The Problem of Estimation: A Seventeenth-Century Controversy and Its Bearing on Modern Statistical Questions, Especially Index-Numbers

As Reviewed by Irving Fisher Mr. Walsh is the author of "The Measurement of General Exchange Value," one of the chief classics on the subject of index numbers. The present book is partly a condensation of "The Measurement of General Exchange Value," and partly an extension of its ideas to other problems than that of index numbers. Nevertheless the chief interest in the present volume centers, I think, in the author's discussion of index numbers. It is largely taken up with a discussion of the geometric mean as contrasted with other means. Mr. Walsh has delved deep in the lore to be found in libraries on the subject and has unearthed an interesting old discussion of Galileo on "the problem of estimation." Mr. Walsh begins his book by giving the arguments of Galileo and his opponent, which are very entertaining. This old problem reads "if a horse worth 100 pounds is estimated by one person at 1000 and by another at 10, which of these two estimates is the less erroneous, or are they equally erroneous?" According to one view 10 is much nearer the truth (100) than is 1000, the difference in the first case being 90 and in the second, 900. According to another view they are equally distant, one being 10 times as large and the other 10 times as little! Galileo took the latter view and Walsh approves. In the part devoted to index numbers, Walsh (page 102) confirms my views expressed in the March Number of the Quarterly Publications of the American Statistical Association on the best formula for an index number although his method of arriving at the result is quite different from my own. It is interesting to observe that Allyn Young of Harvard in the August number of the Quarterly Journal of Economics in an article entitled "The Measurement of Changes of the General Price Level," has reached the same result from a different angle. Walsh's argument is largely that the formula in question comes nearest to satisfying Westergaard's test, that is the so-called circular test that any index number should, if calculated from one year to another and from the second to a third and so on in a chain, give the same result as though calculated directly from the original base to the last year. In my forthcoming book on Index Numbers, to be published by the Pollak Foundation for Economic Research, I propose to show that, while index numbers which come near to satisfying Westergaard's test are better than those which fall far short of satisfying it, nevertheless there is an irreducible minimum of discrepancy which is not only inevitable in good index numbers but commendable. Much of Walsh's work in index numbers has been in the search for a formula which will completely fulfill Westergaard's test. The truth is that no such formula exists; at least not one which has different weights for each year to year comparison. But the search for such fulfillment, while in vain in the sense of being unsuccessful in its object, has nevertheless not been fruitless; for Walsh has laid secure foundations in this subject, which he was enabled to lay by virtue of his search for the impossible.