Measures of Conditional Linear Dependence and Feedback between Time Series

Abstract

Measures of linear dependence and feedback for two multiple time series conditional on a third are defined. The measure of conditional linear dependence is the sum of linear feedback from the first to the second conditional on the third, linear feedback from the second to the first conditional on the third, and instantaneous linear feedback between the first and second series conditional on the third. The measures are non-negative and may be expressed in terms of measures of unconditional feedback between various combinations of the three series. The measures of conditional linear feedback can be additively decomposed by frequency. Estimates of these measures are straightforward to compute, and their distribution can be routinely approximated by bootstrap methods. An empirical example involving real output, money, and interest rates is presented.

Key Words:

• Bootstrap
• Resampling
• Macroeconomics
• Spectral density