Theory and Methodology Estimating and bootstrapping Malmquist indices 1 L eopold Simar a,2 , Paul W. Wilson b, * ,3 a Institut de Statistique and CORE, Universit e Catholique de Louvain, Voie du Roman Pays 20, Louvain-la-Neuve, Belgium b Department of Economics, University of Texas at Austin, Austin, TX 78712-1173, USA Received 1 August 1996; accepted 1 October 1997 Abstract This paper develops a consistent bootstrap estimation procedure for obtaining con®dence intervals for Malmquist indices of productivity and their decompositions. Although the exposition is in terms of input-oriented indices, the techniques can be trivially extended to the output orientation. The bootstrap methodology is an extension of earlier work described in Simar and Wilson (Simar, L., Wilson, P.W., 1998, Management Science). Some empirical examples are also given, using data on Swedish pharmacies. Ó 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: DEA; Productivity; Resampling; Bootstrap; Malmquist indices 1. Introduction Fa Ère et al. (1992) merge ideas on measurement of eciency from Farrell (1957) and on measure- ment of productivity from Caves et al. (1982) to develop a Malmquist index of productivity change. Caves et al. de®ne their input-based Malmquist productivity index as the ratio of two input distance functions, while assuming no tech- nical ineciency in the sense of Farrell. F are et al. extend the Caves et al. approach by dropping the assumption of no technical ineciency and devel- oping a Malmquist index of productivity that can be decomposed into indices describing changes in technology and eciency. We extend the F are et al. approach by giving a statistical interpretation to their Malmquist productivity index and its components, and by presenting a bootstrap algo- rithm which may be used to estimate con®dence intervals for the indices. This work will allow re- searchers to speak in terms of whether changes in productivity, eciency, or technology are signi®- cant in a statistical sense. In other words, our methods can be used to determine whether indicated changes in productivity, eciency, or European Journal of Operational Research 115 (1999) 459±471 * Corresponding author. Tel.: 512 471 3211; fax: 512 471 3510; e-mail: wilson@eco.utexas.edu. 1 Pontus Roos graciously provided the data used in the empirical examples. We alone, of course, are responsible for any remaining errors or omissions. 2 Research support from the contract ``Projet d'Actions de Recherche Concert  ees'' (PARC No. 93/98±164) of the Belgian Government is gratefully acknowledged. 3 Research support from the Management Science Group, US Department of Veterans Aairs, is gratefully acknowledged. 0377-2217/99/$ ± see front matter Ó 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 7 ) 0 0 4 5 0 - 5