Estimating alternative technology sets in nonparametric efficiency analysis: restriction tests for panel and clustered data

Abstract

Nonparametric efficiency analysis has become a widely applied technique to support industrial benchmarking as well as a variety of incentive-based regulation policies. In practice such exercises are often plagued by incomplete knowledge about the correct specifications of inputs and outputs. Simar and Wilson (Commun Stat Simul Comput 30(1):159–184, 2001) and Schubert and Simar (J Prod Anal 36(1):55–69, 2011) propose restriction tests to support such specification decisions for cross-section data. However, the typical oligopolized market structure pertinent to regulation contexts often leads to low numbers of cross-section observations, rendering reliable estimation based on these tests practically unfeasible. This small-sample problem could often be avoided with the use of panel data, which would in any case require an extension of the cross-section restriction tests to handle panel data. In this paper we derive these tests. We prove the consistency of the proposed method and apply it to a sample of US natural gas transmission companies from 2003 through 2007. We find that the total quantity of natural gas delivered and natural gas delivered in peak periods measure essentially the same output. Therefore only one needs to be included. We also show that the length of mains as a measure of transportation service is non-redundant and therefore must be included.

Appendices

Appendix 1: Proof of consistency

A robust approach to obtain corrected standard errors with clustered data is to sub-sample block-wise (Davison and Hinkley 1997). This allows for arbitrary dependence between the observations belonging to the same cross-section unit.

We show that this procedure meets the essential consistency requirements set out in Politis et al. (2001). Let sample size $n_{PD}$ be defined by the number of different cross-section observations. Although we used the more easily interpretable Farrel–Debreu measure so far, for actual calculations it is preferable to use the inverse $\lambda =1/\theta$ because it is truncated only once.

Proposition

Let $n\left( Z\right) ={\scriptstyle {\displaystyle { \sum \nolimits_{i=1}^{n_{PD}}Z_{i}}}}$ where iid random variables $Z_{i}$ give the number of time observations per cross-section unit with distribution function $F_{Z}$ defined on the support $S_{Z}=1,\ldots ,L$ and expectation $e\in \left[ 1,L\right]$, then for the test-statistic $t_{n_{PD}}\left( X,Y,Z\right)$ the asymptotic distribution of $\sqrt{{n_{PD}}}n_{PD}^{2/\left( p+q+1\right) }t_{n_{PD}}\left( X,Y,Z\right)$ is non-degenerate with expectation zero.

Proof

If we reformulate the time subscripts to take only consecutive integers, we can use the following definition:

$$\begin{aligned} t_{i}\left( X,Y\,|\, Z=z\right) =\sum _{t=1}^{z_{i}}\left( \frac{\hat{\lambda }_{full_{it}}\left( X_{i},Y_{i},\,|\, Z_{i}=z_{i}\right) }{\hat{\lambda }_{nested_{it}}\left( X_{i},Y_{i},\,|\, Z_{i}=z_{i}\right) }-1\right) . \end{aligned}$$

It follows from the results of Kneip et al. (2008) that

$$\begin{aligned} n^{2/\left( p+q+1\right) }\left( \frac{\hat{\lambda }_{full_{it}}\left( X_{i},Y_{i},\,|\, Z_{i}=z_{i}\right) }{\hat{\lambda }_{nested_{it}}\left( X_{i},Y_{i},\,|\, Z_{i}=z_{i}\right) }-1\right) \overset{d}{\rightarrow }H_{n} \end{aligned}$$

for any fixed $z_i$, where $H_{n}$ is a random variable with an asymptotic distribution function Q that is non-degenerate and has mean 0 under $H_{0}$. Furthermore we can rewrite $n=n_{PD}e$ , where e is the expectation of Z. Replacing and rearranging yields

$$\begin{aligned} n_{PD}^{2/\left( p+q+1\right) }\left( \frac{\hat{\lambda }_{full_{it}}\left( X_{i},Y_{i},\,|\, Z_{i}=z_{i}\right) }{\hat{\lambda }_{nested_{it}}\left( X_{i},Y_{i},\,|\, Z_{i}=z_{i}\right) }-1\right) \overset{d}{\rightarrow }\frac{1}{e^{2/\left( p+q+1\right) }}H_{n}. \end{aligned}$$

Since the right-hand-side is a scaled version of $H_{n}$, also

$$\begin{aligned} n_{PD}^{2/\left( p+q+1\right) }\left( \frac{\hat{\lambda }_{full_{it}}\left( X_{i},Y_{i},\,|\, Z_{i}=z\right) }{\hat{\lambda }_{nested_{it}}\left( X_{i},Y_{i},\,|\, Z_{i}=z\right) }-1\right) \end{aligned}$$

has a non-degenerate distribution. This implies that the conditional distribution of $n_{PD}^{2/\left( p+q+1\right) }t_{i}\left( X,Y|Z=z\right)$ is non-degenerate. Call this distribution D(z).

Furthermore, we obtain the distribution of $t_{i}\left( X,Y,Z\right)$ by marginalizing out Z: $D(\cdot )=\int _{z\in S_{Z}}D(z)dF_Z$. Obviously, if D(z) is non-degenerate with a given scaling factor, then $D(\cdot )$ must be non-degenerate with the same scaling factor. In order to complete the proof, since $t_{n_{PD}}\left( X,Y,Z\right)$ is an empirical mean of the $t_{i}\left( X,Y,Z\right)$, it follows by using the redefinition $n=n_{PD}c$ that $\tau _{n_{PD}}t_{n_{PD}\left( X,Y,Z\right) }$ with $\tau _{n_{PD}}=\sqrt{{n_{PD}}}n_{PD}^{2/\left( p+q+1\right) }$ is non-degenerate and additionally has an asymptotic expectation equal to zero under $H_{0}$, because the mean associated with the asymptotic distribution Q is zero. As a consequence of this result, the subsampling methods proposed by Politis et al. (2001) are consistent, when subsampling is conducted block-wise along the cross-section dimension. The sub-sampling size $m_{PD}$ is as usually defined as the integer part of $n_{PD}^{k}$ for $0<k<1$. It should be noted that these results include the case of ordinary cross-section data and a balanced panel setting. In the former case $z_{i}=1$ and $n=n_{PD}$ yielding just the formulae in Schubert and Simar (2011). In the latter case $z_{i}=L$ implying that $z_i$ cannot affect the asymptotic distribution because it is non-random.

Appendix 2: Additional test results

In this appendix we report the additional tests performed for length as single output variable. Figure 2a presents the visual analysis in which length is tested against the alternative technology set with length and deliv. At the optimal subsample size of 21 the observed test statistic of 0.468 is very close to the critical value. The corresponding p value of this test is 8 % when the panel structure is accounted for and 1 % otherwise. Panel (b) of this figure shows the visual results for testing length against length and peak. In this case the observed test statistic takes the value 0.593 and clearly exceeds the critical value at the subsample size of 23. The p value is 3 % (0 %) when the subsampling accounts (ignores) the panel structure of the data.

Consequently, we find empirical evidence that the technology set should include either deliv or peak if length is chosen as inital output variable. Therefore, these tests support the previous findings reported in Fig. 1. Also they show that the test procedure does not depend on the order of variable selection.

Fig. 2

Results of restriction tests for length as initial output variable. a Test I: output relevance of deliv. b Test II: output relevance of peak