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 $r=5,6$ .

3.1 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.

4.1 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

4.2.2 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 1–4 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 1–4, 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.