REGRESSIONS WITH SLOWLY VARYING REGRESSORS
REGRESSI ONS WITH asymptotically collinear regressors arise in a number of applications, both in linear and nonlinear settings. Examples are the log -periodogram analysis of long memory (see Robinson, 1995; Hurvich et al., 1998; Phillips, 1999 and references therein), the study of growth convergence (Barro and Sala-i-Martin, 2003), and NLS estimation (Wu, 1981). This chapter is based on Phillips' 2007 paper. His contribution to the theory of regressions with asymptotically collinear regressors can be described as follows:
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He used the properties of SV functions to develop asymptotic expansions of some nonstochastic expressions that arise in regression analysis.
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Based on those asymptotic expansions, he employed Brownian motion to derive central limit results for weighted sums of linear processes where the weights are standardized SV regressors.
For the cases when the conventional scheme does not work (because of asymptotic collinearity of the regressors) he modified it so as to obtain convergence of the OLS estimator in a variety of practical situations.
This chapter is structured accordingly. The main results are contained in Sections 4.2, 4.4, 4.5, and 4.6. Section 4.2, called Phillips Gallery 1, covers his asymptotic expansions of nonstochastic expressions. Not all of them are applied later in the book, but I include them for two reasons: they may be useful in other applications and they have helped me to guess some facts related to L p -approximability.
Section 4.4 is devoted to generalizations of the central limit results established by Phillips. The main point is that, for these sort of results, using L p -approximability and my CLT 3.5.2 is preferable to recourse to Brownian motion. Here the reader will see that some expansions of nonstochastic expressions are also implied by L p -approximability.
In Section 4.5, named Phillips Gallery 2, we return to Phillips' exposition by going over applications. Section 4.6, which is about regressions with two SV