.378 0.586-0.741 0.389 0.700 0.028 0.023 0.747 0.25 4.018 1.419 18.161-0.959 0.443 1.115-1.065 0.466 1.350 0.003 0.007 0.714 0.30 5.761 1.560 35.622-1.341 0.522 2.071-1.545 0.536 2.673-0.001 0.004 0.684 0.35 7.932 1.659 65.672-1.736 0.629 3.409-2.082 0.658 4.766-0.006 0.002 0.665 0.40 10.506 1.883 113.91-2.257 0.759 5.671-2.747 0.768 8.135-0.006 0.001 0.652 0.45 13.833 2.212 196.23-2.949 0.899 9.505-3.590 0.894 13.68-0.006 0.001 0.642 0.50 17.617 2.522 316.73-3.579 1.055 13.92-4.611 1.040 22.34-0.005 0.001 0.640 400 0.00-0.257 1.043 1.155 0.000 0.195 0.038 0.007 0.190 0.036 0.109 0.073 0.962 0.05 0.204 1.124 1.306-0.114 0.220 0.061-0.120 0.218 0.062 0.096 0.065 0.913 0.10 0.924 1.116 2.099-0.242 0.251 0.122-0.278 0.256 0.143 0.044 0.030 0.850 0.15 1.848 1.209 4.875-0.455 0.284 0.287-0.505 0.294 0.342 0.026 0.018 0.793 0.20 3.074 1.176 10.832-0.696 0.331 0.594-0.803 0.348 0.766 0.007 0.007 0.749 0.25 4.674 1.216 23.328-0.996 0.388 1.142-1.156 0.410 1.504-0.005 0.002 0.718 0.30 6.676 1.363 46.431-1.390 0.470 2.152-1.638 0.471 2.903-0.004 0.002 0.692 0.35 9.186 1.539 86.755-1.874 0.563 3.828-2.230 0.556 5.282-0.005 0.001 0.675 0.40 12.278 1.843 154.13-2.475 0.696 6.609-2.990 0.665 9.381-0.005 0.001 0.662 0.45 16.270 2.214 269.61-3.165 0.842 10.72-3.971 0.810 16.42-0.004 0.001 0.656 0.50 20.983 2.802 448.14-4.051 1.038 17.48-5.069 1.040 26.77-0.004 0.000 0.654 10

Table 2. Estimation Results Using HW Moel. N δ Bias SD Bias SD Bias SD Bias Eff Eff Mean Rank 50 0.00-0.407 1.840 3.552 0.035 0.727 0.530 0.013 0.673 0.453 0.304 0.247 0.843 0.05-0.293 1.923 3.785-0.103 0.767 0.598-0.058 0.768 0.594 0.319 0.264 0.827 0.10-0.237 2.138 4.629-0.149 0.813 0.684-0.243 0.817 0.727 0.347 0.288 0.790 0.15 0.069 2.177 4.745-0.390 0.934 1.023-0.353 0.857 0.859 0.332 0.279 0.735 0.20 0.204 2.384 5.727-0.604 1.008 1.380-0.471 0.955 1.135 0.355 0.302 0.682 0.25 0.430 2.513 6.499-0.779 1.064 1.738-0.708 1.083 1.674 0.361 0.310 0.638 0.30 0.525 2.689 7.506-0.930 1.132 2.145-1.012 1.138 2.319 0.376 0.325 0.595 0.35 0.819 2.939 9.312-1.344 1.229 3.317-1.246 1.245 3.102 0.396 0.347 0.560 0.40 1.009 3.194 11.217-1.633 1.376 4.559-1.610 1.346 4.404 0.415 0.371 0.529 0.45 1.068 3.195 11.351-2.109 1.506 6.714-1.811 1.433 5.332 0.451 0.407 0.506 0.50 1.280 3.408 13.250-2.530 1.645 9.103-2.186 1.555 7.199 0.441 0.403 0.489 100 0.00-0.404 1.667 2.943-0.010 0.475 0.226 0.013 0.445 0.198 0.266 0.214 0.895 0.05-0.141 1.708 2.936-0.111 0.500 0.263-0.099 0.472 0.233 0.254 0.206 0.877 0.10 0.106 1.912 3.665-0.286 0.549 0.383-0.213 0.548 0.345 0.263 0.215 0.827 0.15 0.451 2.064 4.463-0.466 0.597 0.574-0.425 0.598 0.538 0.279 0.232 0.773 0.20 0.898 2.285 6.028-0.669 0.679 0.909-0.600 0.673 0.813 0.262 0.223 0.727 0.25 1.236 2.447 7.517-0.954 0.730 1.444-0.847 0.751 1.281 0.264 0.225 0.686 0.30 1.631 2.877 10.939-1.260 0.834 2.282-1.139 0.814 1.961 0.296 0.259 0.652 0.35 1.961 3.030 13.026-1.706 0.888 3.698-1.514 0.892 3.087 0.345 0.311 0.621 0.40 2.442 3.532 18.439-2.186 0.980 5.738-2.038 1.040 5.235 0.379 0.344 0.593 0.45 2.894 3.977 24.189-2.932 1.115 9.837-2.456 1.141 7.332 0.419 0.384 0.574 0.50 3.186 4.343 29.013-3.635 1.243 14.75-3.026 1.254 10.72 0.459 0.419 0.558 200 0.00-0.326 1.299 1.794 0.001 0.296 0.087 0.005 0.295 0.087 0.184 0.140 0.929 0.05-0.071 1.493 2.233-0.125 0.327 0.123-0.115 0.319 0.115 0.202 0.155 0.898 0.10 0.325 1.661 2.865-0.257 0.349 0.188-0.270 0.359 0.202 0.185 0.142 0.839 0.15 0.945 1.894 4.480-0.480 0.379 0.374-0.447 0.412 0.370 0.171 0.138 0.784 0.20 1.665 1.973 6.665-0.761 0.440 0.773-0.693 0.451 0.683 0.154 0.130 0.735 0.25 2.195 2.234 9.808-1.100 0.512 1.473-0.987 0.529 1.255 0.170 0.142 0.697 0.30 3.151 2.575 16.559-1.502 0.569 2.581-1.352 0.583 2.169 0.168 0.146 0.666 0.35 4.103 3.158 26.809-2.052 0.685 4.679-1.870 0.677 3.955 0.213 0.195 0.639 0.40 4.723 3.859 37.198-2.733 0.770 8.060-2.459 0.764 6.631 0.281 0.258 0.616 0.45 6.189 4.586 59.340-3.592 0.932 13.77-3.091 0.903 10.36 0.279 0.253 0.601 0.50 7.199 5.268 79.573-4.663 1.078 22.90-3.949 1.076 16.75 0.291 0.249 0.587 300 0.00-0.349 1.190 1.537-0.005 0.229 0.053-0.006 0.223 0.050 0.167 0.119 0.945 0.05 0.065 1.283 1.650-0.122 0.257 0.081-0.112 0.265 0.083 0.137 0.097 0.908 0.10 0.663 1.413 2.437-0.292 0.289 0.168-0.258 0.287 0.149 0.111 0.082 0.850 0.15 1.336 1.540 4.158-0.523 0.318 0.374-0.484 0.357 0.361 0.101 0.076 0.794 0.20 2.231 1.760 8.077-0.799 0.379 0.782-0.741 0.386 0.698 0.085 0.063 0.752 0.25 3.006 2.233 14.019-1.189 0.434 1.603-1.090 0.448 1.389 0.129 0.107 0.715 0.30 4.444 2.410 25.560-1.679 0.489 3.058-1.523 0.508 2.579 0.098 0.086 0.688 0.35 5.590 3.239 41.740-2.307 0.556 5.632-2.084 0.604 4.709 0.161 0.149 0.663 0.40 7.527 3.669 70.113-3.076 0.674 9.913-2.738 0.711 8.001 0.140 0.131 0.646 0.45 9.236 4.817 108.50-4.181 0.808 18.13-3.630 0.782 13.78 0.173 0.148 0.627 0.50 10.873 6.006 154.29-5.477 0.948 30.90-4.619 0.964 22.26 0.197 0.156 0.617 400 0.00-0.268 1.102 1.287-0.001 0.199 0.040 0.001 0.199 0.040 0.126 0.090 0.952 0.05 0.261 1.162 1.419-0.142 0.219 0.068-0.119 0.232 0.068 0.098 0.068 0.910 0.10 0.857 1.245 2.286-0.297 0.248 0.150-0.288 0.256 0.149 0.076 0.053 0.850 0.15 1.651 1.421 4.743-0.551 0.283 0.384-0.501 0.300 0.341 0.068 0.050 0.798 0.20 2.480 1.737 9.169-0.858 0.332 0.846-0.789 0.346 0.742 0.087 0.064 0.754 0.25 3.942 1.841 18.930-1.264 0.376 1.739-1.141 0.411 1.471 0.055 0.049 0.723 0.30 5.130 2.437 32.254-1.798 0.426 3.413-1.615 0.458 2.818 0.094 0.086 0.695 0.35 7.172 2.872 59.679-2.519 0.523 6.619-2.249 0.549 5.358 0.087 0.081 0.672 0.40 9.656 3.539 105.76-3.406 0.643 12.01-3.046 0.653 9.705 0.080 0.071 0.654 0.45 11.949 4.798 165.80-4.652 0.760 22.21-4.033 0.745 16.81 0.100 0.080 0.637 0.50 13.703 6.749 233.31-6.248 0.917 39.87-5.197 0.913 27.84 0.152 0.110 0.626 11

Table 3. Estimation Results Using HV Moel. N δ Bias SD Bias SD Bias SD Bias Eff Eff Mean Rank 50 0.00-0.124 1.785 3.200 0.081 1.117 1.255-0.070 1.070 1.149 0.208 0.150 0.493 0.05-0.152 1.907 3.660 0.055 1.166 1.363 0.001 1.141 1.301 0.203 0.148 0.518 0.10-0.359 2.007 4.155 0.227 1.236 1.580-0.070 1.249 1.565 0.217 0.157 0.520 0.15-0.267 2.063 4.326 0.239 1.365 1.919-0.046 1.323 1.753 0.190 0.138 0.561 0.20-0.288 2.162 4.758 0.224 1.460 2.183-0.085 1.378 1.905 0.195 0.139 0.575 0.25-0.184 2.319 5.412 0.434 1.562 2.628-0.254 1.539 2.434 0.162 0.113 0.613 0.30-0.382 2.344 5.640 0.590 1.685 3.189-0.350 1.613 2.723 0.168 0.117 0.627 0.35-0.210 2.473 6.160 0.624 1.881 3.928-0.342 1.733 3.122 0.128 0.085 0.664 0.40-0.197 2.619 6.897 0.695 2.004 4.498-0.526 1.824 3.603 0.125 0.084 0.680 0.45-0.258 2.722 7.474 0.775 2.058 4.835-0.462 1.990 4.172 0.116 0.078 0.707 0.50-0.136 2.890 8.370 0.993 2.330 6.416-0.570 2.072 4.617 0.095 0.064 0.731 100 0.00-0.296 1.667 2.868 0.042 0.742 0.552 0.043 0.744 0.555 0.165 0.115 0.552 0.05-0.383 1.764 3.257 0.116 0.798 0.650 0.016 0.797 0.636 0.166 0.116 0.569 0.10-0.578 1.915 4.003 0.201 0.874 0.804 0.033 0.911 0.831 0.147 0.096 0.585 0.15-0.490 1.926 3.951 0.294 0.963 1.013-0.076 0.939 0.887 0.129 0.085 0.630 0.20-0.565 2.023 4.411 0.340 1.055 1.228-0.065 1.014 1.033 0.112 0.070 0.661 0.25-0.669 2.131 4.988 0.561 1.064 1.446-0.158 1.119 1.278 0.094 0.060 0.707 0.30-0.693 2.300 5.770 0.661 1.212 1.907-0.167 1.207 1.485 0.073 0.044 0.736 0.35-0.586 2.461 6.402 0.683 1.280 2.105-0.219 1.328 1.813 0.053 0.029 0.759 0.40-0.508 2.587 6.950 0.972 1.379 2.846-0.424 1.421 2.199 0.042 0.022 0.784 0.45-0.587 2.684 7.550 0.885 1.573 3.257-0.327 1.496 2.345 0.030 0.014 0.804 0.50-0.419 3.052 9.489 1.063 1.681 3.955-0.385 1.693 3.014 0.024 0.012 0.823 200 0.00-0.316 1.442 2.180 0.016 0.534 0.286 0.038 0.526 0.278 0.133 0.087 0.601 0.05-0.554 1.580 2.803 0.125 0.592 0.366 0.020 0.552 0.305 0.130 0.083 0.611 0.10-0.622 1.612 2.984 0.233 0.626 0.446-0.033 0.631 0.399 0.098 0.056 0.661 0.15-0.640 1.696 3.287 0.333 0.635 0.514-0.046 0.680 0.464 0.069 0.039 0.724 0.20-0.722 1.856 3.967 0.495 0.719 0.761-0.098 0.732 0.546 0.045 0.021 0.758 0.25-0.803 1.821 3.963 0.583 0.783 0.953-0.127 0.801 0.658 0.034 0.016 0.794 0.30-0.676 1.984 4.394 0.705 0.813 1.157-0.215 0.859 0.785 0.020 0.007 0.823 0.35-0.505 2.137 4.820 0.777 0.921 1.453-0.312 0.964 1.027 0.014 0.005 0.841 0.40-0.600 2.141 4.945 0.906 1.042 1.906-0.377 1.025 1.193 0.013 0.004 0.854 0.45-0.416 2.447 6.163 0.982 1.124 2.229-0.426 1.148 1.498 0.010 0.004 0.866 0.50-0.477 2.448 6.220 1.168 1.220 2.851-0.531 1.187 1.692 0.008 0.003 0.875 300 0.00-0.382 1.370 2.022 0.041 0.454 0.208 0.006 0.443 0.196 0.123 0.073 0.608 0.05-0.583 1.410 2.328 0.142 0.465 0.236 0.021 0.487 0.238 0.094 0.055 0.674 0.10-0.742 1.500 2.801 0.291 0.493 0.327 0.003 0.535 0.286 0.066 0.037 0.723 0.15-0.816 1.567 3.123 0.399 0.539 0.450-0.044 0.586 0.346 0.041 0.018 0.767 0.20-0.745 1.634 3.223 0.470 0.547 0.520-0.062 0.629 0.399 0.022 0.008 0.817 0.25-0.812 1.746 3.706 0.625 0.647 0.810-0.126 0.673 0.469 0.013 0.003 0.842 0.30-0.702 1.808 3.763 0.689 0.699 0.963-0.210 0.749 0.605 0.011 0.004 0.858 0.35-0.728 1.949 4.330 0.813 0.754 1.231-0.244 0.817 0.728 0.008 0.002 0.872 0.40-0.778 2.096 4.997 1.005 0.836 1.709-0.324 0.880 0.879 0.007 0.002 0.881 0.45-0.732 2.348 6.047 1.098 0.988 2.181-0.402 0.980 1.122 0.008 0.003 0.887 0.50-0.687 2.485 6.650 1.212 1.113 2.708-0.435 1.089 1.375 0.009 0.004 0.892 400 0.00-0.292 1.294 1.760 0.012 0.391 0.153 0.027 0.383 0.148 0.093 0.054 0.669 0.05-0.601 1.335 2.142 0.176 0.403 0.193 0.020 0.433 0.188 0.069 0.036 0.710 0.10-0.694 1.417 2.490 0.286 0.436 0.272-0.022 0.464 0.216 0.047 0.022 0.757 0.15-0.828 1.473 2.855 0.385 0.453 0.353-0.029 0.515 0.266 0.026 0.009 0.801 0.20-0.874 1.534 3.118 0.544 0.508 0.554-0.079 0.538 0.296 0.014 0.004 0.840 0.25-0.838 1.505 2.969 0.631 0.521 0.670-0.129 0.576 0.349 0.008 0.002 0.868 0.30-0.895 1.674 3.604 0.728 0.596 0.886-0.161 0.632 0.425 0.007 0.002 0.878 0.35-0.843 1.883 4.255 0.857 0.676 1.191-0.239 0.736 0.598 0.006 0.001 0.887 0.40-0.838 2.078 5.022 0.974 0.799 1.588-0.252 0.796 0.697 0.006 0.002 0.895 0.45-0.721 2.091 4.890 1.096 0.896 2.005-0.356 0.882 0.906 0.007 0.003 0.900 0.50-0.752 2.311 5.908 1.255 1.062 2.704-0.448 0.942 1.088 0.008 0.004 0.903 12

Table 4. Estimation Results Using HVW Moel. N δ Bias SD Bias SD Bias SD Bias Eff Eff Mean Rank 50 0.00-0.207 1.816 3.341 0.064 1.169 1.371-0.013 1.110 1.233 0.226 0.162 0.467 0.05-0.242 1.880 3.593 0.025 1.153 1.330 0.047 1.180 1.395 0.222 0.162 0.506 0.10-0.303 2.009 4.126 0.047 1.291 1.669-0.011 1.310 1.715 0.235 0.168 0.496 0.15-0.271 2.020 4.153-0.033 1.440 2.074 0.075 1.350 1.828 0.216 0.155 0.528 0.20-0.219 2.190 4.842-0.153 1.548 2.418 0.066 1.446 2.095 0.222 0.160 0.539 0.25-0.169 2.227 4.989-0.106 1.624 2.647-0.057 1.623 2.636 0.202 0.144 0.562 0.30-0.446 2.240 5.216-0.070 1.813 3.291-0.117 1.697 2.892 0.219 0.156 0.567 0.35-0.277 2.447 6.064-0.102 2.012 4.057-0.112 1.800 3.252 0.180 0.125 0.617 0.40-0.317 2.524 6.470-0.238 2.112 4.516-0.141 1.911 3.673 0.185 0.132 0.627 0.45-0.280 2.627 6.979-0.356 2.235 5.122-0.062 2.093 4.385 0.176 0.125 0.652 0.50-0.287 2.676 7.245-0.230 2.506 6.331-0.114 2.174 4.738 0.141 0.096 0.685 100 0.00-0.339 1.680 2.938 0.024 0.787 0.620 0.087 0.756 0.580 0.170 0.118 0.538 0.05-0.359 1.762 3.233 0.035 0.865 0.750 0.054 0.832 0.695 0.176 0.120 0.546 0.10-0.388 1.902 3.770-0.023 0.954 0.910 0.064 0.927 0.864 0.168 0.112 0.565 0.15-0.291 1.953 3.899-0.064 1.075 1.161 0.010 0.966 0.932 0.155 0.103 0.594 0.20-0.346 2.104 4.548-0.118 1.151 1.338 0.040 1.042 1.087 0.151 0.099 0.614 0.25-0.333 2.250 5.174-0.146 1.191 1.440 0.006 1.162 1.351 0.136 0.091 0.653 0.30-0.430 2.390 5.894-0.100 1.388 1.935 0.031 1.256 1.578 0.111 0.072 0.687 0.35-0.416 2.401 5.937-0.264 1.453 2.181 0.093 1.342 1.811 0.094 0.059 0.714 0.40-0.228 2.594 6.779-0.117 1.656 2.757-0.094 1.437 2.073 0.077 0.049 0.749 0.45-0.352 2.595 6.858-0.336 1.795 3.336 0.063 1.537 2.368 0.061 0.035 0.770 0.50-0.367 2.733 7.603-0.205 1.944 3.822 0.082 1.671 2.798 0.048 0.028 0.805 200 0.00-0.309 1.498 2.341 0.006 0.591 0.349 0.047 0.529 0.282 0.135 0.088 0.590 0.05-0.407 1.636 2.841-0.017 0.676 0.457 0.046 0.558 0.314 0.146 0.092 0.584 0.10-0.342 1.725 3.091-0.070 0.719 0.521 0.012 0.656 0.430 0.128 0.077 0.618 0.15-0.301 1.878 3.616-0.062 0.791 0.629 0.032 0.695 0.485 0.095 0.058 0.678 0.20-0.287 1.980 4.001-0.052 0.884 0.785 0.016 0.771 0.595 0.069 0.037 0.713 0.25-0.301 2.000 4.091-0.139 0.952 0.925 0.041 0.838 0.704 0.057 0.031 0.749 0.30-0.286 2.202 4.930-0.088 1.036 1.081 0.024 0.894 0.799 0.046 0.026 0.789 0.35-0.092 2.225 4.960-0.135 1.080 1.185-0.030 1.002 1.005 0.029 0.015 0.820 0.40-0.228 2.224 4.998-0.112 1.225 1.514-0.035 1.053 1.111 0.021 0.010 0.846 0.45-0.026 2.367 5.603-0.165 1.256 1.604-0.009 1.154 1.331 0.014 0.006 0.871 0.50-0.083 2.473 6.124-0.163 1.378 1.926 0.003 1.266 1.603 0.009 0.003 0.887 300 0.00-0.328 1.439 2.178 0.000 0.508 0.258 0.010 0.445 0.198 0.132 0.080 0.598 0.05-0.365 1.504 2.396-0.025 0.561 0.316 0.047 0.495 0.247 0.108 0.063 0.649 0.10-0.366 1.694 3.005-0.019 0.638 0.408 0.045 0.556 0.311 0.088 0.052 0.685 0.15-0.308 1.781 3.267-0.057 0.709 0.505 0.029 0.602 0.363 0.061 0.031 0.731 0.20-0.145 1.880 3.555-0.112 0.716 0.525 0.047 0.650 0.424 0.040 0.021 0.785 0.25-0.199 2.096 4.434-0.091 0.836 0.707 0.026 0.705 0.498 0.028 0.013 0.809 0.30-0.122 2.025 4.117-0.166 0.904 0.844 0.035 0.773 0.599 0.021 0.009 0.834 0.35-0.138 2.168 4.717-0.106 0.923 0.864 0.036 0.849 0.722 0.010 0.003 0.869 0.40-0.108 2.162 4.687-0.092 0.955 0.921 0.017 0.905 0.820 0.007 0.002 0.890 0.45-0.074 2.241 5.027-0.110 1.084 1.187-0.017 1.006 1.012 0.005 0.001 0.904 0.50-0.052 2.206 4.870-0.125 1.139 1.314 0.019 1.050 1.102 0.004 0.001 0.919 400 0.00-0.342 1.367 1.985 0.012 0.461 0.213 0.030 0.384 0.149 0.109 0.064 0.636 0.05-0.380 1.438 2.211 0.024 0.522 0.273 0.037 0.439 0.194 0.079 0.041 0.681 0.10-0.241 1.601 2.621-0.030 0.579 0.336 0.012 0.480 0.230 0.061 0.031 0.727 0.15-0.304 1.717 3.040-0.065 0.606 0.372 0.040 0.525 0.277 0.045 0.022 0.772 0.20-0.230 1.801 3.297-0.025 0.694 0.482 0.028 0.553 0.306 0.022 0.008 0.820 0.25-0.110 1.817 3.314-0.090 0.716 0.521 0.038 0.596 0.357 0.014 0.005 0.853 0.30-0.160 1.917 3.701-0.107 0.742 0.563 0.065 0.658 0.437 0.009 0.003 0.877 0.35-0.018 1.986 3.943-0.116 0.830 0.703 0.016 0.751 0.564 0.006 0.002 0.896 0.40-0.110 2.096 4.407-0.054 0.877 0.772 0.046 0.802 0.646 0.003 0.001 0.914 0.45-0.033 2.022 4.089-0.089 0.898 0.814 0.033 0.861 0.742 0.002 0.000 0.929 0.50-0.057 2.115 4.478-0.061 1.053 1.112-0.007 0.943 0.889 0.002 0.000 0.938 13

Table 5. Bias an Rank Correlation in the Four Moels (homosceastic case) (N=200, δ=0) H0 HW HV HVW -0.364 (1.257) -0.326 (1.299) -0.316 (1.442) -0.309 (1.498) 0.008 (0.292) 0.001 (0.296) 0.016 (0.534) 0.006 (0.591) β 2-0.008 (0.279) 0.005 (0.295) 0.038 (0.526) 0.047 (0.529) Bias in Efficiency 0.186 0.184 0.133 0.135 Rank Correlation Of Effiency 0.944 0.929 0.601 0.590 Table 6. Bias an Rank Correlation in Various Moels (N=200, δ=0.25) H0 HW HV HVW β 0 3.006 (1.802) 2.195 (2.234) -0.803 (1.821) -0.301 (2.000) β 1-0.878 (0.527) -1.100 (0.512) 0.583 (0.783) -0.139 (0. 952) β 2-0.991 (0.516) -0.987 (0.529) -0.127 (0.801) 0.041 (0.838) Bias in Efficiency 0.048 0.170 0.034 0.057 Rank Correlation Of Effiency 0.697 0.697 0.794 0.749 Table 7. Bias an Rank Correlation in Various Moels (N=200, δ=0.50 ) H0 HW HV HVW β 0 13.607 (2.542) 7.199 (5.268) -0.477 (2.448) -0.083 (2.473) β 1-3.112 (1.161) -4.663 (1.078) 1.168 (1.220) -0.163 (1.378) β 2-3.893 (1.150) -3.949 (1.076) -0.531 (1.187) 0.003 (1.266) Bias in Efficiency -0.007 0.291 0.008 0.009 Rank Correlation Of Effiency 0.610 0.587 0.875 0.887 14

Figure 1. Parameter Biases (N=50) 5.00 b 0 4.00 3.00 2.00 1.00 0.00-1.00 (a) 1.50 1.00 0.50 0.00-0.50-1.00-1.50-2.00-2.50-3.00 b 1 (b) 0.50 b 2 0.00-0.50-1.00-1.50-2.00-2.50 (c)h0 HW HV HVW 15

Figure 2. Parameter Biases (N=400) 25.00 b 0 20.00 15.00 10.00 5.00 0.00-5.00 (a) b 1 2.00 1.00 0.00-1.00-2.00-3.00-4.00-5.00-6.00-7.00 (b) 1.00 b 2 0.00-1.00-2.00-3.00-4.00-5.00-6.00 (c) H0 HW HV HVW 16

Figure 3. Efficiency Biases. (N=50) 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 (a) 0.16 (N= 400) 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00-0.02 (b) H0 HW HV HVW 17

Figure 4. Mean Rank Correlation. 1.00 (N= 50) 0.90 0.80 0.70 0.60 0.50 0.40 0.30 (a) (N= 400) 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 (b) H0 HW HV HVW 18