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Journal of Econometrics 46 (1990) 141-163. North-Holland
A GAMMA-DISTRIBUTED STOCHASTIC FRONTIER MODEL
William H. GREENE
New York University, New York, NY 10006, USA
We modify the stochastic frontier model of Aigner, Loveil, and Schmidt to allow the one-sided part of the disturbance to have a two-parameter Gamma distribution rather than the less flexible half-normal distribution. Maximum-likelihood estimation and the estimation of firm-specific ine~ciency estimates require the evaluation of integrals which have no closed form and for which there are no polynomial approximations available. We consider two methods of computing these integrals. We also present a corrected OLS estimator based on the methods of moments. An application is presented for illustration. We find that for these data, the gamma distribution produces results which differ noticeably from those of three alternative formulations.
1. Introduction
A large part of the recent empirical literature on efficiency in production has been carried out in the framework of Aigner, Lovell, and Schmidt’s (1977) [and Meeusen and van den Broeck’s (1977)l composed error or stochastic frontier model. But, the one-sided disturbance as a model of ine~ciency continues to have an appeal. See, for example, Aguilar (1988), Greene (1980), Kopp and Diewert (1982), and Fare, Grosskopf, and Love11 (198.5). The half-normal distribution proposed by Aigner, Lovell, and Schmidt is a bit inflexible in that it is a single-parameter distribution and it does embody the assumption that the density of the disturbances is most concen- trated near zero. In view of this, Stevenson (1980) suggested shifting the half-normal distribution by allowing a nonzero mode, producing a general truncated normal distribution instead. Stevenson also proposed a restricted version of the Gamma model analyzed here. Thus, various methods of relaxing the restrictions implicit in the half-normal have appeared in the literature. Still, the model of Aigner, Lovell, and Schmidt continues to dominate the received empirical analysis.
This paper will report development of an alternative model based on the Gamma distribution instead of the half-normal. The Gamma frontier model was proposed by Greene (1980) in the context of the deterministic frontier model:
y=f(x,j?) -u where u-G(O,P),
0304-4076/90/$3.50 Q 1990, Elsevier Science Publishers B.V. (North-Holland)
142
SO
WH. Greene, A gamma-distributed stochastic frontier model
BP f(u) = q$Pe-@,r, u20, O,P>O.
[The preceding and all of the derivations to follow employ the notation of Aigner, Lovell, and Schmidt (1977) in order to facilitate comparison of the results.] Maximum-likelihood estimation and corrected least-squares estima- tion are discussed in Greene (1980), to which the reader is referred for further details. The salient features of this model are, foremost, its one-sided disturbance, and, second, the useful features of the Gamma distribution. In particular, the asymmet~ of the distribution is contained in the parameter P; 0 is just a scale parameter. The skewness coefficient is larger the smaller is P. The efficiency gain of maximum-likelihood estimation over least-squares estimation in this model is P/(P - 2). The distribution approaches symmetry as P goes to infinity. The OLS constant term remains inconsistent, however.’
In spite of their aesthetic appeal, one-sided disturbances have a serious practical shortcoming. Any measurement error in the dependent variable will have extremely detrimental effects on the anaiysis, and a single errant observation can dominate even a large sample. We propose in this paper to combine the Gamma frontier model with Aigner et al.‘s stochastic frontier specification. Thus, the model to be analyzed here is
y=f(-r,B) +u--UP
where
u = N[0,a2] and u N G[O, P], (2)
This refinement of the stochastic frontier model was suggested by Stevenson (1980), who gives results for a few integer values of P. A formulation of the log-likelihood and its gradient appears in Beckers and Hammond (1987),2 but no practical application of the model has appeared previously.
The plan of this paper is as follows: Section 2 will briefly review the formulation of the stochastic frontier model to provide more detail on the specific context of this model In section 3, we derive a formulation of the probability distribution of the difference between a normally distributed random variabIe and one with a Gamma distribution. In section 4, we present a m~imum-iikeli~~d estimator and procedures for evaluating these integral expressions. This section will also present an estimator of the firm-specific ine~cien~y component. In section 5, we present a corrected ordinary least-
‘For analysis of this point, see Deprins and Simar (1986).
2They present alternative forms of our (29), (331, (35), (37), and (42).
W.H. Greene, A gamma-distributed stochastic frontier model 143
squares estimator that could be used in its own right, though we are primarily interested in a consistent starting value for the maximum-likelihood estima- tor. Some empirical estimates are given in section 6 to illustrate the tech- nique. Conclusions are drawn in section 7.
2. The stochastic frontier model
The stochastic frontier model departs from an idealized production func- tion.
Y* =f(x>P). (3)
For any producer, the frontier is stochastic for a variety of reasons including, for example, simpte good luck. Thus, for any individual firm,
yO=f(x,@) +c. (4)
Inefficiency enters the production model through a positive disturbance, U, that is independent of U. For the ith firm,
Aigner, Lovell, and Schmidt present the probability distribution of E under the assumption that c is distributed as N[O, $1 and u is the absolute value of a variable which is independent of c’ and is distributed as N[O, ~j!]. The log-1ikeIihood for a sample of N observations is
lnL= -(N/2)(ln2~+lna2)+ C[In~[-~,h/a]-:(~~/rr)‘],
(6)
where @p[ .I and & .] (used below) are the CDF and PDF of the standard normal distribution, respectively, a2 = a,? + aU2 and A = ~~,,/a,,.
Once the parameters are estimated, primary interest centers on the estima- tion of inefficiency, u,. E[u,] is a summary measure which, for obvious reasons, is not particularly satisfactory. Jondrow, Lovell, Materov, and Schmidt (1982) suggest a firm-specific estimate,
(7)
144 W H. Greene, A gamma-distributed stochastic frontier model
For our purposes in this study, Stevenson’s (1980) extension of the model is important. He analyzes the case in which U, still truncated at zero, has a truncated normal distribution with parameters CL, which is allowed to differ from zero in either direction, and ai. Details on this model, including the appIi~able log-likelihood function used in estimation, may be found in his paper. The counterpart to (7) (which he does not present) is obtained by replacing E$/U in (7) with
3. Combining the gamma distribution with the stochastic frontier
We now consider a stochastic frontier model, (2), in which ui has the gamma distribution in (1). This specification enjoys essentially the same properties as normal/half-normal model with the additional advantage of the flexibility of a two-parameter distribution. The primary advantage is that it does not require that the firm-specific inefficiency measures be predominately near zero. For values of P between 0 and 1, it has the shape of the exponential (which has also been proposed as an alternative modeJ3 though it has rarely been used empirically). Thus, for these values, the mass of the distribution is still concentrated near zero. But, for values of P greater than 1, the terminal value of f(u) is zero. As such, the distribution of disturbances is concentrated at a point away from zero. The larger is P, the greater is this effect. (The mean is P/O.) Note that in Greene’s (1980) paper, the require- ment that P be greater than 2 implied this second general characteristic. This restriction was needed to ensure regularity of the log-likelihood func- tion. Since the range of the random variable, E, is no longer restricted, there is no such requirement in the model we consider here. We only require that P be positive. The distribution of inefficiency can have many different shapes.
The density of a gamma distributed variable is given in (1). The expected value and variance of the disturbance in the frontier model (2) are
E[&]=E[r;-u]= -E[u]= -P/O (8)
and
(9)
The third central moment for a gamma-distributed variate can be derived by
3See Meeusen and van den Broeck (1977).
WT. Greene, A guava-dist~buted stochastic frontier model 14.5
expanding the raw moment and using the result for the gamma distribution,
E[u’] = (r( P + I),#( P)),‘O’. (10)
Thus, the skewness coefficient for this disturbance is
E[ ( u - E[u])‘] = 2P/03. (111
For purposes of estimation, we require the density of the compound disturbance E = c’ - U. The joint density of E and u is
f(e,u) =f,.(e + U)fU(U). (12)
Now, insert the normal density of u and the gamma density of u to obtain
f( E, U) = (2Trcr2) -‘/2e-l/2(F+u)*/(T* Of
e-Ch4
r(p)
uP-l (13)
We now expand this and collect terms. Thus,
f(E,U) = (2aa2) -‘/2e-“*/2”*e-U2/2”* e -Eu/o= op uP- 1 e-OU
r(p) *
(14)
Gathering terms in U,
= 1y e-FZ/2crZe-rr2/2LI*-ufe/ol+O)UP- 1 > (15)
where K is the leading product of constants. Then,
f(e.+) =li;e-“2/2”*e-1/2[~=+211u2(f/~*+Ot]/a~UP-l. (16)
Completing the square in the second exponent produces
~(E,u) = Ke-“2’2”2e - 1/2[“~+2u(E+OaZf+(E+(3~*f*-(E+Ocr2)*]/o~UP- I
=~e-F~/2~~+l/2(8+Orr~~~/*~e-1/2[U+(F+OC+~~]*/o* ?,F1. (17)
146 W: H. Greene, A gamma-distributed stochastic frontier model
To obtain the marginal distribution of E, we must integrate expression. The first term does not involve U. Carrying part of integral produces
h = juW(27rc2) -‘~2e-1/2[“+‘“+~“‘)]*/“2u~-1~u
u out of this K inside the
(18)
Let QI& be a normally distributed variable with mean -(E f @a’) and variance u2. Then,
I;i(2xc2) -I” e- ~/2[Q+(~+@~Z)~L/~ZdQ z Pr&[Q > ()IE].
As such, after cancelling the constants, we find that
(19)
e-l/2[Q+(~+Ou2)]2/~2
/
m e-‘/2[Q+(~+~~2)]2/,‘dQ
=f(QlQ~o,~). (201
0
This is the density of a random variable with truncated normal distribution. Therefore, the desired integral in (18) can be written as
= Prob[Q > OIE]E[Q~-‘IQ > O,E],
where Q has the truncated normal distribution in (20). Returning to the marginal density, we have
(21)
xProb[Q > OIE.]E[Q’-“IQ > o,E].
Expanding the exponent and collecting terms, we arrive, finally, at
op f(s) = qp) -e@“+“2@2/2Prob[Q > O/E]E[ Qp-‘IQ > O,E],
(22)
(23)
W.H. Greene, A gamma-distributed stochastic frontier model 147
where Q/E is normally distributed with mean and variance
E[QIE] = -(c +@a*) and var[Qlc] =(+2. (24)
If
h(b 4 = E[Q~ IQ > 0,4, (25)
then
f(4 = r(p>e @’ ‘E+u2@Z/2Prob[Q >OIE]h(P- 1,~). (26)
The fatter term in f(c) is a fractional moment of the truncated normal distribution. Since P need not be an integer, there will be no closed form for h(P - 1, E) and thus none for the density.
Finally, for estimation of a cost frontier rather than a production frontier, where u enters positively rather than negatively in (3, we must change the sign of E in the exponent and the mean of Q to E - 0~~.
We are also interested in estimation of the conditional moments of the efficiency component, u. Following Jondrow et al., we obtain the conditional mean function, E[u(F]. To derive E[u/F], we begin with
Inserting our earlier results and cancelling terms in the fraction, we find
(27)
Jondrow et al. present the counterpart to (27) for the special case in which u has an exponential distribution. They find the conditional distribution is a truncated normal distribution where the untruncated structurat normal vari- able has mean -(E + 0cr2) and variance a2 (using our notation). Their result follows after the substitution (T = u(, and 0 = l/o,. The exponential case corresponds to P = 1. If P = 1, in (27), h(P - 1, E) = 1 and the truncated normal distribution emerges.
Multiplying and dividing (27) by Prob[Q > 0 Is] produces
E[u{E] = /muf(ule)du =h(P,&)/h(P- 1,~). 0
(28)
148 W. H. Greene, A gamma-distributed stochastic frontier model
Jondrow et al. (1982, (5) on p. 236) give an expression for E[uIE] for the case in which u has an exponential distribution. With P = 1, h(P - 1, E) equals 1 while h(P, E) is just the expected value EEQIQ > 01 when Q has mean -(E + @a’). After this substitution, as expected, their (5) emerges as the special case.4
4. Maximum-likelihood estimation
The log-likelihood function for a sample of N observations is
In L = Clnf[Ei]
= C[PInO-lnT(P) +02~2/2COEi
+InProb[Q>Ol,j] +Inh[P- l,si]]. (29)
The log-likelihood cannot be evaluated in the form given above. The stan- dard normal integral, Prob[& > 01~ ] = dr[ - (E + Ov*)/u] can be approxi- mated by the familiar polynomial approximation. The one given by Abramovitz and Stegun (1964) is typically used. But this leaves (181, the integral in the expectation function, h(P - 1, E).
For purposes of estimation, we used a U-point Laguerre quadrature.’ This was fast and accurate and replicated the computation of the exponential model (see section 6 below) in which computation of Iz(Y,E) is bypassed, to within a difference of less than 1%. But, we found that in computation of E[u Is], the quadrature formula performed rather poorly in the tails of the distribution (when E[ule] was very small), so in order to calculate the ratio h(P, c)/h(P - 1, E) at the MLE, we used instead a Newton-Cotes quadra- ture with 500 segments. 6, ’ We required two modifications of this rule for our purposes. The range of integration in (18) is [O,m>, so the trapezoid rule is not quite appropriate. But, the normal distribution has finite moments of al1 orders. Thus, in the tails of the distribution, f(u) must fall to zero. As an approximation, we applied the rule over the range 0 to +5 standard devia-
4The same procedure produces au* 1 E] = h(P + 1, e)/h(P - 1, E) which can be used to find the variance. For integer powers, E[u’ 1 E] = h(P + r - 1, .z)/h(P - 1, E).
‘See Abramovitz and Stegun (1964, p. 890).
%or discussion, see Kennedy and Gentle (1980, pp. 87-88).
‘The Monte Carlo method of integration [see, e.g., McFadden (1989) or Manski and Lerman (197711 is another possibility. However, we found the sample sizes needed to obtain stable enough values of the function to use in an iterative procedure were too large (over 5000) to be practical.
W. H. Greene, A ~urnm~-dist~~uted stochastic frontier model 149
tions of the range of Q. If P is less than 1, the termina1 value is not computable when-we evaluate h(P - 1, E). To avoid the division by zero, we used +O.OOOl standard deviations instead of zero for a. For our applications we used 500 segments. For a few known values, this was accurate to six digits. ~though far too slow for estimation, this method was preferable for comput- ing the MLE of E[uIEI.
We now obtain the derivatives of the log-likelihood. Write
lnf(&) =PInO-InT(P) -t~*0~/2+&
+ In 0m(2rc2) /
-'/2e-'/2[U+E+~U2]2/,2Up-1du (30)
Since the limits of integration are not dependent on the parameters, we may pass the differentiation through the integral. Thus,
8 In f a&
I 72 ao2)-1’2[_(u+6+~rr~)/rr~]e-“2~“+~+~~Z12/~2~~-1dU
=@+ O
I 72 RU2) -l/2 e-1/2[U+E+~~2]2/,2Up--ldU
0
(31)
The ratio of integrals is a sum of three terms. But, the second and third do not involve U, so they may be factored out of the ratio, which then evaluates to one. The third term, involving --u , is of the same form we used to obtain E[u[E]. Combining terms and using the results derived earlier, we find
a In f - = -(l/d)
a& & + ,;yyJ
3
= -(l/o”)<& + E[u]E]).
From the original specification, E =y - p’x, so
T = (l,d)[c + h(y2;,)E)]x.
(32)
(33)
The derivatives of the log-likelihood are obtained by summing this expression over the N observations.
150 W.H. Greene, A gamma-distributed stochastic ffontier model
Proceeding in similar fashion,
alnf P -=-
x9 0 +&+&r2
I 72 7io2)-1/2[_(U+F+~~2)],-1/2[u+~+c-,rr2]z/U2~~-1dL~
0 (34)
+
I ?2 7ftT2) -l/2 e-1/2[“+‘+““2]2/rrZUP-l du
0
As before the latter two of the three terms in the integral become constants times one, and the third reduces as previously, so we obtain
alnf P h(P,d
ao =o- h(P_l,E) = E[u] - E[z.J/&]. (35)
Continuing,
dlnf a( p> -=ln@--- dP r(p)
h., u e-1/2[~+~+e~~2]‘/o’UP--1 du
(36) +
/ ?2 TU2) -l/2 e-‘/2[u+r+~cr2]2/a2UP-IdU
0
This has the same form as several earlier expressions. Dividing numerator and denominator by the relevant probability produces
d In f F(P) E[Q’-‘lnQIQ>O,E]
-=in@- F(Y) + l?P h(P- 1,s) . (37)
The latter part of (37) may be recognized as E[ln U/E], which produces a useful formulation of d In f/aP,
T=in@-F(P)/I.(P) +E[lnuI&]. (38)
To differentiate with respect to (T*, it is convenient to define
q = (271-t?) -'/2e-l/2["+E+(")~*1*/~*UP-l. (39)
W.H. Greene, A gamma-distributed stochastic frontier model 151
This is the weighting function in the preceding integrals. We will require
I m
aln f 3q,b2 du
-=02/2+ ” m aa
_
/
(40)
qdu 0
Let w = u + F + Orr*. Then,
aq -4 -=- 2 2a2
+4 W2
i 1 --ow *
au a* 2a2
After some tedious algebra, this reduces to
a4 q 1 E2 @*a2 u* EU -=-- --___ - ad I CT2 2 2a2 + 2 +3-g* 1 143)
In ah f/aa2, the terms in dq/&r2 which do not involve u can be moved outside the integral, and the ratio of integrals will be one. The terms involving u and U* are of the form derived earlier as ratios of the function h(r, ~1. The third term in brackets above integrates simply to - @/2 which cancels the leading term in the derivative. Combining all of these results produces
alnf 1 &2 -=_ 1 1 _ _ 1 + h(P+ I,&) Jr2&h(P,F)
aa 2~~ ffz 2(a2)2h(P- 1,F) ’ (42)
As in (33), the effect of the asymmetry is to displace the familiar result for the classical normal regression model. As before, this can be written as the classical regression counterpart plus a moment of ~1.5:
-=- ---+&~+ 2&44. (43)
The derivatives are useful to improve the search for a maximum-IikeIihood estimator, since the numerical derivatives may have some rounding error. This is likely to be more problematic than usual in the current context because of the need to approximate h(r, E). They will also be usable for estimating the asymptotic covariance matrix of the MLE using the Berndt et a1._(1974) estimator.
152 W.H. Greene, A gamma-distributed stochastic frontier model
5. A consistent method-of-moments estimator based on OLS
As has been documented a number of times in the received literature, in the stochastic frontier model (and the one-sided gamma frontier model as well), the OLS slope estimators are consistent, but the OLS constant term is biased. It is offset by the nonzero mean of the disturbance. Greene (1980) obtained estimators for the parameters of the disturbance distribution, and thus the constant term, in the gamma frontier model by manipulating the OLS residuals. [Waldman (1982) presents a counterpart for the normal/half-normal model. Gabrielsen (1975) and Aguilar (1988) present some general results on the method of proof used in Greene (1980).] A similar procedure can be used here. Note, first,
E[&]=E[v-u]=E[u]-E[u]= -E[u]= -P/O.
The least-squares residuals have mean zero by construction, so the mean residual is not useful. But, higher central moments of F can be consistently estimated by using the counterpart for the OLS residuals. Thus,
s2 = e’e/N
is a consistent estimator of
E[(E-E[c])~] =var[E]=var[u]+var[U]=a2+P/02.
In obtaining the third and fourth moments, it is useful to note that
E[(s-E[B])‘] =E[(L;-(u-E[u])}‘].
Since E[ VI = 0, we can easily expand these functions for values of r using the binomial expansions, and, thereafter, use the well-known central moments for the normal and gamma distributions. Thus, taking the third and fourth powers, we have
E[(E-E[E])~] =E[u3]-3E[u’]E[u-E[u]]
All terms are zero except the last. The third central moment of the gamma
WH. Greene, A gamma-distributed stochastic frontier model 153
distribution is
E[(u - E[u])~] = 2P/03,
so it follows that
piim(l/N) Ce! = -2P/03.
Using the same device, we find that
E[( F - E[E])~] = 3 CT4 + 6dP/@ +* + 3P( P + 2)/04.
For purposes of manipulating the moments, it is simpler to work with
E[ ( E - E[E])~] - 3(var[E])2 = 6P/04.
This moment would be zero if the disturbance were normally distributed. Thus, it may be viewed as reflecting the departure from normality. Using the standard notation, we have
pIim( l/N) Cef = plimm, = CT’ + P/@,
plim(l/N) CeT = plimm, = -2P/O,
plim( m, - 3m2,) = plimm,* = 6P/04.
The method of moments estimators are
6 = -3m,/m,*,
P = - O”m,/2,
(44)
(45)
n ^2 c?2=m2-P/0.
Only the first two are actually needed to correct the OLS intercept, though the variance is also useful for other purposes. Note that the procedure breaks down if the third moment of the residuals is positive while 0 and P will both be zero if the residuals are distributed symmetrically.*
%n this connection, see the results of Waldman (1982).
154 ?VX Greene, A gamma-distributed stochastic frontier model
This gives a complete set of estimates. The OLS estimated standard errors are appropriate for all coefficients save for the intercept. The appropriate asymptotic standard error for the corrected constant term and for the estimates of the nuisance parameters can be obtained using Newey’s (1984) results for GMM estimators. Since these are generally not needed for inference in this model, in the interest of brevity, they are omitted. In principle, we can also use the earlier results to compute conditional estimates of the firm-specific efficiency values, E[u(E]. The corrected OLS estimates can also be used as starting for an iterative maximum-likelihood procedure or to begin the two-step procedure described below.
Since expressions for the first derivatives have been obtained, a single Newton-like step from these initial consistent estimates using the BHHH as~ptotic covariance estimator provides a set of estimates which have the same asymptotic properties as the MLE’s.
6. Application
To illustrate the preceding, we have used the data from Christensen and Greene’s (1976) study of the U.S. electric utility industry. The model to be fit is a cost function rather than a production function.’ We use one of the restricted specifications for the cost function,
In (Cost/Pf) = PO + pi In Q + P2 In2 Q + P3 ln(P,/P_)
+ fiq ln( P,/&) + a.“’ (46)
Output <Q> is a function of three factors, labor (I), capital (k), and fuel (f>. The three factor prices are Pl, Pk, and pr. The restriction of linear homogeneity in the factor prices has been imposed a priori in the cost function. Further details on the model and data construction can be found in Christensen and Greene. The data used to estimate the model appear in the appendix. Since we are estimating a cost frontier rather than a produc- tion frontier, the disturbance in (2) is u + u rather than u - u. Some minor changes are required in several results and derivations. For the convenience of the interested practitioner, those in the estimating equations are as
“See Schmidt and Love11 (1979) for discussion of the two formulations.
‘“Without the factor share equations, estimation of Christensen and Greene’s full translog model in this framework was hampered by a severe problem of multi~llinearity. The correct formulation of the relationship between the disturbances in the share equations and the efficiency component in the cost function remains to be derived. Thus, the estimates presented here are intended only to be illustrative of the differences which arise when the several models are applied to the same data.
W H. Greene, A gamma-distributed stochastic j?ontier model
follows:
15.5
(1) In f(c), the exponent OE is changed to -OF [see (23) and (2911. (2) In all results using h(r, E), the mean of Q is now F - @a’. (3) The sign of E is changed in (34). (4) The last term in the brackets in (41) is now added. (5) The terms 2&h(P, E) in (42) and ~EE[uIE] in (43) are now subtracted.
Table 1 lists the Aigner et al. stochastic frontier and the corrected ordinary least-squares estimates of the parameters in (46). The COLS estimates differ from OLS only in the adjusted constant term, which is the OLS constant plus the estimate of P/O. We have also computed the estimates for the exponen- tial model which results when P is constrained to equal one in the gamma model. With this restriction, the log-likelihood and derivatives simplify con- siderably:
In f( E,) = In 0 + 02cr2/2 - O.5; + in @[ ( Fi - @a*)/~]
(with the sign of ci reversed for a production frontier). Maximization of the log-likelihood is straightforward and does not require computation of h(r, E). For convenience, this is parameterized in terms of u instead of V’ using the formulation suggested in Aigner et al. Here, our 0 equals their l/4. Finally, we have obtained the estimates of Stevenson’s generalization of the model. Table 1 shows the maximum-likelihood estimates for the various formulations for this model. The maximum-Iikelihood estimates of the gamma frontier model were computed using the estimates from the exponential model as the starting point for the DFP iterations.
The estimates of the parameters of the cost function obtained with the four models are roughly similar. Stevenson’s extension of the ALS model is rejected by the Wald statistic. ” It is worth noting, however, the very large changes in the variance parameters which arise with this model. The hypoth- esis of the exponential model is also strongly rejected by the likelihood ratio statistic of 91.28. Likewise, the Lagrange multiplier statistic is 123.00 which leads to the same conclusion. It is interesting that the Wald statistic is so small (1.73), but the power of the test in the presence of such a different value of 0 may be problematic. The fact that the full translog model is known to be a preferable specification for these data” suggests that these specification tests may be picking up some of the effect of the missing variables as well as the differences in the disturbance processes assumed.
“The expressions for E[tr] and var[u] in Stevenson (1980) are in error. The calculations above are based on the moments of the truncated normal distribution.
‘*See Christensen and Greeene (1976).
Tab
le
1
Par
amet
er e
stim
ates
for
th
e st
och
asti
c fr
onti
er m
odel
s (e
stim
ated
as
ympt
otic
sta
nda
rd e
rror
s ar
e gi
ven
in
par
enth
eses
).
Cor
rect
ed
OL
S
-___
- __
____
_
AL
S
Ste
ven
son
G
amm
a
PO p”:
P
3
P4
h cr
u2
P
P
0 Log
-L
E[ul
vadul
var[
u]
var[
E]
var[
u]/
var[
e]
7.21
4 (0
.752
) 0.
3858
(0
.038
3)
0.03
16
(0.0
0269
) 0.
2470
(0
.067
0)
0.07
84
(0.0
617)
0.01
64
2.52
9 0.00
65
0.00
257
0.01
99
0.02
25
0.11
42
- 7.
408
(0.2
92)
- 7.
501
(0.3
13)
0.40
78
(0.0
293)
0.
4292
(0
.038
1)
0.03
059
(0.0
0218
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ed)
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~
. __
-___
_
Exp
onen
tial
KH. Greene, A gamma-distributed stochastic frontier model 1.57
Table 2
Estimated efficiency distributions.
Half-normal Truncated
normal
Gamma models ~~- COLS Gamma Exponential
Mean 0.1234 0.1039 0.0295 0.1051 0.0989 Std. dev. 0.0652 0.0759 0.0315 0.0293 0.0770 Minimum 0.0304 0.0243 0.0073 0.0001 0.0234 Maximum 0.3917 0.4858 0.2253 0.2055 0.5041
Table 2 summarizes the estimates of E[u 1~1 based on (7) for the ALS model and Stevenson’s extension and (27) for the gamma and exponential models. Once again, for the exponential model, the computation is consider- ably simpler:
where y equals E - @a* for a cost frontier and -(E - @a’) for a production frontier.13 The two distributions of disturbances based on the truncated normal distribution are quite similar to each other, but rather different from those based on the gamma model. Based on the half-normal frontier model we find that var[u] accounts for 43.2% of the estimated variance of E. The exponential model is similar; the counterpart is 47.1%. But, the gamma model allocates only (P/02>/(rr2 + P/O21 = 22.4% of the total variance of the disturbance to the inefficiency term.
Fig. 1 shows the estimated density functions for the four models. The gamma model is alone among the four in its nonzero mode. It appears that the estimated inefficiencies based on the gamma model are generally smaller than those based on the half-normal. Inspection of the data suggests this as well. It is noteworthy that the simple correlation of the inefficiency estimates based on the half-normal and the COLS/gamma estimates is 0.906, while that of the MLE/gamma estimates with the ALS estimates is 0.918. We find that in those cases in which the ALS model suggests a very large disturbance, the gamma model follows suit, through with a smaller estimate. For those disturbances not obviously in the tail the gamma model predicts values very similar to the other two.
13The individual computed estimates are contained in the appendix.
J.Econ F
158 WM. Greene, A gamma-d~st~buted stochastic frontier model
EXPNTL
Fig. 1. Efficiency distributions.
7, Conclusions
The gamma model appears to offer a promising alternative to the half-nor- mal and exponential models for the stochastic frontier. For our data, the half-normal, truncated normal, and exponential models give essentially the same distribution. But, the likelihood ratio and LM tests strongly reject the restriction of the exponential model. This suggests that the restriction in both the half-normal and exponential models may have a large influence on the pattern of the estimated inefficiency.
The estimated inefficiencies suggest that the restricted models (and Steven- son’s extension) produce much larger values than the more general gamma distribution for most of the observations in the sample. But, for those data points for which the former two models suggest large inefficiency estimates, the gamma model gives largely the same conclusion. The upshot would seem to be that the single-parameter models are providing a more pessimistic impression than is warranted. Stevenson’s extension of the ALS model bears some similarity to the gamma model in that its second parameter allows some greater flexibility in the shape of the distribution. Nonetheless, this model looks quite similar to the other two for these data, while the gamma model is qualitatively different. The marked difference suggested in fig. 1 is strongly suggestive.
App
endi
x
Cal
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c.
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cost
Tab
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3
Dat
a us
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in t
he
anal
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of
pro
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cost
s.
__-_
_ C
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func
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data
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Est
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Tab
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(c
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6.32
80
.660
21.7
550
20.6
191
0.05
3 0.
072
0.06
2 0.
083
0.12
2 0.
153
0.16
5 0.
190
0.08
1 0.
107
0.05
3 0.
072
0.08
4 0.
113
0.05
0 0.
058
0.11
4 0.
155
0.07
6 0.
050
0.07
8 0.
086
0.04
0 0.
073
0.12
1 0.
069
0.06
1 0.
108
0.10
1 0.
063
0.11
2 0.
052
0.25
2 0.
053
0.09
0 0.
041
0.05
7 0.
135
0.04
7 0.
078
0.09
6 0.
216
0.05
0 0.
057
0.04
6 0.
153
0.20
4 0.
023
0.04
6 0.
121
0.08
6 0.
017
0.09
3 0.
014
0.11
8 0.
016
40.5
281
37.0
666
35.9
651
25.6
208
20.2
790
22.0
330
31.7
601
24.5
804
40.9
692
35.7
882
44.1
571
35.8
083
25.8
240
22.5
536
37.2
477
33.3
944
34.9
616
20.3
0~
0.13
1 0.
032
25.1
686
45.1
827
55.1
764
41.1
798
80.3
593
68.4
800
97.3
859
0.10
3 0.
086
0.04
4 0.
020
0.10
3 0.
014
0.02
2 0.
023
0.01
2
0.09
3 0.
121
0.04
2 0.
058
0.10
7 0.
078
0.07
8 0.
129
0.07
4 0.
065
0.10
5 0.
101
0.16
0 0.
100
0.12
1 0.
098
0.02
1 0.
035
52.7
634
38.8
472
48.1
125
0.08
7 0.
094
0.01
9 0.
116
0.14
6 0.
109
0.14
1 0.
117
0.11
3 0.
017
0.20
9 79
.070
5 11
1.88
63
2321
7.0
77.8
849
1370
2.0
0.06
7 0.
091
0.12
0 0.
151
0.05
6 0.
075
0.09
6 0.
029
0.11
7 0.
018
0.08
9 0.
032
0.15
6 0.
015
0.08
8 0.
096
87.1
015
2770
8.0
57.7
267
1000
4.0
90.7
168
1728
0.0
28.0
959
0.25
9 0.
264
36.8
816
0.05
6 0.
078
183.
2315
27
118.
0 16
9.23
54
3834
3.0
41.7
578
31.5
897
46.0
701
0.09
7 0.
043
0.06
1 0.
145
0.05
0 0.
083
0.10
3 0.
225
0.05
0 0.
059
0.04
9 0.
161
0.21
1 0.
024
0.04
9 0.
129
0.12
8 0.
059
0.08
5 0.
177
0.06
8 0.
111
0.10
9 0.
078
0.09
1 0.
125
0.08
5 0.
103
0.11
1 0.
147
0.09
3 0.
093
0.08
3
0.01
6 0.
026
0.01
2 0.
017
0.04
2 0.
014
0.02
2 0.
228
0.08
0 0.
009
0.01
3 0.
012
76.2
528
134.
2283
16
8.37
77
125.
3356
19
1.56
25
240.
5137
0.
1304
0.
7293
1.
7705
2.
2367
2.
5593
2.
0358
7.
6236
11
.109
1
1166
7.0
1944
5.0
9829
.32
67.5
80
38.8
027
3421
2.0
5683
.83
80.3
85
40.5
286
2400
1.0
8047
.35
74.3
72
33.0
932
3095
8.0
9810
.10
69.5
41
36.3
076
2961
3.0
9312
.93
81.7
50
41.8
872
0.13
5 0.
241
0.05
3 0.
073
0.06
4 0.
17%
0.
216
0.03
0 0.
065
0.15
5
4.0
6009
.70
92.6
50
33.1
990
60.0
79
72.7
1 75
.464
33
.707
0 15
3.0
1096
3.90
57
.612
46
.160
0 19
8.0
7046
.50
64.9
45
38.~
~
243.
0 70
00.8
1 81
.750
33
.436
0 61
7.0
9547
.72
81.7
50
18.4
900
1340
.0
6438
.42
79.2
20
38.1
483
1961
.0
6964
.81
82.4
85
28.2
634
0.13
2 0.
146
0.03
6 0.
001
0.05
0 0.
083
0.12
1 0.
007
0.01
3
Tab
le 3
(c
~)n
tin
u~
d)
Cos
t fu
nct
ion
dat
a E
stim
ated
in
e~ci
enci
es
cost
Q
PL
P
K
PF
T
RN
&b.
S
er. N
ew M
ex.
9.46
74
2233
.0
6717
.18
59.9
57
22.6
933
0.08
4 N
evad
a P
ower
13
.826
2 25
82.0
83
34.8
9 77
.531
34
.181
1 0.
055
Ora
nge
& R
ock
ln.
17.2
895
2763
.0
8058
.91
80.6
60
45.6
636
0.04
9 K
entu
cky
Uti
ls.
16.1
704
2863
.0
7509
.37
81.5
50
30.2
374
0.11
5 H
awai
ian
Ele
c.
24.3
975
3490
.0
8477
.23
75.4
20
38.0
000
0.14
0 T
oled
o E
diso
n
27.2
443
4568
.0
9619
.76
76.1
40
35.6
645
0.08
0 C
olm
s.&
So.
O
hio
29
.487
0 52
92.0
81
76.3
3 76
.140
22
.837
4 0.
280
Day
ton
Pw
r.&
Lt.
35
.109
1 56
99.0
79
88.4
8 80
.370
42
.635
1
0.07
0 L
ouis
vill
e G
.&E
. 25
.481
4 57
02.0
69
47.1
1 78
.225
24
.259
6 0.
114
Cin
ci.
Gas
& E
l.
42.4
117
8650
.0
7146
.67
72.9
67
27.8
693
0.12
0 C
leve
lan
d E
LI.
72
.035
5 12
724.
0 80
37.8
4 74
.025
37
.517
4 0.
082
Flo
rida
Pw
r.&
L.
146.
9890
25
147.
0 99
00.7
6 75
.725
33
.474
2 0.
104
New
En
glan
d E
l.
51.7
415
1036
1.0
9578
.63
68.0
16
28.1
423
0.09
4 N
ew E
ng.
G.&
E.
Ass
. 21
.558
7 38
86.0
95
38.6
8 63
.569
30
.889
4 0.
090
Eas
t. U
tl.
Ass
. 14
.271
0 19
01.0
98
22.6
4 69
.411
36
.000
0 0.
146
Nrt
h.
Sts
. Pw
r.
66.1
032
1183
7.0
8709
.43
75.3
79
31.3
321
0.12
3 U
nio
n E
lec.
Co.
11
3.25
55
2252
2.0
9500
.78
76.7
32
25.0
289
0.14
1 A
lleg
hen
y P
r.
90.3
718
2195
6.0
7954
.47
83.3
38
22.9
115
0.09
2 S
outh
ern
Co.
24
0.48
58
5391
8.0
6068
.87
78.3
80
31.1
954
0.05
9 A
mer
. E
lec.
Pr.
27
7.29
65
7224
7.0
7419
.92
56.3
01
25.0
914
0.04
8 G
ener
al P
ub.
U.
107.
9776
18
455.
0 66
90.2
3 76
.300
32
.965
4 0.
174
Com
mon
. E
diso
n
269.
7728
46
870.
0 97
61.3
8 69
.541
33
.199
9 0.
091
Nor
thea
st U
til.
79
.620
7 16
508.
0 94
04.9
7 78
.044
42
.208
6 0.
032
Pu
b. S
er.
Col
a.
33.8
814
7382
.0
7512
.72
72.3
62
25.9
001
0.10
3 M
inn
esot
a P
.&L
. 14
.952
9 23
25.0
85
68.2
3 53
.890
38
.916
1 0.
078
Del
mar
va P
.&L
. 33
.973
3 57
08.0
10
024.
20
78.1
02
42.1
660
0.05
1 O
hio
Edi
son
Co.
78
.702
8 17
132.
0 81
60.8
0 78
.899
25
.537
8 0.
105
Uta
h P
ower
& L
t.
19.4
391
4560
.0
8558
.37
76.4
64
23.7
777
0.07
4 S
’wes
tern
El.
Pr.
20
.883
6 62
59.0
66
97.0
2 69
.764
21
.292
0 0.
054
New
Orl
ean
s P
.S.
21.7
792
6746
.0
9419
.27
49.7
78
20.1
000
0.04
4
AL
S
__-
0.11
0 0.
075
0.06
7 0.
143
0.16
9 0.
108
0.27
8 0.
095
0.14
2 0.
150
0.11
0 0.
135
0.12
4 0.
120
0.17
5 0.
153
0.16
9 0.
118
0.07
9 0.
066
0.19
9 0.
120
0.04
3 0.
132
0.10
7 0.
071
0.13
4 0.
098
0.07
3 0.
061
EX
P
GM
A
OL
S
0.07
8 0.
103
0.03
1 0.
052
0.08
7 0.
020
0.04
6 0.
083
0.01
5 0.
107
0.11
6 0.
014
0.13
0 0.
123
0.02
8 0.
075
0.10
1 0.
037
0.27
4 0.
160
0.02
1 0.
065
0.09
6 0.
107
0.10
6 0.
116
0.01
9 0.
112
0.11
8 0.
028
0.07
6 0.
102
0.03
0 0.
097
0.11
2 0.
022
0.08
8 0.
108
0.02
7 0.
083
0.10
5 0.
024
0.13
7 0.
125
0.02
3 0.
114
0.11
8 0.
038
0.13
2 0.
125
0,03
2 0.
086
0.10
8 0.
036
0.05
6 0.
091
0.02
2 0.
046
0.08
3 0.
016
0.16
5 0.
134
0.01
3 0.
085
0.10
7 0.
050
0.03
0 O
.OU
O
0.02
4 0.
096
0.11
2 0.
010
0.07
2 0.
099
0.02
6 0.
048
0.08
4 0.
021
0.09
8 0.
113
0.01
4 0.
069
0.09
9 0.
026
0.05
1 0.
087
0.01
8 0.
042
0.07
9 0.
014
W. H. Greene, A gamma-distributed stochastic frontier model 163
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