of grQ~nds by Just and Pope (1978). In par~icul~r~ they object to the impl!cit ~@sumption that marginal risk defi~ as the partial. ~the variance of output with ...
USING TIME-SERIES AND CROSS-SECTION DATA TO ESTIMATE A PRODUCTION FUNCTION WITH POSITIVE AND NEGATIVE MARGINAL RISKS W.E. Griffiths and J.R. Anderson No.ll - December IS80
ISSN 0157-0188 ~SBN 0 85834 352 5
~sin~ Time-Series and Cross-Section Data ~Q Estimate a Production Function with Positive and o Negative Marginal Risks
WILLIAM E. GRIFFITHS and JOCK R. ANDERSON
1.
INTRODUCTION
~ ,~ne trad~tiQnal multiplicative stochastic @p@cification in the .es~i~tion of production functions has been cri~iqiz~d on a number
of grQ~nds by Just and Pope (1978). In par~icul~r~ they object to the impl!cit ~@sumption that marginal risk defi~ as the partial ~the variance of output with respect ~to the k-th input, ~$!l~@it~ve for all inputs It is not dlff!~ul~ to think of particularly in agricultural ~exa~es, where one would the ~@ri~nce of output to decline as th~ level of some inputs is. incre~sed~.Consequently, using the CobbvDo?~as function as an
(l.i)
Xk . !:
(1.2)
where, . ~ ¯ y represents::. ,.: ~. the level of output; Xk is the level of the k-th and 8 , k=l, 2, ..., K areparameters; and v is a
2 stochastic term with mean zero and variance o . In equation (i.i), it is obvious that both ~Var(y)/~ and ~E(y)/~.will have the same sign as ek which, in general, one would expect to be positive. However, in equation (1.2), the sign of 3Var(y)/~ will depend upon that of 8k ¯ while ~k will determine the sign of ~E(y)/~. The aim in this study is to estimate a function such as (1.2) for the Pastoral Zone of Eastern Australia’and hence, to ascertain whether both positive and negative marginal risks are likely to exist. However, if data from a single cross-section of firms at one point in time, or a single time-series on one firm, were used, it is highly likely that estimation would be unsuccessful. Over time the levels of inputs in a given firm are unlikely to vary sufficiently to permit accurate estimation of the ek’S. On the other hand, if a single cross-section of firms was used, it is unlikely that the 8k’S could be estimated accurately. Much of the variation in y can be attributed to weather variation and, if the increased useage of some inputs is to have a mitigating effect on output variation, then observations over time are likely to be needed to capture this effect. ~~us, it would be desirable to use both timeseries and cross-section data to estimate (1.2) and it is to this question that we now turn. In Section 2 we specify a fo~n of (1.2) which has both time and firm components and indicate how the parameters c~n be estimated. In Section 3 we present the results obtained from application of the model and estimation techniques to the Pastoral Zone of Eastern Australia.
2.
q~IE MODEL AND ESTIMATORS
Assuming there are T time series observations on N firms t]~e proposed model can be written as
r~k 711 Yit = Xkit + k
(2 .i)
~’lt ’
where 8k = +~ ~it (Bi Its+ vit) H Xkit k
(2.2)
(2.3)
E(~i) = E(It) = E(vit) = 0, and E( 2
2
2t
02
~i) = 0 , E(I ) = o21
E(v~t) =
v ,
(2.4)
In addition we ass~ne that the ~i’ Itand vit are mutually uncorrelated for all i and t. ~us iJ~e model combines the error component assumptions used bymany authors, for example Fuller and Battese (1974), .with the nonlinear heteroscedastic error model suggested by Just and Pope (1978). If we let o = 0~2 + 2 ol +2Ov 2 and
zit
= Hk
~k ~it’
then the above assumptions
imply that £it has ~e following properties (2.5) E(eit) = 0, 2 2 E(£it) = 0 z.lt
’
(2.6)
E(Sit ~. is)
2 = o z. z , t ~ s, ~ it is
(2.7)
E(£it~jt)
2 = olzitz. 3t,
(2.8)
i ~ j,
(2.9) E(£it£js) = 0, i ~ j and t ~ s. Equations (2.6) to (2.9) also represent d~e variance and covariance properties of Yit" In (2.6), the variance of output is a function of input levels, while equations (2.7) and (2.8) allow for the existence of nonzero correlation between outputs from the same firm in different time periods, or between outputs from different firms in the same time period. Outputs from different firms in different time periods are assumed to be uncorrelated. It will be convenient to put equations (2.6) to (2.9) in matrix notation.
Defining
~’i = (ell’ ~i2’ .... IxT
TxT
eiT)’ ~’ = (£{’ ~’ "’’’ ~} , IxNT
= diag.(zil, zi2, ..., ZiT), and Z = diag.(ZI, Z2 ..... NTxNT
ZN)
enables us to write
E(~’) = V = Z~Z,
where
(2. i0)
(e.g.~ Fuller and Battese 1974) @ = ~2(iN ~ jT) + s2(JN ~ IT) + ~2vINT ;
(2.11)
IN, IT and INT are identity matrices of orders N, T and NT, respectively; and JN and JT are, respectively, (NxN) and (TxT) matrices with all their elements equal to unity. As a first step in the estimation procedure we suggest obtaining the nonlinear least squares estimator for s’ = (~i’ ~2’ ..., ~K) and ¥. q~lis estimator minimizes N T~ ~,e = Z 7. (yi~ -y H Xkit)ek 2 , k i=l t=l
(2 .12)
^
and we shall denote it by (e’, 7) ’
Providing appropriate conditions
(Malinvaud 1970, p.330) are satisfied, it will be consistent, where consistency is defined in terms of increasing N ~%d T. However, it will not be asymptotically efficient because information about the covariance structure of E has not been used. To estimate the other parameters in the model and eventually to obtain a more efficient estimator for s, we follow Just and Pope (1978) and first consider the logarithm of the square of equation (2.2).
This
yields K 2 )2} log(£it ) = log{(Hi + I t + vit + 2 7. ~klOg Xkit. k=l
(2.13)
Now if wit
2 ) _ E[log(£~t)] = log(£it = l°g[(ui + it + vit)2} - 8o’
(2.14)
= E[log{(~i + It + vit) 2}], equation (2.13) can be written as
where K + l°g(£~t) = 80 2 X 8klOg ~it + wit " k=l
(2.15)
~is equation would be a convenient one for estimation of the 8k if the Citwer~observable, and if we knew the properties o[ wit. qhe first ^
problem can be overcome by noting that, if ~’, Y) is a consistent estimator, then the nonlinear least~squares residuals ^ ~k cit = Yit - ~ H ~it k
(2.16)
will converge in distribution to the tit and so, asymptotically, we are ^
justified in replacing£.tin (2.15) with cit. Furthermore, if we add the assumption float the components ~i’ It and vit are normally distributed then qit ¯ (Hi + It + vit)/o has a normal distribution with zero mean and unit variance and 2 = log[ (~i + I + Vi.t)2] - log 02 log qi t t
=
wit + 8o
- log ~ 2
(2.17)
2 is distributed as t!]e logarithm of a X random variable with one degree of freedom. Such a random variable has mean and variance given by (Harvey 1976;
Just and Pope 1978) 2 2 E(log qit) = -1.270% and Var(log qit) = 4.9348.
(2.18)
This implies that 2 8o - log ~ = -1.2704, and Var(wit) = 4.9348. Also, because qit and qjs will be correlated when i = j or t = s, 2 and log 2 log qit qjs will be correlated when i = j or t = s and this latter correlation will be equal to that between wit and Wjs.
(2.19)
we have
(2.20)
E (wit) = 0 2 E(wit) =
¯2
E(witWis)
E(witwjt)
4.9348
(2.21)
= ~2
t ~ s
(2.22)
= T~
i ~ j
(2.23)
=
and (2.24)
E (witwj s ) = 0 t ~ s and i ~ j
where the subscripts ~ and I have been used to indicate that ~e covariances 2 2 ~ and TI will depend, respectively, on E(qitqis) = o2/o2~ and E(~itqjt) ~ c 2 2 oi/o . This dependence can be made explicit by using an expression in Johnson and Kotz (1972, p.226), and by noting that, within ~is expression, 7.i=0 (i + .~)-2 = 4.9348.
Specifically, 2h h : F 09 h2 F (h+-~.)
h=l
(2.25)
and the expression for T~ is similar. We are now in a position to discuss estimation of the ~k’S. If, ^
in equation (2.15), we replace Eit by Eit and, correspondingly, wit by wit, ^
then because £it converges in distribution to Eit’ wit will converge in distribution to a random variable wit wit!] the properties given in equations (2.20) to (2.24). These properties are precisely those of the disturbance in the standard linear model with time and firm error components and so, providing we can obtain estimates of T~]
2 and 2 YI’ techniques
such as those outlined by Fuller and Battese (1974) can be used, and these will be asymptotically justified.
8
~;J2’ o~ and o2,
We approach estimation of .[2U and ~2X by estimating.
and using these estimates in equation (2.25) and its l-counterpart. Providing the appropriate matrices of second order moments of the log ~it’s approach positive definite limits (e.g., ~]eil 1971), the ^
^
~
ordinary least squares estimator ~’ = (8 , ’~ .... , o 1
~
), obtained from K
equation .K ^ = 8o + 2 E 8klOg Xkit + wit, k=l
^2 l°g(~it)
(2.26)
will be a consistent estimator for 8’ = (8o’ 81’ "’’’ 8K)" ^
Using ~ and equation (2.19) we obtain o ^ = exp(~ + 1.2704) o
^2 o
2 as a consistent estimator for o . ¯ To estimate ti~e components 0
(2.27) 2
and
2 ~ we let
zit = H ~it k
(2.28)
and then, from equations (2.7) and (2.8), the estimators
(2.29)
and ^2 °l
1 T N N sit £jt = NT(N-I)/2 Z Z Z ^ " t=l i=j+l j=l zit zjt
are obvious choices.
(2.30)
2
Finally o can be estimated from v
^2 0 V
=
^2 ^2 ^2 0 - 0 - O_ ~
(2.31)
^2 ^2 Let ~ and ~[ be the estimates obtained from equation (2.25) and ^2 ^2 its l-counterpart, respectively, where o , o and
~2have replaced the
corresponding unknown parameters. A more (asymptotically) efficient estimator of ~ and, perhaps more importantly, a consistent estimator of the covariance matrix for that estimator, can now be obtained by applying the computational procedure suggested by Fuller and Battese (1974).
Let ^2 ~it = log ~it’ uit = l°g(git) ’
^2 ~v
4.9348
^2 ~
^2 Y1 ’
~i
bI
=
1 - ¯
b3
=
bI + b2 - 1 + ~v/~3
i’
b2
=
T~2 H
^2 ^2 v
1 - ~v/T2, and ^
Then, a feasible generalized least squares
estimator for 8 is obtained
by applying ordinary least squares to the transformed equation , K , ^, uit = ~0(l-bl-b2+b3) + 2 ~ ~kXkit + wit k=l
(2.32)
*
where, using uit as an example, a transformed observation is given by *
uit
=
-,T with u.i. = Et=l uit/T’
uit -
_ b2u.t + b3u
-N -T N u.t = Zi=l uit/N’ and u .. = Et=l [i=l
(2.33)
uit/NT-
i0
Let this estimator be denoted by ~ ~ ^2 ^2 . The remaining problem is how to use 8, oI , o and o to obtain
^2
an asymptotically efficie~nt estimator for ~. We achieve this by finding tJ~e nonlinear least squares estimator for (~ which minimizes (2.34)
where v, Z and Care the matrices V, Z ~nd ’~ with 8, o~, 02 and 0~ replaced by ~ ~, ~2 and ~2 A " convenient way to carry out this ’ ~1 9.
**
**
~inimization is to minimize £ ’£
A typical element of & is obtained
by a two-step transformation procedure. First define £it = £it/zit’ --* ~T *_ --* Z~=I ~t/N and--*~ = then calculate ~ = ¯ ¯ ¯ i. t=l sit/T, g . t= ** * ’zT z’N Then, £it is given by git/NT. t=l i=l **
*
Eit = ~it -
__* al~i.
--* -
a2~.t
_* (2.35) + a3~..
where aI = i - Ov/OI,
a2 = i - Ov/O2,
= T~ + o a3 = aI + a2 - i + ^Ov/O ^ 3, o I U ^2 ^2 ^2v
^2 ^2 ^2 ^2 ^2 ^2 ^2 ~2 = NO1 + Ov’ and 03 = TO~ + NOl + Ov.
We now summarize the above estimation procedure. Step 1 :
Use nonlinear least squares to find (~, y) Which minimizes ~’~ and obtain the corresponding residuals
Step 2 :
^2 Regress log £it on the log ~it’s to obtain estimates ^
^
^
80, 281, 282, .... 28K.
ii
Step 3:
2 Estimate 2 from o = exp (~0 + 1.2704) and o , o ^2
and ov 2
from equations (2.29) to (2.31). Step 4 :
Use the estimates obtained in Step 3 to obtain estimates
^2 TZ’ S tep 5 :
^2 ^2 "IX and "Iv
^2 on tl~u Use generalized luast squares to regress log ~it log Xkit’s to obtain estimates 80, 2gI, 282, ..., 2 .
Step 6 :
Use nonlinear least squares to find (e’, y) which minimizes
3.
~~E APPLICATION
The procedures just described were used to estimate a production function for the Pastoral Zone of Eastern Australia. Individual farm data for 38 farms for a ten year period from 1964/65 to 1973/74 were made available by the Bureau of Agricultural Economics. Output was defined in terms of annual "wool equivalents" (tonnes) while the inputs used were labor (man weeks), "sheep equivalents" (number), water, fencing, plant, and buildings and land (these last four as annual real dollar flows). The estimates obtained are presented in Tables i and 2. In addition to the model with time and firm components, a m~del with time component only and a model with firm component only were also estimated. If one notes Fuller and Battese’s (1973) computational suggestions for the error components model, it is straightforward to modify the above procedure for these two cases. ~e various estimates for s are given in Table 1 while those for 8 are given in Table 2. The first row in Table 1 contains the unweighted nonlinear least squares estimates for ~ and the first row in Table 2 contains the ordinary least squares estimates for 8. Each of the
12
remaining rows in Table 1 gives a set of weighted nonlinear least squares estimates for a with weights dependent on the generalized least squares 8 estimates given in the corresponding row of Table 2. The nonlinear least squares computer package which we used finds estimates via the Gauss-Newton algorithm and, in every case, the same point of convergence was obtained from three or four different starting values. The estimates of the ~’s appear reasonable with the exception of some negative coefficients on water and plant. It is unlikely that additional expenditure on watering facilities and on plant will reduce mean output. However, this need not detract from the usefulness of ~del if we note that, in the more realistic case where both time and firm components appear, the standard errors are relatively large. In estimation of the 8’s the goodness-of-fit is not outstanding, and most of the standard erros are fairly high, but the signs are quite plausible. It is reasonable to expect that additional labor, and additional expenditure on watering and fencing facilities, will reduce the variance of output. On the other hand, properties which are larger in the sense that they carry more sheep, and have more invested in buildings and land, are, other things equal, likely to suffer from a greater variance in output. Note that the estimates of the 8’s are not ^2 ^2 very sensitive to the ass[uned model. This is because ~ and T1 are very small relative to T^2 It appears the d~e correlation between £-t~ and ~0 for i=j or t=s does not lead to a very high correlation between 3s ’ wit and Wjs, and that little would have been lost if Steps 4 and 5 of the estimation procedure were omitted.
i3
If £it is indeed heteroscedastic with variance dependent upon the inputs, the null hypothesis 81 = 82 = "’" = ~6 = 0 should be rejected. In Table 2 we present values of the F statistic commonly used to test this hypothesis and, in all cases, it is significant at the 5% level. Note that, except for the first row, the F statistic (correctly) uses
the transforn~d variables and that, strictly speaking, because the results are asymptotic, we should be using a X2 test. However, the F statistic can be justified on the grounds that it converges in distribution to a multiple of the X2statistic and that it is (appropriately) more conservative than the X2 test (~eil 1971, p. 402). Finally, note that the estimates of o 2are remarkably close to 4.9348 w
which is the "asymptotic variance", of wit when lli, )~t and vit are normally distributed.
4.
SU~4A RY
We have suggested a model and estimators which can be utilized if combined time-series and cross-section data are available for production function estimation. The disturbance term of the proposed model includes both firm-specific and time-specific components and permits the variance of output to increase or decrease as the k-th input is increased. It is tl~is last property which differentiates the model and estimators from those used in previous empirical work. When the model was used to represent a production f~ction for the ’Pastoral Zone of Eastern Australia, our estimates were a little disappointing in tez]ns of statistical significance. However, most of our results were quite plausible amd did suggest that production function modelling should allow for both positive and negative marginal risks instead of imposing the conventional positive marginal-risk restriction.
co 00
,--I O4
0o
z.-f [’--,
~.~ , ~ 00
cO 0
c; o
0 CO
d°
co ~
00 0
!! o~ 0
0 0
o 0
0 0 0,-’1
o ~
0 I
0 ~-~
,-I 0
o~ o~
o
(H 0
0 0
o o
0 0
0 ¯u~~ 00
o
Fuller, W.A., and Battese, G.E. (1973), "Transfcrmations for Estimation of Linear Models with Nested Error Structure", Journal of the American Statistical Association, 68, 626-636. Fuller, W.A., and Battese, G.E. (1974), "Estimation of Linear Models with Cross-Error Structure", Journal of Econometrics, 2, 67-78. Ha~vey, A.C., (1976), "Estimating Regression Models with Multiplicative Heteroscedasticity, Econometrica, 44, 461-465. Johnson, N.L., and Kotz, S. (1972), .Distributions in Statististics: Continuous Multivariate Distributions, New York: John Wiley and Sons, Inc. Just, R.E., and Pope, R.D. (1978), "Stochastic Specification of Production Functions and Economic Implications", Journal of Econometrics, 7, 67-86. Malinvaud, E. (1970), Statistical Methods in Econometrics, 2nd Edition, Amsterdam: North Holland. Theil, H. (1971), Principles of Econometrics, New York: Wiley.
