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1、Stochastic Frontiers In this section we take the maximum likelihood approach and apply it to a fairly useful and powerful tool - stochastic frontier estimation. Whats the basic idea? How to estimate economic relationships that ought to be modeled as upper or lower frontiers rather than averages. For

2、 example: Consider a demand curve. From the theory of demand, the demand curve is a frontier which tells the fi rm the most it can charge for the marginal unit. Econometrics via traditional OLS would gather Price and Quantity data and estimate an average demand curve. At any Q0the model predicts P0

3、but the fi rm could actually change PA. The average OLS approach might over or under predict price. The model the underpredicts costs the fi rm money. The model the overpredicts might be catastrophic. Frontier estimation tries to fi x this problem. However, not all data are conduce to SFA. Uses: Pro

4、duction functions, cost functions, demand models, test of union eff ectiveness, agency costs, reservation wages, school outcomes, profi tability, survivorship, merger and acquisition analysis, eff ect of shadow inputs such as corruption. Consider the traditional production function : 1 This is the s

5、ingle input case where q = f(X). The slope of the ray from the origin is a measure of productivity q/X. Note that q/X at Point A is less than at Point B and Point C. We can imagine technological change over time which would be a shift in the production frontier Using multiple inputs, the picture cha

6、nges a little: Point P is ineffi cient relative to Point M. Point A is both technically and allocatively effi ciency. We can measure Technical Effi ciency as TE = OM/OP and Technical Ineffi ciency as 1 TE = 1 OM/OP. Allocative effi ciency is measured as ON/OM 1. Overall effi ciency is measured as (O

7、M/OP)(ON/OM) = (ON/OP) 2 History Farrell (1957, Journal of the Royal Statistical Society, Series A): Derives a production function approach and identifi es two sources of fi rm ineffi ciency/effi ciency 1. Technical Effi ciency: Produce the most output with a given level of inputs 2. Allocative Effi

8、 ciency: Produce a given output as cheaply as possible. Most of the time we focus on technical effi ciency in the explanation To determine if a fi rm is effi cient, we have to know the production function of the fully effi cient fi rm. However, we never know the fully effi cient production function

9、Farrell suggested estimating a fully effi cient production function. There are two ways to do this: 1. Non parametric techniques: Data envelopment analysis. This technique assumes that all deviations from the effi cient frontier is a realization of ineffi ciency 2. Parametric techniques: Stochastic

10、Frontier Analysis. This technique assumes that deviations from the effi cient frontier can be either a realization of ineffi ciency or a random shock. Aigner, Lovell and Schmidt (1977) and van den Broeck (1977) both introduced a way to deal with SFA and production functions. Basic Setup Consider a p

11、roduction function qi= f(xi;) where xiis a vector of inputs, qiis output, and is a k 1 vector of parameters to be estimated. 3 We can think of effi ciency being measured as imultiplied by the theoretical norm where i 0,1 such that qi= f(xi;)i If i = 1 then the fi rm is fully effi cient and produces

12、the most it can. If i 1 then the fi rm is not fully effi cient. We can let qi= f(xi;) be the level of output that should happen. Let q0be the observed output where q0 qF because of ineffi ciency and other factors. As q0 qF= f(xi;), Aigner and Chu (1968) suggested adding a non-negative random variabl

13、e to f(xi ) which would capture the technical ineffi ciency of fi rm i: qo= f(xi;) ui To estimate this type of model, one could use a fi xed eff ects model where uiwas treated as the fi rm fi xed eff ects. Lets assume: f(xi;)=0X1 1 X2 2 Xk k lnf(xi;)=ln0+ 1lnX1+ 2lnX2+ klnXk lnqi=1lnX1+ 2lnX2+ klnXk

14、 ui lnqi=X ui Aigner and Chu (1968) suggested a measure of technical effi ciency of ObservedOutput FrontierOutput = qi exp(xi) = exp(xi ui) exp(xi) 4 where 0 0 A00might be observed output: q = exp(xi + vi ui) For Firm j: B is deterministic output level B0might be frontier output: q = exp(xj + vj) wh

16、0.000-2.373671-1.105077 /lnsig2u |.5756768.15694043.670.000.2680792.8832745 -+- sigma_v |.4190827.0678132.3051855.5754871 sigma_u |1.333542.10464331.1434381.555251 sigma2 |1.953964.2448871.4739942.433934 lambda |3.182049.15815072.872083.492019 - Likelihood-ratio test of sigma_u=0: chibar2(01) = 15.1

17、1Prob=chibar2 = 0.000 We fi nd that only lnRBI is signifi cant, the other variables are not. /lnsig2v is the log(2 v) Here it is 1.739 2 v = 0.175 or v= 0.419 12 /lnsig2u is the log(2 u) Here it is 0.575 2 v = 1.77 or u= 1.33. Combining the results, we fi nd that 2=2 v + 2 u = 0.175 + 1.777 = 1.945

18、=u/v= 1.33/0.419 = 3.17 The LR test that u= 0 is rejected. The second specifi cation includes runs scored as an additional input. . frontier lnsal lnrbi lnhr lnk lnruns Stoc. frontier normal/half-normal modelNumber of obs=292 Wald chi2(4)=227.24 Log likelihood = -356.78414Prob chi2=0.0000 - lnsal |C

20、v |-2.301862.3545424-6.490.000-2.996753-1.606972 /lnsig2u |.6088014.12879074.730.000.3563763.8612265 -+- sigma_v |.3163421.0560784.2234928.4477654 sigma_u |1.355812.0873081.195051.538201 sigma2 |1.938299.21843751.5101692.366429 lambda |4.285905.12790214.0352214.536589 - Likelihood-ratio test of sigm

21、a_u=0: chibar2(01) = 26.34Prob=chibar2 = 0.000 Now = 4.28 and u= 0 is still rejected. 13 We grab the technical effi ciency score and the technical ineffi ciency measure u (45 missing values generated) . predict u1, u (45 missing values generated) . sort u1 . gen rank1 = _n Third specifi cation: Here

22、 we think that there might be heteroscedasticity in vibased on the log of homeruns, we use the command frontier y x1 x2 x3, vhet(varlist) . frontier lnsal lnrbi lnhr lnk lnruns, vhet(lnhr) Stoc. frontier normal/half-normal modelNumber of obs=292 Wald chi2(4)=213.59 Log likelihood = -352.99795Prob ch

24、50812.50763 -+- lnsig2v| lnhr |-.6033235.2166761-2.780.005-1.028001-.1786462 _cons |-1.304674.4176328-3.120.002-2.123219-.4861282 -+- lnsig2u| _cons |.5851313.11859724.930.000.352685.8175775 -+- sigma_u |1.339861.07945191.1928471.504994 - . predict te2, te (45 missing values generated) 14 . predict

25、u2, u (45 missing values generated) . sort u2 . gen rank2 = _n Note: We can also model heteroscedasticity in uivia the uhet(varlist) option. Now, all inputs are statistically signifi cant with the correct signs. More homeruns correlate with smaller v- do homeruns provide information about quality of

26、 player? Here we grab the TE and u from this specifi cation as well. Rankings: We compare the Top 10 to the Bottom 10 in overall rankings. . list name salary hr te1 u1 rank1 te2 u2 if !missing(te1) ) + iwhere i= vi ui, vi iidN(0,2 v), and ui are non-negative random variables that represent technical

27、 ineffi ciency. From the graph above, the technical effi ciency is measured as OS/OR 1, allocative effi ciency is OC/OS 1, and overall effi ciency is OS/OR OC/OS = OC/OR. 20 They assume ui= Zi+ wiwhere wi iid half normal with variance 2 w and mean zero. They include in X: cost of hired labor, cost o

28、f fertilizer (organic), cost of fertilizer (non- organic), land area, soil maintenance cost, value of subsidy. They include in Z with anticipated impacts on effi ciency: 1. Visits by an extension offi cer (+) 2. Number of Training classes (-) 3. Other income sources to the farmer (+) 4. The slope of

29、 the land (-) 5. Farmer experience (-) 6. Farmer age (none) 7. Farmer education (+) 8. Diversity of spices grown (-) The average technical effi ciency is 84% - not bad? Depken (2000, Journal of Sports Economics): Uses SFA to estimate fan loyalty in profes- sional baseball. The estimated fan loyalty (fan ineffi ciency scores) is then related to the outcomes of public referenda for new stadiums. As 0 more fan loyalty 21 AS 1 less fan loyalty 22

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