Sobolev Estimation of Approximate Regressions
Marc Ivaldi () and
Sophie Larribeau ()
Econometric Theory, 1996, vol. 12, issue 05, 753-772
This paper focuses on the estimation of an approximated function and its derivatives. Let us assume that the data-generating process can be described by a family of regression models null, where a is a multi-index of differentiation such that D α null(x i ) is the αth derivative of null( x ) with respect to x i . The estimated model is characterized by a family D α f(X i |θ), where D α f(X i |θ) is the αth derivative of f(x i ,|θ) and θ is an unknown parameter. The model is in general misspecified; that is, there is no θ such that D α f(X i |6) is equal to D α null(X i ). Three different problems are discussed. First, the asymptotic behavior of the seemingly unrelated regression estimator of θ is shown to achieve the best approximation, in the Sobolev norm sense, of null by an element of (f(X i |θ)|θ ε Θ). Second, in the case of polynomial approximations, the expected derivatives of the limit of the estimated regression 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 α are introduced, and the different limits of estimated 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.
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