A Common Weight Approach to Construct Composite Indicators: The Evaluation of Fourteen Emerging Markets

Abstract

This study presents an ongoing project, emerging market (EM) evaluation project, of the Taiwan Institute of Economic Research (TIER). The purpose of this project is to construct a composite indicator (CI) named as growth potential index (GPI) for selecting the promising EMs, in which to begin new or expand existing business is attractive to governments, firms, and investors. However, weight determination is one of the most difficult tasks in the construction process of a CI. A new approach inspired by the Z score and rooted in data envelopment analysis (DEA) is proposed to objectively determine the common weights for constructing the GPI without requiring data normalisation beforehand. The same dataset is used to compare the proposed common weight approach with the equal weighting method (currently used by the TIER), the widely used DEA-CI model, and the first common weight DEA-CI model. Spearman’s rank correlation test revealed a high positive correlation between the GPIs obtained by the proposed approach and each considered method. The major findings include: (1) China is the most promising EM; (2) Argentina, China, Malaysia, Poland, and Russia are above-average EMs; (3) India, Indonesia, Saudi Arabia, South Africa, and Thailand are below-average EMs; and (4) of the so-called BRIC countries (Brazil, Russia, India, and China), China is the best EM, and India is the worst EM.

Appendices

Appendix 1: DEA-CI Model

Assume that n DMUs are evaluated. Each DMU _j (j = 1, 2, …, n) with m indicators, Y _rj (r = 1, 2, …, m), which are known, positive, and the larger the better. The CI score of DMU _j denoted as CI _j can be expressed by

$${\text{CI}}_{j} = \mathop \sum \limits_{r = 1}^{m} w_{r} Y_{rj} ,$$

(2)

with a positive weight set w _r (r = 1, 2, …, m). When calculating the CI score of DMU _j , the typical approach is to choose a favorite set of weights for each DMU in order to maximize its CI score. This concept is in accordance with that of DEA and can be formulated as follows (see Cherchye et al. 2007):

$$\begin{aligned} {\text{Maximize}} & \quad CI_{j} = \mathop \sum \limits_{r = 1}^{m} w_{r} Y_{rj} \\ {\text{subject to}} & \quad \mathop \sum \limits_{r = 1}^{m} w_{r} Y_{rj} \le 1,\quad j = 1,2, \ldots ,n, \\ & \quad w_{r} \ge \varepsilon > 0,\quad r = 1,2, \ldots ,m, \\ \end{aligned}$$

(3)

where w _r is the decision variable and ε is the non-Archimedean small number. The CI sore (${\text{CI}}_{j}^{*}$) can range from 0 (worst) to 1 (best). If ${\text{CI}}_{j}^{*} = 1$, then DMU _j is a relative best practice; otherwise, the DMU is relatively inefficient.

Appendix 2: Classifying the Higher, Lower, and Overlapped DMUs

The output-oriented stratification DEA model (Seiford and Zhu 2003) and output-oriented super-efficiency DEA model (Andersen and Petersen 1993; Zhu 2003) are useful to classify n DMUs into hD _i , lD _i , and oD _i in terms of their positions on the projected space relative to the μ.

Let DMU _j (j = 1, 2, …, n) be n DMUs that convert m inputs x _ij (i = 1, …, m) into s outputs y _rj (r = 1, …, s). If the set of all DMUs is defined as J ¹ and the set of efficient DMUs in J ¹ is defined as E ¹, the sequences of J ^z and E ^z can be defined recursively as J ^z+1 = J ^z − E ^z. The output-oriented stratification DEA model is formulated as follows:

$$\begin{aligned} & \eta_{k}^{z} = \hbox{max} \eta \\ & {\text{subject to}} \\ & \mathop \sum \limits_{{j \in J^{z} }} \lambda_{j} x_{ij} \le x_{ik} \\ & \mathop \sum \limits_{{j \in J^{z} }} \lambda_{j} y_{rj} \ge \eta y_{rk} \\ & \lambda \ge 0, \\ \end{aligned}$$

(4)

where x _ik and y _rk are i-th input and r-th output of DMU k. The DMUs in set E ¹ define the first-level efficient frontier. When z = 2, model (4) gives the second-level efficient frontier after excluding the first-level efficient DMUs. In this manner, several levels of efficient frontiers are defined. Then E ^z consists of the z-th level efficient frontier. Additionally, the output-oriented super-efficiency DEA model is shown as follows:

$$\begin{aligned} {\text{Maximize}} & \quad \emptyset_{k}^{\sup } \\ {\text{subject to}} & \quad \mathop \sum \limits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} \lambda_{j} x_{ij} + s_{i}^{ - } = x_{ik} ,\quad i = 1, \ldots ,m \\ & \quad \mathop \sum \limits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} \lambda_{j} y_{rj} - s_{r}^{ + } = \emptyset_{k}^{\sup } y_{rk} ,\quad r = 1, \ldots ,s \\ & \quad \mathop \sum \limits_{{\begin{array}{*{20}c} {j = 1} \\ {j \ne k} \\ \end{array} }}^{n} \lambda_{j} = 1 \\ & \quad \lambda_{j} ,s_{i}^{ - } , s_{r}^{ + } \ge \varepsilon , \\ \end{aligned}$$

(5)

where s ⁻ _i is the input excess of i-th input and s ⁺ _r is the output shortfall of r-th output. λ _j is a non-negative variable to construct a convex combination of other DMUs to compare evaluating DMU, and ε is a non-Archimedean element in order to optimize s ⁻ _i and s ⁺ _r and ensure that these variables are considered in solution. $\emptyset$ ^sup _k is the super efficiency value of DMU _k .

In the first step, the output-oriented stratification DEA model that can partition all evaluated DMUs into various efficiency levels is applied to identify hD _i , lD _i , and oD _i . Technically, DEA is a frontier efficiency method because all DMUs located on the efficient frontier have 100% efficiency (note that, here, DEA is directed towards effectiveness rather than efficiency because all this paper discusses the situation, where DEA with a dummy input and multiple outputs). Withdrawing the DMUs located on the first efficiency level results in the situation in which the remaining DMUs construct the second efficiency level. When the DMUs located on the second efficiency level are withdrawn, the remaining DMUs construct the third efficiency level until no DMUs remain. As a result of this stratification process, the DMUs on the i-th efficiency level are dominated by the DMUs on the j-th efficiency level if i > j (Seiford and Zhu 2003). Because of this property, the DMUs located on the h-th efficiency level and the l-th efficiency level can be respectively recognized hD _i and lD _i if the μ is located on the c-th efficiency level and l > c > h.

The second step further deals with the DMUs which are located on the same efficiency level with the μ. Note that the second step can be omitted if no DMU locates on the same efficiency level with the μ. In this step, the output-oriented super-efficiency DEA model is used to classify the DMUs located on the same efficiency level with the μ. The output-oriented super-efficiency DEA model ranks the efficient DMUs by excluding a DMU from its own reference set in the envelopment model (Andersen and Petersen 1993). The distance between the excluded DMU and the new facet constructed by the remaining efficient DMUs is the so-called super efficiency. This property enables the use of the μ for “paired comparison” with another DMU so that their relative positions on the projected space can be clearly identified. The paired comparison determines (1) whether the output-oriented super efficiency of the DMU is smaller (or larger) than that of the μ, which indicates that the position of this DMU is relatively higher (or lower) than that of the μ on the projected space and that the DMU is the hD _i (or lD _i ); and (2) whether the output-oriented super efficiency of the DMU is equal to that of the μ, which indicates that the DMU overlaps the μ on the projected space and that this DMU is the oD _i . After the above two steps, n DMUs can be completely classified into hD _i , lD _i , and oD _i .