335

DUMMY VARIABLE ESTIMATORS OF TECHNICAL EFFICIENCY: A COMMENT

Nigel Hall and Peter Bardsley

Bureau of Agricultural Economics, Australia*

Dawson (1985) in a paper in this Journal discussed the effects of management bias and technical efficiency on the estimation of production functions and suggested a method of avoiding the bias and obtaining a farm-level estimate of technical efficiency. Briefly, the method proposed (which is due to Timmer, 1971) is to add year dummies and an individual dummy for each farm into the production function. This explicitly estimates the effects on farm production of each season and of individual characteristics in addition to the effects of continuous economic variables such as capital and land. The model is described in more detail in Dawson and Lingard (1982).

The method proposed is straightforward and the objectives are very desirable: unbiased estimates of elasticity, avera e seasonal effects on

particularly interesting because it opens up the possibility of modelling technical efficiency itself as well as the possibility of focusin extension programmes on the least technically efficient farms by estimation o B production functions from survey data.

The production function is specified as follows (slightly modifying the original notation):

production and the technical efficiency of each P arm. The last item is

Y X i = l . . . n t = 1 . . . T Djit

D s i t

Uit

is annual output of a farm is a vector of variables such as land, labour and capital

is a farm-specific dummy variable is a year-specific dummy variable is a random error term.

* Bureau of Agricultural Economics, Macarthur House, Macarthur Avenue, Lyneham, GPO Box 1563,Canberra,ACT2601.

336 NIGEL HALL AND PETER BARDSLEY

Hence in Dawson’s model there are three year dummies, 55 farm dummies and five continuous variables: wages, livestock and crop costs, machinery costs, general costs and rent of land.

Comparisons of production funtions estimated over a four-year period for a sample of dairy farms are presented and it is shown that the use of individual farm dummies has a substantial effect on the magnitude of the coefficients of the estimated production function. These differences are inferred to be the result of bias caused by varyin technical efficiencies. It is also indicated in

with farm size and labour use, implying that larger farms were more technically efficient.

A similar technique was used by Lingard, Castillo and Jayasuriya (1983) in analysing rice farms in the Phili pines. They studied 32 farms over three years,

coe8icients were shown to be strongly correlated with a number of variables such as the farmer’s age, credit access and t pe of tenure. Again, it is inferred

example, the education or tenure conditions of their operators. Dawson’s finding that larger farms in terms of area and labour use are more

technically efficient is not im lausible but the production function on which the

variables and so their full effect on production should have been ex ressed in the production function itself. There is something wrong if the resi c f ual errors from a regression are correlated with one or more of the independent variables. This problem does not appear in Lingard et aZ‘s paper but in this case it is not clear why the variables such as farmer’s age and education were not included in the origmal production function. What is being calculated is a two-stage

rocess - a production function based on economic variables and another Eased on social variables. The problem with doing this is that it necessarily excludes any interactions between these two sets of variables and hence the coefficients produced in both equations may be biased because of the exclusion of significant variables.

In addition to the problems of correlation and excluded variables there are two other difficulties with Dawson’s results. The first is the wide spread of technical efficiencies found: 25 per cent of farms were less than 30 per cent efficient. This seems to imply a very large degree of market imperfection if farms can survive for four years obtaining only one-third of the otential output from given inputs. Our view is that such a wide range, althoug R not absolutely impossible, is not very plausiMe. Timmer reports a similar wide ranee of estimates of technical efficiency from his analysis and describes it as ‘unreasonable’ (Timmer, p.788).

The second problem with the method is the level of significance of the individual farm dummy coefficients. These are not quoted in Dawson’s paper but the similar analysis by Lingard et a1 gives only three out of 31 dummies significantly different from the base farm. This indicates that, although it is possible that the magnitude of the coefficients of the dummies gives some information about the distribution of technical efficiency, there is no sound basis on which to say that individual farms attain any particular level of technical efficiency. It would be equally consistent with the estimates to say that most of the farms are at the same level of technical efficiency, although a small number are distinctly higher or lower than the rest. Given the excluded variables, it seems likely that the different values could be explained in terms of major differences in unmeasured resource inputs.

Dawson (1985) that technical e b ‘ciency in the sample was strongly correlated

usin both year and individua P farm dummies. The individual farm dummy

that technical efficiency is the result of cy ifferenges between farms in, for

estimates were based invo P ved both area and labour use as explanatory

DUMMY VARIABLE ESTIMATORS OF TECHNICAL EFFICIENCY-A COMMENT 337

Our concern with these problems led us to reconsider what is being estimated in this t pe of cross section and time series analysis and we reason as follows.

A panel of N farms is observed for T periods. The output of farm i in period t

We fin dY it convenient to reformulate the mode! in matrix form as follows.

is Yi,. The complete vector of outputs will be written

where the prime denotes transpose. Let Xkit be the use of input k on farm i in period 1. The complete vector of the use of input k will be written

The complete set of inputs will be written in matrix notation as

x = [XI, . . . , X,].

Let j = [I, . . . ,1] be a Txl column matrix of ones, and let P be an NTxN block diagonal matrix which has j for the diagonal blocks and zeros elsewhere. P is a matrix €or which the columns are the usual dummy variables, one for each farm.

Dawson’s farm dummy model can now be written

y = Pd + Xb + v, (1)

where d are the coefficients of the dummy variables (Dawson interprets these as indices of technical efficiency) and b are the coefficients of the regressors in X..Dawson includes time dummies but, for simplicity, these are absorbed here into the matix X. v is an error term.

The next step is to explain the indices of technical efficiency in terms of additional factors by regressing exp(d) against a number of variables. The nonlinearity introduced by exponentiation complicates the analysis somewhat without changing the main properties of the estimation method. These can be seen more clearly by considering a linearised version in which one simply regresses d rather than exp(d) against the explanatory factors, as follows:

d = za + u, (2)

where Z is a matrix of explanators and u is an error term independent of v.

338 NlGEL HALL AND PETER BARDSLEY

Substituting (2) into (1) gives the following implied model

y = PZa + Xb + Pu + v (3) = Aa + Xb + w+v.

Since premultiplying by P has the effect of averaging out any time variability, A is a set of variables which is constant through time and w is a random variable which is constant through time but varying from farm to farm. This is a standard components of variance model described, for example, in Judge, Hill, Griffiths, Lutkepohl andLee (1982, ch. 16).

It is clear from looking at the model that it does not make a great deal of sense to distinguish between factors of production X and efficiency explanators Z. The difference is in the way they are estimated. Factors X enter the production function only through their time variation (any cross-sectional variation is absorbed by the dummies). This can be seen quite clearly by lookin at the algorithm for estimating dumm variable models given by Judge et al. $ariables z enter the model only throug i their cross-sectional variability (because they are premultiplied by P). If (3) is the correct model, the two-step estimation via (1) and (2) IS unbiased for b and a, but is inefficient because it throws away the cross-sectional variation in z when estimating the production function and any time variability in x, and also ignores the structure of the error term. Because of the misspecification of the error term, the t statistics are unreliable and the model is likely to be miss ecified by including the wrong variables, since it is impossible to decide whic f are the significant variables on the basis of unreliable t tests.

If d* is the OLS estimate of d from equation (l), then it can be shown that

d* = Za + u + Qv , where Q is a non-stochastic matrix.* Thus the usual estimator of residual variance from equation (2) is an estimator of

V(u + Qv) = V(u) + Tr(QQ‘)V(v).

Clearly this is an overestimate of V(u). Thus the two-step estimates will over- estimate the spread of efficiency in the population. This applies equally if z is a null matrix and the variance of d* is used directly as a measure of the spread of efficiency. This is consistent with the findings in both Dawson and Lingard et a1 of extremely wide ranges of technical efficiency. In summary, this analysis suggests that there are problems in principle with the approach used by Dawson, and makes it clear why the spread of efficiency is overestimated and why there is correlation between the technical efficiency measures and farm size even though measures of farm size were included in the production function. * The OLS estimates d’ and b’ of d and b in equation ( 1 ) are found by solving the normal

equations (in matrix form)

Using a partitioned matrix inversion, this implies that

where M = I - (In) P‘P. Note that MP = 0. Substituting from equation (3) for y gives the desired form, with

Q = (In) P’(1- X(X’MX)-IX’M)

d’ = (IR) (P’ - P’X(X’MX)-’ X’M) y

DUMMY VARIABLE ESTIMATORS OF TECHNICAL EFFICIENCY - A COMMENT 339

It can be seen from equation 3 that the two-stage method of estimation means that the production function has picked up only the size effects associated with changes in size over time. That is, the positive coefficient in the production function with respect to farm size indicates merely that farms which grew in size over time increased their production. The cross-sectional association between size of farm and production is concealed by the technical efficiency dummies. Thus it is not surprising that these dummies are strongly correlated with farm area and labour use, even though these variables may have been included in the production function. This correlation simply reflects the cross sectional association between size and production which was excluded from the production function. However, it is not correct to conclude (as does Dawson) that larger farms are technically more efficient, merely that larger farms produce more output. The nonlinearity introduced by exponentiation into Dawson’s algorithm complicates the analysis, but we do not see that it could alleviate in any way the problems which have been outlined above.

References

Dawson, P. J . (1985). Measuring Technical Efficiency from Production Functions: Some Further

Dawson, P. J . and Lingard, J. (1982). Management bias and returns to scale in a Cobb-Douglas

Fuller, W. A. and G. E. Battese (1974). Estimation of Linear Models with Crossed Error

Judge, G. G . , R. C. Hill, W. Griffiths, H. Lutke oh1 andT. Lee (1982). Inrroduction rothe Theory

Lingard, J., L. Castillo and S. Jayasuriya (1983). Com arative Efficiency of Rice Farms in Central

Timmer. C. P. (1971). Using a probablistic frontier roduction function to measure technical

Estimates, Journal of Agricultural Economics, 36,31-40.

production function for agriculture, European Review of Agricultural Economics, 9, 7-24.

Structure, Journal of Econometrics, 2,67-78.

and Practice of Econometrics, Wiley, New qork.

Luzan, the Philippines, Journal of Agricultural Lonomics, 34, 163-74.

efficiency, Journal of Political Economy, 79, 776-94.

DUMMY VARIABLE ESTIMATORS OF TECHNICAL EFFICIENCY: A REPLY P. J. Dawson and J. Lingard

University of Newcastle upon Tyne*

Measuring farm-specific technical efficiencies from a set of dummy variables in a production function is not new and can be traced back as far as Mundlak (1961). Subsequently, it has been used fre uently in the literature, for example, Hoch (1962, 1976) and Timmer (1370, 1971) and perhaps most recently by ourselves, Lingard et af (1983) and Dawson (1985) in this Journal. We therefore find it somewhat surprising that only now have Hall and Bardsley (1987) commented on the technique.

The thrust of Hall and Bardsley’s comment is of a theoretical, econometric nature. Both Lingard eta1 and Dawson estimate a production function in which farm-specific dummies are included. These dummies are included to ractically

In common with the literature (see, for example, Timmer 1970, p.104) take into account the input of management which is essentially uno f servable.

* Department of Agricultural Economics, The University, Newcastle upon Tyne NEI 7RU.