Analysis of trends in technical and allocative efficiency: An application to Texas public school districts

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1 European Journal of Operaional Research 154 (2004) Analysis of rends in echnical and allocaive efficiency: An applicaion o Texas public school disrics Rajiv D. Baner, Surya Janairaman, Ram Naarajan * The Universiy of Texas a Dallas, School of Managemen, Richardson, TX , USA Absrac We presen a new DEA based mehod o analyze efficiency rends over ime and differences across subgroups in a panel daa seing. We employ a resul ha he aggregae echnical and allocaive inefficiency score equals he echnical inefficiency when inpu quaniies are aggregaed ino a single oal inpu cos variable, and develop es procedures o evaluae he presence of allocaive inefficiency. We apply hese mehods o es for he presence of allocaive inefficiency in Texas school disrics over , and analyze shifs and rends in boh echnical and allocaive inefficiencies over ime for differen regions. Our empirical resuls indicae he exisence of saisically significan allocaive inefficiencies. While echnical inefficiency increased over he six year sample period, allocaive inefficiency remained relaively sable during his period. These resuls for he full sample obain also when we repea he analysis for differen regions. Ó 2003 Elsevier B.V. All righs reserved. Keywords: Daa envelopmen analysis; Allocaive efficiency; Technical efficiency; Texas school disrics; Trend analysis 1. Inroducion In weny years since is incepion, daa envelopmen analysis (DEA) has become a widespread analyical ool for evaluaing he relaive efficiency of comparable organizaions (Seiford, 1996). While he original DEA models specify he producion se relaing inpus o oupus only in erms of properies such as convexiy and monooniciy and do no impose any parameric srucure on he disribuion of he inefficiency of individual observaions, saisical properies can be derived for he DEA esimaor and a variey of saisical ess * Corresponding auhor. address: naaraj@udallas.edu (R. Naarajan). can be devised if addiional srucure is specified (Baner, 1993, 1996). When daa are available in panel form wih several observaions for each decision maing uni (DMU) in he sample over several ime periods, he researcher may be ineresed in evaluaing efficiency rends over ime as well as across DMUs. Techniques such as window analysis (Charnes e al., 1985) and Malmquis index (Fare e al., 1994) have been used for his purpose. In his paper, we presen an alernaive way of analyzing efficiency rends over ime and efficiency differences across DMUs in a panel daa seing. To illusrae our mehod, we analyze echnical and allocaive efficiency rends and differences over a six year period using daa on 585 Texas Public School disrics. Real expendiures for public K-12 educaion in he Unied Saes increased from $ /$ - see fron maer Ó 2003 Elsevier B.V. All righs reserved. doi: /s (03)

2 478 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) billion in 1890 o nearly $190 billion in 1990 (Hanushe e al., 1994). During his period, he growh in educaion expendiures was more han hree imes he growh in he Gross Naional Produc. As a resul, considerable aenion has cenered on wheher he public school sysem efficienly uilizes resources. DEA had been used in he pas o analyze he educaion producion funcion (Bessen e al., 1982; Ahn e al., 1988; Ray, 1991) and more recenly in Grossopf e al. (1999). Many of hese sudies focus on he cross-secional relaionship beween expendiures and suden enrollmen or expendiure per suden and oucomes such as es scores. We conribue o his lieraure by providing a saisical analysis of efficiency scores and heir rends for a large number of school disrics over an exended period of ime. We focus on wo differen dimensions of efficiency in a muliple oupu muliple inpu producion environmen and examine he echnical and allocaive efficiencies for school disrics locaed in hree differen geographical regions of Texas over a six year period. We use he saisical ess developed in Baner (1993, 1996) o examine he rends in echnical and allocaive inefficiencies. We adap he saisical ess developed in Baner e al. (1999) o examine he exisence of allocaive inefficiencies associaed wih inpu uilizaion. We employ he resul ha he DEA echnical inefficiency measure using a single aggregae cos variable, consruced from muliple inpus weighed by heir uni coss, reflecs he aggregae echnical and allocaive inefficiency. Thus, our analysis provides an inuiive inerpreaion for he echnical inefficiency esimaed by sudies ha use aggregae cos expendiure informaion. Based on his analysis, we consruc saisical ess of he null hypohesis of no allocaive inefficiency analogous o hose of he null hypohesis of no scale inefficiency described in Baner (1996). Our empirical resuls indicae saisically significan allocaive inefficiencies. While echnical inefficiency increased over he six year sample period, allocaive inefficiency remained relaively sable during his period. These resuls for he whole sample obain also when we repea he analysis for differen regions. Pairwise comparison of he performance of he hree regions indicaes ha Texas school disrics in he Norh region operae a he lowes levels of echnical inefficiency followed by he Souh-Eas and finally he Wes region. The res of he paper is organized as follows. In Secion 2, we describe he DEA mehodology used in he esimaion of echnical and allocaive inefficiencies using panel daa. In Secion 3, we describe our daa and, in Secion 4, we discuss he resuls of our empirical analysis. In Secion 5, we presen a summary and concluding remars. 2. Evaluaing shifs in echnical and allocaive inefficiencies using DEA Le Y j ¼ðy 1j ;...; y rj ;...; y Rj Þ P 0 and X j ¼ ðx 1j ;...; x ij ;...; x Ij Þ P 0, j ¼ 1;...; N, ¼ 1;...; T, be he observed oupu and inpu vecors generaed from an underlying producion possibiliy se S ¼ fðx ;Y Þj oupus Y can be produced from inpus X g for a sample of N firms in he same indusry over T ime periods. 1 Each inpu i, i ¼ 1;...; I, is bough by all firms in he same compeiive mare a a price p i. Le P ¼ðp 1 ;...; p I Þ be he vecor of inpu prices. The cos of inpu i for firm j in period is hen c ij ¼ p i x ij. Denoe he oal cos of inpus for firm j in period as c j ¼ P i p ix ij. The inpu-oriened echnical inefficiency h B j P 1 of an observaion ðx j ; Y j Þ2S for he period, measured radially by he reciprocal of ShephardÕs (1970) disance funcion, is given by h B j hb ðx j ; Y j Þ¼supfhjðX j =h; Y j Þ2Sg. Assume ha he producion se S is monoonically increasing [i.e. ðx j ; Y j Þ2S; X P X j, Y 6 Y j )ðx ; Y Þ2S] and convex [i.e. ðx j ; Y j Þ; ðx ; Y Þ2S ) ðx j ; Y j Þþ ð1 ÞðX ; Y Þ2S for all ] and ha he probabiliy densiy funcion f B ðh B Þ is such ha f B ðh B Þ¼0 if h < 1 and R 1þd f B ðh B Þdh B > 0 for 1 d > 0. 2 Then, following Baner (1993), a consis- 1 For each observaion, a leas one oupu y rj and one inpu x ij are assumed o be sricly posiive wih a posiive price. 2 The daa generaing process comprises he generaion of he oupu vecor Y j, he direcion vecor represening he relaive mix of inpus, and he scalar h B j ha deermines he disance raversed by he observed inpu vecor X j along he direcion vecor represening he relaive inpu mix.

3 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) en esimaor of h B j is obained as ^h B j by solving he following Baner e al. (BCC 1984) model: ^h B j ¼ Max h ð1:0þ subjec o y r P y rj 8r ¼ 1;...; R; ð1:1þ x i 6 x ij =h 8i ¼ 1;...; I; ð1:2þ ¼ 1; ð1:3þ h; P 0 8 ¼ 1;...; K; ¼ 1;...; T : ð1:4þ 2.1. Measuring and evaluaing allocaive inefficiency Baner and Maindiraa (1988) describe he calculaion of aggregae, echnical and allocaive inefficiency using DEA-based linear programs o ascerain wheher he se of observed oupu, inpu and price daa is no consisen wih profi maximizaion (or cos minimizaion) for a leas one firm in he sample. They define allocaive inefficiency as aggregae inefficiency divided by echnical inefficiency. Here, we address he siuaion when informaion abou inpu prices is no available, excep for he nowledge ha he firms procure he inpus in he same compeiive mare place. We show how an aggregae echnical and allocaive inefficiency measure, equivalen o he one described by Baner and Maindiraa (1988) can be calculaed when he available daa consiss of only quaniies of oupu and oal coss of various inpus. We measure allocaive inefficiency as aggregae inefficiency divided by echnical inefficiency as in Baner and Maindiraa (1988). More imporanly, we develop saisical ess, along he lines of hose developed for he revenuemaximizing case in Baner e al. (1999), o deermine he presence of allocaive inefficiency in he observed sample of DMUs. We firs define a new producion se S derived from he original producion se S, for a given inpu price vecor P. This derived se is based on he aggregae cos, c ¼ P i p ix i > 0, as a single inpu and he same muliple oupus as in S. Formally, we define S ¼fðc ; Y Þj here exiss some ðx ; Y Þ2 S such ha c ¼ P X ¼ P i p ix i g. I immediaely follows ha if S is a monoone increasing and convex se, hen so is S. We nex consider he wo measures defined by h A j ¼ MaxfP i p ix ij = P p i x i jðx j ; Y j Þ2Sg and h Z j ¼ Maxfhjðc j =h; Y j Þ2Sg. Here, h A j is he aggregae echnical and allocaive inefficiency, for he observaion ðx j ; Y j Þ2S, as defined by Baner and Maindiraa (1988), relaive o he producion se S, while h Z j is he echnical inefficiency for he observaion ðc j ; Y j Þ2S, as defined by Baner e al. (1984), relaive o he producion se S. Also, since ðx j =h B j ; Y jþ 2S, h B j ¼ P i p ix ij = P i p iðx ij =h B j Þ 6 ha j. I is sraighforward o adap he proof by Baner e al. (1999) o show ha h Z j ¼ ha j P hb j and he probabiliy densiy funcion f Z ðh Z Þ is such ha f Z ðh Z Þ¼0 for all h Z < 1 and R 1þd f Z ðh Z Þdh Z > 0 1 for all d > 0. The allocaive inefficiency of an observaion ðx j ; Y j Þ relaive o he se S is hen given by h V j ¼ h A j =hb j ¼ hz j =hb j P 1. The resul hz j ¼ ha j P hb j holds rue for any monoone increasing and convex se S and he corresponding derived se S. In paricular, i holds for he DEA free disposable convex hull of sample observaions and he corresponding derived se based on aggregae cos observaions. Specifically, consider he following linear program, used by Baner and Maindiraa (1988), o esimae aggregae echnical and allocaive efficiency: Max ^h A j ¼ X, X p i x ij p i x i ð2:0þ i i subjec o y r P y rj 8r ¼ 1;...; R; ð2:1þ x i 6 x ij 8i ¼ 1;...; I; ð2:2þ ¼ 1; ð2:3þ h; P 0 8 ¼ 1;...; K; ¼ 1;...; T : ð2:4þ

4 480 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) Consider also he linear program in (1), afer replacing he I consrains in (1.2) by a single consrain, P P c 6 c j =h for he aggregae cos. Tha is, consider he following modified version of (1): ^h Z j ¼ Max h ð3:0þ subjec o y r P y rj 8r ¼ 1;...; R; ð3:1þ c 6 c j =h; ¼ 1; ð3:2þ ð3:3þ h; P 0 8 ¼ 1;...; K; ¼ 1;...; T : ð3:4þ Applying he resul h Z j ¼ ha j o he DEA esimaors ^h A j and ^h Z j we observe ha he esimaed aggregae echnical and allocaive inefficiency ^h A j, calculaed using (2), is idenical o ^h Z j, he esimaed echnical inefficiency relaive o he derived producion se wih a single aggregae oupu, calculaed using (3). Furher, using a proof analogous o ha in Baner e al. (1999), i can be shown ha ^h Z j is a consisen esimaor of h Z j, he echnical inefficiency of observaion j in period relaive o he derived producion se S wih he same muliple oupus in he original producion se S and he aggregae cos as a single inpu. Combining he earlier resul ðh A j ¼ hz jþ and he above resul, i can be seen ha ^h Z j is a saisically consisen esimaor of h A j, he aggregae echnical and allocaive inefficiency measure. Since he consrain (3.2) can be obained by summing he I consrains in (1.2) weighed by he corresponding prices, p i, and he remaining srucure of he program in (3) is idenical o ha in (1), i is eviden ha ^h B j is a feasible soluion o (3), and herefore ^h B j 6 ^h Z j. The esimaor for he allocaive inefficiency, ^h V j, is hen calculaed as ^h Z j =^h B j. Based on Baner (1993) and analogous o he ess of no scale inefficiency described in Baner (1996), we develop a variey of procedures o es he null hypohesis of no allocaive inefficiency agains he alernaive of he presence of such inefficiency in a paricular period based on assumed srucure for he disribuion of sðhþ, where sðþ is an appropriae ransformaion funcion. The following illusrae he es procedures for he case when sðþ ¼ lnðþ and sðþ ¼ ðlnðþþ 2 : ii(i) i(ii) (iii) If lnðh B jþ is disribued as exponenial over ½0; 1Þ, hen under he null hypohesis of no allocaive inefficiency, he es saisic is calculaed as P N j¼1 lnð^h Z j Þ= P N j¼1 lnð^h B j Þ and evaluaed relaive o he criical value of he half-f disribuion wih (2N, 2N) degrees of freedom. If lnðh B jþ is disribued as half-normal over he range ½0; 1Þ, hen under he null hypohesis of no allocaive inefficiency, P he es N P saisic is calculaed as j¼1 ðlnð^h Z j ÞÞ2 = N j¼1 ðlnð^h B j ÞÞ2 and evaluaed relaive o he criical value of he half-f disribuion wih (N, N) degrees of freedom. If no such assumpions are mainained abou he probabiliy disribuion of inefficiency, a non-parameric SmirnovÕs es saisic given by he maximum verical disance beween F ðlnð^h Z j ÞÞ and F ðlnð^h B jþþ, he empirical disribuions of lnð^h Z j Þ and lnð^h B jþ, respecively, is used. This saisic, by consrucion, aes values beween 0 and 1 and a high value is indicaive of he exisence of allocaive inefficiency. However, as Baner (1993, 1996) poins ou, while lnð^h Z j Þ and lnð^h B jþ are asympoically independen of each oher, hey need no be independenly disribued for finie samples. For finie samples, hey need no also follow he rue disribuion of lnðh Z j Þ and lnðhb jþ. An imporan cavea, herefore, is ha he above ess are designed for large samples, heir small sample performance needs o be evaluaed wih Mone Carlo experimenaion Evaluaing shifs in inefficiencies over ime and across groups To examine shifs in inefficiencies over ime or across groups in he sample we use es saisics similar o hose described in Baner (1993) and Baner and Chang (1995) o evaluae he null

5 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) hypohesis of no difference in he inefficiency disribuions of wo sub-samples. For N 1 and N 2, DMUs in subgroups G 1 and G 2, respecively, he null hypohesis of no difference in echnical inefficiency beween he wo subgroups can be esed using he following procedures: ii(i) If lnðh B jþ is disribued as exponenial over ½0; 1Þ, hen under he null hypohesis ha here is no difference beween he wo groups, he es saisic is calculaed as, X X ðlnð^h B j Þ=N 1Þ ðlnð^h B j Þ=N 2Þ j2g 1 j2g 2 and evaluaed relaive o he criical value of he F disribuion wih (2N 1,2N 2 ) degrees of freedom. i(ii) If lnðh B jþ is disribued as half-normal over he range ½0; 1Þ, hen under he null hypohesis ha here is no difference beween he wo groups, he es saisic is calculaed as, X X ððlnð^h B j ÞÞ2 =N 1 Þ ððlnð^h B j ÞÞ2 =N 2 Þ j2g 1 j2g 2 and evaluaed relaive o he criical value of he F disribuion wih (N 1, N 2 ) degrees of freedom. (iii) If no such assumpions are mainained abou he probabiliy disribuion of inefficiency, a non-parameric SmirnovÕs es saisic given by he maximum verical disance beween F G 1 ðlnð^h B j ÞÞ and F G 2 ðlnð^h B jþþ, he empirical disribuions of lnð^h B j Þ for he groups G 1 and G 2, respecively, is used. This saisic, by consrucion, aes values beween 0 and 1 and a high value for his saisic is indicaive of significan differences in echnical inefficiency beween he wo groups. 3 3 In our empirical ess we repor he value for he p asympoic ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi saisic KSa. This saisic is given by 0.5D N 1 þ N 2 where D is he maximum verical disance and N 1 and N 2 are DMUs in subgroups G 1 and G 2, respecively. The above ess can be suiably modified o examine shifs in inefficiencies beween wo differen ime periods and m for DMUs in a paricular subgroup, say G 1. Finally, while he above ess were described for esimaed echnical inefficiencies, analogous ess can be consruced for esimaed allocaive efficiencies. We use hese various ess in our empirical analysis ha follows o examine rends and changes in he efficiency of Texas school disrics. Addiional ess can be consruced based on he mulivariae disribuion of inefficiency scores rerieved empirically from DEA esimaion. We can specify a suiable srucure o represen he evoluion of inefficiency scores over ime o reflec poenial serial correlaion and covariaion, and esimae is parameers in a second sage procedure based on he empirical disribuion of he esimaed inefficiency scores. While differences over ime and across groups can be evaluaed in his manner, in his paper we shall resric our aenion o relaively simple es procedures employed in prior DEA lieraure. 3. Educaion producion funcions and Texas school disrics daa Many sudies have been underaen over he las hree decades o esimae cos and producion funcions in educaion. Hanushe (1979, 1986) provides comprehensive reviews of he lieraure on he educaional producion funcion. Unil he early 1980s single oupu educaional producion funcions were esimaed using ordinary leas squares (Polache e al., 1978). More recen wor recognizes he need o specify a producion funcion wih muliple oupus (Hanushe, 1979, 1986; Bessen e al., 1982; Chizmar and Za, 1983; Callan and Sanerre, 1990; Ray, 1991). To esimae muliple oupu producion echnologies, sudies in educaion have employed he ranslog cos funcion (Jimenez, 1986; Gyimah-Brempong and Gyapong, 1992) or DEA (Bessen e al., 1982; Ahn e al., 1988; Fare e al., 1989; Ray, 1991) mehodologies. Research on educaional producion funcions can be divided ino wo broad groups. The firs group includes several sudies, such as Buler and Mon (1985), Ahn e al. (1988), and

6 482 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) Cohn e al. (1989), which involve he esimaion of he inpu oupu producion correspondence o evaluae he relaive efficiency of educaional insiuions. The second group includes Gyimah- Brempong and Gyapong (1992), Ray (1991) and ohers ha focus on he inpu and oucome (raher han oupu) relaion o evaluae effeciveness. Inpu measures used ypically include insrucional expendiures, full ime equivalens of eachers, and oher physical measures such as number of square fee of space. Oupu measures generally comprise full ime suden equivalens. Oucome measures include scores from sandard achievemen ess, or oher proficiency ess. From an applicaion poin of view, our sudy belongs o he firs group of sudies such as Buler and Mon (1985), Ahn e al. (1988), and Cohn e al. (1989). In he spiri of hese sudies, we concenrae on he producion correspondence beween oupus (suden enrollmen) and inpus (operaing expendiures). More imporanly, we use he esimaed producion correspondence o analyze shifs and rends in inefficiencies across regions as well as over ime. The daa for his sudy is colleced from he AEIS (Academic Excellence Indicaor Sysem) repors provided o he public by he Texas Educaion Agency (TEA). These repors are an inegral par of he school accounabiliy sysem in Texas. Texas was one of he five saes o earn a spo on he Thomas B. Fordham FoundaionÕs honor roll in 1999 because of is solid academic sandards and srong school accounabiliy sysem. TEA has been maing available o he public deailed daa on sudens, saff, performance, revenue and expense daa since Six years of daa, for he period o , boh academic years inclusive, are currenly available from TEA in elecronic form. We use hese six years of daa in our analysis. In , Texas had 1042 independen school disrics. The disrics are serviced by 20 regional educaion service ceners. Toal suden enrollmen in Texas schools was 3.95 million and he school disrics spen a oal of $20.5 billion in operaing expendiures. We collec and analyze daa a he school disric level because resource allocaion decisions are generally made a his level. TEA menions in is SnapshoÕ98 repor ha while he SBOE (Sae Board of Educaion) and he commissioner of educaion provide leadership for educaion, much of he conrol of public schools resides wih he local disrics and charers. We measure hree oupus: oal suden enrollmen in elemenary schools (K 5), middle schools (6 8) and high schools (9 12) in every disric. These oupu measures are denoed as ELE- MENTARY, MIDDLE and HIGH. We use hree ypes of operaing expendiures as our inpu variables. All expendiures are convered ino dollars using he CPI (U) daa for he Souhern Region for he sample period available from he Bureau of Labor Saisics (BLS). TEA provides disric-level operaing expendiures for various groups of funcions. We condense hese various caegories ino hree broad groups of expendiures comparable over he six years of sample daa. The hree inpus are (a) INSTRUCTION: measured as he expendiures associaed wih all aciviies dealing direcly wih he ineracion beween eachers and sudens, including insrucion aided wih compuers (b) ADMINISTRATION: measured as he sum of expendiures relaed o insrucional, campus and cenral adminisraion and daa processing services and (c) SUPPORT: measured as he sum of expendiures corresponding o insrucion relaed services, suden suppor services, suden ransporaion, food services, school-sponsored cocurricular or exracurricular aciviies, plan mainenance and operaions and securiy and monioring services. Our choice of oupus and inpus is in line wih measures used in prior research (Buler and Mon, 1985; Jimenez, 1986; Ahn e al., 1988; Callan and Sanerre, 1990; Ray, 1991). We impose he following resricions o arrive a our final sample: (i) All oupu and inpu daa for he disric should be available for all six years of he sample period. (ii) There should be a leas one suden enrolled in each of he grades K 12. (iii) There should be a leas 50 sudens in boh he 6h and 10h grades during he academic year This resricion is idenical o ha used by Grossopf e al. (1999) o avoid sampling problems. Our final sample consiss of a balanced panel of daa for 585 Texas school disrics over a six year period from 1993 o We assign he school

7 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) Fig. 1. The hree regions of Texas school disrics. disrics in our sample o one of hree regions, SOUTH-EAST, NORTH or WEST, described in he map of Texas in Fig. 1, by consolidaing he geographical regions served by he 20 regional educaion service ceners (ESCs). For his purpose, ESCs 1 6, 7 12 and 13 20, respecively were assigned o he regions SOUTH-EAST, NORTH and WEST, respecively. Based on suden enrollmen, he hree regions are of similar size accouning for 38%, 34% and 28% of oal sudens in he sae. Wih a oal enrollmen of 3.64 million sudens and oal operaing expendiures of $17.2 billion, he 585 school disrics accoun for 92% of he oal sudens in he sae and 84% of he oal expendiures. Table 1 provides descripive saisics on he oupu and inpu measures for he year The disribuions were similar for oher years excep for growh in suden enrollmen as well as real expendiures which we will discuss shorly. The represenaive median school in he sample had a oal suden enrollmen of 1880 and oal operaing expendiures of $8.11M. Elemenary, Middle and High school enrollmens were 47%, 25% and 28% of he oal suden body, respecively. Insrucion accouned for 59% of he operaing expendiures. Suppor expenses wih 30% and adminisraion wih 11% accouned for he remainder. In , he Texas school disrics spen $4419 per suden. Table 2 provides six years of daa for saewide averages for he various measures. The oal suden enrollmen grew by a oal of 9.0% during he sample period. Toal expendiures, hough, grew faser, by 16.5%. As a resul, he cos per suden in real dollars grew by 9.2% and was $4901 in High school suden enrollmen grew a a much faser pace compared o he oal suden Table 1 Descripive saisics on suden enrollmen and expendiures (based on daa, N ¼ 585) Mean Sd. Dev. 25% Median 75% Toal sudens , Elemenary Middle High Elemenary (%) Middle (%) High (%) Toal expenses a $25.28 M $60.34 M $4.82 M $8.11 M $19.49 M Insrucion a $14.83 M $35.61 M $2.84 M $4.75 M $11.33 M Adminisraion a $3.24 M $7.74 M $0.65 M $1.10 M $2.40 M Suppor a $7.21 M $17.09 M $1.39 M $2.40 M $5.68 M Insrucion (%) Adminisraion (%) Suppor (%) Expense per suden $4490 $601 $4104 $4419 $4814 a The symbol M indicaes ha he amouns are presened in millions of dollars.

8 484 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) Table 2 Trends in suden enrollmen and expendiures Toal sudens Elemenary Middle High Elemenary (%) Middle (%) High (%) Toal expenses a $25.28 M $25.85 M $27.01 M $26.58 M $28.30 M $29.46 M Insrucion a $14.83 M $15.15 M $15.84 M $15.42 M $16.36 M $16.95 M Adminisraion a $3.24 M $3.31 M $3.50 M $3.17 M $3.44 M $3.62 M Suppor a $7.21 M $7.39 M $7.67 M $8.00 M $8.50 M $8.89 M Insrucion (%) Adminisraion (%) Suppor (%) Expense per suden $4490 $4490 $4647 $4527 $4764 $4901 a The symbol M indicaes ha he amouns are presened in millions of dollars. enrollmen. This may have been a conribuing facor o he higher cos per suden in Efficiency analysis We apply DEA o he full panel of 3510 observaions for 585 disrics over six years. We calculaed wo inefficiency measures for each school disric for every year. The firs is he echnical inefficiency esimaor, ^h B j, calculaed using he program in (1). The program is based on he hree suden oupus and he hree inpus. We also calculae he aggregae inefficiency esimaor, ^h Z j, using he program in (3). We replace he hree individual inpu consrains by a single consrain using he oal expendiure as a single inpu variable. As described in Secion 2, he echnical inefficiency esimaor for he hree oupu and he single aggregae inpu case, ^h Z j, is idenical o he aggregae echnical and allocaive inefficiency for he hree oupu and hree inpu case. We calculae he allocaive inefficiency ^h V j for each school disric j in year as ^h Z j =^h B j. Table 3 provides he disribuion of esimaed echnical and allocaive inefficiencies on a year by Table 3 Disribuion of echnical and allocaive inefficiencies Mean Sd. Dev. 25% Median 75% Panel A: Technical Panel B: Allocaive

9 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) Table 4 Tess of exisence of allocaive inefficiency in Texas school disrics based on full sample Year BanerÕs sum raio es BanerÕs sum of squares raio es Kolmogorov Smirnov es N ¼ 585. Tes saisics significan a 5% (10%) level are mared wih ( ). year basis. The disribuions are quie revealing. The school disrics in Texas appear o be operaing under significan echnical inefficiency. The median school disric was operaing a an inefficiency level of (his corresponds o 78% efficiency) in There also seems o have been a seady increase in echnical inefficiency over ime. In , he median echnical inefficiency had increased o (73% efficiency). Lower quarile values of echnical inefficiency of he order of 1.2 for all he six years sugges ha he echnical inefficiency is quie widespread. The allocaive inefficiencies are much lower. The median school disric was operaing a an inefficiency level of (98% allocaive efficiency) in There appears o be no significan rend in he allocaive inefficiency levels over ime. Finally, here also appears o be a significan deparure from he seady increase in echnical inefficiency in While he disribuions are quie informaive, addiional assumpions on he inefficiency variables are necessary o mae inferences abou shifs, rends and changes in efficiencies. Table 4 describes he saisical analysis esing for he exisence of allocaive inefficiency in school disrics. Recall from Secion 2 ha hese ess are based on a comparison of he observed disribuion of aggregae and echnical inefficiencies because he rue disribuions for he wo are idenical under he null hypohesis ha here is no allocaive inefficiency. We provide deails on hree differen ess. These ess, namely, BanerÕs (1993, 1996) sum raio and sum of squares raio ess and he Kolmogorov Smirnov es were described earlier in Secion 2. The ess conclusively rejec he null hypohesis of no allocaive inefficiency for all six years. Togeher, hey indicae ha alhough he allocaive inefficiencies are small in magniude hey are saisically significan. Table 5 Tess of shifs over ime in echnical and allocaive inefficiency in Texas school disrics Year BanerÕs sum raio es BanerÕs sum of squares raio es Kolmogorov Smirnov es Panel A: Technical inefficiency vs vs vs vs vs vs Panel B: Allocaive inefficiency vs vs vs vs vs vs N ¼ 585. Tes saisics significan a 5% (10%) level are mared wih ( ).

10 486 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) Table 6 Trends in regional averages of suden enrollmen and expendiures Panel A: Souh-Eas (N ¼ 182) Toal sudens Elemenary Middle High Toal expenses a $32.90 M $33.65 M $35.30 M $34.02 M $36.25 M $37.58 M Insrucion a $19.06 M $19.51 M $20.49 M $19.54 M $20.71 M $21.41 M Adminisraion a $4.16 M $4.25 M $4.53 M $4.04 M $4.35 M $4.54 M Suppor a $9.69 M $9.89 M $10.27 M $10.44 M $11.18 M $11.63 M Expense per suden $4668 $4653 $4808 $4614 $4844 $5000 Panel B: Norh (N ¼ 233) Toal sudens Elemenary Middle High Toal expenses a $19.85 M $20.42 M $21.41 M $21.74 M $23.21 M $24.38 M Insrucion a $11.79 M $12.08 M $12.68 M $12.73 M $13.52 M $14.13 M Adminisraion a $2.57 M $2.64 M $2.78 M $2.58 M $2.89 M $3.09 M Suppor a $5.49 M $5.70 M $5.95 M $6.42 M $6.80 M $7.16 M Expense per suden $4248 $4247 $4419 $4357 $4583 $4683 Panel C: Wes (N ¼ 170) Toal sudens Elemenary Middle High Toal expenses a $24.56 M $24.93 M $25.80 M $25.26 M $26.76 M $27.73 M Insrucion a $14.46 M $14.67 M $15.20 M $14.67 M $15.59 M $16.04 M Adminisraion a $3.18 M $3.23 M $3.37 M $3.04 M $3.20 M $3.37 M Suppor a $6.92 M $7.03 M $7.23 M $7.55 M $7.97 M $8.32 M Expense per suden $4633 $4649 $4787 $4666 $4928 $5094 a The symbol M indicaes ha he amouns are presened in millions of dollars. We nex urn our aenion o esing for shifs and rends in inefficiencies over ime. Table 5 provides deails on he ess used o examine rends in echnical and allocaive inefficiencies over ime. The increases in echnical inefficiency beween consecuive years excep he change from o , are all saisically significan. The decline in echnical inefficiency from o is also significan. 4 Overall, 4 TEA changed is caegories of operaing expendiures in I is possible ha, as a resul, some iems of expendiures were classified as non-operaing expendiures in I is worhwhile noing ha operaing expendiures per suden declined from $4647 in o $4527 in however, he echnical inefficiency of Texas school disrics, over a six-year period, has increased considerably. Allocaive inefficiency has remained relaively sable from year o year, excep for a significan increase from o This appears o be he main reason for he overall increase in allocaive inefficiency from o Nex we focus our aenion on analysis a he regional level. Our objecive is o examine wheher region-specific paerns exis in school disric performance. Table 6 provides deails on rends in regional averages of suden enrollmen and expendiures. The school disrics in he Souh-Eas region are considerably bigger han in he oher wo regions and Norh has he lowes number of

11 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) sudens per school disric. Norh also appears o have a much lower cos per suden. While cos per suden in real dollars has increased over ime for all hree regions, all of hem show a decline in cos per suden from o This paern is consisen wih ha observed for he full sample. Table 7 provides deails on rends in mean inefficiencies by region. The changes over ime observed for he whole sample also obain a he regional level. Addiionally, i is seen ha Norh has he lowes echnical inefficiency of he hree regions followed by Souh-Eas and hen Wes. No specific paern could be observed for changes in allocaive inefficiency across regions. Table 8 provides deails on ess of exisence of allocaive inefficiency by region over ime. The es resuls confirm he widespread presence of allocaive Table 7 Trends in mean echnical and allocaive inefficiencies Panel A: Technical Full sample Souh-Eas Norh Wes Panel B: Allocaive Full sample Souh-Eas Norh Wes Table 8 Tess of exisence of allocaive inefficiency regional analysis Year BanerÕs sum raio es BanerÕs sum of squares raio es Kolmogorov Smirnov es Panel A: Souh-Eas (N ¼ 182) Panel B: Norh (N ¼ 233) PANEL C: Wes (N ¼ 170) Tes saisics significan a 5% (10%) level are mared wih ( ).

12 488 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) Table 9 Tess of shif in echnical efficiency over ime regional analysis Year BanerÕs sum raio es BanerÕs sum of squares raio es Kolmogorov Smirnov es Panel A: Souh-Eas (N ¼ 182) vs vs vs vs vs vs Panel B: Norh (N ¼ 233) vs vs vs vs vs vs Panel C: Wes (N ¼ 170) vs vs vs vs vs vs Tes saisics significan a 5% (10%) level are mared wih ( ). Table 10 Tess of shif in allocaive efficiency over ime regional analysis Year BanerÕs sum raio es BanerÕs sum of squares raio es Kolmogorov Smirnov es Panel A: Souh-Eas (N ¼ 182) vs vs vs vs vs vs Panel B: Norh (N ¼ 233) vs vs vs vs vs vs Panel C: Wes (N ¼ 170) vs vs vs vs vs vs Tes saisics significan a 5% (10%) level are mared wih ( ).

13 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) Table 11 Comparison of regions echnical inefficiency Year BanerÕs sum raio es BanerÕs sum of squares raio es Kolmogorov Smirnov es Panel A: Souh-Eas (N ¼ 182) vs. Norh (N ¼ 233) Panel B: Norh (N ¼ 233) vs. Wes (N ¼ 170) Panel C: Wes (N ¼ 170) vs. Souh-Eas (N ¼ 182) Tes saisics significan a 5% (10%) level are mared wih ( ). Table 12 Comparison of regions allocaive inefficiency Year BanerÕs sum raio es BanerÕs sum of squares raio es Kolmogorov Smirnov es Panel A: Souh-Eas (N ¼ 182) vs. Norh (N ¼ 233) Panel B: Norh (N ¼ 233) vs. Wes (N ¼ 170) Panel C: Wes (N ¼ 170) vs. Souh-Eas (N ¼ 182) Tes saisics significan a 5% (10%) level are mared wih ( ).

14 490 R.D. Baner e al. / European Journal of Operaional Research 154 (2004) inefficiencies in all he regions. One can, herefore, conclude ha he allocaive inefficiency is prevalen across he sae wih no single region conribuing in a singular way. We repea he ess for examining shifs over ime a he regional level. These ess are repored in Tables 9 and 10. The resuls repored in Table 9 indicae ha he saisically significan increase in echnical inefficiency observed a he sae level is also a regional phenomenon. I also confirms he blip in o The overall increase for he six year period appears o be more in Norh compared o he oher regions. In oher words, he more efficien Norh region appears o be moving down o cach up wih is less echnically efficien counerpars! Table 10 resuls indicae ha while allocaive inefficiency has increased over ime in all hree regions, he increase in he Souh-Eas region is more pronounced han in he oher regions. The year o year increases in allocaive inefficiency are also much smaller han he increases observed in echnical inefficiency. Tables 11 and 12 provide deails on wheher he regions are differen in he way hey uilize heir resources. We compare each region wih each of he oher wo. The ess for echnical inefficiency differences across regions confirm he paern observed in Table 7. Norh has he lowes echnical inefficiency for all six years. Souh-Eas, in a relaive sense, has decreased is echnical inefficiency in he las wo years, and Wes, however, has consisenly railed he oher wo regions wih high echnical inefficiency. Table 12 indicaes ha here is no significan difference in allocaive inefficiency beween Norh and Souh- Eas. Wes has a lower allocaive inefficiency when compared o boh Norh and Souh-Eas. 5. Summary and conclusion This paper conribues o he relaively sparse lieraure in DEA on he analysis of echnical and allocaive inefficiency for panel daa seings. Firs, we adaped he ess presened in Baner e al. (1999) o assess he presence of allocaive inefficiency in inpu consumpion. We employed a resul ha he DEA echnical inefficiency measure using a single aggregae cos variable, consruced from muliple inpu quaniies weighed by heir uni coss, equals he aggregae echnical and allocaive inefficiency for he deailed muliple inpu model. We also presened new es procedures for researchers ineresed in examining ime rends and iner-group differences in inefficiencies presen in panel daa. We hen demonsraed an applicaion of he ess developed in his paper o Texas school disric daa for he period , analyzing he rends in inefficiency in resource uilizaion by public schools in differen regions. Our empirical resuls indicaed he presence of saisically significan allocaive inefficiencies. We found ha median echnical inefficiency increased from o over he six year period. Median allocaive inefficiency remained relaively sable a around over he six year period. We also discovered ha Texas school disrics in he Norh region operae a he lowes levels of echnical inefficiency followed by he Souh-Eas and finally he Wes region. References Ahn, T., Charnes, A., Cooper, W.W., Some saisical and DEA evaluaions of relaive efficiencies of public and privae insiuions of higher learning. Socio-Economic Planning Science 22 (6), Baner, R.D., Maximum lielihood, consisency and daa envelopmen analysis. Managemen Science 39 (Ocober), Baner, R.D., Hypohesis ess using daa envelopmen analysis. Journal of Produciviy Analysis 7, Baner, R.D., Chang, H., A simulaion sudy of hypohesis ess for differences in efficiencies. Inernaional Journal of Producion Economics 39, Baner, R.D., Maindiraa, A., Nonparameric analysis of echnical and allocaive efficiencies in producion. Economerica 56, Baner, R.D., Charnes, A., Cooper, W.W., Models for he esimaion of echnical and scale inefficiencies in daa envelopmen analysis. Managemen Science 30, Baner, R.D., Chang, H., Naarajan, R., Efficiency of public accouning firms, Woring paper, The Universiy of Texas a Dallas. Bessen, A., Bessen, W., Kenningon, J., Reagan, B., An applicaion of mahemaical programming o assess produciviy in he Houson independen school disrics. Managemen Science 28,

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