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

Pointer to an array of length 2 × lagmax + 1 containing the cross-correlations between the time series x and y. The k-th element of this array contains the cross-correlation between x and y at lag(k-lagmax) where k = 0, 1, …, 2*lagmax. To release this space, use imsls_free. If no solution can be computed, NULL is returned.

Function imsls_f_crosscorrelation estimates the cross-correlation function of two jointly stationary time series given a sample of n = n_observations observations {Xt} and {Yt} for t = 1, 2, …, n.

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

where K = lagmax. Note that

is equivalent to the sample variance x_variance. 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 se_option for the optional argument IMSLS_SE_CCF. 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. Function imsls_f_crosscorrelation is used to compute the cross-covariances and cross-correlations between time series X and Y with lags from  -10 through lag  10. In addition, the estimated standard errors of the estimated cross-correlations are computed. The standard errors are based on the additional assumption that all cross-correlations for X and Y are zero.