Moment preserving approximations of multivariate probability distributions for decision and policy analysis: Theory and applications
Abstract
In various applications from applied economics, it is often necessary to evaluate the expectations of functions of random variables. Frequently, it is impossible or computationally too expensive to derive analytic solutions, or compute exact values, for these expectations. Thus, approximations of expectations of functions of random variables are used in models of economic phenomena. Commonly employed methods may either not accurately approximate these expectations, may be computationally prohibitive or both. The theory of Multivariate Gaussian quadrature (MGQ) is developed, and MGQ is used to approximate expectations in this study. Gaussian quadrature approximations are found by choosing a set of points and probability weights which preserve the lower ordered moments of the distribution of the random variables relevant to the expectation. The function whose expectation is to be computed is evaluated at each point in the approximation. The results are weighted by the appropriate probability and then summed. This weighted sum forms a Gaussian quadrature approximation to the expectation. Univariate Gaussian quadrature approximations (UGQ) are used by numerous authors and the properties of UGQ approximations are well established. This study develops a heuristic algorithm for generating MGQ approximations for arbitrary joint probability distributions. This algorithm appears to be reliable in practice. It is demonstrated that multiple solutions may exist matching for matching any given set of moments from a multivariate distribution. It is argued that MGQ approximations exist such that the points of the approximation always lie within the support of the original distribution. This property is important since there are some applications for which points which lie outside the support of the distribution are nonsensical. For example, some elasticities are restricted by theory to be positive. Therefore, the points of the approximation of the distribution of those elasticities should be restricted to positive values. The relative accuracy of the Gaussian quadrature method is established through two examples of its application.
Degree
Ph.D.
Advisors
Preckel, Purdue University.
Subject Area
Agricultural economics
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