SOBOLEV ESTIMATION OF APPROXIMATE REGRESSIONS

This paper focuses on the estimation of an approximated function and i ts derivatives. Let us assume that the data-generating process can be described by a family of regression models y(i)(alpha) = D-alpha phi(x (i)) + u(i)(alpha), where alpha is a multi-index of differentiation su ch that D-alpha phi(x(i)) is the alpha th derivative of phi(x(i)) with respect to x(i). The estimated model is characterized by a family D-a lpha f(x(i)\theta), where D-alpha f(x(i)\theta) is the alpha th deriva tive of f(x(i)\theta) and theta is an unknown parameter. The model is in general misspecified; that is, there is no theta such that D-alpha f(x(i)\theta) is equal to D-alpha phi(xi). Three different problems ar e discussed. First, the asymptotic behavior of the seemingly unrelated regression estimator of theta is shown to achieve the best approximat ion, in the Sobolev norm sense, of phi by an element of {f(x(i)\theta) \theta is an element of theta}. Second, in the case of polynomial appr oximations, the expected derivatives of the limit of the estimated reg ression and of the true regression are proved to be equal if and only if the set of explanatory variables has a normal distribution. Third, different sets of alpha are introduced, and the different limits of es timated regressions characterized by these sets are proved to be equal if and only if the explanatory variables have a normal distribution. This result leads to a specification test.