Computes the sample cross‑correlation function of two stationary time series.

Routine CCF estimates the cross‑correlation function of two jointly stationary time series given a sample of n = NOBS observations {Xt} and {Yt} for t = 1, 2, …, n.

Let

be the estimate of the mean μX of the time series {Xt} where

The autocovariance function of {Xt}, σX(k), is estimated by

where K = MAXLAG. Note that

is equivalent to the sample variance XVAR. The autocorrelation function ρX(k) is estimated by

Note that

by definition. Let

be similarly defined.

The cross‑covariance function σXY(k) is estimated by

The cross‑correlation function ρXY(k) is estimated by

The standard errors of the sample cross‑correlations may be optionally computed according to argument ISEOPT. One method is based on a general asymptotic expression for the variance of the sample cross‑correlation coefficient of two jointly stationary time series with independent, identically distributed normal errors given by Bartlett (1978, page 352). The theoretical formula is

For computational purposes, the autocorrelations ρX(k) and ρY(k) and the cross‑correlations ρXY(k) are replaced by their corresponding estimates for ∣k∣ ≤ K, and the limits of summation are equal to zero for all k such that ∣k∣ > K.

A second method evaluates Bartlett’s formula under the additional assumption that the two series have no cross‑correlation. The theoretical formula is

For additional special cases of Bartlett’s formula, see Box and Jenkins (1976, page 377).

An important property of the cross‑covariance coefficient is σXY(k) = σYX(‑k) for k ≥ 0. This result is used in the computation of the standard error of the sample cross‑correlation for lag k < 0. In general, the cross‑covariance function is not symmetric about zero so both positive and negative lags are of interest.

Consider the Gas Furnace Data (Box and Jenkins 1976, pages 532–533) where X is the input gas rate in cubic feet/minute and Y is the percent CO2 in the outlet gas. Routine CCF is used to computed the cross‑covariances and cross‑correlations between time series X and Y with lags from ‑MAXLAG = ‑10 through lag MAXLAG = 10. In addition, the estimated standard errors of the estimated cross‑correlations are computed. In the first invocation with ISEOPT = 1, the standard errors are based on the assumption that autocorrelations and cross‑correlations for lags greater than MAXLAG or less than ‑MAXLAG are zero. In the second invocation with ISEOPT = 2, the standard errors are based on the additional assumption that all cross‑correlations for X and Y are zero.