Academic journal article
By Blankmeyer, Eric
Atlantic Economic Journal , Vol. 37, No. 4
If the prices and the corresponding quantities of K commodities are observed in a cross section of R regions, the data may be arrayed in two matrices, P and Q respectively, each having R rows and K columns. Theil [Econometrica, 1960] defines the cross-value matrix C = PQ' as a positive-valued array whose typical element [c.sub.rt] measures the aggregate monetary value of the quantities in region t at the prices in region r. While the rank of C is usually the smaller of R and K, Theil proposes to approximate C by a matrix of rank one in the orthogonal regression model
C = pq' + E, (1)
where the R-element column vectors p and q are respectively the regional price and quantity indexes, and E is a matrix of independently distributed random errors. By minimizing the trace of E'E, Theil shows that the orthogonal least-squares estimate of the price index is the eigenvector corresponding to [lambda], the largest eigenvalue of CC'. Likewise, the quantity index is estimated by the eigenvector for the largest eigenvalue of C'C , which is also [lambda]. The algebraic properties of these best linear index numbers are examined in detail by Theft and by Kloek and DeWit [Econometrica, 1961].
However, these authors do not discuss the problem of inference, an understandable omission since the sampling properties of the eigenvectors may be intractable unless CC' and C'C are covariance matrices [Bat et al., Annals of Probability, 2007]. This is problematic because all the elements of C are positive. In any case, only large-sample results could be obtained analytically. On the other hand, resampling methods are a practical way to compute standard errors and test hypotheses for Theil's price and quantity indexes. …