Statistical Hypothesis Testing for Returns to Scale Using Data Envelopment Analysis

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1 Statstcal Hypothess Testng for Returns to Scale Usng Data nvelopment nalyss M. ukushge a and I. Myara b a Graduate School of conomcs, Osaka Unversty, Osaka , apan (mfuku@econ.osaka-u.ac.p) b Graduate School of conomcs, Osaka Unversty, Osaka , apan (cg087m@srv.econ.osaka-u.ac.p) bstract: Data envelopment nalyss (D) s a knd of non-parametrc and non-statstcal tools for measurng technologcal effcences. Recently, several researchers adopt ths method to evaluate the effcences of frms, publc utltes, and so on. ecause ths method s not based on the statstcal setup, we cannot test hypothess where the producton functon should be assumed. In ths paper, we propose several tests for returns to scale usng dfferent D models. We construct test statstcs for the hypothess about the returns to scale n producton technology wthout any assumpton on the true dstrbuton of the techncal neffcences. We conduct a smulaton study on the sze and power of the proposed test statstcs under several condtons: Cobb-Douglas producton functon wth half-normally and eponentally dstrbuted error terms. Keywords: Data nvelopment nalyss; Test for returns to scale; Monte Carlo smulaton 1. INTRODUCTION Data nvelopment nalyss (D) has been appled to evaluate the effcences n the economc actvtes lke publc or prvate servce producton. Ths s a knd of non-parametrc and non-statstcal tools, so the estmated neffcences depend on the assumpton whether the producton technology s constant returns to scale or not. rom the researchers or polcy makers pont of vew, whether the producton technology s ncreasng or decreasng returns to the scale s one of the most mportant characterstcs to make decsons. However, only few attempts have been made at the D-based statstcal tests for the returns to scale. anker (1996) surveys statstcal tests usng D but these tests are not enough to use n an emprcal research because ther asymptotc dstrbuton and fnte sample propertes of them are not clear. Generally, D s a knd of non-parametrc and non-statstcal tools, so t s dffcult to obtan ther asymptotc dstrbuton. Therefore, n the present paper, we propose some testng procedures and conduct a smulaton study to nvestgate ther fnte sample propertes. The paper conssts as follows. Secton ntroduces the new testng procedures for the returns to scale. Secton 3 eplans the setups of the Monte Carlo smulaton. Secton 4 presents smulaton results. Secton 5 s a concludng remark.. NW TSTS OR RTURNS TO SCL USING DIRNT D MODLS.1. D models wth dfferent assumpton of returns to scale There are some models wth dfferent assumpton n D;,,, and CC. (See Cooper, Seford and Tone (000).) model s proposed by Charnes, Cooper and Rhodes (1978) and s assumed the fronter to be constant returns to scale. The nput orented model s wrtten as the followng lnear programmng problem: s.t. 1 mn y y m 1 y 0 m 1,, n 1,, M N 1,, m n (1) where =1,, (number of unts), m=1,,m (number of outputs), n=1,,n (number of nputs)

2 , and CC models have slghtly dfferent assumptons n quaton 1; they nclude a constrant on the multpler,. model assumes that the fronter ehbts ncreasng returns to scale: 1 1. model assumes the fronter to be decreasng returns to scale: 1 1. CC model, developed by anker, Charnes and Cooper (1984), assumes the fronter 1 to be varable returns to scale: 1... Relatonshps between the true producton functon and the effcency scores n D models We often obtan dfferent effcency scores by applyng dfferent D models because there ests the dfference between the returns to scale n the true producton technology and those assumed by the D model that we appled. In the present paper, to smplfy the eplanaton, we fgure the relatonshps between the true producton technology and the estmated effcency scores wth one nput and one output case as an eample. rstly, let us see the case that the true fronter producton functon, y f (), s ncreasng returns to scale. See gure 1. ll ponts from to represent productons. Pont,, C and D are on the true fronter functon. The estmated fronters by D models are as follows. The estmated fronter by model s a straght lne Ol passng through pont D, the model fronter s ODd, the model fronter s al1, and the CC model fronter s a lnear envelop, add. The effcency score of each pont vares accordng to the D models. or eample, the effcency score of pont estmated by or that estmated by epresses rato of b1 to b1b, that estmated by or CC model s rato of b to bb. The former s lower than the latter. Then, let be the mean of the estmated effcences from model (for =,,, CC), the followng nequalty tends to hold; CC. () The mean of CC scores s the hghest and the mean of s the lowest among all of means because of the neffcent producton such as pont or. Inequalty between the mean of scores and that of scores holds ecept n some cases; there are some unts wth large neffcences whch are mproved n model. b1 y O b a b f () gure 1. Increasng returns to scale technology and D models. Secondly, let us see the case that the true fronter producton functon s decreasng returns to scale as gure. The estmated fronters n D models are as follows. The model fronter s a straght lne Ol passng through pont, the model fronter s al, the model fronter s OCDd, and the CC model fronter s a lnear envelop, aocdd. mong the means, the followng nequalty tends to hold; C CC. (3) It s same as ncreasng returns to scale fronter that the hghest mean s the CC mean and the lowest s the mean. However, nequalty between the and means s dfferent; the mean s hgher than the mean ecept n some crcumstances. O y a l C D D l1 l f () gure. Decreasng returns to scale technology and D models. Lastly, let us see the case that the true fronter functon s constant returns to scale. In ths case, the followng nequalty tends to hold; CC (, ). (4) If all producton unts are effcent, all of the means are same and equal to one. When there are neffcent unts, the hghest and the lowest mean

3 are same as other cases: the mean of CC scores s hghest and the mean of scores s lowest and the relatonshp between the and means s not clear. ut, we can epect all the means take closer values n case of constant returns to scale. In summary, we can see the followng ponts from the eamples above. rstly, t holds always that the CC mean s the hghest and the mean s the lowest. Secondly, the mean s hgher than the mean n case of ncreasng returns to scale ecept some crcumstances. Thrdly, the mean s larger than the mean. nally, all the means take closer value n case of constant returns to scale..3. New tests for returns to scale usng D We propose some testng procedures for returns to scale usng relatonshp among the means of scores estmated by dfferent models. We wll test the equalty of and CC usng the test of equalty of means or sgn test. The reason why we adopt sgn test s that means can be easly affected by outlers. When the alternatve hypothess that s accepted, the true fronter functon can be regard as ncreasng returns to scale. When the hypothess that s accepted, the true fronter functon can be regard as decreasng returns to scale. When the CC hypothess that or s not reected, the true fronter functon can be regard as constant returns to scale. We suggest the followng new four testng procedures. ) Test of equalty of mean of scores and that of scores. Case of : H 0 : vs. H 1 : The test statstcs s: z 1 (5) S S n n where, S stands for standard devaton and n s the number of producton unts. Reect the null hypothess when the test statstcs s larger than 1.645, the upper 5% crtcal value from standard normal dstrbuton. Then accept the alternatve hypothess (ncreasng returns to scale). Case that : H 0 : vs. H 1 :. n quaton 5 s replaced wth. ) Sgn test of the probablty that number of unts, whose scores are mproved n than, s more than half of all unts Case of # ( ) n : H 0 : P( ) 0. 5 H 1 : P( ) 0.5. The test statstcs s: z X ( n ) (6) n 4 where X s the number of unts whose score s larger than score, n s the number of unts. Reect the null hypothess when the test statstcs s larger than 1.645, the upper 5% crtcal value from standard normal dstrbuton. Case that : H 0 : P( ) 0. 5 H 1 : P( ) 0. 5 and X s replaced wth the number of unts whose score s larger than score. C) Test of equalty of the CC score mean and the score mean and the equalty of the scores mean and the scores mean. 1 st step: Test the equalty of the mean of CC scores and scores. CC CC H 0 : vs. H 1 :. The test statstcs s: z 3 CC (7) S CC S n n Reect the null hypothess when the test statstcs s larger than 1.645, the upper 5% crtcal value from standard normal dstrbuton, then proceed net step. If not reected, accept the null hypothess (constant returns to scale). nd step: Test the equalty of the mean of scores and scores. It s same as ()

4 D) Test of equalty of the CC score mean and the score mean and sgn test of the probablty that number of unts, whose scores are mproved n than, s more than half of all unts. 1 st step: Ths step s same as frst step of (C). When the null hypothess s reected, then proceed to net step. If not reected, accept the null hypothess (constant returns to scale). nd step: Ths step s same as (). 3. STTING OR MONT CRLO SIMULTIONS We conduct Monte Carlo smulaton to study the fnte sample propertes of the proposed tests Producton Technology Generally, the D can be appled to multple nputs and multple output producton technology. ut, for ts smplcty, accordng to anker (1996), we specfy the followng Cobb-Douglas producton technologes; Homogenous producton functon: Q 10 Concave producton functon: (8a) Q 10 (8b) XP(0.05) (1) XP() represents the eponental dstrbuton wth mean. The random varable of (9) and (10) s generated from half normal dstrbuton. oth of means of (9) and (11) are same, lso both of the means of (10) and (1) are same, The dstrbuton of (9) and (11) are same as anker (1996) adopted Sample Sze and Smulated Observaton s noted above, the value of nputs vector for each observaton are generated from a unform dstrbuton. The correspondng value of the output for each observaton, Y, s obtaned usng the producton functon (8) and the smulated value of the neffcency term. We generate Y as follows: Y Q / f ( X ) / (13) We consder the case that the sample sze s 100. Totally we consder 1 setups: 3 producton technologes, 4 neffcency dstrbutons and 1 sample sze. We replcate 500 drawngs for each setup. We use GUSS 5.0 to generate random varables, solve lnear programmng for the D estmates, and construct test statstcs. Conve producton functon: 4. RSULTS O SIMULTIONS Q 10 (8c) The frst In ths paper, we compare the proposed tests above and the tests ntroduced by anker (1996). efore the dscusson of smulatons results, we wll eplan the tests for returns to scale usng D that are ntroduced by anker (1996). where 1 and are each drawn randomly and ndependently from unform probablty dstrbuton over the nterval [5, 15]. technology ehbts constant returns to scale for all nput value. The second producton functon s concave, ehbtng decreasng returns to scale for all nputs. The thrd producton functon s conve, ehbtng ncreasng returns to scale for all nputs. These settngs are same as anker (1996) adopted. 3.. Ineffcency Dstrbutons The dstrbuton of the neffcency for any observaton must be above 1, we wrte 1, 0. We consder four dfferent dstrbutons as follows: N(0,0.0) (9) ) Test of goodness of ft t N 1 4 N 1 ( ( CC 1) 1) whch follows the dstrbuton wth (N,N) f the CC dstrbuton of and are same. If not reected, accept the constant returns to scale hypothess. ) Kolmogorov-Smlrnov Test CC ma ( ) ( ) 1, N, N(0,0.063) (10) where (.) s the emprcal dstrbuton functon. XP(0.159) (11) If not reected, accept the constant returns to scale hypothess

5 Table 1: Summary of statstcal test results (sample sze=100) Panel :True producton technology s constant return to scale Ineffcency dstrbuton Type of Tests Hypothess ponetal Half Normal Null lt. CRS ( ( ( ( CRS ( ( ( ( CRS CRS CC CRS, C CRS CRS CC CRS, D CRS CRS CC CRS, CRS, CC CRS, CRS, Panel :True producton technology s decreasng returns to scale Ineffcency dstrbuton Type of Tests Hypothess ponetal Half Normal Null lt. CRS ( ( ( ( CRS ( ( ( ( CRS CRS CC CRS, C CRS CRS CC CRS, D CRS CRS CC CRS, CRS, CC CRS, CRS, Panel C:True producton technology s ncreasng returns to scale Ineffcency dstrbuton Type of Tests Hypothess ponetal Half Normal Null lt. CRS ( ( ( ( CRS ( ( ( ( CRS CRS CC CRS, C CRS CRS CC CRS, D CRS CRS CC CRS, CRS, CC CRS, CRS,

6 Table 1 presents the results of s knds of tests for returns to scale. The percentage n each cell represents the percentage of that the null hypothess s reected. rom Panel 1, sze of the test () ntroduced n anker s 0% regardless of the neffcency dstrbuton. On the other hand, sze of test () s 100%. Sze of the proposed test (), (C) and (D) are less than 6% n the case of eponental dstrbuted neffcency. When the true producton technology s decreasng or ncreasng returns to scale, the percentage of reecton represents the power of test. s for the decreasng returns to scale, the power of test () s 100% n all cases. The power of test () s over 85% regardless of dstrbuton. The power of test () and (D) are over 95% ecept the case of half normal dstrbuton wth larger mean. s for the case of eponental dstrbuton, the power of test () and (D) are over 99%. Half normal dstrbuton has larger varance than eponental dstrbuton wth same mean as half normal dstrbuton. Therefore one of reasons why the power of test usng half normal dstrbuted neffcency s lower may be occurrence of outlers. s for the case of ncreasng returns to scale technology, the power of all tests s over 99% regardless of dstrbuton n contrast wth decreasng returns to scale technology. Ths result may be brought by that nput value range s narrow. or ncreasng returns to scale technology, the dfference between the effcency of and s larger as nput value s larger (See gure1). 7. RRNCS anker, Rav D., Hypothess Tests Usng Data nvelopment nalyss, ournal of Productvty nalyss, 7, , anker, Rav D., Charnes,. and Cooper, W. W., Some models for estmatng techncal and scale neffcences n data envelopment analyss, Management Scence, 30, , Charnes,., Cooper, W. W. and Rhodes,, Measurng the effcency of decson makng unts, uropean ournal of Operatonal Research,, 13-, Cooper, Wllam W., Seford, Lawrence M. and Tone, Kaoru, Data nvelopment nalyss: Comprehensve Tet wth Models, pplcatons, References and D-Solver Software, CONCLUSIONS In ths paper, we propose new tests for returns to scale wth some dfferent D models. To nvestgate the sze and powers of the test statstcs, smulatons are conducted under several condtons: Cobb-Douglas wth eponental and half normal dstrbuted error. rom results of smulatons, test (), (C) and () have good property n the case of the eponental dstrbuted neffcency. However, there s no reason that the neffcency follows only eponental dstrbuton. One of reasons that the test statstcs do not work well wth the half normal dstrbuted neffcency can be effects of outler for the mean. We have to pay attenton to use these tests n case of neffcency dstrbuted half normal. 6. CKNOWLDGMNT Ths research was supported by the Mnstry of ducaton, Scence, Sports and Culture, Grant-n- ds for Scentfc Research (C),