Lagged relationships are characteristic of many natural physical systems. Lagged
correlation refers to the correlation between two time series shifted in time relative to one
another. Lagged correlation is important in studying the relationship between time series for two
reasons. First, one series may have a delayed response to the other series, or perhaps a delayed
response to a common stimulus that affects both series. Second, the response of one series to the
other series or an outside stimulus may be “smeared” in time, such that a stimulus restricted to
one observation elicits a response at multiple observations. For example, because of storage in
reservoirs, glaciers, etc., the volume discharge of a river in one year may depend on precipitation
in the several preceding years.
Or because of changes in crown density and photosynthate
storage, the width of a tree-ring in one year may depend on climate of several preceding years.
The simple correlation coefficient between the two series properly aligned in time is inadequate
to characterize the relationship in such situations. Useful functions we will examine as
alternative to the simple correlation coefficient are the cross-correlation function and the impulse
response function. The cross-correlation function is the correlation between the series shifted
against one another as a function of number of observations of the offset. If the individual series
are autocorrelated, the estimated cross-correlation function may be distorted and misleading as a
measure of the lagged relationship. We will look at two approaches to clarifying the pattern of
cross-correlations. One is to remove the persistence from, or prewhiten, the two series before
cross-correlation estimation. In this approach, the two series are essentially regarded on an
“equal footing”. An alternative is the “systems” approach, in which one series is viewed as input
to and the other series as output from a dynamic linear system. The lagged response is estimated
through the “impulse response function”, which is the response of the output at current and future
times to a hypothetical “pulse” of input restricted to the current time. Estimation of the impulse
response function involves application of the same filter to the input and output series, followed
by cross-correlation of the filtered series. The filter is designed such that it whitens the input
series.