This paper advances the theory and methodology of signal extraction
by introducing asymptotic and finite sample formulas for optimal
estimators of signals in nonstationary multivariate time series.
Previous literature has considered only univariate or stationary
models. However, in current practice and research, econometricians,
macroeconomists, and policy-makers often combine related series -
that may have stochastic trends--to attain more informed
assessments of basic signals like underlying inflation and business
cycle components. Here, we use a very general model structure, of
widespread relevance for time series econometrics, including
flexible kinds of nonstationarity and correlation patterns and
specific relationships like cointegration and other common factor
forms. First, we develop and prove the generalization of the
well-known Wiener-Kolmogorov formula that maps signal-noise
dynamics into optimal estimators for bi-infinite series. Second,
this paper gives the first explicit treatment of finite-length
multivariate time series, providing a new method for computing
signal vectors at any time point, unrelated to Kalman filter
techniques; this opens the door to systematic study of near
end-point estimators/filters, by revealing how they jointly depend
on a function of signal location and parameters. As an illustration
we present econometric measures of the trend in total inflation
that make optimal use of the signal content in core inflation.
General
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