5 Estimation of Technical Efficiency: A Stochastic Approach
MODEL SPECIFICATION
In the estimation, our production function consists not only of labour and capital but energy as well. A proper estimation of technical efficiency hinges crucially on the correct model specification, which in turn depends on the characteristics of enterprises covered by the data. This section therefore focuses on model specification and the adjustment of data.
Defining Technical Efficiency in the Stochastic Production Frontier
Let the deterministic frontier production function be written as follows:
y*„=f(x,) (5.1)
where fix J is the appropriate production function, y^ represents the production level of the firm in the time period and x-, is a vector of core inputs used in production. In the production firontier model, y/, is understood to be the maximum possible output the firm can obtain in the period from the input set using the best techniques available; in other words, it is output at the production frontier. In the case of the stochastic production fi-ontier, equation (5.1) can be written as follows:
r „ = m , ) e x p ( v , ) (5.2) where v^, is a random variable which is assumed to be normally distributed as N(0,aj)
independently of input vector x-,. This variable does not differ from the statistical white noise attached to the conventional production function (Zellner, Kmenta and Dreze
1966). The last variable is supposed to capture the influence of factors outside the control of enterprises that cause the actual output of enterprises to vary around some mean level (chapter 2).
If the production process were purely the engineering relationship between a set of inputs X,, and observed output J,,, then a well-defined production function would describe the process accurately and any variation in inputs would result in a corresponding change in output. In reality, observed output is often the result of a series of economic decisions which are realised through the production function, and so the variables associated with the relevant economic institution will also play an important
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part in an enterprise's output. For this reason alone, some enterprises may be producing not on but inside the frontier defined by equation (5.2) as a result of non-price or organisational factors. For example, in pre-reform Chinese state enterprises, lack of incentive, the soft budget constraint, the inefficient transmission of information about production processes to government decision-makers and ineffective govemment control over enterprises! could all cause deviation of realised output from the frontier level. Hence we can model the realised production frontier as follows:
= / ( x , ) e x p ( £ J = /(x,)exp(v„ - u,) (5.3) where is a non-negative random variable representing specific productive
characteristics of the enterprise influencing the technical aspects of production in the rth period.2 The enterprise fully realises its potential or produces at the production frontier in the period if and only if u-, equals zero. This specification implies that enterprises cannot produce more than a theoretically possible level and is more consistent with economic theory than the average production frontier approach. The greater the observed value of u-^, the further is the enterprise from the production frontier and the less efficiently it produces. An objective of economic reform was to motivate state enterprises to achieve the optimal technical relationship between inputs and output or, in other words, to achieve maximum output at any input level. If reform had been effective, firms would be operating increasingly closer to their frontiers. This means that as t increased, u-, would decline. If the reform was fully successful in the enterprise, then we should observe u-, = 0 for all f's.
Now, the level of technical efficiency in the enterprise in the t^^ period can be measured using equation (5.4):
TE, =
% (5.4)J.
where y.^ is observed or realised output level and yl is in theory maximum output evaluated at input vector x-,. This measure is consistent with the definition of technical efficiency illustrated in Figures 2.1 and 2.3 in chapter 2. Given the stochastic production function (5.3), the technical efficiency of the firm in the period can be expressed as:
^The last two problems were proposed under the so-called "command economy hypothesis" (Danilin et al. 1985).
^For convenience of presentation, u is defmed so as to be non-negative in equation (5.3), which is different from equation (2.3) in chapter 2.
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TE„ = e x p ( - « J = r (5-5)
/(x,)exp(v„) = >'„
An interesting feature of this definition of technical efficiency is that technical efficiency is evaluated at the stochastic frontier. In the stochastic model, the random variation of output captured by v., is excluded from the technical efficiency index. In both the deterministic frontier model and the total factor productivity approach, such variation is included in the efficiency index.
Distribution of Technical Efficiency
In order to estimate the stochastic production frontier, our approach is to assume a distribution for the random variable w,,. Specifying an appropriate distribution depends not only on the form of the data (for example time series, cross-sectional or panel), but on the question being investigated.
In the early literature, the stochastic production frontier was mainly applied to cross- sectional data (Meeusen and Van den Broeck 1977; Aigner, Lovell and Schmidt 1977; Kalirajan and Flinn 1983); Jondrow et al. 1982). In applying the stochastic production frontier approach to panel data, it is necessary to consider the assumption of technical efficiency and specification of the model on a time-series dimension. We can imagine the following three cases for u-, in equation (3):
a. u^/s remain constant over time or are time-invariant This means that the technical efficiency of enterprises follows a particular distribution during a particular period of time. This distribution carries over to other periods and remains unchanged over time. Furthermore, the efficiency level of each enterprise remains constant over time. Any deviation from the production function results from random shock rather than inefficiency.
b. 's are correlated over time for a particular enterprise. Technically we can express this as ) = a„ for all i and ) = 0 for aU j. Under this specification, any variation from technical efficiency follows a traceable pattern.
c. 's are uncollated with other units across enterprises at any point in time as well as over time for any particular enterprise. In the latter case, levels of technical efficiency are determined by the specific production or instiuitional characteristics of the enterprise during the r^^ period of time, and differ not only across enterprises but also over time.
115 In the case of (a), can be expressed as U^ without losing any information in the technical efficiency index. This model was used by Pitt and Lee (1981) in their estimation of the average technical efficiency of the Indonesian weaving industry (model I). The authors adopted the maximum likelihood estimation method on the assumption that v, and follow respectively normal and half-normal distribution. Schmidt and Sickles (1984) proposed three approaches, namely the within estimator. Generalised Least Square Estimation and Hausman-Taylor Estimator, for estimating panel data. They discussed the statistical properties of each of these methods when applied to panel data of different lengths either in their longitudinal or cross-sectional dimensions. Battese, Coelli and Colby (1989) estimated panel data for Indian farms using maximum likelihood estimation and a relaxed assumption for the distribution of u.^, which was still normally distributed but not necessarily truncated at zero. However, despite the numerous advantages of this model from the statistical point of view (Schmidt and Sickle 1984), this time-invariant model is clearly inappropriate for examining the impact of economic reform on the technical efficiency of enterprises in our sample, because efficiency is, by definition, invariant over time.
Case (b) is a generalisation of case (a) that attracts considerable interest from economists. The criticisms of the time-invariant model concem its implicit violation of economic rationality and its inability to deal with dynamic institutional change. Economists argue that it is not possible for firms to be unaware of inefficiency if the period of investigation is sufficientiy long. If inefficiency is detected, then it would be economically irrational and unrealistic for a profit-maximising decision-maker not to attempt to deal with it (Kumbhakar 1990). Again, if an economy undergoes institutional change — during the course of deregulatory transition, for example — the time-invariant model would inevitably fail to reflect the consequences for efficiency (ComweU, Schmidt and Sickles 1990). To deal with these problems, a time-variant model for is necessary.
So far, three time-variant models have been proposed. In the simplest and most straightforward specification, proposed by Battese and Coelli (1992), w, is assumed to be an exponential function of time. Specifically:
= = { e x p [ - r | ( ^ - T)]}u^ (5.6)
where T is length of time and t refers to the t^^ period. In this specification, it is implicidy assumed that variation of all firm effects (technical efficiency) is monotone throughout time periods and that one rate of change applies to all firms in the sample. Technical efficiency either increases, remains constant or decreases depending on whether Tj > 0 , r | = 0 or r | < 0 . This is, as the authors rightiy point out, "a rigid parameterisation" (p. 154).
116 In a relatively flexible model was proposed by Comwell, Schmidt and Sickles (1990), is assumed to be a quadratic function of time. Specifically, the frontier production function is defined as:
+ and + (5.7) where t is time trend. Similarly Kumbhakar (1990) proposed the following time-variant
model for u :
u,=y{t)x^ and yit) = [l + cxp{bt + ct')Y\ (5.8)
The latter two models are mathematically superior to the first in the sense that, in addition to the direction of the time trend, the concavity or convexity of the behaviour of technical efficiency over time can also be determined, by the parameter in (5.7) and c in (5.8). In both cases, is a monotonic function of time.
But serious problems still remain even in the latter two models, particularly in situations in which economic institutions have changed very rapidly, as was the case with Chinese state enterprises in our sample. First, the functional forms in equations (5.7) and (5.8) lack a theoretical foundation to explain why the dynamics of technical efficiency should behave monotonically and continuously. Second, and probably more serious, it is unrealistic to assume that the technical efficiency of different enterprises would vary over time following exactiy the same locus. This point is particularly pertinent if we are trying to work out the impact of institutional change on efficiency at enterprise level in (Zhina's context. We cannot rule out the possibility that the response of one enterprise to new institutions will differ from the response of others. Early study on economic decision- makers' responses to institutional change suggested that its effects were likely to be discontinuous.^ In chapters 3 and 4 we discussed the fact that economic reforms were introduced into different state enterprises at different times and that policies were not adopted uniformly in enterprises in different localities or in different sectors. In addition, our data showed tiiat tiie intensity of the implementation of reform varied enormously among sample enterprises (chapters 3 and 4), and that this may well have caused enterprises to respond to reform in vastiy different ways. Against this background, case (b) can be considered inappropriate for our purpose. Our interest lies not only in comparing technical efficiency across enterprises during a particular period and in
-Refer to Lin (1987) for farmers' responses to the newly instituted Household Responsibility System in the late 1970s. In that paper, the author found a significant jump in the productivity of Chinese agriculture as it moved towards quasi-privatisation.
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gauging overall efficiency trends but also in observing the behaviour of each enterprise within its somewhat unique policy environment during the reform period.
To capture the dynamic and varying quality of reform, the distribution of technical efficiency is assumed here to follow case (c). The specification of w,, in this study differs widely from its specification in all the previous studies. Specifically, the non-negative component of equation (5.3) is assumed to vary across all enterprises in any of the six time periods and during the years 1980 and 1984-88 according to the specifications of a multivariate distribution. Moreover, technical efficiency («,,) follows a truncated half- normal distribution, and the point of truncation is not pre-determined at zero. All the related assumptions are given in equation (5.9.1 to 5.9.7):
U, - and > 0 (5.9.1)
- N{0,(5]) (5.9.2) Eiu^^u.^.) = 0 for all i ^ i' and t ^^ t' (5.9.3)
) = g] for i = and t = f (5.9.4) £ ( V. V. ) = 0 for /' or t^r (5.9.5)
v^^v, ) = a ' for / = i and t = t' (5.9.6) Eiu^^v. ) = 0 , for aU i and t, (5.9.7) where r=l, 2, ...,6 and /=1, 2, ..., N and N is the number of enterprises in a particular
industrial sector, |i is the mean of the half-normal distribution and a j and o j are respectively the variances of the distributions of and v^,.
This specification differs from that of Jondrow et al. (1982) and Kalirajan and Flinn (1983) in the sense that the mean of the truncated half-normal distribution is not restricted to zero-^ and the distribution is defined on panel rather than cross-sectional data. It is more general than those of Pitt and Lee (1981), Schmidt and Sickles (1984) and Battese, Coelli and Colby (1989) in that technical efficiency is allowed to vary not only across firms in a particular time period but also over time for a particular firm. Unlike the specifications of Battese and Coelli (1992), Kumbhakar (1990) and Comwell,
•^Kumbhakar (1991) made a similar attempt to relax this restriction for the point (instead of average) estimates of technical efficiency using different model specifications.
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Schmidt and Sickle (1990), it does not impose a unifomi pattern of change over time in technical efficiency for all firms in the sample. Since no such model exists in the literature, it is necessary to derive formulas for estimation. The following model is a modification of those described in the literature on the stochastic production frontier approach.5
The density function of u^^ can be expressed as:
v ^ a [ l - / ( - | i / a ) for u > 0 (5.10)
where / ( * ) is the distribution function for a standard normal variable. It can be shown that the mean and variance of U . are:
E{u„) = n + [f{-\i / )/[! - / )]} (5.11)
Var{u,) = C5l{\-/ ( - ^ / a j u , / ( - ^ / a j + ) a l - F ( - n / a J
(5.12)
where ) is the density function of a standard normal distribution.
It is weU known that the best (minimum square error) predictor of an unobservable random variable, conditional on the value of a known random variable, is the conditional expectation of the first random variable, conditional on the value of the second random variable. Therefore the best predictor of the technical efficiency of the firm in the r^^ period, e x p ( - ^ , ) , conditional on the value of the random variable ^
£ [ e x p ( — / e j . We define e^^ = v^, The joint density function for u.^ and e.^ can be expressed as:
= (5.13,
By integrating equation (5.13) with respect to U.^, where > 0 , we obtain the following density function for e^^:
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(5.14)
where f i ' = - + and a ' = + a ' ) . The density
function of conditional on ^ is then as follows:
(5.15)
Equation (5.15) looks exactly the same as the expression of density function of the positive truncation of normal distribution with as its mean and a ' as variance. The expected technical efficiency expressed by equation (5.5) can be obtained by straightforward integration of equation (5.15):
TE, = e x p ( - M , ) = j e x p ( - M , ) / ( « , /e , ) d u .
- ( i i ' / g ) (5.16)
In this study, we use maximum likelihood estimation to estimate the production frontier. If /(x.^) in equation (5.3) is in Cobb-Douglas or translog form, the density function of y.^ can be obtained by substituting (y.^ for ^u in equation (5.14). The log-likelihood function is then expressed as (5.17):
Ud'-,y) = -HN. X T) Xln(27c) -i[(7V, xTX^^l + o])] - (N, x 7:)ln[l - F(-\x / c j
N T
+ (N, X 7:)ln[l - F{-ii' / G')] - ^ ^ [ ( J , , - x„py (y, -x„(3) / a,^)
- H N , X 7:)(^ / a j ^ +HN, xTM' / a ) (5.17)
where 6 ' = (p' ,aJ,a^,|Ll)', N, refers to the number of enterprises in the t^^ period, T
is the number of periods for enterprise / and therefore N^xT is total observations in the estimation.^ Following the parameterisation of Battese and Corra (1977), we fmd:
^If the panel is unbalanced in the sense that the lengths of time series differ among enterprises, then ( N ^ x T ) in equation (5.17) and equations thereafter should be replaced by , where N.^ is number of observations in the t'^ period and T is the total number of periods.
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+ (5.18.1)
Y = (5.18.2)
Equation (5.17) can be rearranged as follows:
L\Q\y) = -HN, xT,)(]n2n + \na')-(N, x7;)ln[l-F(-z)]-i(N, x T y F{-zl)]
I r
+ i i i - i i i (J. - ^.P)' (}'„ - / (1 - Y)o^ (5.19)
where
= [|Li(l - Y) - Y(X, - - y)c', (5.19.1)
e = (p^a^Y,|n)' (5.19.2)
z = [ L / 4 W (5.19.3)
The parameter 7 has interesting significance in the stochastic frontier model. By definition, it is the ratio of the variance of the truncated normal distribution to the variance of the residual term E^^ in equation (5.3) and varies between 0 and 1. If this parameter turns out to be zero or close to zero, then this means that the residual terms are dominated by the true random noise v.^. This suggests that the random variable u^^ and the full frontier model has little explanatory power over the technical efficiency of individual enterprises in the sample. Conversely, if the parameter is close to 1, then the residual is dominated by the random variable U.^, which implies that the model has strong explanatory power.
Given that both U.^ and V, are independent of X, in the stochastic production function (5.3), a system equation can be formed by taking the partial derivatives of equation (5.19) with respect to 9 and making them equal to zero (see Appendix 5.1). Although the system has an equal number of parameters and equations, it is impossible to work out a numerical solution. We therefore use the Davidon-Fletcher-Powell algorithm
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recommended by Pitt and Lee (1981) to approximate the solution to the system and calculate the maximum likelihood estimates.
Using the estimated parameters and substituting (x^-x^p) for in |J.', the technical efficiency of each enterprise in each period of time can be obtained readily by straightforward calculation using equation (5.16). To estimate above model specification in this study, I use TEALEC, a Fortran program developed by the Division of Economics in the Research School of Pacific Studies of the Australian National University.
Functional Form
Sample enterprises are divided into two industrial sectors, namely light industry and heavy industry, for which separate estimations are made. The former sector comprises
122 enterprises and the latter 174.
The following unconstrained Cobb-Douglas type of model is used in the estimation of enterprise-specific technical efficiency for the years 1980 and 1984-1988:
= a + p, ln(LJ + p, + P3 ln(£„) + p,(/y„)
2 9 5 (-5 20^ :=1 ;=5
where /=1, 2 (enterprises); r=1980, 1984,..., 1988; j=l, 2, 3 ,4 ,5 (city); and 5=2,...,S (industries). The variables y^^, L^, K^^ and E^^ represent output, and inputs of labour, capital and energy in the enterprise and the r^^ period of time. The variable / y is investment in the upgrading of technology accumulated since 1980 of the enterprise in the r*^^ period of time. D^ is a size dummy variable: 1 for large and zero for medium-sized enterprises. This variable is intended to capture the effect on productivity of scale (see chapter 2). DJ's are locality dummies for Shanghai, Wuhan, Chongqing, Guangzhou and Shenyang. D/'s are dummy variables for industries in light or heavy