Using Weighted Least Squares to Deflate Input–Output Tables

Abstract

This article proposes a balancing procedure for the deflation of input–output (I-O) tables from the viewpoint of users. This is a ‘subjective’ variant of the Weighted Least Squares (WLS) method, already known in the literature. It is argued that it is more flexible than other methods, and it is shown that SWLS subsumes the first-order approximation of RAS as a special case. Flexibility is due to the facts that (a) users can attach differential ‘reliability’ weights to first (unbalanced) estimates, depending on the confidence they have in the different parts of their pre-balancing work, (b) differently from RAS, one is not bound to take any row or column total as exogenously given, and (c) additional constraints can be added to it. The article describes also how SWLS was utilised to estimate a yearly (1959–2000) series of constant-price I-O tables for the Italian economy.

Keywords:

• I-O tables
• deflation
• balancing
• weighted least squares

Acknowledgements

The author is indebted to Italo Lavanda for helpful observations. Many thanks are also due to the Editor of this journal and to two anonymous referees for comments and suggestions that helped, hopefully, to clarify many previously obscure points. Any remaining errors are the author's.

Notes

By this we mean, as usual, studying changes in input (both domestic and imported) and labour coefficients, changes in the composition of total output and final demand, and changes in all indicators derived from those coefficients/shares.

There are some exceptions to this. For instance, if one is prepared to assume product homogeneity along each row of the I-O tables, it is not difficult to show that vertically integrated labor requirements (i.e. labor contents in final demand items) do not depend on the price-index vector, that is current-price and constant-price computations give the same results.

It would be a different matter, of course, if a user had privileged and very reliable information on some of the items a-d above for the whole period and for all sectors to be treated: see, for example, Dietzenbacher and Hoen (1998).

Alternative routes to double deflation were proposed, for example, by David (1962), Sato (1976) and Durand (1994); the first one is more simple-minded, assuming equality of output and value added price indexes. From our current perspective, they all share with double deflation the feature that one starts with exogenously given price indexes, and ‘solves for’ deflated value added.

All the methods quoted when introducing double deflation above, apart from David (1962), could in principle be ‘inverted’ to obtain something similar to the second route we are discussing. Route II is more in the vein of, but still different from, a linear version of Sato (1976), fixed capital being not considered. See also Febrero and De Juan (2002) for critical comments on Sato (1976), which are, however, out of the scope of the present paper.

Applications also exist in the field of estimation of social accounting matrices (Robinson et al. 2001) and of multiregional I-O models (Canning and Wang 2005).

More generally, one could take also co-variances into account, so passing from a ‘weighted’ to a ‘generalized’ version of the method. We will, however, limit ourselves to the weighted case.

From the numerical point of view, in order to avoid terms going to infinity in the loss function, these variances should be set equal to some very small number ϵ.

See also Lahr (2001) and Dalgaard and Gysting (2004).

As regards this point, if a ‘generalised’ method were used considering co-variances as well, one might assume positive (resp. negative) co-variances between sectoral inventory changes and total output (resp. total demand).

V is diagonal, so we are using weighted, not generalized, LS. But neither ˆ r nor V is a scalar multiple of the identity matrix, which would lead to OLS instead of WLS. Recall also that the solution (12) is homogeneous of degree zero in V: only relative reliabilities and variances matter. Finally, the variances of null elements are set equal to zero (see, however, note 8 above): this prevents the procedure from producing unwanted negative results.

Interpreting V as a Bayesian ‘prior’ on variances, one can show that the corresponding ‘posterior’, i.e. the covariance matrix of balanced estimates, is the positive definite matrix (V − VG′(GVG′)⁻¹ GV), which is ‘lower’ (in the positive-definite-matrix ordering) than V.

Dietzenbacher and Hoen (1999) prove that these difficulties are, as expected, grounded on aggregation problems.

In the RAS tradition, the ‘first estimate’ might be directly a table of an adjacent year, or one built using the coefficients thereof and the gross outputs of the target year.

Of course, if the user has reason to believe that row and column sums are perfectly known, and if reliabilities are all equal among themselves, then SWLS collapses to the first-order approximation of RAS.

De Mesnard (2006), being interested in a different problem − namely, measuring structural change by means of biproportional, among which RAS, filters − proves that if no aggregation problem is present, then results are unaffected by using data in currency units instead of physical ones (or current-price data instead of constant-price ones). This is the ‘theoretical’ side of the story: ‘Nevertheless, a slight complication occurs because of aggregation’ (De Mesnard, 2006, p. 467). Since users live just in these slightly more complicated worlds, it follows that, as recommended by Dietzenbacher and Hoen (1998, 1999), RAS is to be preferred to double deflation to compute constant-price tables. And, if the arguments put forward above are acceptable, SWLS should be a welcome route.

For instance, De Mesnard (2004), Jackson and Murray (2004), and Oosterhaven (2005) compare balanced tables with ‘true’ ones, obtaining performance indicators of different balancing methods. This exercise was prevented in our case by the unavailability of official constant-price Italian I-O tables.

The aggregation adopted in the I-O dataset counts 42 sectors, i.e. the standard NACE-44 classification, with the three sectors of non-market services summed together; from 1959 to 1964, the number of sectors is 38, some private service sectors being aggregated.