Factor analysis applied to developed and developing countries • u Tilburg Studies on Economics 1 Edited by the Tilburg Institute of Economics of the Tilburg School of Economics Members of the Board P.A. Verheijen, Chairman Th.C.M.J. van de Klundert H.W.J. Bosman Director of Research J.J.J. Dalmulder A study on Econometrics Factor analysis applied to developed and developing countries J.R.F. Schilderinck Department of Econometrics, Tilburg School of Economics Preface by J.J.J. Dalmulder Professor in Econometrics and Mathematical Economics, Head of the Department of Econometrics, Tilburg School of Economics 1970 Rotterdam University Press Wolters -Noordhoff Publishing, Groningen The Netherlands Distributors: Rotterdam University Press, P.O. Box 1474, Rotterdam, The Netherlands. ISBN 978-90-237-2901-3 ISBN 978-94-015-7202-6 (eBook) DOI 10.1007/978-94-015-7202-6 Copyright © 1970 by Universitaire Pers Rotterdam No part of this work may be reproduced in any form, by print, photoprint, microfilm or any other means, without written permission from the publisher. Preface The Tilburg Institute of Economics - Institute of the Economic Faculty of the Tilburg University - proposes itself to publish results of economic research taking part in the F acul ty . To facilitate the choice of the potential reader, every publication will be marked by the department, where the publication took its origin. As Mr. Schilderinck's 'Factor Analysis' applied to developed and developing countries, is a result of research in the Econometric Department, it is mark ed Econometrics. Every publication will be published under the supervision of the head of the department. For this reason this preface is written by the head of the Econometric Department. Mr. Schilderinck's study forms an introduction to a larger project of research, which proposes itself to develop methods of analysis, which try to eliminate the difficulties of multi-collinearity and the arbitrariness of the introduction of lags in regression analysis. This study applies the method of factor analysis to statistical material collected by the Institute of Development Problems of our University. Prof. Dr. J. J. J. Dalmulder Head of Department of Econometrics v Contents PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. v INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .. IX 1. AIM OF FACTOR ANALYSIS . . . . . . . . . . . . . . . . . . . . . 2. THE METHOD OF FACTOR ANALYSIS. 3 2.1 Normalisation of the variables . . . 3 2.2 Correlation and variance in factor analysis . 4 2.3 The model of the factor analysis 5 2.4 Solution of the model ..... . 9 2.5 Interpretation of the final aspects . 17 3. APPLICATION OF FACTOR ANALYSIS TO DATA FROM DEVELOPING COUNTRIES . . . . . . . . . . . . . . . . . . . .. 22 3.1 The variables . . . . . . . . . . . . . . . . . . 22 3.2 Working out the factor analysis . . . . . . . . 26 3.3 Interpretation of the results according to areas 33 3.4 Interpretation of the results according to final aspects. 43 SUMMARY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51 VII Introduction With statistical research on a fairly large scale, one is constantly establishing relationships between the variables under examination without being in a position to state with certainty the difference in significance between these relationships of the variables. This problem is not peculiar to economic research; it is also present, for example, in sociological, psychological and biological research. The problem is statistical; various methods have been developed to reveal the most important of the many possible relationships between the variables. One of these methods is factor analysis. Factor analysis is based on the assumption that there are a number of general causal factors which give rise to the various relationships between the variables under examination. The number of general causal factors will on the whole be considerably smaller than the number of relationships. Many relationships between variables are, for the most part, due to the same general causal factor. These general causal factors are referred to in the literature as factors, components, conditions, or dimensions. This difference in defmition arises in fact from the different technical ways of developing factor analysis. The method of factor analysis can now therefore be defined as follows. Factor analysis is the attempt, based on statistical observations, to deter mine the quantitative relationships between variables where the relation ships are due to separate conditioning factors or general causal factors. By a relationship is meant a certain pattern of motion between two or more of the variables under examination. Such a pattern of motion is expressed in coefficients or percentages which indicate to what extent the variances of the variables in question are influenced by a certain general causal factor. This factor is common to the variables which form part of a specific pattern of motion. IX 1. Aim of factor analysis The aim of factor analysis is to group by means of a kind of transformation the unarranged empirical data of the variables under examination in such a way that: (a) a smaller whole is obtained from the original material, whereby all the information given is reproduced in summarised form; (b) Factors are obtained which each produce a separate pattern of motion between the variables; (c) the pattern of motion can be interpreted logically. The number of possible patterns of motion is dependent on the number of variables involved in the examination. If there are n variables in a factor examination, then n2 different relationships between variables (factors) are theoretically possible, of which only n are possible at the same time. A selection must be made from this theoretically possible number. Factor analysis does not only perform this selection of important rela tionships but also aims at interpreting the relationship which results from each of the separate factors. As each choice and each interpretation is to a greater or less extent subjective, factor analysis is open to criticism on this point. Having obtained the results of a factor analysis, one cannot claim to have established the only possible true relationships. Despite this apt criticism, the method of factor analysis is an important aid in observing the mutual relationships between the variables under exami nation. These relationships can, furthermore, improve the establishment of a certain theory. Conversely, a certain theory can also be tested by means of a factor analysis. If, for example, one assumes certain relationships between variables, one must be able to justify this assumption by means of a factor analysis. If one determines the relationships to be observed by means of a system of regression equations, then a factor analysis indicates how many equations are necessary before a solution can be reached. Factor analysis can produce other important results in comparative ana lysis. For example, if one typifies a number of enterprises by various rele- vant index numbers, then one can appraise the enterprises concerned by comparing step by step the corresponding factors. Factor analysis is used most of all in comparative analysis. 2 2. The method of factor analysis 2.1 NORMALISATION OF THE VARIABLES In factor analysis, one tries to discover the general factors which cause the variables in question to show a relationship between each other. This is done by defining a number of vectors which fully describe the variables. Each vector represents another general causal factor, condition, aspect or which ever other name it is given. This kind of a factor can be regarded as a theoretical or hypothetical variable. In general, factor analysis does not begin with the original observations of the variables. It sets about normalising them in a certain way in order to make a mutual comparison possible. Normalisation is done by expressing the deviations from the original observations with regard to their arithme tical mean in their standard deviations. If the number of observations ranges from 1 to N and the number of variables from 1 to n, ani Xi represents a variable for which the observations have been normalised, then the follow ing formula is obtained: Xig z. =- (1) 19 UXj where Xig = Xig - Xi (i = 1, 2, ... n; g = 1, 2, ... N) :EX. - g 19 X=-- (2) 1 N Fa! - 2 2 ~(X.-X.) :Ex . U = a = y. /' g 19 I =y. ./ g 19 (3) Xi Xi N N The expected value (Le. the mean) of a normalised variable like this equals 0 and its variance equals 1. 3