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Empirical Economics

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01.09.2006 | Original Paper

Linear approximations to the quadratic almost ideal demand system

verfasst von: Toshinobu Matsuda

Erschienen in: Empirical Economics | Ausgabe 3/2006

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Abstract

This paper investigates linear approximations to the recently popular quadratic almost ideal demand system (QUAIDS) by proposing a new composite variable and conducting a simulation study. The linear approximations are especially useful when one uses nonstationary time series, to which nonlinear systems are difficult to apply properly. The new composite variable performs well in combination with the price indices appropriate for linearizing the almost ideal demand system. The QUAIDS can be linearly approximated on a practical basis if the appropriate combinations of composite variables and elasticity formulas are employed and the base point is set to the point of evaluation.

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Employed in Nicol (1991) and Moosa and Baxter (2002), log P ^C is equivalent to \( {\sum\nolimits_i {\overline{w} ^{0}_{i} } }\log {\left( {{p_{i} } \mathord{\left/ {\vphantom {{p_{i} } {p^{0}_{i} }}} \right. \kern-\nulldelimiterspace} {p^{0}_{i} }} \right)} \) because \( {\sum\nolimits_i {\overline{w} ^{0}_{i} } }\log p^{0}_{i} \) is constant and unimportant in estimation.

If the model of Blundell et al. (1993) is to be built on utility maximization, then the ratio of η_i to β_i is forced to be the same for all i, that is, the total number of free parameters is reduced to that of the AIDS. If η_i and β_i are to be independent of each other for all i, therefore, then their model cannot be derived from utility maximization.

Estimation results of the nonlinear QUAIDS given by Eq. 12 are found to be virtually unchanged whether α₀ is estimated or fixed at any particular value including zero.

With parameter restrictions (5)–(10) imposed and the values of inconsequential parameters α₀ and β₀ preset to zero, the total number of free parameters of the QUAIDS is (n+6)(n−1)/2, which is n−1 more than the number required for parsimonious flexible functional forms (Pollak and Wales 1992) such as the AIDS. The additional n−1 parameters are necessary for quadratic logarithmic expenditure terms.

A covariance matrix for five goods, Ω is given based on the analogy of the covariance matrix used by Moschini (1995), who conducted an investigation into linear approximations to the AIDS in a four-good case.

The simulation is performed using SHAZAM 10.0.

Otherwise there is no difference between the linear approximate and original nonlinear formulas at the sample mean.

As expected, using P* yields unambiguously poor results, as in the case of the AIDS shown by Moschini (1995).

Our simulation exercise is almost as limited in scope and depth as the simulation study of the AIDS by Moschini (1995). Buse and Chan (2000) further investigated the linearized AIDS by performing a more comprehensive simulation. Applying their type of Monte Carlo design to the QUAIDS is a possible direction for future research.

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Titel: Linear approximations to the quadratic almost ideal demand system
verfasst von: Toshinobu Matsuda
Publikationsdatum: 01.09.2006
Verlag: Physica-Verlag
Erschienen in: Empirical Economics / Ausgabe 3/2006
Print ISSN: 0377-7332
Elektronische ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-005-0044-3

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