Vector Regression : a bridge between micro and macro ?

iii Contents Abstract v 1. Introduction 1 2. Optimal aggregation 5 3. From micro to macro 9 4. From macro to semi-and non-parametric 11 5. Statistical learning 17 6. Conclusion 23 Appendix. The dual quadratic program 23 References 27 v Abstract Support Vector (SV)-regression, a common tool in statistical learning, estimates functions as linear combinations of given (nonlinear) kernel functions. To this representation corresponds through duality a weighted, possibly infinite sum of generally unknown eigenfunctions. The paper proposes to start from given eigenfunctions, as these can be thought of as representing a known micro based model, say, the sum of known Marshallian demand functions of all individual consumers and applies SV-regression to estimate a more compact and macro representation to replace this micro-model. The method applies to any single valued and bounded, possibly discontinuous eigenfunction and we show that it can accommodate multiple equations as well as constraints on parameters and function derivatives. The existing SV-algorithm can solve this micro-based problem at moderate computational cost, even when the number of individuals is very large, say, running in the millions because it can store all the necessary information in a Gram-matrix whose dimensions do not depend on this number. Among various problems in the range between micro and macro, optimal aggregation is the most micro. It amounts to finding an optimal, as sparse as possible vector of weights, so that the individuals with positive weights fit the true aggregate model with sufficient accuracy. Next, comes the competition between the micro-model and a given macro-model, and a we gradually attribute a more modest to the micro a priori, we eventually reach pure kernel approaches, the Gram-matrix is postulated as a covariance-like measure of the distance between observations and is applied to the macro-model directly. Finally, beyond the macro models lay the essentially descriptive non-parametric data enveloping techniques that only postulate a Gram-matrix. While the same SV-algorithm of dual quadratic programming applies throughout, it appears that the consistency properties of the estimators become weaker along this path, and whereas stochastic quasigradient methods achieve convergence almost surely, SV-regression only reaches convergence in probability.