Testing for (efficiency) catching-up.

1. Introduction

Economic growth research has received substantial recognition in recent years. In particular, two major strands of research have dominated the literature. One approach uses the cross-sectional type of regressions found in Baumol (1986), which seek to determine whether there is a tendency for the world's economies to converge over time (poor catching up with the rich). The other strand decomposes growth into components attributable to capital deepening and technological progress going back to Solow (1957). However, there is a third strand of research that has become increasingly popular; a method based on Malmquist productivity indexes (Caves, Christensen, and Diewert 1982), computed via the data envelopment analysis (DEA) estimator. Beginning with Fare et al. (1994), this strand has introduced a third component into economic growth: efficiency, the ability of a given country to fully exploit its available resources in producing total output. While much of the mainstream research suggests making adjustments to the input mix, if the DEA approach shows that efficiency is found to affect the growth of labor productivity, then perhaps policymakers should also address methods that would improve efficiency. Conceptually, the efficiency component is nothing but the residual, somewhat like the "Solow residual," that proxies for the aggregated effect of various factors, other than technology and standard inputs on producing total output. This efficiency component can also be understood through Leibenstein's (1966) "X-efficiency" concept, related to the internal and external motivation of an agent. In our case, X-efficiency would be related to the aggregate result of influence by local and international institutions on each particular country. X-efficiency is an abstract concept, which of course is unobserved, and in practice is often proxied via the Debreu (1951)-Farrell (1957) measure of technical efficiency, which is usually estimated using the DEA estimator (e.g., see Leibenstein and Maital 1992).

The DEA method of estimating efficiency has gained its popularity because of several advantages over other methods. Perhaps the main advantage of DEA-type estimators is that they are nonparametric, in the sense that they do not require any parametric assumption on the structure of technology (e.g., Cobb-Douglas) or on the inefficiency term. Another important advantage is that, as long as inputs and outputs are measured in the same units of measurement, an assumption about complete homogeneity of considered economic agents is not needed. This means that the population of economic agents can potentially consist of different subpopulations governed by different distributional laws on the generation of the input-output mix and on inefficiency. For our purpose, this means that certain groups of countries (developed vs. developing) can potentially have different distributions of efficiency scores and different group efficiencies, which is what we aim to investigate in this study.

While there are a number of other advantages of the DEA method, there are also some drawbacks, and, as a result, it has received some opposition. There are those who believe that the entire world does not follow one unique production frontier. This controversy is reconciled with the notion of the so-called best practice frontier, which could be considered as an envelope of all possible frontiers for the production process feasible at a particular time.

There are also others who believe that DEA has inherent flaws. One of those flaws is that traditional (or old paradigm) DEA methods did not have a solid statistical foundation behind them. This, however, has been changed with seminal works on consistency of the DEA estimator by Korostelev, Simar, and Tsybakov (1995) and Kneip, Park, and Simar (1998), and on limiting distribution and consistency of bootstrap by Kneip, Simar, and Wilson (2003).

One of the most common critiques of the DEA approach is that it assumes away any measurement error and so could potentially suffer from outliers. For example, Koop, Osiewalski, and Steel (1999) state that "the sensitivity of DEA to outliers is no doubt one of the weaknesses of the DEA approach. In particular, it is difficult to present some measure of uncertainty (e.g. confidence intervals) using DEA methods." To combat comments such as these, Simar and Wilson (1998, 2000) and others have introduced bootstrapping into the DEA framework. Their methods, based on statistically well-defined models, allow for consistent estimation of the production frontier, corresponding efficiency scores, as well as standard errors and confidence intervals. Although advances were made to DEA, these have not been included in many recent papers that examine macroeconomic growth. Recently, Kumar and Russell (2002) employed standard DEA production-frontier methods to analyze convergence by decomposing labor productivity growth into components attributable to technological change, technological catch-up (changes in efficiency), and physical capital accumulation. They find the main factor driving economic growth to be capital accumulation. Henderson and Russell (2005) extend Kumar and Russell (2002) by adding human capital accumulation into the decomposition and show that about one-third of the productivity growth attributed by Kumar and Russell to physical capital accumulation should instead be attributed to the accumulation of human capital. Further, they show that the qualitative shift from a unimodal to a bimodal distribution in labor productivity (over a 25-year period) is accounted for primarily by efficiency changes. However, both of these papers are subject to the same scrutiny as Fare et al. (1994). If research is going to continue in this area, it needs to take notice of the advancements in DEA that address the current concerns. (1) In this paper we will focus on circumventing the drawbacks of DEA in an empirical context. Specifically, we will use the recently developed techniques in the statistical analysis of DEA estimates to check for robustness of efficiency estimates for a sample of 52 developed and developing countries, We will also investigate the issue of convergence/divergence in terms of efficiency across countries.

Various empirical studies on economic growth have brought convincing evidence that the world consists of at least these two groups: developed and developing countries. These groups are indeed distinct in their performance as well as in the key factors determining them (especially in institutional development). Quah (1996) has theoretically justified the possibility of the existence of two clubs in the world, with convergence within them and divergence between them, claiming empirical tendency for such phenomenon to be true. In our work, we will employ the notion of 'catching-up' first discussed in the seminal paper of Abramovitz (1986). Initially envisioning this phenomenon, Abramovitz's argument is based on the discovery of the considerable reduction in the coefficient of variation of growth rates within a group of 16 industrialized countries. Later, Fare et al. (1994) re-formalized the notion of catching-up as the decrease over time in the distance between the actual performance of a country and its potential, according to the best-practice frontier (i.e., as the decrease in inefficiency of the countries over time). In the spirit of F/ire et al. (1994), we will consider three types of (efficiency) catching-up: (i) within the entire sample, (ii) within distinct groups in the sample, and (iii) between these groups. We have two distinct groups in mind: developed and developing countries. (2,3)

Specifically, we first use the study of Henderson and Russell (2005) as a stepping-stone and compare our bootstrap bias-corrected efficiency scores with the results of their study. We then break the sample into two groups to see if the efficiency scores exhibit club convergence. Section 2 of this paper describes the theory of efficiency measurement and gives a brief description of the current advances in the literature. The third section describes the data, while the fourth section presents the results of the experiment.

2. Methodology

In this section we discuss the background of efficiency measurement as well as the latest research advances that will help us obtain more accurate measures for our problem. Although these procedures can be used to analyze any number of (macro or micro) decision-making units with multiple inputs and outputs, here we will describe the special case related to our example. For each country i (i = 1, 2, ..., n) we will use the period-t input vector [x.sup.t.sub.i] = ([K.sup.t.sub.i], [H.sup.t.sub.i] x [L.sup.t.sub.i]), where [K.sup.t.sub.i] is physical capital, and [H.sup.t.sub.i] x [L.sup.t.sub.i] is human capital ([H.sup.t.sub.i]) augmented labor ([L.sup.t.sub.i]). Further, [y.sup.t.sub.i] is a single output (gross domestic product--GDP) for country i in period t (all inputs and outputs are assumed to be positive). The technology of converting inputs into GDP for each country i, in each time period t, can be characterized by technology set

[T.sup.t.sub.i] [equivalent to] {([x.sup.t.sub.i], [y.sup.t.sub.i]) [[x.sup.t.sub.i] can produce [y.sup.t.sub.i]}.

Equivalently, the same technology can be characterized by the output sets

[P.sup.t.sub.i]([x.sup.t.sub.i]) [equivalent to] ([y.sup.t.sub.i] | [x.sup.t.sub.i] can produce [y.sup.t.sub.i]), [x.sup.t.sub.i] [member of][R.sup.2.+].

Here we assume that the technology follows standard regularity assumptions, under which the Shephard (1970) output oriented distance function