Explaining inefficiency in nonparametric production models: the state of the art

Abstract

The performance of economic producers is often affected by external or environmental factors that, unlike the inputs and the outputs, are not under the control of the Decision Making Units (DMUs). These factors can be included in the model as exogenous variables and can help to explain the efficiency differentials, as well as improve the managerial policy of the evaluated units. A fully nonparametric methodology, which includes external variables in the frontier model and defines conditional DEA and FDH efficiency scores, is now available for investigating the impact of external-environmental factors on the performance.

In this paper, we offer a state-of-the-art review of the literature, which has been proposed to include environmental variables in nonparametric and robust (to outliers) frontier models and to analyse and interpret the conditional efficiency scores, capturing their impact on the attainable set and/or on the distribution of the inefficiency scores. This paper develops and complements the approach of Bădin et al. (2012) by suggesting a procedure that allows us to make local inference and provide confidence intervals for the impact of the external factors on the process. We advocate for the nonparametric conditional methodology, which avoids the restrictive “separability” assumption required by the two-stage approaches in order to provide meaningful results. An illustration with real data on mutual funds shows the usefulness of the proposed approach.

Appendix: The bootstrap algorithm

The bootstrap algorithm can be described by the following steps:

• [1] Based on the sample ${\mathcal{S}}_{n} =\{(X_{i},Y_{i},Z_{i})|\; i=1,\ldots ,n\}$ compute the n efficiency scores $\widehat{\lambda}(X_{i}, Y_{i})$ and the conditional efficiency scores $\widehat{\lambda}(X_{i}, Y_{i}|Z_{i})$. For the conditional efficiency scores, compute the optimal bandwidth h _n,i , attached to the ith observation, via the LSCV procedure proposed in Bădin et al. (2010). Compute the n ratios $\widehat{R}(X_{i}, Y_{i}|Z_{i})$.

• [2] Select a fixed grid of values for Z, say {z ₁,…,z _k } to evaluate the regression. We compute the nonparametric regression by one of the methods described in (4.21): this provides $\hat{\tau}^{z_{j}}_{n}$ for j=1,…,k. The bandwidth $h_{n}^{z}$ is selected by least-squares crossvalidation.

• [3] For a given value of m<n and a large B (e.g. B=2000), repeat steps [3.1] to [3.3] for b=1,…,B.

  • [3.1] Draw a random sample ${\mathcal{S}}^{*}_{m,b}=\{ (X^{*,b}_{i},Y^{*,b}_{i},Z^{*,b}_{i})|\;i=1,\ldots,m\}$ without replacement from ${\mathcal{S}}_{n}$. By doing so, we keep also the value of the bandwidth $h^{*,b}_{n,i}$ computed at step [1] attached to the corresponding selected data $(X^{*,b}_{i},Y^{*,b}_{i},Z^{*,b}_{i})$.

  • [3.2] Compute the m ratios $\widehat{R}^{*,b}(X^{*,b}_{i}, Y^{*,b}_{i}|Z^{*,b}_{i})$, i=1,…,m by the same techniques as in [1]. Note that here we have to rescale the corresponding bandwidths $h^{*,b}_{n,i}$ at the appropriate size, so we use the bandwidths $h^{*,b}_{m,i} = (n/m)^{1/(r+4)} h^{*,b}_{n,i}$ for computing the conditional scores in the bootstrap sample ${\mathcal{S}}^{*}_{m,b}$.

  • [3.3] By the same nonparametric method as in [2], estimate the regressions $\hat{\tau}^{*,b,z_{j}}_{m}$ at the fixed points z _j , for j=1,…,k. One can use here the same bandwidth computed in [2], but rescaled to the appropriate size.¹³ So we use here $h_{m}^{z}=(n/m)^{1/(r+4)} h_{n}^{z}$ and obtain $\hat{\tau}^{*,b,z_{j}}_{m}$ for j=1,…,k.

• [4] For each j=1,…,k, compute $(q^{*,z_{j}}_{m;\alpha /2},q^{*,z_{j}}_{m;1-\alpha/2})$, the α/2 and 1−α/2 quantiles of the B bootstrapped values of $\hat{\tau}^{*,b,z_{j}}_{m} -\hat{\tau}^{z_{j}}_{n}$. This provides the k confidence intervals of $\tau^{z_{j}}(P)$ at each fixed z _j :

  

  (A.1)

The selection of m is done as follows. We redo the steps [3] to [4] over a grid of L values of m, say, m ₁<m ₂<⋯<m _L and we obtain for each m _ℓ , the k resulting confidence intervals (A.1).¹⁴ Then we compute the volatility of the quantity of interest seen as a function of m. Here the two bounds of the confidence intervals (A.1) are of the quantities of interest, Politis et al. (2001) suggest in this case to take $c^{z_{j}}(m) = (1/2)[\mathrm{low}_{m}^{z_{j}} + \mathrm{up}_{m}^{z_{j}}]$, where the notation is implicit. The volatility is measured by the “moving” standard deviation of 3 adjacent values of $c^{z_{j}}(m)$ centred at the current value of m _ℓ , ℓ=2,…,L−1. As explained in Politis et al. (2001), a reasonable value for $m^{z_{j}}$ should correspond to the value that minimises this volatility. Intensive Monte Carlo experiments in Simar and Wilson (2011a) and Daraio et al. (2010), in similar setups of nonparametric frontier estimation, indicate that this procedure provides very good results in terms of coverage, size of tests, power of tests, etc.

A simpler alternative is to select a common value of m for the different values of z _j . Is possible, for instance, to select the m equal to the average of all the m ^z. One could also use the same approach as above, but then, the volatility would be measured on an average value $c(m)= (1/k) \sum_{j} c^{z_{j}}(m)$. This approach could provide a more stable behaviour of c(m) as a function of m.