# GALTONIAN REGRESSION ACROSS COUNTRIES AND THE CONVERGENCE OF PRODUCTIVITY†

 Published date 01 August 1995 Date 01 August 1995 DOI http://doi.org/10.1111/j.1468-0084.1995.mp57003001.x
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 57,3(1995)
0305-9049 S3.00
GALTONIAN REGRESSION ACROSS
COUNTRIES AND THE CONVERGENCE OF
PRODUCTIVITYt
Peter E. Hart
I. INTRODUCTION
Many economists have used regression analysis to measure whether the
productivity levels of different economies tend to converge over time. See
Baumol (1986), De Long (1988), Baumol, Blackman and Wolff (1989),
Dowrick and Nguyen (1989), Barro (1991), Barro and Sala-i-Martin
(1991,1992). These results are alleged to be tainted with a statistical fallacy
associated with Galton's original regression towards the mean, Friedman
(1992), Quah (1993). The present paper uses the Galton model to explain
why these allegations are unfounded.
II. THE GALTON MODEL
Galton's (1889, 1892) initial hypothesis was that the average height of the
sons of one father was the same as the father's height. But if this were true, the
dispersion of men's height would increase over time because some of the sons
of the tallest father would exceed his height and some of the sons of the
shortest father would be even shorter. Since the dispersion stays the same,
extreme values must 'regress' towards the mean height.
Formally, let y( j, t - 1) denote the height of the ith father (measured in
terms of deviations from fathers' mean height) and suppose that each father
has one son with height y( i,t), (measured in terms of deviations from sons'
mean height). Suppose that their joint distribution is bivariate normal so that
the regression (in the modern sense) of y(t) on y(t 1) is linear and
homoscedastic. We have
y(i,t)=ßy(i,t-1)+e(i,t) (1)
I should like to thank Robert Barro, Simon Burke, John Cantwell, Mark Casson, Milton
Friedman and Danny Quah for their comments on an earlier and longer version of this paper,
Hart (1994), but they must not be held responsible for any of its shortcomings.
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