# The conditions for eliminating first order aggregation bias in input-output models when technical coefficients are estimated by a member of the k-class

Chapter

## Abstract

Over the past twenty-five years, the problem of aggregation bias in input-output analysis has been investigated by many scholars.^{1} This interest has arisen largely because aggregation bias has important consequences for input-output forecasting: In particular, it may cause the estimates of total output to be inaccurate. However, judging from the current literature on this subject, there is little that can be done to remove these inaccuracies. As has been pointed out repeatedly, the conditions which have been established for zero aggregation bias are quite restrictive and unlikely to be satisfied in practical situations.

### Keywords

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### Notes

- 1.For example, see Kenjiro Ara, ‘The Aggregation Problem in Input-Output Analysis,’
*Econometrica*, XXVII, April, 1959; Michio Hatanaka, ‘A Note on Consolidation Within a Leontief System,’*Econometrica*, XX, April, 1952; Yoshinori Morimoto, ‘On Aggregation Problems in Input-Output Analysis,’*Review of Economic Studies*, XXXVII (1), January, 1970; and Henri Theil, ‘Linear Aggregation in Input-Output Analysis,’*Econometrica*, XXV, January, 1957. For a more complete listing of the better known works in this area, consult the bibliography.Google Scholar - 2.Theil, ‘Linear Aggregation.’ It might be noted that elimination of first order aggregation bias does not constitute elimination of total aggregation bias. The distinction between these two concepts will be made explicit shortly.Google Scholar
- 3.Theil, ‘Linear Aggregation’ and Morimoto, ‘On Aggregation Problems.’Google Scholar
- 4.Recall that a more complete discussion of random disturbances in input-output relations are given in Chapters 2 and 3.Google Scholar
- 5.Note that equation (5.2) contains no statement about which technique ought to be used to estimate the
*α*_{th}. As was mentioned, in the introduction, both the ratio and the*k*-class techniques will be considered, but this will be done at a later point in the chapter. In addition, equation (5.2) will produce ‘columns only’ estimates of the*α*_{th}.The aggregation bias arising from ‘rows only’ estimates are ignored in this chapter.Google Scholar - 6.In the analysis to follow, the subscript refers to the order of the vector or matrix in question. For example, in equation (5.5),
*X*_{N}is*Nx*1 and*A*_{NN}is*NxN.*Google Scholar - 7.The discussion in the remainder of this section closely follows Theil, ‘Linear Aggregation,’ p. 114. It should be noted that Theil’s focus was exclusively upon deriving an expression for aggregation bias.Google Scholar
- 8.Fredrick Waugh, ‘Inversion of the Leontief Matrix by a Power Series,’
*Econometrica*, XVIII, April, 1950.Google Scholar - 9.This definition of first order aggregation bias was first given in Theil, ‘Linear Aggregation,’p. 117.Google Scholar
- 10.It should be recalled that the ratio estimator for the macro coefficients was discussed in Section 2 of Chapter 3.Google Scholar
- 11.To economize on notation, in the remainder of this chapter denote where
*n*_{j}is the sample size in sector*j*, is simplified as Σ_{r}.Google Scholar - 12.No distinction is made as to the micro sectors from which these firms are drawn.Google Scholar
- 13.Henceforth, it is assumed that once the micro sectors have been aggregated into macro sectors, the firms in question are appropriately renumbered. This implies that variables such as ζ
_{th}may be summed over*r*.Google Scholar - 14.For a closely related theorem, Yoshinori Morimoto, ‘On Aggregation Problems,’ p. 121.Google Scholar
- 15.Ibid.Google Scholar
- 16.Actually, the condition for eliminating first order aggregation bias in the ratio estimator as given in equation (5.31) was stated and proved as a theorem by Theil, ‘Linear Aggregation.’Google Scholar
- 17.Recall that in Chapter 3,
*WS*_{j}^{(r)}and*PG*_{j}^{(r)}denoted, respectively, wages and salaries and payments to government. Since both variables are exogenous variables in (3.16), they are used in the auxiliary regression just described.Google Scholar - 18.Roberto S. Mariano and Takamitsu Sawa, ‘The Exact Finite Sample Distribution of the Limited Information Maximum Likelihood Estimator in the Case of Two Endogenous Variables,’
*Journal of the American Statistical Association*, LXVII, March, 1972, p. 161–62.Google Scholar

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© H. E. Stenfert Kroese B. V., Leiden 1976