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Inference for DEA estimators of malmquist productivity indices: an overview, further improvements, and a guide for practitioners

Published online by Cambridge University Press:  13 March 2025

Valentin Zelenyuk
Affiliation:
School of Economics and Centre for Efficiency and Productivity Analysis (CEPA), University of Queensland, Brisbane, QLD, Australia
Shirong Zhao*
Affiliation:
School of Finance, Dongbei University of Finance and Economics, Dalian, China
*
Corresponding author: Shirong Zhao; Email: shironz@163.com
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Abstract

Rigorous methods have recently been developed for statistical inference of Malmquist productivity indices (MPIs) in the context of nonparametric frontier estimation, including the new central limit theorems, estimation of the bias, standard errors and the corresponding confidence intervals. The goal of this study is to briefly overview these methods and consider a few possible improvements of their implementation in relatively small samples. Our Monte-Carlo simulations confirmed that the method from Simar et al. (2023) is useful for the simple mean and aggregate MPI in relatively small sample sizes (e.g., up to around 50) and especially for large dimensions. Interestingly, we also find that the “data sharpening” method from Nguyen et al. (2022), which helps in improving the approximation in the context of efficiency is not needed in the context of estimation of productivity indices. Finally, we provide an empirical illustration of the differences across the existing methods.

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Articles
Copyright
© The Author(s), 2025. Published by Cambridge University Press

1. Introduction

The Malmquist productivity index (MPI) (Caves et al. Reference Caves, Christensen and Diewert1982) typically estimated using data envelopment analysis (DEA) is often employed to measure the productivity change of firms over time. For some examples of applications of MPI, see Färe et al. (Reference Färe, Grosskopf, Norris and Zhang1994), Ray and Desli (Reference Ray and Desli1997), Kumar and Russell (Reference Kumar and Russell2002), Casu et al. (Reference Casu, Ferrari and Zhao2013), Ramakrishna et al. (Reference Ramakrishna, Ramulu and Kumar2016), Kevork et al. (Reference Kevork, Pange, Tzeremes and Tzeremes2017), Pastor et al. (Reference Pastor, Lovell and Aparicio2020), Simar and Wilson (Reference Simar and Wilson2023), to name a few.Footnote 1

Recently, based on the seminal work of Kneip et al. (Reference Kneip, Simar and Wilson2015) for technical efficiency, Kneip et al. (Reference Kneip, Simar and Wilson2021) established theoretical results for the individual MPI and geometric mean of MPIs, and also for these two measures in log terms. Based on Kneip et al. (Reference Kneip, Simar and Wilson2015, Reference Kneip, Simar and Wilson2021) and Simar and Zelenyuk (Reference Simar and Zelenyuk2018), Pham et al. (Reference Pham, Simar and Zelenyuk2024) further established the theoretical results for the weighted harmonic-type mean aggregation of MPIs, by taking the economic weight of each individual into account. These recent theoretical frameworks of MPI (developed by Kneip et al. Reference Kneip, Simar and Wilson2021 and Pham et al. Reference Pham, Simar and Zelenyuk2024), have enabled researchers and policy makers for the first time to make theoretically well-grounded statistical inferences on the DEA-estimated productivity changes for a group of firms from various interesting economic questions.

However, it is observed from the Monte-Carlo (MC) simulations that for the simple mean of log MPIs (as evident in our MC results in Section 4) and the aggregate of log MPIs (Pham et al. Reference Pham, Simar and Zelenyuk2024), the confidence intervals constructed using the developed central limit theorems (CLTs) typically under-cover the true values in relatively small sample sizes, especially in large dimensions. This under-covering phenomenon is also well observed in the other related nonparametric frontier efficiency estimators, such as the simple mean (or unweighted) efficiency (Kneip et al. Reference Kneip, Simar and Wilson2015) and the aggregate (or weighted) efficiency (Simar and Zelenyuk, Reference Simar and Zelenyuk2018). This under-covering phenomenon mainly comes from the remaining bias in the estimation of the first and second moments. It is also amplified by the widely known “curse of dimensionality” problem that the nonparametric methods typically suffer from, which states that the estimation errors are larger in finite sample sizes and in large dimensions, due to the slower convergence rates of the nonparametric methods compared to parametric methods.

To improve the finite sample approximation of these nonparametric frontier efficiency estimators, Simar and Zelenyuk (Reference Simar and Zelenyuk2020), Nguyen et al. (Reference Nguyen, Simar and Zelenyuk2022), Simar et al. (Reference Simar, Zelenyuk and Zhao2023, Reference Simar, Zelenyuk and Zhao2024) have proposed various improving methods for the simple mean and aggregate (both input-oriented and output-oriented) efficiency. Their MC results suggest that the data sharpening method (Nguyen et al. Reference Nguyen, Simar and Zelenyuk2022) and the method of obtaining variance estimates through the bias-corrected individual efficiency estimates (Simar et al. Reference Simar, Zelenyuk and Zhao2023) are particularly useful in terms of larger coverages of the true values while also maintaining the developed CLTs for the simple mean and aggregate efficiency.

In this paper, our main objective is to examine whether the improving methods in Nguyen et al. (Reference Nguyen, Simar and Zelenyuk2022) and Simar et al. (Reference Simar, Zelenyuk and Zhao2023) developed specifically for the context of the simple mean and aggregate efficiency, are also useful to improve the finite sample approximation of CLTs for the simple mean and aggregate MPI, established by Kneip et al. (Reference Kneip, Simar and Wilson2021) and Pham et al. (Reference Pham, Simar and Zelenyuk2024), respectively. Through Monte-Carlo simulations, we find that the method adapted from Simar et al. (Reference Simar, Zelenyuk and Zhao2023) could provide a better performance for the simple mean and aggregate MPI for relatively small samples (e.g., up to around $50$ , perhaps $100$ ) and after that the original methods from Kneip et al. (Reference Kneip, Simar and Wilson2021) and Pham et al. (Reference Pham, Simar and Zelenyuk2024) are recommended.

Moreover, we find that the better performance of the data sharpening method (proposed by Nguyen et al. Reference Nguyen, Simar and Zelenyuk2022) observed in the simple mean and aggregate efficiency by Nguyen et al. (Reference Nguyen, Simar and Zelenyuk2022), and Simar et al. (Reference Simar, Zelenyuk and Zhao2023, Reference Simar, Zelenyuk and Zhao2024), is not observed in the simple mean and aggregate MPI, as the MPI estimates involve the ratios of technical efficiency estimates and so the magnitude of the bias seems to be partially cancelled out in practice. Since our Monte-Carlo evidence shows that the data sharpening method cannot provide significant improvements for MPIs, we suggest not to use it in the context of MPIs, as it could potentially bring some adverse effects without significant gains.Footnote 2 The disadvantages of using the data sharpening method might include: (1) it adds more complexity for practitioners; (2) it might be context-dependent, which means the tuning parameter in the data sharpening method could dependent on the functional form of the production technology, the distribution of inefficiency, the dimensions of inputs and outputs, etc.; (3) it could cause some potential distortions when applied to MPIs.

Finally, using the well-known Penn World Table data set from 1990 to 2019 as an example, we illustrate the differences of all these methods in the estimated standard deviations and the significant levels for the simple mean and aggregate MPI for the same data set as Pham et al. (Reference Pham, Simar and Zelenyuk2024), where we consider for the whole 84 countries, 27 developed and 57 developing countries (separately), and for pairs of years at 5-year intervals and the overall period 1990–2019.

We organize the rest of the paper as follows. In Section 2, we introduce the theoretical background on technical efficiency, individual MPI, the simple mean and aggregate MPI, and the estimators of these measures. Section 3 discusses how to adapt the improving methods from Nguyen et al. (Reference Nguyen, Simar and Zelenyuk2022) and Simar et al. (Reference Simar, Zelenyuk and Zhao2023) to the context of the simple mean and aggregate MPI. Section 4 performs extensive MC experiments to examine whether these improving methods are effective. In Section 5, we use one real empirical data set to illustrate the differences of these methods in the estimated standard deviations and the significance levels. We conclude in Section 6. Additional results can be found in the Supplementary Document. Specifically, Appendix A briefly summarizes the main theoretical results for the simple mean and aggregate MPI established by Kneip et al. (Reference Kneip, Simar and Wilson2021) and Pham et al. (Reference Pham, Simar and Zelenyuk2024), respectively. The inference and improvement methods for the geometric means of MPI are provided in Appendix B. Additional simulation results are available in Appdices C–D.

2. The theoretical background

2.1 The Production Economics Model

Denote $x\in {\mathbb {R}}_+^p$ and $y\in {\mathbb {R}}_+^q$ column vectors as inputs and outputs, respectively. The typical production set is

(2.1) \begin{equation} \Psi ^t = \{(x,y)\mid x\text { can produce } y \text { at time } t\}, \end{equation}

which describes the set of physically attainable points $(x,y)$ in the relevant input-output space at time $t$ . We impose the common regularity assumptions on the production set $\Psi ^t$ , which are described in Appendix A in Pham et al. (Reference Pham, Simar and Zelenyuk2024). The upper boundary of $\Psi ^t$ , is called the frontier, which is given by

(2.2) \begin{equation} \Psi ^{t\partial }\,:\!=\left \{ (x,y)\mid (x,y)\in \Psi ^t,\;(x/\gamma, \gamma y)\notin \Psi ^t \text { for any } \gamma \in (1,\infty )\right \}. \end{equation}

The technical efficiency for a particular point $(x, y)$ is computed by the distance between the point of $(x, y)$ in $\Psi ^t$ and the technology $\Psi ^{t\partial }$ . Technical efficiency can be computed in various orientations or directions. The output-oriented efficiency measure (Farrell, Reference Farrell1957) is

(2.3) \begin{equation} \lambda (x,y \ | \ \Psi ^t)\,:\!= \mathrm {sup}\{ \lambda \ | \ (x, \lambda y) \in \Psi ^t \}, \end{equation}

which gives the maximal proportion by which all outputs can be increased, while holding the inputs and technology fixed. For simplicity, in this paper, we focus on the MPI in the output orientation. The results can be extended to the other directions.

The conical closure of the production set $\Psi ^t$ is given by

(2.4) \begin{equation} \mathcal{C}(\Psi ^t)\,:\!=\left \{ (\widetilde {x}, \widetilde {y}) \mid \widetilde {x}=ax, \ \widetilde {y}=ay, \ \forall \ a \in {\mathbb {R}}_+^1, \forall \ (x,y) \in \Psi ^t \right \}. \end{equation}

If $\Psi ^{t\partial }$ exhibits globally constant returns to scale (CRS), then $\mathcal{C}(\Psi ^t)=\Psi ^t$ ; otherwise, $\Psi ^t \subset \mathcal{C}(\Psi ^t)$ . The corresponding conical Farrell output-oriented efficiency measure is given by

(2.5) \begin{equation} \lambda _C(x,y \ | \ \Psi ^t)\,:\!=\lambda (x,y \ | \ \mathcal{C}(\Psi ^t))= \mathrm {sup}\{ \lambda \ | \ (x, \lambda y) \in \mathcal{C}(\Psi ^t) \} . \end{equation}

2.2 The simple mean and aggregate MPI

Now consider a sample ${\mathcal S}_n=\{(X_i^1,Y_i^1), (X_i^2,Y_i^2)\}_{i=1}^n$ of input-output combinations for $n$ firms observed in periods $t=1$ and $2$ . To simplify the notation, let $Z_i^1=(X_i^1,Y_i^1)$ , $Z_i^2=(X_i^2,Y_i^2)$ , ${\mathcal S}_n^1=\{Z_i^1\}_{i=1}^n$ and ${\mathcal S}_n^2=\{Z_i^2\}_{i=1}^n$ . We assume each firm $i$ has access to the technology $\Psi ^{t\partial }$ in period $t$ , although potentially not efficient with respect to this technology. Following Caves et al. (Reference Caves, Christensen and Diewert1982), the productivity change for firm $i$ from $t=1$ to $t=2$ can be defined as

(2.6) \begin{equation} \mathcal{M}_i\,:\!= \Bigg ( \frac {\lambda _C(Z_i^2 \ | \ \Psi ^1)}{\lambda _C(Z_i^1 \ | \ \Psi ^1)} \times \frac {\lambda _C(Z_i^2 \ | \ \Psi ^2)}{\lambda _C(Z_i^1 \ | \ \Psi ^2)} \Bigg )^{-1/2}. \end{equation}

Clearly, $\mathcal{M}_i\gt 1,\,=1, \text {or} \ \lt 1$ , indicates the $i$ th firm’ productivity has increased, remained unchanged or decreased from $t=1$ to $t=2$ .

In addition to estimating the productivity change for individual firms, applied researchers often are interested in whether the productivity change for a group, such as the geometric means of the individual productivity change,

(2.7) \begin{equation} \mathcal{M} \,:\!= \Bigg ( \prod _{i=1}^{n} \mathcal{M}_i \Bigg )^{1/n}, \end{equation}

is significantly greater or less than 1. Note that $\mathcal{M}$ uses the equally weighted geometric mean. Now consider the log MPI for individual firm as

(2.8) \begin{equation} \begin{split} \log \mathcal{M}_i = -\frac {1}{2} \Big [ & \log \lambda _C(Z_i^2 \ | \ \Psi ^1) + \log \lambda _C(Z_i^2 \ | \ \Psi ^2) \\ & -\log \lambda _C(Z_i^1 \ | \ \Psi ^1) - \log \lambda _C(Z_i^1 \ | \ \Psi ^2) \Big ], \end{split} \end{equation}

and denote the mean value as

(2.9) \begin{equation} \mu _\mathcal{M}\,:\!=E(\log \mathcal{M}_i) . \end{equation}

The log MPI for a group of firms is

(2.10) \begin{equation} \mu _{\mathcal{M},n}\,:\!=\log \mathcal{M}=\frac {1}{n} \sum _{i=1}^n \log \mathcal{M}_i, \end{equation}

which is an estimate of $\mu _\mathcal{M}$ if the true value for individual MPI, $\mathcal{M}_i$ , is known.

Another alternative is to take individual economic importance (such as the revenues) into account and consider the aggregate MPI (Zelenyuk, Reference Zelenyuk2006), defined as

(2.11) \begin{equation} \overline {M}\,:\!= \Bigg ( \frac {\sum _{i=1}^n \beta _i^2 \lambda _C(Z_i^2 \ | \ \Psi ^1)}{\sum _{i=1}^n \beta _i^1 \lambda _C(Z_i^1 \ | \ \Psi ^1)} \times \frac {\sum _{i=1}^n \beta _i^2 \lambda _C(Z_i^2 \ | \ \Psi ^2)}{\sum _{i=1}^n \beta _i^1 \lambda _C(Z_i^1 \ | \ \Psi ^2)} \Bigg )^{-1/2}, \end{equation}

where

(2.12) \begin{equation} \beta _i^t = \frac {w^tY_i^t}{\sum _{i=1}^n w^tY_i^t}, \end{equation}

is the revenue weight for firm $i$ in period $t$ , and $w^t \in {\mathbb {R}}_{++}^q$ is the row vector of output prices, assumed to be the same for different firms in the same period $t$ .

Now, the log version of $\overline {M}$ can be defined as

(2.13) \begin{equation} \begin{split} \xi _n = \log \overline {M} = & -\frac {1}{2} \Bigg [ \log \Big (\sum _{i=1}^n \beta _i^2 \lambda _C(Z_i^2 \ | \ \Psi ^1)\Big ) + \log \Big (\sum _{i=1}^n \beta _i^2 \lambda _C(Z_i^2 \ | \ \Psi ^2)\Big ) \\ & - \log \Big (\sum _{i=1}^n \beta _i^1 \lambda _C(Z_i^1 \ | \ \Psi ^1) \Big ) - \log \Big ( \sum _{i=1}^n \beta _i^1 \lambda _C(Z_i^1 \ | \ \Psi ^2) \Big ) \Bigg ] \\ =& -\frac {1}{2} \Bigg [ \log \Big (\sum _{i=1}^n \lambda _C(Z_i^2 \ | \ \Psi ^1)w^2Y_i^2\Big ) + \log \Big (\sum _{i=1}^n \lambda _C(Z_i^2 \ | \ \Psi ^2)w^2Y_i^2\Big ) \\ & - \log \Big (\sum _{i=1}^n \lambda _C(Z_i^1 \ | \ \Psi ^1)w^1Y_i^1 \Big ) - \log \Big ( \sum _{i=1}^n \lambda _C(Z_i^1 \ | \ \Psi ^2)w^1Y_i^1 \Big ) \Bigg ] \\ & + \log \Big (\sum _{i=1}^n w^2Y_i^2 \Big ) - \log \Big ( \sum _{i=1}^n w^1Y_i^1 \Big ) . \end{split} \end{equation}

As shown by Pham et al. (Reference Pham, Simar and Zelenyuk2024), $\xi _n$ is a consistent estimate of

(2.14) \begin{equation} \xi = -\frac {1}{2} (\log \mu _1+ \log \mu _2 - \log \mu _3 - \log \mu _4) + \log \mu _5 - \log \mu _6, \end{equation}

where $\mu _s=E(U_{s,i})$ , $s=1,2,\ldots, 6$ , and

(2.15) \begin{equation} \begin{split} U_{1,i} & = \lambda _C(Z_i^2 \ | \ \Psi ^1)w^2Y_i^2, \\ U_{2,i} & = \lambda _C(Z_i^2 \ | \ \Psi ^2)w^2Y_i^2, \\ U_{3,i} & = \lambda _C(Z_i^1 \ | \ \Psi ^1)w^1Y_i^1, \\ U_{4,i} & = \lambda _C(Z_i^1 \ | \ \Psi ^2)w^1Y_i^1, \\ U_{5,i} & = w^2Y_i^2, \\ U_{6,i} & = w^1Y_i^1. \\ \end{split} \end{equation}

However, all the above quantities of $ \lambda _C(Z_i^2 \ | \ \Psi ^1)$ , $\lambda _C(Z_i^2 \ | \ \Psi ^2)$ , $\lambda _C(Z_i^1 \ | \ \Psi ^1)$ , and $\lambda _C(Z_i^1 \ | \ \Psi ^2)$ are the so-called true quantities of interest, derived and based on economic theory reasoning, which are usually unobserved in practice. Hence, all the above quantities must be estimated from the sample data, as discussed in the next subsection.

2.3 DEA estimators

In the empirical analysis, we do not observe $ \lambda _C(Z_i^2 \ | \ \Psi ^1)$ , $\lambda _C(Z_i^2 \ | \ \Psi ^2)$ , $\lambda _C(Z_i^1 \ | \ \Psi ^1)$ , and $\lambda _C(Z_i^1 \ | \ \Psi ^2)$ , and thus we do not observe $ \mu _{\mathcal{M},n}$ and $\xi _n$ and hence they must be estimated from the sample data.

Given a random sample ${\mathcal S}_n$ , the conical Farrell output efficiency $\lambda _C(x,y\mid \Psi ^t)$ estimated by the DEA estimator is

(2.16) \begin{equation} \widehat \lambda _C(x,y\mid {\mathcal S}_n^t)= \max _{\lambda, s_1,\ldots, s_n} \Big \{ \lambda \mid \lambda y \le \sum _{i=1}^{n}s_i Y_i^t,\; x \ge \sum _{i=1}^{n}s_i X_i^t,\; \forall \; s_i \ge 0 \Big \}. \end{equation}

The simple mean MPI, $\mu _\mathcal{M}$ , can then be estimated by

(2.17) \begin{equation} \widehat{ \mu}_{\mathcal{M},n} =\frac {1}{n} \sum _{i=1}^n \log \widehat {\mathcal{M}}_i, \end{equation}

where

(2.18) \begin{equation} \begin{split} \log \widehat {\mathcal{M}}_i = -\frac {1}{2} \Big [ & \log \widehat {\lambda} _C(Z_i^2 \ | \ {\mathcal S}_n^1) + \log \widehat \lambda _C(Z_i^2 \ | \ {\mathcal S}_n^2) \\ & -\log \widehat \lambda _C(Z_i^1 \ | \ {\mathcal S}_n^1) - \log \widehat \lambda _C(Z_i^1 \ | \ {\mathcal S}_n^2)\Big ]. \end{split} \end{equation}

Similarly, the aggregate MPI, $\xi$ , can be estimated by

(2.19) \begin{equation} \widehat \xi _n = -\frac {1}{2} (\log \widehat \mu _1+ \log \widehat \mu _2 - \log \widehat \mu _3 - \log \widehat \mu _4) + \log \widehat \mu _5 - \log \widehat \mu _6, \end{equation}

where $\widehat \mu _s=\frac {1}{n}\sum _{i=1}^n \widehat U_{s,i}$ , for $s=1,2,3,4$ , and

(2.20) \begin{equation} \begin{split} \widehat U_{1,i} & = \widehat \lambda _C(Z_i^2 \ | \ {\mathcal S}_n^1)w^2Y_i^2, \\ \widehat U_{2,i} & = \widehat \lambda _C(Z_i^2 \ | \ {\mathcal S}_n^2)w^2Y_i^2, \\ \widehat U_{3,i} & = \widehat \lambda _C(Z_i^1 \ | \ {\mathcal S}_n^1)w^1Y_i^1, \\ \widehat U_{4,i} & = \widehat \lambda _C(Z_i^1 \ | \ {\mathcal S}_n^2)w^1Y_i^1, \end{split} \end{equation}

and where $\widehat \mu _r=\frac {1}{n}\sum _{i=1}^n U_{r,i}$ , for $r=5,6$ .

3. Further improvements of finite sample approximation of CLTs

The statistical properties of the estimators $\widehat \mu _{\mathcal{M},n}$ and $\widehat \xi _n$ have been thoroughly established by Kneip et al. (Reference Kneip, Simar and Wilson2021) and Pham et al. (Reference Pham, Simar and Zelenyuk2024), respectively.Footnote 3 These findings are succinctly summarized in Appendix A to provide a foundation for introducing our enhanced methodologies.

It is observed from the simulations that for the simple mean (Kneip et al. Reference Kneip, Simar and Wilson2021) and aggregate MPI (Pham et al. Reference Pham, Simar and Zelenyuk2024), the estimated confidence intervals constructed using the developed CLTs typically under-cover the true values in finite sample sizes and large dimensions. For example, Table EC.9 in Pham et al. (Reference Pham, Simar and Zelenyuk2024) presents the coverage of the estimated confidence intervals when $p=4$ , $q=1$ , $\delta =0.04$ , which shows that when the nominal coverage is $95\%$ , the estimated confidence intervals based on (A.23) for $n=10,20,50,100$ is only $0.684,0.835,0.902,0.932$ , respectively.

This under-covering phenomenon is also observed in the simple mean and aggregate efficiency (Kneip et al. Reference Kneip, Simar and Wilson2015, Simar and Zelenyuk, Reference Simar and Zelenyuk2018). Recently, Simar and Zelenyuk (Reference Simar and Zelenyuk2020), Nguyen et al. (Reference Nguyen, Simar and Zelenyuk2022), and Simar et al. (Reference Simar, Zelenyuk and Zhao2023, Reference Simar, Zelenyuk and Zhao2024) propose various methods to improve the finite sample approximation for the simple mean and aggregate (input-oriented and output-oriented) efficiency. In this section, to improve the finite sample approximation of CLTs for the simple mean (unweighted mean of) and aggregate (weighted mean of) MPI, we adapt the methods from Nguyen et al. (Reference Nguyen, Simar and Zelenyuk2022) and Simar et al. (Reference Simar, Zelenyuk and Zhao2023).

3.1 Improvements via data sharpening

It is well known that the empirical distribution of nonparametric efficiency estimators can be problematic due to discretization near the boundary, especially where efficiency values are close to one. Specifically, many estimates end up being one, even though the data-generating process suggests these values should have a zero probability by continuity. These misleading estimates are referred to in the literature as ‘spurious’ ones. A potential improvement could be achieved through a simplified smoothing approach that focuses on adjusting only the values near this efficient boundary, thereby enhancing the accuracy of the estimates.

Specifically, we adapt the idea of the data sharpening method in Nguyen et al. (Reference Nguyen, Simar and Zelenyuk2022) to the output-oriented MPI. First, we sharpen $Z_i^1=(X_i^1,Y_i^1)$ as follows,

(3.1) \begin{equation} \widetilde \lambda _C (Z_i^1 \mid {\mathcal S}_n^1) = \begin{cases} \widehat \lambda _C (Z_i^1 \mid {\mathcal S}_n^1), & \text {if}\ 1/\widehat \lambda _C (Z_i^1 \mid {\mathcal S}_n^1) \lt 1- \tau, \\ \\ \widehat \lambda _C (Z_i^1 \mid {\mathcal S}_n^1) / \varepsilon _i, & \text {otherwise}, \end{cases} \end{equation}

where the sharpening parameter $\tau =n^{-\gamma }$ , while $\varepsilon _i \overset {\text {iid}} {\sim } \text {Uniform}(1 - \tau, 1)$ and we set $\gamma =\kappa$ .Footnote 4 It can be shown that $\widetilde \lambda _C (Z_i^1 \mid {\mathcal S}_n^1) = \widehat \lambda _C ( X_i^1,\widetilde Y_i^1 \mid {\mathcal S}_n^1)$ , where

(3.2) \begin{equation} \widetilde Y_i^1 = \begin{cases} Y_i^1, & \text {if}\ 1/\widehat \lambda _C (Z_i^1 \mid {\mathcal S}_n^1) \lt 1- \tau, \\ \\ Y_i^1 \times \varepsilon _i, & \text {otherwise}. \end{cases} \end{equation}

After the data sharpening, $Z_i^1=(X_i^1, Y_i^1)$ becomes $\widetilde Z_i^1=(X_i^1,\widetilde Y_i^1)$ . Moreover, we have

(3.3) \begin{equation} \widetilde \lambda _C (Z_i^1 \mid {\mathcal S}_n^2) = \widehat \lambda _C (\widetilde Z_i^1 \mid {\mathcal S}_n^2)= \begin{cases} \widehat \lambda _C (Z_i^1 \mid {\mathcal S}_n^2), & \text {if}\ 1/\widehat \lambda _C (Z_i^1 \mid {\mathcal S}_n^1) \lt 1- \tau, \\ \\ \widehat \lambda _C (Z_i^1 \mid {\mathcal S}_n^2) / \varepsilon _i, & \text {otherwise} . \end{cases} \end{equation}

Similarly, we do the data sharpening for $Z_i^2=(X_i^2,Y_i^2)$ as follows,

(3.4) \begin{equation} \widetilde \lambda _C (Z_i^2 \mid {\mathcal S}_n^2) = \begin{cases} \widehat \lambda _C (Z_i^2 \mid {\mathcal S}_n^2), & \text {if}\, 1/\widehat \lambda _C (Z_i^2 \mid {\mathcal S}_n^2) \lt 1- \tau, \\ \\ \widehat \lambda _C (Z_i^2 \mid {\mathcal S}_n^2) / \epsilon _i, & \text {otherwise}, \end{cases} \end{equation}

where the sharpening parameter $\tau =n^{-\kappa }$ , while $\epsilon _i \overset {\text {iid}} {\sim } \text {Uniform}(1 - \tau, 1)$ . It can be shown that $\widetilde \lambda _C (Z_i^2 \mid {\mathcal S}_n^2) = \widehat \lambda _C ( X_i^2,\widetilde Y_i^2 \mid {\mathcal S}_n^2)$ , where

(3.5) \begin{equation} \widetilde Y_i^2 = \begin{cases} Y_i^2, & \text {if}\, 1/\widehat \lambda _C (Z_i^2 \mid {\mathcal S}_n^2) \lt 1- \tau, \\ \\ Y_i^2 \times \epsilon _i, & \text {otherwise}. \end{cases} \end{equation}

After the data sharpening, $Z_i^2=(X_i^2, Y_i^2)$ becomes $\widetilde Z_i^2=(X_i^2,\widetilde Y_i^2)$ . Moreover, we have

(3.6) \begin{equation} \widetilde \lambda _C (Z_i^2 \mid {\mathcal S}_n^1) = \widehat \lambda _C (\widetilde Z_i^2 \mid {\mathcal S}_n^1)= \begin{cases} \widehat \lambda _C (Z_i^2 \mid {\mathcal S}_n^1), & \text {if}\, 1/\widehat \lambda _C (Z_i^2 \mid {\mathcal S}_n^2) \lt 1- \tau, \\ \\ \widehat \lambda _C (Z_i^2 \mid {\mathcal S}_n^1) / \epsilon _i, & \text {otherwise}. \end{cases} \end{equation}

Combining together, after the data sharpening, the input-output pair for observation $i$ , $(X_i^1,Y_i^1,X_i^2,Y_i^2)$ becomes $(X_i^1,\widetilde Y_i^1,X_i^2,\widetilde Y_i^2)$ . We then use the sharpened sample $\{ (X_i^1,\widetilde Y_i^1,X_i^2,\widetilde Y_i^2) \}_{i=1}^n=\{ (\widetilde Z_i^1,\widetilde Z_i^2) \}_{i=1}^n$ to obtain the estimates of the simple mean and aggregate MPI as well as their confidence intervals. Note that here, the estimates of the various efficiency components for the data sharpening methods are computed for the observations in $\{ (\widetilde Z_i^1,\widetilde Z_i^2) \}_{i=1}^n$ , while the reference set is still same as the case without data sharpening methods. More specifically, the estimates of the various efficiency components for the data sharpening methods are $\widehat \lambda _C (\widetilde Z_i^2 \mid {\mathcal S}_n^1)$ , $\widehat \lambda _C (\widetilde Z_i^2 \mid {\mathcal S}_n^2)$ , $\widehat \lambda _C (\widetilde Z_i^1 \mid {\mathcal S}_n^1)$ , and $\widehat \lambda _C (\widetilde Z_i^1 \mid {\mathcal S}_n^2)$ , where we recall ${\mathcal S}_n^1=\{Z_i^1\}_{i=1}^n=\{X_i^1, Y_i^1\}_{i=1}^n$ and ${\mathcal S}_n^2=\{Z_i^2\}_{i=1}^n=\{X_i^2, Y_i^2\}_{i=1}^n$ .

For the simple mean MPI, we can use the sharpened sample $\{ (X_i^1,\widetilde Y_i^1,X_i^2,\widetilde Y_i^2) \}_{i=1}^n$ with the reference set $\{ (X_i^1, Y_i^1,X_i^2, Y_i^2) \}_{i=1}^n$ to obtain the corresponding estimates for the mean, bias and standard deviations, denoted as $\widehat {\widehat \mu }_{\mathcal{M},n}$ , $\widehat {\widehat B}_{\mathcal{M},n,\kappa, K}$ , and $\widehat {\widehat \sigma }_{\mathcal{M}}$ , respectively. The asymptotic $100(1-\alpha )\%$ confidence intervals for $\mu _{\mathcal{M}}$ can be constructed as

(3.7) \begin{equation} \bigg [ \widehat {\widehat \mu }_{\mathcal{M},n} - \widehat {\widehat B}_{\mathcal{M},n,\kappa, K} \ \pm \ \Phi _{1-\alpha /2}^{-1} \ \widehat {\widehat \sigma }_{\mathcal{M}}/ \sqrt {n} \bigg ], \end{equation}

and

(3.8) \begin{equation} \bigg [ \widehat {\widehat \mu }_{\mathcal{M},n_\kappa } - \widehat {\widehat B}_{\mathcal{M},n,\kappa, K} \ \pm \ \Phi _{1-\alpha /2}^{-1} \ \widehat {\widehat \sigma }_{\mathcal{M}}/\sqrt {n_\kappa } \bigg ], \end{equation}

for $\kappa \geq 2/5$ and $\kappa \lt 1/2$ , respectively.

Similarly for the aggregate MPI, we can use the sharpened sample $\{ (X_i^1,\widetilde Y_i^1,X_i^2,\widetilde Y_i^2) \}_{i=1}^n$ with the reference set $\{ (X_i^1, Y_i^1,X_i^2, Y_i^2) \}_{i=1}^n$ to obtain the corresponding estimates for the mean, bias and standard deviations, denoted as $\widehat {\widehat \xi }_{n}$ , $\widehat {\widehat B}_{\xi, n,\kappa, K}$ , and $\widehat {\widehat \sigma }_{\xi }$ , respectively. The asymptotically $100(1-\alpha )\%$ confidence intervals for $\xi$ can be constructed as

(3.9) \begin{equation} \bigg [\widehat {\widehat \xi }_n - \widehat {\widehat B}_{\xi, n,\kappa, K} \ \pm \ \Phi _{1-\alpha /2}^{-1} \ \widehat {\widehat \sigma }_{\xi }/ \sqrt {n} \bigg ], \end{equation}

and

(3.10) \begin{equation} \bigg [\widehat {\widehat \xi }_{n_\kappa } - \widehat {\widehat B}_{\xi, n,\kappa, K} \ \pm \ \Phi _{1-\alpha /2}^{-1} \ \widehat {\widehat \sigma }_{\xi }/\sqrt {n_\kappa } \bigg ], \end{equation}

for $\kappa \geq 2/5$ and $\kappa \lt 1/2$ , respectively.

3.2 Improvements via bias-corrected individual efficiency estimates

Recently, Simar et al. (Reference Simar, Zelenyuk and Zhao2023) propose estimating the variance for the simple mean efficiency through using the bias-corrected individual efficiency estimates, which is further extended by Simar et al. (Reference Simar, Zelenyuk and Zhao2024) for the aggregate efficiency. As the simple mean and aggregate MPI is constructed using various technical efficiency, the method in Simar et al. (Reference Simar, Zelenyuk and Zhao2023) can also be extended to here.

Simar et al. (Reference Simar, Zelenyuk and Zhao2023) have shown that by using the bias-corrected individual efficiency estimate, we are able to keep the asymptotic properties established by Kneip et al. (Reference Kneip, Simar and Wilson2015), including their CLTs. This result also holds for the MPIs. While asymptotically equivalent, the improvements of the bias correction method can be substantial in finite samples for MPIs. This is due to the fact that in finite samples there are large estimation errors for the MPI estimates, especially in large dimensions. The bias correction method for the variance proposed by Simar et al. (Reference Simar, Zelenyuk and Zhao2023) is helpful in reducing the estimation error and gives a more accurate estimation for the variance. See Simar et al. (Reference Simar, Zelenyuk and Zhao2023) for more details.

First, recall that when we estimate the bias for the simple mean MPI estimate $\widehat \mu _{\mathcal{M},n}$ or aggregate MPI estimate $\widehat \xi _n$ , we can also obtain the bias for each individual technical efficiency estimate $\widehat \lambda _C (Z_i^s \mid {\mathcal S}_n^t)$ , where $s,t \in \{1,2\}$ . Specifically, when splitting the sample ${\mathcal S}_{n}$ into ${\mathcal S}_{1,n/2,k}$ and ${\mathcal S}_{2,n/2,k}$ , we know that the observation $i$ with the input-output pair $(Z_i^1, Z_i^2)$ must lie in either ${\mathcal S}_{1,n/2,k}$ or ${\mathcal S}_{2,n/2,k}$ with equal probability. Without loss of generality, we assume $(Z_i^1, Z_i^2) \in {\mathcal S}_{1,n/2,k}$ . Then we compute

(3.11) \begin{equation} \widehat B_{i,s,t,k}^* = \widehat \lambda _C (Z_i^s \mid {\mathcal S}_{1,n/2,k}^t) - \widehat \lambda _C (Z_i^s \mid {\mathcal S}_n^t) . \end{equation}

Repeating the above process $K$ times, we end up with the estimate of the bias term for $\widehat \lambda _C (Z_i^s \mid {\mathcal S}_n^t)$ given byFootnote 5

(3.12) \begin{align} \widehat B_{i,s,t} = \frac {1}{K} \sum _{k=1}^{K}(2^{\kappa }-1)^{-1}( \widehat B_{i,s,t,k}^*). \end{align}

Extending the idea of Simar et al. (Reference Simar, Zelenyuk and Zhao2023) to the simple mean and aggregate MPI, for the original variance estimator of the simple mean and aggregate MPI expressed in equations (A.7) and (A.19), respectively, we propose replacing $\widehat \lambda _C (Z_i^s \mid {\mathcal S}_n^t)$ by $\widehat \lambda _C (Z_i^s \mid {\mathcal S}_n^t) -\widehat B_{i,s,t}$ at every place, where $s,t \in \{1,2\}$ , and $\widehat B_{i,s,t}$ is the estimated individual efficiency bias for $\widehat \lambda _C (Z_i^s \mid {\mathcal S}_n^t)$ as discussed above.

To be more specific, the estimate of the variance for the simple mean MPI using this method is given by

(3.13) \begin{equation} \widetilde \sigma ^2_{\mathcal{M}}= \frac {1}{n} \sum _{i=1}^n (\log \widetilde {\mathcal{M}}_i - \widetilde {\mu} _{\mathcal{M},n})^2, \end{equation}

where

(3.14) \begin{equation} \widetilde {\mu} _{\mathcal{M},n} = \frac {1}{n} \sum _{i=1}^{n} \log \widetilde {\mathcal{M}}_i, \end{equation}

and where

(3.15) \begin{equation} \begin{split} \log \widetilde {\mathcal{M}}_i = -\frac {1}{2} \Big [ & \log \big (\widehat \lambda _C(Z_i^2 \ | \ {\mathcal S}_n^1) - \widehat B_{i,2,1} \big )+ \log \big (\widehat \lambda _C(Z_i^2 \ | \ {\mathcal S}_n^2) - \widehat B_{i,2,2} \big ) \\ & -\log \big (\widehat \lambda _C(Z_i^1 \ | \ {\mathcal S}_n^1) - \widehat B_{i,1,1} \big ) - \log \big (\widehat \lambda _C(Z_i^1 \ | \ {\mathcal S}_n^2) - \widehat B_{i,1,2} \big ) \Big ]. \end{split} \end{equation}

Then the asymptotic $100(1-\alpha )\%$ confidence intervals for $\mu _{\mathcal{M}}$ can be constructed as

(3.16) \begin{equation} \bigg [ \widehat \mu _{\mathcal{M},n} - \widehat B_{\mathcal{M},n,\kappa, K} \ \pm \ \Phi _{1-\alpha /2}^{-1} \ \widetilde \sigma _{\mathcal{M}}/ \sqrt {n} \bigg ], \end{equation}

and

(3.17) \begin{equation} \bigg [ \widehat \mu _{\mathcal{M},n_\kappa } - \widehat B_{\mathcal{M},n,\kappa, K} \ \pm \ \Phi _{1-\alpha /2}^{-1} \ \widetilde \sigma _{\mathcal{M}}/\sqrt {n_\kappa } \bigg ], \end{equation}

for $\kappa \geq 2/5$ and $\kappa \lt 1/2$ , respectively.

Similarly, using the bias-corrected individual efficiency estimates, we can obtain the estimate of the variance for the aggregate MPI as

(3.18) \begin{equation} \widetilde \sigma ^2_{\xi } = [\nabla \widetilde \xi _n]' \widetilde \Sigma [\nabla \widetilde \xi _n], \end{equation}

where $\nabla \widetilde \xi _n$ is the column vector of the gradient of $\widetilde \xi _n$ with respect to $\widetilde \mu _s$ . Further,

(3.19) \begin{equation} \widetilde \xi _n = -\frac {1}{2} (\log \widetilde \mu _1+ \log \widetilde \mu _2 - \log \widetilde \mu _3 - \log \widetilde \mu _4) + \log \widetilde \mu _5 - \log \widetilde \mu _6, \end{equation}

where $\widetilde \mu _s=\frac {1}{n}\sum _{i=1}^n \widetilde U_{s,i}$ , for $s=1,2,3,4$ , and

(3.20) \begin{equation} \begin{split} \widetilde U_{1,i} & = \big (\widehat \lambda _C(Z_i^2 \ | \ {\mathcal S}_n^1) - \widehat B_{i,2,1} \big )w^2Y_i^2, \\ \widetilde U_{2,i} & = \big (\widehat \lambda _C(Z_i^2 \ | \ {\mathcal S}_n^2) - \widehat B_{i,2,2} \big )w^2Y_i^2, \\ \widetilde U_{3,i} & = \big (\widehat \lambda _C(Z_i^1 \ | \ {\mathcal S}_n^1) - \widehat B_{i,1,1} \big )w^1Y_i^1, \\ \widetilde U_{4,i} & = \big (\widehat \lambda _C(Z_i^1 \ | \ {\mathcal S}_n^2) - \widehat B_{i,1,2} \big )w^1Y_i^1, \end{split} \end{equation}

and where $\widetilde \mu _r=\widehat \mu _r=\frac {1}{n}\sum _{i=1}^n U_{r,i}$ , for $r=5,6$ . Formally, $\nabla \widetilde \xi _n= [\frac {\partial \widetilde \xi _n}{\partial \widetilde \mu _1}, \frac {\partial \widetilde \xi _n}{\partial \widetilde \mu _2}, \frac {\partial \widetilde \xi _n}{\partial \widetilde \mu _3}, \frac {\partial \widetilde \xi _n}{\partial \widetilde \mu _4}, \frac {\partial \widetilde \xi _n}{\partial \widetilde \mu _5}, \frac {\partial \widetilde \xi _n}{\partial \widetilde \mu _6}]'$ , where

(3.21) \begin{equation} \begin{split} & \frac {\partial \widetilde \xi _n}{\partial \widetilde \mu _1}=-\frac {1}{2 \widetilde \mu _1 },\ \frac {\partial \widetilde \xi _n}{\partial \widetilde \mu _2}=-\frac {1}{2 \widetilde \mu _2 }, \ \frac {\partial \widetilde \xi _n}{\partial \widetilde \mu _3}=\frac {1}{2 \widetilde \mu _3 }, \\ & \frac {\partial \widetilde \xi _n}{\partial \widetilde \mu _4}=\frac {1}{2 \widetilde \mu _4 },\ \frac {\partial \widetilde \xi _n}{\partial \widetilde \mu _5}=\frac {1}{ \widetilde \mu _5 }, \ \frac {\partial \widetilde \xi _n}{\partial \widetilde \mu _6}=\frac {1}{ \widetilde \mu _6 }. \end{split} \end{equation}

Further, $\widetilde \Sigma$ is the covariance matrix of $[\widetilde U_{1,i},\widetilde U_{2,i},\widetilde U_{3,i},\widetilde U_{4,i}, U_{5,i}, U_{6,i}]'$ . Formally,

(3.22) \begin{equation} \begin{split} \widetilde \Sigma _{s,s^*} & = \frac {1}{n} \sum _{i=1}^{n} (\widetilde U_{s,i}-\widetilde \mu _s)(\widetilde U_{s^*,i}-\widetilde \mu _{s^*}), \; \text {for } s, s^* \in \{1,2,3,4\}, \\ \widetilde \Sigma _{s,r} & = \frac {1}{n} \sum _{i=1}^{n} (\widetilde U_{s,i}-\widetilde \mu _s)( U_{r,i}-\widetilde \mu _r), \; \text {for } s \in \{1,2,3,4\},\; r \in \{5,6\},\\ \widetilde \Sigma _{r,r^*} & = \frac {1}{n} \sum _{i=1}^{n} (U_{r,i}- \widetilde \mu _r) ( U_{r,i}- \widetilde \mu _{r^*}), \; \text {for } r, r^* \in \{5,6\}. \end{split} \end{equation}

Then the asymptotically $100(1-\alpha )\%$ confidence intervals for $\xi$ can be constructed as

(3.23) \begin{equation} \bigg [\widehat \xi _n - \widehat B_{\xi, n,\kappa, K} \ \pm \ \Phi _{1-\alpha /2}^{-1} \ \widetilde \sigma _{\xi }/ \sqrt {n} \bigg ], \end{equation}

and

(3.24) \begin{equation} \bigg [\widehat \xi _{n_\kappa } - \widehat B_{\xi, n,\kappa, K} \ \pm \ \Phi _{1-\alpha /2}^{-1} \ \widetilde \sigma _{\xi }/\sqrt {n_\kappa } \bigg ], \end{equation}

for $\kappa \geq 2/5$ and $\kappa \lt 1/2$ , respectively.

3.3 Improvements via data sharpening and bias-corrected individual efficiency estimates

Similar to Simar et al. (Reference Simar, Zelenyuk and Zhao2023) and Simar et al. (Reference Simar, Zelenyuk and Zhao2024), we can also combine the data sharpening method in the Subsection 3.1 and the method using the bias-corrected individual efficiency estimates in the Subsection 3.2 together to examine whether this combined method could improve the finite sample approximation for the CLTs for the simple mean and aggregate MPI.

For the sharpened sample $\{ (X_i^1,\widetilde Y_i^1,X_i^2,\widetilde Y_i^2) \}_{i=1}^n=\{ (\widetilde Z_i^1,\widetilde Z_i^2) \}_{i=1}^n$ , we have the estimates of the various efficiency components as $\widehat \lambda _C (\widetilde Z_i^2 \mid {\mathcal S}_n^1)$ , $\widehat \lambda _C (\widetilde Z_i^2 \mid {\mathcal S}_n^2)$ , $\widehat \lambda _C (\widetilde Z_i^1 \mid {\mathcal S}_n^1)$ , and $\widehat \lambda _C (\widetilde Z_i^1 \mid {\mathcal S}_n^2)$ . Moreover, using the same procedure as the Subsection 3.2, we can obtain their corresponding bias, denoted as $\widetilde B_{i,2,1}$ , $\widetilde B_{i,2,2}$ , $\widetilde B_{i,1,1}$ , and $\widetilde B_{i,1,2}$ . Extending the idea of Simar et al. (Reference Simar, Zelenyuk and Zhao2023) to the simple mean and aggregate MPI, for the original variance estimator of the simple mean and aggregate MPI expressed in equations (A.7) and (A.19), respectively, we propose replacing $\widehat \lambda _C (Z_i^s \mid {\mathcal S}_n^t)$ by $\widehat \lambda _C (\widetilde Z_i^s \mid {\mathcal S}_n^t) -\widetilde B_{i,s,t}$ at every place, where $s,t \in \{1,2\}$ .

To be more specific, the estimate of the variance for the simple mean MPI can be obtained as

(3.25) \begin{equation} \widehat {\widetilde {\sigma }}^2_{\mathcal{M}}= \frac {1}{n} \sum _{i=1}^n (\log \widehat {\widetilde {\mathcal{M}}}_i - \widehat {\widetilde {\mu }}_{\mathcal{M},n})^2, \end{equation}

where

(3.26) \begin{equation} \widehat {\widetilde {\mu }}_{\mathcal{M},n} = \frac {1}{n} \sum _{i=1}^{n} \log \widehat {\widetilde {\mathcal{M}}}_i, \end{equation}

and where

(3.27) \begin{equation} \begin{split} \log \widehat {\widetilde {\mathcal{M}}}_i = -\frac {1}{2} \Big [ & \log \big (\widehat \lambda _C(\widetilde Z_i^2 \ | \ {\mathcal S}_n^1) - \widetilde B_{i,2,1} \big )+ \log \big (\widehat \lambda _C(\widetilde Z_i^2 \ | \ {\mathcal S}_n^2) - \widetilde B_{i,2,2} \big ) \\ & -\log \big (\widehat \lambda _C( \widetilde Z_i^1 \ | \ {\mathcal S}_n^1) - \widetilde B_{i,1,1} \big ) - \log \big (\widehat \lambda _C( \widetilde Z_i^1 \ | \ {\mathcal S}_n^2) - \widetilde B_{i,1,2} \big ) \Big ]. \end{split} \end{equation}

Then the asymptotic $100(1-\alpha )\%$ confidence intervals for $\mu _{\mathcal{M}}$ can be constructed as

(3.28) \begin{equation} \bigg [ \widehat {\widehat \mu }_{\mathcal{M},n} - \widehat {\widehat B}_{\mathcal{M},n,\kappa, K} \ \pm \ \Phi _{1-\alpha /2}^{-1} \ \widehat {\widetilde {\sigma }}_{\mathcal{M}}/ \sqrt {n} \bigg ], \end{equation}

and

(3.29) \begin{equation} \bigg [ \widehat {\widehat \mu }_{\mathcal{M},n_\kappa } - \widehat {\widehat B}_{\mathcal{M},n,\kappa, K} \ \pm \ \Phi _{1-\alpha /2}^{-1} \ \widehat {\widetilde {\sigma }}_{\mathcal{M}}/\sqrt {n_\kappa } \bigg ], \end{equation}

for $\kappa \geq 2/5$ and $\kappa \lt 1/2$ , respectively.

Similarly, the estimate of the variance for the aggregate MPI can be obtained as

(3.30) \begin{equation} \widehat {\widetilde {\sigma }}^2_{\xi } = [\nabla \widehat {\widetilde {\xi }}_n]' \widehat {\widetilde {\Sigma }}[\nabla \widehat {\widetilde {\xi }}_n], \end{equation}

where $\nabla \widehat {\widetilde {\xi }}_n$ is the column vector of the gradient of $ \widehat {\widetilde {\xi }}_n$ with respect to $ \widehat {\widetilde {\mu }}_s$ . Further,

(3.31) \begin{equation} \widehat {\widetilde {\xi }}_n = -\frac {1}{2} (\log \widehat {\widetilde {\mu }}_1+ \log \widehat {\widetilde {\mu }}_2 - \log \widehat {\widetilde {\mu }}_3 - \log \widehat {\widetilde {\mu }}_4) + \log \widehat {\widetilde {\mu }}_5 - \log \widehat {\widetilde {\mu }}_6, \end{equation}

where $ \widehat {\widetilde {\mu }}_s=\frac {1}{n}\sum _{i=1}^n \widehat {\widetilde {U}}_{s,i}$ , for $s=1,2,3,4$ , and

(3.32) \begin{equation} \begin{split} \widehat {\widetilde {U}}_{1,i} & = \big (\widehat \lambda _C(\widetilde Z_i^2 \ | \ {\mathcal S}_n^1) - \widetilde B_{i,2,1} \big )w^2Y_i^2, \\ \widehat {\widetilde {U}}_{2,i} & = \big (\widehat \lambda _C(\widetilde Z_i^2 \ | \ {\mathcal S}_n^2) - \widetilde B_{i,2,2} \big )w^2Y_i^2, \\ \widehat {\widetilde {U}}_{3,i} & = \big (\widehat \lambda _C(\widetilde Z_i^1 \ | \ {\mathcal S}_n^1) - \widetilde B_{i,1,1} \big )w^1Y_i^1, \\ \widehat {\widetilde {U}}_{4,i} & = \big (\widehat \lambda _C(\widetilde Z_i^1 \ | \ {\mathcal S}_n^2) - \widetilde B_{i,1,2} \big )w^1Y_i^1, \end{split} \end{equation}

and where $ \widehat {\widetilde {\mu }}_r=\widehat \mu _r=\frac {1}{n}\sum _{i=1}^n U_{r,i}$ , for $r=5,6$ . Formally, $\nabla \widehat {\widetilde {\xi }}_n= [\frac {\partial \widehat {\widetilde {\xi }}_n}{\partial \widehat {\widetilde {\mu }}_1}, \frac {\partial \widehat {\widetilde {\xi }}_n}{\partial \widehat {\widetilde {\mu }}_2}, \frac {\partial \widehat {\widetilde {\xi }}_n}{\partial \widehat {\widetilde {\mu }}_3}, \frac {\partial \widehat {\widetilde {\xi }}_n}{\partial \widehat {\widetilde {\mu }}_4}, \frac {\partial \widehat {\widetilde {\xi }}_n}{ \partial \widehat {\widetilde {\mu }}_5}, \frac {\partial \widehat {\widetilde {\xi }}_n}{\partial \widehat {\widetilde {\mu }}_6}]'$ , where

(3.33) \begin{equation} \begin{split} & \frac {\partial \widehat {\widetilde {\xi }}_n}{\partial \widehat {\widetilde {\mu }}_1}=-\frac {1}{2 \widehat {\widetilde {\mu }}_1 },\ \frac {\partial \widehat {\widetilde {\xi }}_n}{\partial \widehat {\widetilde {\mu }}_2}=-\frac {1}{2 \widehat {\widetilde {\mu }}_2 }, \ \frac {\partial \widehat {\widetilde {\xi }}_n}{\partial \widehat {\widetilde {\mu }}_3}=\frac {1}{2 \widehat {\widetilde {\mu }}_3 }, \\ & \frac {\partial \widehat {\widetilde {\xi }}_n}{\partial \widehat {\widetilde {\mu }}_4}=\frac {1}{2 \widehat {\widetilde {\mu }}_4 },\ \frac {\partial \widehat {\widetilde {\xi }}_n}{\partial \widehat {\widetilde {\mu }}_5}=\frac {1}{ \widehat {\widetilde {\mu }}_5 }, \ \frac {\partial \widehat {\widetilde {\xi }}_n}{\partial \widehat {\widetilde {\mu }}_6}=\frac {1}{ \widehat {\widetilde {\mu }}_6 }. \end{split} \end{equation}

Further, $ \widehat {\widetilde {\Sigma }}$ is the covariance matrix of $[ \widehat {\widetilde {U}}_{1,i}, \widehat {\widetilde {U}}_{2,i}, \widehat {\widetilde {U}}_{3,i}, \widehat {\widetilde {U}}_{4,i}, U_{5,i}, U_{6,i}]'$ . Formally,

(3.34) \begin{equation} \begin{split} \widehat {\widetilde {\Sigma }}_{s,s^*} & = \frac {1}{n} \sum _{i=1}^{n} ( \widehat {\widetilde {U}}_{s,i}- \widehat {\widetilde {\mu }}_s)( \widehat {\widetilde {U}}_{s^*,i}- \widehat {\widetilde {\mu }}_{s^*}), \; \text {for } s, s^* \in \{1,2,3,4\}, \\ \widehat {\widetilde {\Sigma }}_{s,r} & = \frac {1}{n} \sum _{i=1}^{n} ( \widehat {\widetilde {U}}_{s,i}- \widehat {\widetilde {\mu }}_s)( U_{r,i}- \widehat {\widetilde {\mu }}_r), \; \text {for } s \in \{1,2,3,4\},\; r \in \{5,6\},\\ \widehat {\widetilde {\Sigma }}_{r,r^*} & = \frac {1}{n} \sum _{i=1}^{n} (U_{r,i}- \widehat {\widetilde {\mu }}_r) ( U_{r,i}- \widehat {\widetilde {\mu }}_{r^*}), \; \text {for } r, r^* \in \{5,6\}. \end{split} \end{equation}

Then the asymptotically $100(1-\alpha )\%$ confidence intervals for $\xi$ can be constructed as

(3.35) \begin{equation} \bigg [\widehat {\widehat \xi }_n - \widehat {\widehat B}_{\xi, n,\kappa, K} \ \pm \ \Phi _{1-\alpha /2}^{-1} \ \widehat {\widetilde {\sigma }}_{\xi }/ \sqrt {n} \bigg ], \end{equation}

and

(3.36) \begin{equation} \bigg [\widehat {\widehat \xi }_{n_\kappa } - \widehat {\widehat B}_{\xi, n,\kappa, K} \ \pm \ \Phi _{1-\alpha /2}^{-1} \ \widehat {\widetilde {\sigma }}_{\xi }/\sqrt {n_\kappa } \bigg ], \end{equation}

for $\kappa \geq 2/5$ and $\kappa \lt 1/2$ , respectively.

4. Monte-Carlo evidence

4.1 Details on Monte-Carlo simulations

Our MC experiments follow that in Pham et al. (Reference Pham, Simar and Zelenyuk2024), and we briefly restate it here for the sake of being self-contained. Interested readers can see Appendix EC.3 in Pham et al. (Reference Pham, Simar and Zelenyuk2024) for more details. The true technology in the first period is

(4.1) \begin{equation} y_i^{1\partial }=\prod _{j=1}^{p}(x_{ij}^1-1)^{\beta _j}, \end{equation}

while the true technology in the second period is

(4.2) \begin{equation} y_i^{2\partial }=(1+\delta )\prod _{j=1}^{p}(x_{ij}^2-1)^{\beta _j+\delta }, \end{equation}

where $\delta$ controls the changes of the technology from period 1 to period 2. Denote $x_i^t=(x_{i1}^t,x_{i2}^t,\ldots, x_{ip}^t)$ , for $t=1,2$ and $i=1,\ldots, n$ , then $(x_i^1,y_i^{1\partial })$ and $(x_i^2,y_i^{2\partial })$ are the points on the technology in (4.1) and (4.2), respectively.

The inputs and technical efficiency between the two periods are typically correlated. To account for the correlations of inputs, following Pham et al. (Reference Pham, Simar and Zelenyuk2024), for each $j=1,\ldots, p$ , we generate the observed inputs as

(4.3) \begin{equation} x_{ij}^1, x_{ij}^2, \; \overset {\text {iid}} {\sim } \; \text {Unif}(1,10), \forall \; i=1,\ldots, n, \end{equation}

where $corr(x_{ij}^1,x_{ij}^2)=0.5$ , for each $j=1,\ldots, p$ .Footnote 6 Similarly, to account for the correlations of efficiency, we generate the true efficiency as

(4.4) \begin{equation} \lambda (X_i^1, Y_i^1 \mid \Psi ^1 ), \lambda (X_i^2, Y_i^2 \mid \Psi ^2 ) \; \overset {\text {iid}} {\sim } \; 1+ |N(0,0.25^2)|, \forall \; i=1,\ldots, n, \end{equation}

where $corr(\lambda (X_i^1, Y_i^1 \mid \Psi ^1 ), \lambda (X_i^2, Y_i^2 \mid \Psi ^2 )) \neq 0$ and $N(0,0.25^2)$ is the normal distribution with the variance $0.25^2$ so that $|N(0,0.25^2)|$ is the half normal distribution. The observed outputs then can be computed as

(4.5) \begin{equation} y_i^1=\prod _{j=1}^{p}(x_{ij}^1-1)^{\beta _j}/\lambda (X_i^1, Y_i^1 \mid \Psi ^1 ), \end{equation}

and

(4.6) \begin{equation} y_i^2=(1+\delta )\prod _{j=1}^{p}(x_{ij}^2-1)^{\beta _j+\delta }/\lambda (X_i^2, Y_i^2 \mid \Psi ^2 ), \end{equation}

i.e., we project the efficient observations from the corresponding technologies to the production set to obtain a simulated sample ${\mathcal S}_n=\{(x_i^1,y_i^1,x_i^2,y_i^2)\}_{i=1}^{n}$ .

We set $\delta =0.04$ , the output prices $w_y^1=w_y^2=1$ , and the values of $\beta _j$ are presented in Table 1. The number of MC trials for each experiment consisting of $(n,p,q)$ is $1,000$ . Moreover, we consider both the simple mean and aggregate Malmquist productivity indices. The true values of the simple mean and aggregate MPI are computed by following the steps in EC.3.8 of Pham et al. (Reference Pham, Simar and Zelenyuk2024). Before presenting our MC results, we use the notations in Table 2 for the simple mean and aggregate MPI.

Table 1. The values of $\beta _j$ and $w_j$

Table 2. List of notations used for MC experiments

4.2 Monte-Carlo results

4.2.1 General remarks

We notice that when the nominal coverage is 99%, the coverage of the estimated confidence intervals for both the simple mean and aggregate MPI using either of (ii)–(v) is generally close to 99%, and thus we only focus on the cases where the nominal coverage is 90% and 95%. Further, the MC results are robust to the values of $\delta$ . The empirical coverages from the simulation results for $\delta =0.02$ and $\delta =0.10$ are reported in Appendix C, which have a similar pattern as those for $\delta =0.04$ . Moreover, we also perform MC experiments on the performance of the estimated confidence intervals discussed in Appendix B for $\exp (\mu _\mathcal{M})$ and $\exp (\xi )$ , which are presented in Appendix D.

4.2.2 Main results

Figures 1 and 2 present the results for the coverage of the estimated confidence intervals for the simple mean MPI when the nominal coverage is 90% and 95% (respectively), while Figures 3 and 4 present similar results for aggregate MPI.Footnote 7 We notice that as the sample size $n$ increases, the coverage of (ii) in Figures 14 shows a gradual improvement in the approximation of the respective nominal coverage across different dimensions, which supports the developed theories for the simple mean MPI in Kneip et al. (Reference Kneip, Simar and Wilson2021) and the aggregate MPI in Pham et al. (Reference Pham, Simar and Zelenyuk2024).

Figure 1. Coverages of various estimated confidence intervals for the simple mean malmquist productivity indices for the 90% nominal coverage, with $\delta =0.04$ .

Figure 2. Coverages of various estimated confidence intervals for the simple mean malmquist productivity indices for the 95% nominal coverage, with $\delta =0.04$ .

Figure 3. Coverages of various estimated confidence intervals for the aggregate malmquist productivity indices for the 90% nominal coverage, with $\delta =0.04$ .

Figure 4. Coverages of various estimated confidence intervals for the aggregate malmquist productivity indices for the 95% nominal coverage, with $\delta =0.04$ .

Comparing (iii) with (ii), across different nominal coverage, we see that the coverage using (iii) is larger than or equal to that using (ii) for the simple mean MPI. For the aggregate MPI, this is also true in general, except for the three cases $p=q=1$ , $n=20, 50, 200$ with $90\%$ nominal coverage, the one case $p=q=1$ , $n=100$ with $95\%$ nominal coverage, and the one case $p=q=1$ , $n=20$ with $99\%$ nominal coverage.Footnote 8 However, the goal should not be just larger per se, but closer to the nominal level. The improvements (measured by the closeness to the nominal level) of (iii) over (ii) are mainly observed in high dimensions ( $p \ge 2$ ) and relatively small sample sizes ( $n \le 50$ ). For example, for the simple mean MPI, when $p=3$ , $n=20$ and the nominal coverage is $95\%$ , the coverage using (iii) is $0.949$ , which is much closer to the $95\%$ nominal coverage than $0.881$ obtained using (ii). Moreover this difference is especially substantial in high dimensions. For example, for the simple mean MPI, holding $n=20$ and the nominal coverage $95\%$ , the difference of the coverage between (iii) and (ii) increases from $0.068$ ( $0.949-0.881$ ) to $0.121$ ( $0.959-0.838$ ) when the number of input increases from $p=3$ to $p=7$ . However, it is observed that (iii) often starts overshooting after $n=100$ (and sometimes from around $n=50$ ), while at around $n=100$ , (ii) starts approximating the nominal levels relatively well and the additional improvements seem to not be needed, especially if they can add noise and overshoot with the nominal coverage.

The comparison between (iii) and (ii) suggests that the method in Simar et al. (Reference Simar, Zelenyuk and Zhao2023) generally could improve the finite sample approximation of the CLTs for the simple mean and aggregate MPI in high dimensions ( $p \ge 2$ ) and relatively small sample sizes ( $n \le 50$ ). To some extent, our results are consistent with Simar et al. (Reference Simar, Zelenyuk and Zhao2023) and Simar et al. (Reference Simar, Zelenyuk and Zhao2024) who also find the better performance of the method in Simar et al. (Reference Simar, Zelenyuk and Zhao2023) for improving the finite sample approximation of CLTs for simple mean efficiency and aggregate efficiency, respectively. However, our result is also different from the context of CLTs for efficiency aggregates, where the improvement was really needed, even for $n=300$ and especially below that. This is likely due to the ratio nature of the measurement where the magnitude of the bias shrinks substantially.

Comparing (iv) with (ii), for both simple mean and aggregate MPI, the coverage using (iv) is generally smaller than that using (ii), except in high dimensions ( $p\ge 4$ ) and small sample sizes ( $n \le 100$ ); in other words, the improvements of (iv) over (ii) are only observed in high dimensions and relatively small sample sizes. For example, for the simple mean MPI, when $p=5$ and the nominal coverage is $95\%$ , the difference of the coverage between (iv) and (ii) is $0.049$ ( $0.912-0.863$ ) when $n=20$ and it decreases to $-0.004$ ( $0.942-0.946$ ) when $n=1000$ . Our results here are different from Nguyen et al. (Reference Nguyen, Simar and Zelenyuk2022), Simar et al. (Reference Simar, Zelenyuk and Zhao2023), and Simar et al. (Reference Simar, Zelenyuk and Zhao2024) where they all find that the data sharpening method could improve the finite sample approximation of the CLTs for simple mean and aggregate efficiency; i.e., they all find the improved performance of the data sharpening method compared to the original methods (Kneip et al. Reference Kneip, Simar and Wilson2015, Simar and Zelenyuk, Reference Simar and Zelenyuk2018). The reason might come from the fact that the MPI estimates involve ratios of technical efficiency estimates and so the magnitude (i.e., the constant part) of the bias seems to be partially cancelled out in practice.Footnote 9

As both (iii) and (iv) have better performance over (ii) in high dimensions and relatively small sample sizes, next we compare these two methods. From Figures 14, we see that the improvements of (iii) are always more substantial than (iv) in all the high dimensions and relatively small sample sizes. Combining the results until now, the estimated coverage using (iii) seems to come closer to the nominal coverage than those using (ii) and (iv) in high dimensions ( $p \ge 2$ ) and relatively small sample sizes ( $n \le 50$ ), while at around $n=100$ , the method (ii) starts approximating the nominal levels relatively well.

Comparing (v) with (iv) again indicates the better performance of Simar et al. (Reference Simar, Zelenyuk and Zhao2023), as for both the simple mean and aggregate MPI, the coverage using (v) is larger than that using (iv), especially in high dimensions ( $p \ge 2$ ) and small sample sizes ( $n \le 50$ ), (v) seems to often give slightly (and sometimes significantly) closer coverage than (iv). However, when comparing (v) and (iii), the former seems to often give slightly (and sometimes significantly) closer coverage than the latter, yet not always and the difference is often too small to justify the additional complexity. So, for simplicity reasons, (iii) might be preferred over (v).

To conclude for this section, both (iii) and (v) are very similar and help to improve the coverage for relatively small samples, such as around $n=50$ and less, and sometimes up to around $100$ , but they also often start to overshoot after that (and sometimes from around $50$ ). Fortunately, and unlike the context of efficiency measurement, already at around $n=100$ , the original methods from Kneip et al. (Reference Kneip, Simar and Wilson2021) and Pham et al. (Reference Pham, Simar and Zelenyuk2024) start approximating the nominal levels relatively well and the additional improvements seem to be not needed, especially if they can add noise and overshoot with the coverage (i.e., rejecting a hypothesis less than they should). This is indeed very different from the context of CLTs for efficiency aggregates, where the improvement was often needed even for $n=300$ and especially below that. This is likely due to the ratio nature of the measurement in the MPI framework where the magnitude of the bias shrinks substantially. Hence, the bottom line conclusion is that the use of (iii) or (v) is advisable for relatively small samples (e.g., up to around $50$ , perhaps $100$ ) and after that just use (ii), i.e., the original methods from Kneip et al. (Reference Kneip, Simar and Wilson2021) and Pham et al. (Reference Pham, Simar and Zelenyuk2024). Finally, comparing the MC experiments from Appendix D with those in Appendix C, suggests that the estimated confidence intervals for $\exp (\mu _\mathcal{M})$ and $\exp (\xi )$ (discussed in Appendix B) are fairly similar to those for $\mu _\mathcal{M}$ and $\xi$ , implying that both approaches can be relied upon.

5. Empirical illustration

Our illustration closely follows that in Pham et al. (Reference Pham, Simar and Zelenyuk2024), which employs the widely used Penn World Table data (PWT 10.0, Feenstra et al. Reference Feenstra, Inklaar and Timmer2015) to study the MPI of countries/regions in the world from 1990 to 2019. For the related literature using PWT, see also Färe et al. (Reference Färe, Grosskopf, Norris and Zhang1994), Kumar and Russell (Reference Kumar and Russell2002), and Badunenko et al. (Reference Badunenko, Henderson and Zelenyuk2008, Reference Badunenko, Henderson, Zelenyuk, Badunenko, Henderson and Zelenyuk2018).

Same as Pham et al. (Reference Pham, Simar and Zelenyuk2024), the production of countries is modelled using labor ( $emp$ ) and capital stock ( $cn$ ) to produce GDP ( $rgdpo$ ). The output price is the same for all countries/regions. To illustrate the evolution of the simple mean and aggregate MPI from 1990 to 2019, we will use the same sub-set of 84 countries/regions as Badunenko et al. (Reference Badunenko, Henderson and Zelenyuk2008) and Pham et al. (Reference Pham, Simar and Zelenyuk2024). Different from Pham et al. (Reference Pham, Simar and Zelenyuk2024), we present results for pairs of years at 5–year intervals and the overall period 1990–2019 in Table 3 for the simple mean MPI and in Table 4 for aggregate MPI. Moreover, same as Pham et al. (Reference Pham, Simar and Zelenyuk2024), we conduct the analysis for the all the countries, 27 developed countries, and 57 developing countries, separately.

Table 3. Estimation results for the simple mean MPI of countries/Regions

Notes: Statistical significance (difference from $1$) for the bias-corrected estimate (i.e., $\exp {(\widehat \mu _{\mathcal{M},n} - \widehat B_{\mathcal{M},n,\kappa, K})}$) of the true mean of MPI at the 10%, 5%, or 1% is denoted by 1, 2, or 3 asterisks, respectively, while “–” indicates insignificance at the ten percent level.

Table 4. Estimation results for the aggregate MPI of countries/Regions

Notes: Statistical significance (difference from $1$ ) for the bias-corrected estimate (i.e., $\exp {(\widehat \xi _n - \widehat B_{\xi, n,\kappa, K})}$ ) of the true aggregate of MPI at the 10%, 5%, or 1% is denoted by 1, 2, or 3 asterisks, respectively, while “–” indicates insignificance at the ten percent level.

First, from Tables 3 and 4, we can see that our results for the simple mean and aggregate MPI estimates from 1990 to 2019 are similar to those in Table 1 of Pham et al. (Reference Pham, Simar and Zelenyuk2024).Footnote 10 In terms of the estimates of the variance for the simple mean and aggregate MPI, we find that (iii) generally yields slightly larger estimates than (ii). For example, among the $21$ cases in Table 3 ( $7$ cases for each of the entire sample, developed and developing countries), the estimates of $\sigma _{\mathcal{M}}$ using (iii) are larger than that using (ii) for $19$ cases; similarly, Table 4 shows that the estimates of $\sigma _{\xi }$ using (iii) are larger than that using (ii) for $16$ cases out of $21$ cases. This result suggests that the estimates of confidence intervals based on (iii) generally will be slightly larger than that using (ii). This is consistent with our MC results which also suggested that (iii) has a better performance (in terms of covering the true values) than (ii) in relatively small sample sizes (e.g., up to around $50$ ) and large dimensions. Similarly, comparing (v) and (iv), we find that (v) generally yields slightly larger estimates of the variance than (iv). Table 3 shows that the estimates of $\sigma _{\mathcal{M}}$ using (v) are larger than that using (iv) for $19$ cases out of $21$ cases and Table 4 shows that the estimates of $\sigma _{\xi }$ using (v) are larger than that using (iv) for $13$ cases out of $21$ cases. This result suggests that the estimates of confidence intervals based on (v) generally will be slightly larger than that using (iv). Thus, our illustration confirms again our main takeaways from the MC results that the use of (iii) or (v) is advisable for relatively small samples (e.g., up to around $50$ , perhaps $100$ ).

From Table 3, we see that the simple mean MPI for the entire sample is significantly different from 1 for 1990–2005; Further, this result is robust as the $95\%$ CI constructed using either of (ii), (iii), (iv), and (v) does not contain 1, suggesting that the productivity growth of these 84 countries significantly decreased from 1990 to 1995, significantly increased from 1995 to 2005 and remained unchanged for the other remaining periods. For the developed countries, the simple mean MPI is significantly larger than 1 in the periods 1990–1995, 1995–2000, 2000–2005, and 2015–2019, while it is significantly smaller than 1 in the periods 2005–2010 and 2010–2015. This result is robust across different methods, suggesting that the productivity growth for the developed countries continued increasing from 1990 to 2005, then continued decreasing from 2005 to 2015 (possibly due to the global financial crisis) and increased again from 2015 to 2019. However, for the whole period 1990–2019, (ii) and (iii) suggest that productivity remained unchanged while (iv) and (v) suggest that productivity increased significantly. The productivity growth for the developing countries significantly decreased from 1990 to 1995, continued increasing from 2000 to 2015, and increased over the whole period 1990–2019. Thus, the results indicate that from 1990 to 2019 only developing countries achieved about $17.72\%$ percent increase in productivity, while there is no evidence supporting the changes in productivity for the full sample of 84 countries and mixed evidence for the developed countries. Recall that these estimates ignore the economic weights of each country in the averaging, e.g., weighting the productivity estimate for the USA with the same weight as for any other country. To address this we estimate the aggregate MPI as per our discussions above and report the corresponding results in Table 4.

Table 4 shows that the aggregate MPI for the entire sample is not significantly different from 1 in most of the considered periods, except for 2005–2010, where only (ii) and (iv) find that the productivity decreased at the $10\%$ significance level. For the developed countries, the aggregate MPI is significantly larger than 1 in the periods 1990–1995, 1995–2000, 2000–2005, 2015–2019 and over the whole period 1990–2019, while it is significantly smaller than 1 from 2005 to 2010. This result is robust across different methods, suggesting that the productivity growth for the developed countries continued increasing from 1990 to 2005, decreased from 2005 to 2010 (when recall the global financial crisis occurred), increased again from 2015 to 2019, and also increased over the whole period 1990–2019. The productivity growth for the developing countries continued decreasing from 1995 to 2005, continued increasing from 2005 to 2015. Thus, the results in Table 4 indicate that only developed countries achieved about $15.83\%$ percent increase in productivity, while productivity for the whole sample of all countries and for the sample of developing countries remained unchanged. This result is different from the simple mean MPI presented in Table 3, where we find that only developing countries achieved about $17.72\%$ percent increase in productivity, while productivity for the full sample of all countries remained unchanged and the productivity change for the developed countries is mixed.

To conclude this section, this empirical illustration shows fairly different results between the weighted and nonweighted methods of MPIs, illustrating the importance of deploying both approaches to check the changes of productivity growth. Moreover, our illustration confirms again our main takeaways from the MC results that the use of (iii) or (v) is advisable for relatively small samples (e.g., up to around $50$ ). To enable researchers to choose and use any of the considered methods, we provide the programming code for this empirical illustration.Footnote 11

6. Conclusions

The CLT results for the simple mean (Kneip et al. Reference Kneip, Simar and Wilson2021) and aggregate MPI (Pham et al. Reference Pham, Simar and Zelenyuk2024) estimated via DEA are important recent developments of the statistical theory for productivity analyses, complementing those for efficiency analyses (Kneip et al. Reference Kneip, Simar and Wilson2015, Reference Kneip, Simar and Wilson2016, Simar and Wilson, Reference Simar and Wilson2020, etc). However, for relatively small sample sizes and large dimensions, the coverages of the estimated confidence intervals constructed using the CLT results for the simple mean and aggregate MPI are below the nominal coverage. This under-covering phenomenon in relatively small sample sizes was also observed for the simple mean and aggregate efficiency.

Some of the recent improving methods were proposed by Simar and Zelenyuk (Reference Simar and Zelenyuk2020), Nguyen et al. (Reference Nguyen, Simar and Zelenyuk2022), Simar et al. (Reference Simar, Zelenyuk and Zhao2023), and Simar et al. (Reference Simar, Zelenyuk and Zhao2024) for the simple mean and aggregate efficiency. However, whether these methods are effective to improve the finite sample performance of CLT results for the simple mean and aggregate MPI remains unknown. This paper fills this gap in the literature of efficiency and productivity analyses by thoroughly examining the performance of these two methods in Nguyen et al. (Reference Nguyen, Simar and Zelenyuk2022) and Simar et al. (Reference Simar, Zelenyuk and Zhao2023) for the simple mean and the aggregate MPI through extensive simulations and we also use one empirical data set to illustrate their differences.

In the Monte-Carlo simulations, we find that the method adapted from Simar et al. (Reference Simar, Zelenyuk and Zhao2023) to the MPI context could provide a better performance for the simple mean and the aggregate MPI for relatively small samples (e.g., up to around $50$ , perhaps $100$ ) and after that the original methods from Kneip et al. (Reference Kneip, Simar and Wilson2021) and Pham et al. (Reference Pham, Simar and Zelenyuk2024) are recommended.

Supplementary material

The supplementary material for this article can be found at https://doi.org/10.1017/S1365100525000094.

Acknowledgments

Valentin Zelenyuk acknowledges the support from the Australian Research Council (FT170100401) and The University of Queensland. Shirong Zhao acknowledges the support from National Natural Science Foundation of China (grant number 72401056). We thank Zhichao Wang for his feedback and assistance in checking and replicating the computations for this paper. We also thank Paul Wilson as the programming codes used in this paper involve some earlier codes from Paul Wilson. Feedback from Léopold Simar and Evelyn Smart is appreciated. These individuals and organizations are not responsible for the views expressed here. These authors contributed equally: Valentin Zelenyuk, Shirong Zhao.

Competing interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and publication of this article.

Footnotes

1 For a comprehensive review, see Ch4 and Ch7 of Sickles and Zelenyuk (Reference Sickles and Zelenyuk2019).

2 We appreciate the reviewer for highlighting this point.

3 Also see some earlier developments in the PhD dissertation, Pham (Reference Pham2019).

4 Simar et al., (Reference Simar, Zelenyuk and Zhao2024) show that the main asymptotic results hold for any $\gamma \gt \text {min}(\kappa /2,1/4)$ , and their extensive MC results suggest that choosing $\gamma$ equal to or near $\kappa$ , usually provides the best coverage. In our MC evidence presented in Section 4, we also try $\gamma =0.75\kappa$ as a robustness check. The results indicate that $\gamma = \kappa$ performs better in MPIs compared to $\gamma = 0.75\kappa$ , which aligns with the findings of Simar et al. (Reference Simar, Zelenyuk and Zhao2024). While these results are not included in the paper, they are available upon request.

5 Note that $\widehat B_{i,s,t}$ depends on $\kappa$ and K, but they are dropped to simplify the notation.

6 See EC.3 in Pham et al. (Reference Pham, Simar and Zelenyuk2024) for more details on how to generate the correlated numbers.

7 Tables C.1 and C.2 in Appendix C present the values for the coverage of estimated confidence intervals for the simple mean and aggregate MPI, respectively.

8 Note that the standard CLT (i) is also correct for the case $p=q=1$ .

9 For example, suppose the true efficiency scores are 1.4 and 1.5, but their DEA estimates are 1.3 and 1.4, respectively: the absolute value of the bias for both cases is 0.1, however if the interest is about the ratio of the true efficiency scores, i.e., 1.5/1.4 = 1.0714, it is still fairly well approximated by the ratio of their estimates, even though those estimates are biased (according to the same estimator). Indeed, 1.4/1.3 = 1.0769, implying the absolute value of the bias is only 0.0055.

10 For example, we find that the bias-corrected simple mean MPI estimate ( $\exp {(\widehat {\mathcal{M}}_n - \widehat {B}_{\mathcal{M},n})}$ ) is $1.0579$ , while it is $1.0640$ in Pham et al. (Reference Pham, Simar and Zelenyuk2024). The small difference comes from the estimated bias for the simple mean and aggregate MPI, which uses the generalized jackknife method by randomly splitting the sample into two sub-samples many times.

11 See https://github.com/srzhao89/zz-mpi for more details.

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Figure 0

Table 1. The values of $\beta _j$ and $w_j$

Figure 1

Table 2. List of notations used for MC experiments

Figure 2

Figure 1. Coverages of various estimated confidence intervals for the simple mean malmquist productivity indices for the 90% nominal coverage, with $\delta =0.04$.

Figure 3

Figure 2. Coverages of various estimated confidence intervals for the simple mean malmquist productivity indices for the 95% nominal coverage, with $\delta =0.04$.

Figure 4

Figure 3. Coverages of various estimated confidence intervals for the aggregate malmquist productivity indices for the 90% nominal coverage, with $\delta =0.04$.

Figure 5

Figure 4. Coverages of various estimated confidence intervals for the aggregate malmquist productivity indices for the 95% nominal coverage, with $\delta =0.04$.

Figure 6

Table 3. Estimation results for the simple mean MPI of countries/Regions

Figure 7

Table 4. Estimation results for the aggregate MPI of countries/Regions

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