Centre for Efficiency and Productivity Analysis

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1 Centre for Effcency and Prodctty Analyss Workng Paper Seres No. WP/7 Ttle On The Dstrbton of Estmated Techncal Effcency n Stochastc Fronter Models Athors We Sang Wang & Peter Schmdt Date: May, 7 School of Economcs Unersty of Qeensland St. Lca, Qld. 47 Astrala ISSN No

2 ON THE DISTRIBUTION OF ESTIMATED TECHNICAL EFFICIENCY IN STOCHASTIC FRONTIER MODELS We Sang Wang Mchgan State Unersty Peter Schmdt Mchgan State Unersty May, 7 1

3 1. INTRODUCTION In ths paper we consder the stochastc fronter model ntrodced by Agner, Loell and Schmdt (1977) and Meesen and an den Broeck (1977). We wrte the model as (1) y = X β + ε, ε =,. Here typcally y s log otpt, X s a ector of npt measres (e.g., log npts n the Cobb-Doglas case), s a normal error wth mean zero and arance, and represents techncal neffcency. Techncal effcency s defned as TE = exp( ), and the pont of the model s to estmate or TE. A specfc dstrbtonal assmpton on s reqred. The papers cted aboe consdered the case that s half normal (that s, t s the absolte ale of a normal wth mean zero and arance ) and also the case that t s exponental. Other dstrbtons proposed n the lteratre nclde general trncated normal (Steenson (198)) and gamma (Greene (198a, 198b, 199) and Steenson (198)). In ths paper we wll consder only the half normal case, bt smlar reslts wold apply to the other cases. Also, or exposton s for the cross-sectonal case, bt we cold also consder panel data as n Ptt and Lee (1981). Defne ˆβ to be the MLE of β, and ˆ ˆ ε = y X β. Then the sal estmate of, sggested by Jondrow et al. (198), s E( ε ), ealated at ε ˆ = ε. We can estmate TE by TE = exp( ˆ ) bt a preferred estmate s TE = E{exp( ) ε } ealated at ε = ˆ ε. See Battese and Coell (1988), who also show how to defne ˆ and TE n the case of panel data.

4 In ths paper we dere the dstrbton of ˆ. (The same method of deraton wold also apply to TE, thogh we do not ge the detals.) It s mportant to realze that ths s not, and shold not be expected to be, the same as the dstrbton of. In other words, f one assmes that the are half normal, t s temptng to look at the ˆ and see f ther dstrbton looks half normal. It shold not, nless s ery small. We show that the dstrbton of ˆ becomes the same as the dstrbton of as on the pont E( ) as (wth fxed), and that the dstrbton of ˆ collapses. We also graph the dstrbton for ntermedate ales of. One way to nderstand the dfference between the dstrbtons of ˆ and s to realze that ˆ s a shrnkage of toward ts mean. Ths reflects the famlar prncple that an optmal (condtonal expectaton) forecast s less arable than the thng beng forecast. The sal breakdown of arance nto explaned and nexplaned parts says: () ar( ) = ar[ E( ε )] + E[ar( ε )] so that ar( ) s greater than ar( ˆ ) by the amont E[ar( ε )]. 1 An mplcaton of shrnkage s that on aerage we wll oerestmate when t s small, and nderestmate when t s large. To see the exact sense n whch ths s tre, we also dere the dstrbton of ˆ condtonal on. We show that as (wth fxed), the dstrbton of ˆ condtonal on collapses on, 1 The expectaton s oer the dstrbton of the condtonng arable, ε. 3

5 whle as, the dstrbton of ˆ condtonal on does not depend on (t collapses on the pont E()). Once agan we graph the dstrbton for ntermedate ales of, for aros ales of. The plan of the paper s as follows. Secton consders the dstrbton of ˆ. Secton 3 consders the dstrbton of ˆ condtonal on. Secton 4 ges or concldng remarks. There s also an Appendx whch contans some of the deratons.. THE DISTRIBUTION OF û In ths secton we dere and dscss the dstrbton of ˆ = E( ε ). Ths s a random arable becase t s a fncton of ε, whch s a random arable, and ts dstrbton follows from the dstrbton of ε. Or dscsson wll gnore estmaton error n β. That s, we consder ˆ = E( ε ), whereas n practce ˆ = E( ε ) ealated at ε ˆ = ε. The dfference between ε and ˆ ε s that ε = y X β whereas ε = y X ˆ β ; that s, the dfference s jst the contrbton of estmaton ˆ error n β. The jstfcaton for gnorng ths s that, n any applcaton we can enson, the ntrnsc randomness n E( ε ) de to ts beng a fncton of ε wll dwarf the randomness de to estmaton error n β. More formally, the former s O p (1) whle the latter s O p (1/ N ). Also, for notatonal smplcty, we wll henceforth omt sbscrpt from ˆ,, and ε. Snce ˆ = E( ε ) t s a fncton of ε, and we can wrte ˆ = h( ε ). The fncton h was gen by Jondrow et al. (198): 4

6 (3) ˆ = h( ε ) = [ ε λ( ε / )] +, where = ( + ) /, + λ( s) = φ( s) /[1 Φ ( s)], and where φ and Φ are the standard normal densty and cdf, respectely. The fncton h s a monotonc (strctly decreasng) fncton, so t can be nerted. That s, we can formally wrte (4) ε = h 1 ( ˆ ) = g( ˆ ). We cannot express the fncton g analytcally, bt t s well defned and we can calclate t. For example, Fgre 1 shows the fncton g for the case that = = 1. Let f ε and f û represent the denstes of ε and û. Then makng the smple change of arables n (4), we hae (5) f ˆ ˆ ˆ ˆ ( ) = f ( g( )) g ( ). ε The densty of ε s gen by Agner, Loell and Schmdt: (6) fε ( ε ) = ( / a) φ ( ε / a) Φ( εb / a), a = +, b = /. Also, we can calclate the Jacoban term g ( ˆ ). We show n Appendx A that (7) a g ( ˆ ) = [ 1 + λ ( g( ˆ ) / )], where λ ( s) = sλ( s) + λ ( s). Ths notaton s slghtly dfferent from Agner, Loell and Schmdt. Or a s ther and or b s ther λ. Bt we hae already sed λ for the nerse Mll s rato, and there are enogh dfferent s already wthot ntrodcng another one. 5

7 So, sbstttng (6) and (7) nto (5), we obtan aφ ( g( ˆ ) / a) Φ( g( ˆ ) b / a) (8) f ˆ ( ˆ ) = 1 + λ ( g( ˆ ) / ). Clearly ths s not the same as f, the half normal densty. The followng reslt shows what happens n the lmt as approaches zero and nfnty, respectely. The proof s gen n Appendx B. THEOREM 1: (1) As, ( ˆ ). () As, f û f (pontwse). (3) As, ˆ E( ). p p (4) As, π π ˆ E N. [ /( )] ( / ) ( ( )) d (,1) These reslts make sense f we realze that we are treatng ε = as or obserable qantty. If =, so that, we effectely obsere, and so n the lmt û = and the dstrbton of û eqals the dstrbton of. Conersely, when =, ε contans no sefl nformaton abot, and the best estmate of s smply ˆ = E( ). Part (4) says that, for large, û s approxmately normally dstrbted arond E( ), wth arance π π. 4 [( ) / ] ( / ) 6

8 For ales of between zero and nfnty, the densty of û represents the shrnkage of towards ts mean, whch s ( / π ), or abot.8. 3 Fgre dsplays the half normal densty of, wth = 1; ths corresponds to the densty of û when =. Fgres 3, 4, 5 and 6 ge the densty of û when =.1, 1, 1 and 1, respectely. None of these denstes looks mch lke the half normal. Comparng the denstes n the dfferent fgres reqres some care, snce the axes are scaled dfferently. Howeer, t s clearly the case that, as ncreases, the densty of û becomes more peaked and concentrated more tghtly abot the mean of.8. As becomes large, the dstrbton of û collapses onto the pont E(), as ndcated n part (3) of Theorem 1. The approxmate normalty of the dstrbton of û for large s edent n Fgre 6. Fnally, Fgre 7 contans all of the fe graphs that were n Fgres throgh 6. The se of a common set of axes makes t hard to see the detal n any one of the graphs, bt seeng them all together does make clear what happens as changes. 3. THE DISTRIBUTION OF û CONDITONAL ON In the preos secton, we saw that the dstrbton of û s a shrnkage toward the mean of the dstrbton of. Inttely, ths means that we shold expect that on aerage we wll oerestmate small realzatons of and nderestmate large ones. To see the precse sense n whch ths s tre, n ths secton we dere and graph the densty of û condtonal on. The densty of û condtonal on s gen by the followng eqaton. 3 Note that, by the law of terated expectatons, the mean of û s the same as the mean of. 7

9 (9) f ( ˆ ) = aexp[ (1/ )( g( ˆ ) + ) ] π ˆ + λ g ( ) 1 ( ( ) / ) The deraton s gen n Appendx C. Theorem 1 aboe ges some gdance as to what we shold expect ths densty to look lke. As, the dstrbton of û condtonal on shold collapse onto the pont. Conersely, as collapses onto the pont E()., the dstrbton of û condtonal on no longer depends on ; t The followng reslt shows that, approxmately normalzed, û condtonal on s asymptotcally normal both as, and as n the two cases.) The proof s gen n Appendx D.. (The normalzaton obosly mst dffer THEOREM : (1) As, ˆ d N(,1). π () As, ( ) ( ) ( ˆ ) (,1) d N π. Reslts (1) and () hold treatng as fxed. That s, they deal wth the dstrbton of û condtonal on. Reslt () s, howeer, the same as the ncondtonal reslt gen n reslt (4) of Theorem 1. Fgres 8 13 ge the densty of û condtonal on, for =.1, = 1, and =.1,.1,.1, 1, 1 and 1. The ale =.1 s a small ale (n the left tal of the dstrbton) and so we expect to oerestmate t, on aerage. Ths does occr except perhaps for the ery smallest 8

10 ale of. We do not hae a strct shrnkage to the mean, n the sense that there s probablty mass for û to the left of the tre ale of, bt except when s ery small the ast majorty of the probablty mass s to the rght of. For the larger ales of most of the probablty mass s near the mean, E(). The approxmate normalty of the dstrbton of û condtonal on for small and for large can be seen n Fgres 8 and 13, respectely. For ntermedate ales of the dstrbton does not look normal. Fgres ge the same reslts, bt now for the case that =. The ale = s a large ale (n the rght tal of the dstrbton) and so we expect to nderestmate t, on aerage. Ths does occr, and agan the amont of shrnkage to the mean s small when s small and large when s bg. Fgre llstrates the pont that, when s large enogh, the densty of û condtonal on no longer depends on. In Fgre we hae = 1and = 1, and we dsplay the densty of û condtonal on for =.1,.5, 1 and. These denstes are not mch dfferent. Wth enogh nose, the data are no longer ery releant n estmatng, or eqalently the estmate s not ery dfferent dependng on the tre ale of that generated the data. 4. CONCLUDING REMARKS Ths paper dered the dstrbton of the techncal effcency estmate ˆ = E( ε ), and also the dstrbton of û condtonal on. We sed these dstrbtons to make two man ponts. The frst pont s that the dstrbton of û s not, and shold not be expected to be, the same as that of. So, for example, f we assme a half normal dstrbton for, and we plot the 9

11 dstrbton of û, we shold not be dstrbed when t does not look half normal. The second pont s that û s (n a probablstc sense) a shrnkage of toward the mean. On aerage, we wll oerestmate the smaller realzatons of and nderestmate the larger realzatons. We stress that nether of these facts means that there s anythng wrong wth û. It s the optmal (ratonal, condtonal expectaton, mnmm mean sqare error, ) forecast of. What the paper llstrates s jst the sense n whch statstcal nose s nconenent. 1

12 APPENDIX A. Deraton of the Jacoban n eqaton (7) From eqaton (3), we hae ˆ = h( ε ) = k[ ε + λ( ε / )], where k = /( + ). So dˆ dε [ 1 ( / ) (1/ )] [ 1 ( / )]. Then (A1) = k + λ ε = k + λ ε (A) 1 dε dˆ 1 g ( ˆ ) = = dˆ = dε k[ 1 + λ ( ε / )] + = [ 1 + λ ( g( ˆ ) / )] and the Jacoban s jst the absolte ale of ths expresson. B. Proof of Theorem 1 Frst we ge some facts abot the nerse Mll s rato λ( s) = φ( s) /[1 Φ ( s)]. As s, () λ( s), () sλ( s), () ( s) = s ( s) + ( s). (Note that () and () follow from λ λ λ (), and () follows from the exstence of the ntegral defnng the mean of the standard normal.) Now we start wth the expresson for û, as gen aboe. As, k 1, ε p (snce as, p ),, and λ( ε / ) λ( ) =. Therefore ˆ p (n the sense that the dfference between û and goes to zero). Ths proes part (1) of Theorem 1. To proe part (), consder the densty of û as gen n eqaton (8) of the text. As, we hae a, a, g( ˆ ) / 1/, φ( g( ˆ ) / a) φ( / ) = φ( / ), and Φ( g( ˆ ) b / a) Φ( ) = 1. Also the Jacoban term 1becase λ ( ) =. Therefore f ˆ ( ˆ ) (1/ ) φ( ˆ / ), whch s the half normal densty.

13 To proe part (3), of the Theorem, we retrn to the expresson for û gen aboe, whch we wrte as ˆ = kε + k λ( ε / ). As, k,, k and λ( ε / ) λ() = ( / π ). Therefore ˆ ( / π ) = E( ). p To proe part (4), we wrte ε (A3) ( ˆ E( )) = kε + k λ( ). π The frst term on the r.h.s. of (A3) eqals (A4), + + where A s N (,1). B means that A B wth probablty one as The second term on the r.h.s. of (A4) s ε (A5) k λ( ) π. Note that / ε = λ( ) + π ε λ( ). π Now se the mean ale theorem (delta method) to wrte (A6) ε λ λ + λ ε ' ( ) () () ( ) = ε π + π

14 and so the term n (A5) becomes (A7) ε. π Also (A8) ε = ( + ) π π = π π +. π Combnng (A8) wth (A4), we hae (A9) ( ˆ E( )) ( 1 + ) π whch s dstrbted as N(,[( π -) / π ] ). C. Deraton of f ( ˆ ) n eqaton (9) We begn by notng that the jont densty of (, ε ) s f (, ε ) = f ( ) f ( ε + ). Now transform to (, ˆ) where as before ε = g( ˆ ). The Jacoban of ths transformaton s g ( ˆ ) as gen n eqaton (7) of the text. Therefore the jont densty of (A1) f (, ˆ ) = f ( ) f ( g( ˆ + ) Jacoban and the condtonal densty of û gen s (, ˆ) s (A11) f ( ˆ ) = f (, ˆ ) / f ( ) = f ( g( ˆ ) + ) Jacoban. Sbstttng the normal densty for f and the Jacoban expresson n (7), we arre at the expresson n eqaton (9) of the text.

15 D. Proof of Theorem To proe part (1) of the Theorem, we wrte ˆ (A1) k ( k 1) k = + + λ( ). As, k 1, so the frst term on the r.h.s. /. The second term s: k 1 (A13) = =. + (Remember s fxed n ths calclaton.) The thrd term s (A14) k λ( ) λ( ) = where we hae sed the facts that, as, k 1 and / 1. Therefore ( ˆ ) / / whch s N (,1). The proof of part () s essentally the same as the proof of part (4) of Theorem 1, and s therefore omtted.

16 REFERENCES Agner, D.J., C.A.K. Loell, and P. Schmdt (1977), Formlaton and Estmaton of Stochastc Fronter Prodcton Fncton Models, Jornal of Econometrcs 6, Battese, G.E., and T.J. Coell (1988), Predcton of Frm-Leel Techncal Effcences wth a Generalzed Fronter Prodcton Fncton and Panel Data, Jornal of Econometrcs 38, Jondrow, J., C.A.K. Loell, I.S. Matero, and P. Schmdt (198), On the Estmaton of Techncal Effcency n the Stochastc Prodcton Fncton Model, Jornal of Econometrcs 19, Meesen, W., and J. Van Den Broeck (1977), Effcent Estmaton from Cobb-Doglas Prodcton Fnctons wth Composed Error, Internatonal Economc Reew 18, Ptt, M.M., and L.F. Lee (1981), The Measrement and Sorces of Techncal Ineffcency n the Indonesan Weang Indstry, Jornal of Deelopment Economcs 9, Steenson, R.E. (198), Lkelhood Fnctons for Generalzed Stochastc Fronter Estmaton, Jornal of Econometrcs 13, Greene, W.H. (198a), Maxmm Lkelhood Estmaton of Econometrc Fronter Fnctons, Jornal of Econometrcs 13, Greene, W.H. (198b), On the Estmaton of a Flexble Fronter Prodcton Model, Jornal of Econometrcs 13, Greene, W.H. (199), A Gamma-Dstrbted Stochastc Fronter Model, Jornal of Econometrcs 46,

17 Fgre 1 The relatonshp between ε and wth = = 1 ε û

18 Fgre A half normal dstrbton, ~ N(,1) + f ( ) Half normal densty wth = 1

19 Fgre 3 Densty of wth = 1 and =.1 f ( ) û

20 Fgre 4 Densty of wth = 1 and = 1 f ( ) û

21 Fgre 5 Densty of wth = 1 and = 1 f ( ) û

22 Fgre 6 Densty of wth = 1 and = 1 f ( ) û

23 Fgre 7 The combned graph for fgre -fgre 6 Denstes of a half normal and f( ) and f( ) and

24 Fgre 8 Densty of =.1 wth = 1 and =.1 f ( =.1).1 =

25 Fgre 9 Densty of =.1 wth = 1 and =.1 f ( =.1).1 =

26 Fgre 1 Densty of =.1 wth = 1 and =.1 f ( =.1).1 =

27 Fgre 11 Densty of =.1 wth = 1 and = 1 f ( =.1).1 =

28 Fgre 1 Densty of =.1 wth = 1 and = 1 f ( =.1).1 =

29 Fgre 13 Densty of =.1 wth = 1 and = 1 f ( =.1).1 =

30 Fgre 14 Densty of = wth = 1 and =.1 f ( = ) =

31 Fgre 15 Densty of = wth = 1 and =.1 f ( = ) =

32 Fgre 16 Densty of = wth = 1 and =.1 f ( = ) =

33 Fgre 17 Densty of = wth = 1 and = 1 f ( = ) =

34 Fgre 18 Densty of = wth = 1 and = 1 f ( = ) =

35 Fgre 19 Densty of = wth = 1 and = 1 f ( = ) =

36 Fgre Densty of for =.1,.5, 1 and wth = 1 and = 1 Densty of wth = 1 and = 1 f ( )