More on Log-Change Index Numbers - Theil, Henri (The Review of Economics and Statistics, 1974)
The logarithmic discrepancy of the factor reversal test, to be written DA for (2) and DB for (3), is of the fifth order of smallness (05), but the leading 05 term of DB is closer to zero than that of DA, which is the reason why Sato prefers (3). If the expenditure shares do not change very much, xiyi being close to zero for each i, the discrepancy D is very close to zero for most reasonable choices of f ( ). This applies to the vast majority of annual time series. Tornqvist's choice of the arithmetic average of x and y is then adequate; it has the additional advantage that the sum over i of f(xi, yi) equals 1, which simplifies the index formulas. My own interest in pursuing this topic further was stimulated by C. M. Walsh's proposal to specify f ( ) as the geometric mean of x and y. It can be shown that the leading 03 term of the discrepancy (1) for the geometric mean is equal to minus onehalf of the leading 03 term of that of the arithmetic mean. My immediate reaction was to take the weighted arithmetic mean, with weights equal to 2/3 and 1/3, of the geometric mean and the arithmetic mean so as to eliminate the leading 03 term. This is precisely Sato's choice (3). But I rejected this choice, because D is unbounded under the specification (3) for 0 -O x, y 1. Suppose that x 0, y > 0, so that fB(X, y) 1/6(y) > 0. If this applies to some subscript i, the discrepancy (1) becomes infinitely large. Sato