EMPIRICAL ESTIMATION OF NON-LINEAR INPUT–OUTPUT MODELS: AN ENTROPY ECONOMETRICS APPROACH

† This article is a revised version of the paper that won the Wassily Leontief Memorial Prize 2014, for the best paper by authors younger than 40 submitted to the 22nd International Input–Output Conference, in Lisbon, Portugal.

Abstract

Despite theoretical advances, non-linear input–output models have been empirically applied only to a limited extent. This is mainly due to the fact that the number of parameters to be estimated is much higher than the number of available data points. Taking advantage of the recent proliferation of input–output databases and by applying an estimation strategy that relies on entropy econometrics, this paper suggests a way to estimate the parameters that characterize non-linear relationships between inputs and output. This non-linear modelling allows for considering time-specific input coefficients, instead of fixed ones. Several types of multipliers can be derived from this non-linear model, and the proposed generalized maximum entropy (GME) estimator allows estimating them from time series or cross-sectional datasets of input–output tables. The proposed GME technique is illustrated by means of an empirical application that estimates the parameters that characterize a non-linear input–output model for the Spanish economy over the period 1995–2011.

KEYWORDS:

• Non-linear input–output modelling
• flexible coefficients
• generalized maximum entropy
• Spain

Acknowledgements

The author would like to thank editor Bart Los for the critiques and suggestions that have helped to improve the clarity and focus of this paper. The comments from the participants in the IV Workshop on Input–Output Analysis (Albacete, 25 and 26 of September, 2014) are highly appreciated as well.

Notes

† This article is a revised version of the paper that won the Wassily Leontief Memorial Prize 2014, for the best paper by authors younger than 40 submitted to the 22nd International Input–Output Conference, in Lisbon, Portugal.

¹Assuming that ${\beta }_{ij}=1$ necessarily implies that ${\alpha }_{ij}=a_{ij}$.

²Note that the structure of the variance of the estimators in Equation 25 leads to smaller variances than a least squares estimator, given that the term ${\sigma }_v^2$ is always non-negative.

³See Dietzenbacher et al. (2013) and Timmer et al. (2015) for details.

⁴This expression is included as a constraint in the GME program directly.

⁵This implies that the estimation of the non-linear I–O model without the sample information contained in Equation 32 will yield the traditional linear I–O model.

⁶These deviations could be used also to derive confidence intervals around our point estimates, which would allow for calculating confidence intervals for the matrices of multipliers as well.

⁷See Dietzenbacher and Hoen (2006) or Wood (2011) for examples of research where the assumption of stability of input–output coefficients is empirically investigated.

⁸Note that the results in this application are partially conditioned by the fact that the Spanish I–O table for 2011 available in WIOD is a projection of the 2005 survey-based I–O table compiled by INE, based on a variant of RAS (see Dietzenbacher et al., 2013). The comparison with a survey-based 2011 I–O table probably would produce results more favorable to the GME estimation than those reported in Table 3, given that a biproportional adjustment of the 2005 table tries to minimize the changes in the initial matrix and only modifies its cells to make them consistent with some row and column totals. This implies that the 2011 Spanish table in WIOD probably contains coefficients closer to the 2005 table than those potentially observed in a survey-based I–O table for 2011.