The most simple rectangular input-output models use two rectangular matrices: R a market coefficient matrix, A* a production coefficient matrix. A given exogenous demand Xo determines the sectorial activity levels X* = [I — RA*]-1Xo. We assume that A* is random with expectation A. We study the distribution of the "error" X* — X with X = [I — RA]-1Xo.
(1) For the statistically independent elements of A*, we analytically prove that X < EX*.
(2) In the more realistic case of statistically dependent elements of A*.
(a) One submatrix of A* with T non zero elements is chosen. The probabilistic model which generates the T coefficients is as follows: a* = (1 — μ)a + μ(S/n) b* où a* is the vector of the T random elements, a is the expectation of a* whose components are observed values of a real input-output model, S is the sum of components of a, μ is a parameter between zero and one, b* is a multinomial random vector with T components and parameters n, number of drawings during an experiment, and a/S, the corresponding probabilities.
We control the variability of a* through μ and n. For a given experiment, we get a realisation of A* and we compute X*. K independent experiments allow us to estimate the expectation and the variance-covariance matrix of X*, simultaneous confidence intervals for the expectation of the components of X*, and also a few global measures of errors on X*.
The Canadian model for 1961 (16 productive sectors, 40 commodities), is tested with that model.
The main result is: the relative errors, measured according to the variation coefficients, are greatly reduced when we pass from the "errors" on a* to the corresponding "errors" on X*.
(b) The same random model is also simultaneously applied to 2 or 3 sub-matrices of A*.
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