Estimating Technical Efficiency in Micro Panels Estimating Technical Efficiency in Micro Panels

Bootstrap confidence intervals on fixed-effects efficiency estimates in micro panels exhibit low coverage probabilities. We propose an alternative efficiency measure involving the mean of the firm effects. With the same estimated efficiency ranks as the traditional measure, its corresponding bootstrap confidence intervals have better coverage probabilities.


Introduction
In the panel stochastic frontier literature, fixed-effects estimation (Schmidt and Sickles, 1984) has the advantage of not requiring distributional assumptions on the error components. Consequently, the model is often employed in empirical research (e.g., Kumbhakar, 1987 andBauer andHancock, 1993). Confidence intervals for technical efficiency may be based on asymptotic normality, when the sample size is large (Horrace and Schmidt, 1996). However, in micro panels with small time dimensions, the preferred method of constructing confidence intervals without distributional assumptions is to perform the bootstrap. Simar (1992), Simar and Wilson (1998), Hall et al. (1993), and Kim et al. (2007) investigate confidence interval coverage rates for various bootstrap techniques based on the fixed-effects estimation, and find that the coverage is generally poor when the time dimension of the data is small. As discussed in Kim et al. (2007), the coverage erosion is due to the bias induced by the estimated maximum of the firm effects involved in the traditional efficiency estimate (Schmidt and Sickles, 1984). The received literature encourages techniques that improve coverage rates by accounting for this bias. However, the improved coverage of bootstrap confidence intervals is still not satisfactory in micro panels with small time dimensions and large numbers of firms. Even worse, when ''ties'' among firms occur (i.e., several firms are equally best) no bootstrap confidence intervals are reliable (Kim et al., 2007). Recently, Satchachai and Schmidt (2010) show that removing the ''max'' operator bias is possible using the generalized panel jackknife, but the resulting estimate has a large variance.
We approach this inference problem from a different perspective. The traditional efficiency measure defined in Schmidt and Sickles (1984) shifts the firm-specific effects by their maximum. We propose an alternative measure which shifts the firm effects by their mean. This efficiency measure yields the same efficiency rank of firms as the traditional measure. However, unlike the estimated maximum of firm effects, the estimated mean is unbiased, so the corresponding bootstrap confidence intervals for the new measures have better coverage probabilities. Thus, when the inference based on the traditional efficiency measure is inaccurate in micro panels or in the case that the identity of the best firm is uncertain, the proposed measure can be used as an alternative and reliable way to estimate firms' technical efficiencies. It should be made clear at the outset that the proposed measure is not a substitute for the traditional measure; it is an additional metric that should be a part of the empiricist's toolbox.

Fixed-effects efficiency measures
In a stochastic frontier model for panel data, (1) The technical inefficiency of firm i is characterized by the time- . Usually, technical efficiency is defined as r i = exp(−u i ) under a logarithmic specification of the Cobb-Douglas production function. The slope parameter β can be estimated consistently using fixed-effects estimation. Call this estimate  β. The firm-specific production level (or firm effect) where y i and x i are withingroup averages. Since u i cannot be identified without additional assumptions, Schmidt and Sickles (1984) suggest the relative efficiency measures, with fixed-effects estimates, Per Schmidt and Sickles (1984),  β converges to β for large N or T , while  α i converges to α i for large T only. Therefore, in micro panels with small T , asymptotic approximations for confidence intervals on functions of α i are inappropriate, and a bootstrap method should be employed. Kim et al. (2007) provide a detailed discussion of methods for bootstrap confidence intervals for technical efficiency r * i = exp(α i − max j α j ), including percentile intervals, bias-corrected with acceleration (BC a ) intervals, bias-corrected (BC ) percentile intervals and a parametric method. Overall, BC a and BC percentile seem preferred considering the coverage and width of the intervals. However, in their simulation study (Kim et al., 2007, Table 3), even the BC a intervals and BC percentile intervals have low coverage rates in the usual panel case. For example, the coverage rate of the BC a intervals with nominal 90% significance is only 68.7% when N = 100, T = 10, γ = 0.1. 3 The corresponding BC percentile intervals have slightly higher coverage than BC a intervals, but with bigger width. As we will see, the coverage rate is even lower when T < 10 and N > 100. The problem arises from the ''max'' operators in  u * i and  r * i . Per Kim et al. (2007), the ''max'' operator induces an upward bias in  α and  u * i , and hence a downward bias in  r * i .
The ''max'' operator favors positive noise. When T is large,  α j → α j , max j  α j → max j α j , and the ''max'' operator bias is mitigated.
However, this bias cannot be ignored in micro panels with small T and large N, which is the focus of this paper. Since the ''max'' operator bias is the main source of the inaccurate inference for technical efficiency in micro panels, removing the bias could be a solution. A successful attempt of bias correction is made by Satchachai and Schmidt (2010) using the generalized panel jackknife, but the resulting estimate has a large variance. Alternatively, Feng and Horrace (2012) propose two efficiency measures involving min j α j , whose corresponding fixed-effects estimates preserve the same ranking as r * i . Although the bias issue persists for the fixed-effects estimates of their two measures, they show that bias is mitigated in certain situations related to the skew of the efficiency distribution.
2 The relationship between inefficiency and unobserved heterogeneity is not considered here. For a detailed discussion of this issue, see Greene (2004Greene ( , 2005. 3 γ is defined as the variance ratio Var(u)/(σ 2 v + Var(u)). A small γ means that the main source of variation of the composite error −u i + v i comes from the noise v i . This paper approaches this inference problem from a different perspective. Since the sample ranking of estimators r * , it seems possible to construct an alternative efficiency measure containing the same information as r * i but without the ''max'' operator bias and its related issues. The traditional measure r * i is a monotonic transformation of α i − max j α j , a shift of α i by max j α j . Using the maximum can be regarded as a normalization to produce efficiency 1 for the firm with the largest α i in the sample. Similar in spirit to r * i , we propose an efficiency measure, where h is a monotonically increasing transformation. E i has the same efficiency rank of firms as r * i in the sample. Unlike r * i , our measure E i shifts α i by the mean 1 In general, E i can be arbitrarily restricted to (0, 1), so that efficiency can be expressed as a percentage. 4 There are an unlimited number of functions one might use. For example, we consider the function form of standard normal cumulative distribution function Φ, with estimate The advantage of using a symmetric and smooth cumulative distribution function is that E i = 0.5 is a meaningful reference point corresponding to a firm with the mean value of α i . Alternative distributions (e.g., Cauchy, student, logistic) can be used for this purpose.
As discussed above, the estimated maximum, max jαj , is biased. This is the main source of the coverage erosion of r * i 's bootstrap confidence intervals. By contrast, the estimated mean 1 N  jα j is unbiased as the estimated firm effects  α i are unbiased. Since  α i − 1 N  jα j is unbiased, the percentile bootstrap confidence intervals for α i − 1 N  j α j have no coverage erosion and thus have high coverage even in micro panels. According to Efron and Tibshirani (1993), the percentile bootstrap confidence intervals are transformation respecting, implying that the percentile intervals for E i will have the same coverage rate as those for α i − 1 Therefore, we can always construct percentile intervals for E i by transforming the intervals for α i − 1 N  j α j . Hence, simple bootstrap confidence intervals like the percentile bootstrap are enough for accurate inference. For empiricists, inference on E i will be considerably easier to implement than inference on r * i . Moreover, the bootstrap confidence intervals for E i are more reliable than those for the traditional measure r * i , no matter how many firms there are, and whether or not the identity of the best firm is in serious doubt. E i = 0.5 for the firm with mean value of α i in (2), so we call the firm with E i = 0.5 the mean firm. The efficiencies of all firms 4 The usual measure of technical efficiency, r i = exp(−u i ), implies that the production function is Cobb-Douglas. However, if the true data-generating process is not Cobb-Douglas, then r i = exp(−u i ) is not technical efficiency, per se, but merely an indicator of technical efficiency. The proposed measure is simply another such indicator. are distributed around the mean firm. The efficiency measures are above 0.5 for the firms with α i > 1 N  j α j and below 0.5 for α i < 1 N  j α j . Rather than using the maximum as the reference point, the measure E i uses the mean firm as the reference point, providing an interesting view for the firms' efficiencies. The reference point  E i = 0.5 divides firms into two groupings: ''efficient'' firms with  E i above 0.5 and ''inefficient'' firms with  E i below 0.5. If the price of products is related to the cost of the mean firm in a relatively new industry, then the mean firm may have zero profit. Accordingly, the ''efficient'' firms may have positive profit, while the ''inefficient'' firms may earn negative profits and be inclined to leave the market.

Simulations and empirical example
To better understand the accuracy of inference for the traditional measure r * i = exp(α i − max j α j ) and our proposed mea- to compare the coverage rate of bootstrap confidence intervals in micro panels. The data-generating process used here follows the procedure in Kim et al. (2007): . We set α = 1 and define γ = Var(u)/(σ 2 v + Var(u)). Small γ indicates a particularly noisy experiment. A large γ (close to 1) means there is little variation in the idiosyncratic error v it . The Monte Carlo parameters are (γ , N, T ). When T is large, the bias issue of traditional efficiency estimate is mitigated. Here, we consider the interesting case when the inference of the traditional measure r * i is inaccurate in micro panels with T = 4, 6, 10. Experiments are conducted with N = 20, 50, 100 and 150, and γ = 0.1, 0.3 and 0.5, with 1000 bootstrap replications. We repeat the Monte Carlo experiments 1000 times. Kim et al. (2007) show that the BC a bootstrap confidence intervals and BC percentile intervals for r * i are better than other intervals, so we report these two intervals. The percentile intervals are also presented for comparison. Since  α i − 1 N  j  α j is unbiased, only percentile intervals for E i are presented. Simulated 90% nominal coverage rates for intervals for r * i and E i are in Table 1. The coverage of percentile intervals for r * i declines dramatically in N and is, therefore, unreliable for panels with small T and large N. For example, the coverage of the 90% nominal percentile interval (column 4) is 24.6% for T = 6, N = 150 and γ = 0.5. As expected, the coverage rates for both BC a (column 6) and the bias-correct percentile (column 8) intervals improve, but are still far from satisfactory. The coverage rates of BC a and the BC percentile intervals for r * i are 41.0% and 68.5% respectively for T = 6, N = 150 and γ = 0.5. 81% and 85% for T = 4, 6 and 10, respectively. Therefore, based on coverage rates, the confidence intervals for E i are much more informative and robust than those for r * i . A quintessential example of a micro panel in the stochastic frontier literature is the Indonesian rice farm data set with N = 171 and T = 6. See Schmidt (1996, 2000) for a detailed description. Table 2 contains the point estimates and bootstrap intervals for r * i and E i . We report results for the best farm, the 99th, 75th, median, 25th, 1th quantile, and the worst farm in rank order. The first column is the farm number. The third column contains the estimates of the firm effects α i = α − u i . It is interesting to Note: r * 's BC a and BC percentile bootstrap confidence intervals are obtained using the indirect method in Kim et al. (2007). note that the point estimator  E i (column 5) is similar in magnitude to r * i (column 4) at the bottom end of the order statistic. Compare 0.336-0.366 for farm 45 and 0.349-0.379 for farm 117. If one is interested in inference on the least efficient farms in the sample, thenÊ i gives point estimates similar to r * i but with higher coverage rates (but for different measures). Fig. 1 contains the graphs of  r *