ESTIMATING LINEAR FILTERS WITH ERRORS-IN-VARIABLES USING THE HILBERT TRANSFORM

In this paper we present a consistent estimator for a linear filter (d istributed lag) when the independent variable is subject to observatio nal error. Unlike the standard errors-in-variables estimator which use s instrumental variables, our estimator works directly with observed d ata. It is based on the Hilbert transform relationship between the pha se and the log gain of a minimum phase-lag linear filter. The results of using our method to estimate a known filter and to estimate the rel ationship between consumption and income demonstrate that the method p erforms quite well even when the noise-to-signal ratio for the observe d independent variable is large. We also develop a criterion for deter mining whether an estimated phase function is minimum phase-lag.