NEW TOOLS FOR UNDERSTANDING SPURIOUS REGRESSIONS

Some new tools for analyzing spurious regressions are presented. The t heory utilizes the general representation of a stochastic process in t erms of an orthonormal system and provides an extension of the Weierst rass 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 tr ends on time polynomials, and regressions among independent random wal ks. It is shown that such regressions reproduce in part and in whole t he underlying orthonormal representations.