New tools for understanding spurious regressions

Some new tools for analyzing spurious regressions are presented. The theory utilizes the general representation of a stochastic process in terms of an orthonormal system and provides an extension of the Weierstrass theorem to include the approximation of continuous functions and stochastic processes by Wiener processes. The theory is applied to two classic examples of spurious regressions: regression of stochastic trends on time polynomials, and regressions among independent random walks. It is shown that such regressions reproduce in part and in whole the underlying orthonormal representations.