Fixed-Effect Estimation of Technical Efficiency with Time-Invariant Fixed-Effect Estimation of Technical Efficiency with Time-Invariant Dummies Dummies

“ Within ” estimation of the fixed-effect stochastic frontier model does not identify parameters on time-invariant explanatory variables. If time-invariant variables are important production inputs, then standard efficiency estimates are biased. This note details bias correction, when time-invariant inputs are dummy variables. © 2006 Elsevier B.V. All rights reserved.


Introduction
Standard fixed-effect estimation of the stochastic frontier model for panel data was introduced by Schmidt and Sickles (1984). Since then, the specification has been implemented extensively in empirical exercises. Very often, these models have included time-invariant explanatory variables, which the within transformation eliminates. Moreover, the removed time-invariant variables may have been dummy variables. For example, Horrace and Schmidt (2000) estimate a production function for 171 Indonesian rice farms on the island of Java, that includes five dummy variables for six different villages on the island. 1 Economics Letters 95 (2007) 247 -252 www.elsevier.com/locate/econbase Their random-effects estimation identifies the village dummy coefficients, but their within estimation does not. This causes their within estimates of farm technical efficiency (unobserved heterogeneity) to be biased. In the words of Greene (2005) the fixed-effect estimator forces "any time invariant cross unit heterogeneity into the same term that is being used to capture the inefficiency." There are variety of potential solutions to this problem, but all of them effectively move away from the pure simplicity of the fixed-effect model to achieve a solution. Random-effects estimation is possible, but a parametric assumption of the distribution of technical efficiency is required. Non-linear least squares or GMM solutions (e.g., Ahn et al., 2001) add several degrees of computational complexity over within estimation. Here, we detail a simple solution to the problem that remains within the framework of within estimation. Our solution exploits the fact that technical efficiency measures are relative comparisons based on differencing; this differencing can be used to remove the bias caused by unidentified parameters on the time-invariant dummy variables. This is accomplished by making technical efficiency measurements within particular categories of the dummy variable (as opposed to across the entire sample), thereby sweeping out the common bias in each dummy variable category. This is how technical efficiency is effectively identified in this note. It should be noted that this method only applies to cases in which the number of time-invariant dummy categories is small relative to the number of firms in the sample, and it does not apply to cases where time-invariant variables are continuous or nearly continuous (countable). We revisit the empirical analysis of Schmidt (1996, 2000), and show that although their estimates are biased, their technical efficiency ranks are accurate, when compared to the ranks from the techniques detailed herein. We demonstrate theoretically that this is not an artifact of the data.

Fixed-effect estimation with dummies
The standard fixed-effect stochastic frontier model with a set of time-invariant dummies is: where Y and X are a single output and multiple inputs (respectively) of a common production function, parameterized by α, β and δ; α i = η − u i is a fixed-effect parameter that captures time-invariant technical inefficiency u i ≥ 0 (η is the intercept parameter); and ν it is an iid zero-mean error term with constant conditional variance, uncorrelated with X.
representing J categories, such that only one element of D i equals 1 (a complete partition of the data). In our example of Indonesian rice farms, D i represents 5 dummies for six different villages on the island of Java. The within transformation is: Ordinary least squares provides an unbiased and consistent estimate β. Then inconsistent estimates of α i are:â It turns out that: plim so if the i th farm is in the j th village, then the probability limit of α i is α i + δ j . This is often reported as a consistent estimate of α i . 2 Therefore, any attempt to estimate relative inefficiency u i asû i ¼ max k¼1; N ;nâk −â i will necessarily be inconsistent, as will the usual normalization of technical efficiency, TE i = exp{−û i } ∈ [1,0). We can, however, consistently estimate relative technical efficiency within each of the J categories. Let N = {1,…, n}, and partition N into, N j , j = 1,…, J, so that N j ⊂ N, and N ¼ [ J j¼1 N j . Then a consistent estimate is: for all iaN j ; and for all j ¼ 1; N ; J : This estimate "differences out" the bias, δ j . Furthermore, a consistent normalization of relative technical efficiency is: Therefore, relative technical efficiency across all farms is not identified. However, it is identified, if we simply compare farms within each village.
Empirical implementations typically report the technical efficiency scores, TE i 1 = 1,…, N, in a relative rank statistic, with a single efficient firm having an efficiency score equal to unity. Analogously, we report J rank statistics on TE i ⁎ i ∈ N j , for all j = 1,…, J, so the technique produces J firms having efficiency score equal to unity. It turns out that the within category ranks of the TE i i ∈ N j , are identical to the ranks of TE i ⁎ i ∈ N j . To see this, we first note that the exponent operator, exp{.}, is monotonic in its argument, so we need only compare the ranks of the arguments, û i and û i ⁎ . Now, consider some farm s ∈ N j , then The equivalence of these expressions implies that the rank of s ∈ N j , relative to i ∈ N j , remains the same regardless of which technique is employed.

Indonesian rice farms
This particular data set of 171 Indonesian rice farms over six growing seasons has been analyzed a number of times, starting with Erwidodo (1990) and, most recently with Ahn et al. (2005). See Horrace and Schmidt (2000) for a description of the data. The estimation results of the fixed-effect regression of rice production on the explanatory variables (land, labor, fertilizer, variety, pesticide, season, and a village dummy) are contained in Horrace and Schmidt (1996 , Table 3). (We do not present the results of the within estimation here.) The only time-invariant regressor is the village dummy, so its coefficient is not identified, creating a bias in the estimate TE i . Ranked estimation results for biased TE i and unbiased TE i ⁎ are presented in Table 1 for villages 1, 2, and 3 and in Table 2 for villages 4, 5, and 6. As expected, the within village ranks are the same for either TE i or TE i ⁎ . The tables highlight the magnitude differences between the two estimators of efficiency; these are presumably due to differences in bias. Also as expected, there is a single efficient farm in each village as measured by TE i ⁎ = 1.000, but only one as measured by TE i , which is in village 6 ( Table 2). Notice that in the last two columns of Table 2, the efficiency scores, TE i and TE i ⁎ , are identical for village 6. This is because the most efficient farm with TE i = 1.000 happened to be in this village (top of the second to last column). This is not the case for the farms in other villages, where TE i is always less than unity. Since the most efficient farm based on TE i is in the village 6, the biased TE i is equal to unbiased TE i ⁎ . This is not the case for the farms in other villages.