CCOVF

This function computes the cross-covariance function between two univariate series, quantifying linear co-movement across lag offsets.

For lag k, the cross-covariance is:

\gamma_{xy}(k) = \operatorname{Cov}(x_{t+k}, y_t)

The result helps identify lagged lead-lag structure before normalization into cross-correlation.

Excel Usage

=CCOVF(x, y, adjusted, demean, fft)

x (list[list], required): First time series as a 2D range.
y (list[list], required): Second time series as a 2D range.
adjusted (bool, optional, default: true): Use denominator n-k instead of n in covariance estimation.
demean (bool, optional, default: true): Subtract sample means from both series before covariance estimation.
fft (bool, optional, default: true): Use FFT-based convolution.

Returns (list[list]): 2D table with columns lag and ccovf.

Example 1: Cross-covariance with default options

Inputs:

x						y						adjusted	demean	fft
1	2	3	4	5	6	2	3	4	5	6	7	true	true	true

Excel formula:

=CCOVF({1,2,3,4,5,6}, {2,3,4,5,6,7}, TRUE, TRUE, TRUE)

Expected output:

Result
0	2.91667
1	1.75
2	0.25
3	-1.58333
4	-3.75
5	-6.25

Example 2: Cross-covariance without demeaning

Inputs:

x						y						adjusted	demean	fft
3	5	4	6	5	7	1	2	2	3	3	4	true	false	true

Excel formula:

=CCOVF({3,5,4,6,5,7}, {1,2,2,3,3,4}, TRUE, FALSE, TRUE)

Expected output:

Result
0	13.6667
1	12.2
2	11.75
3	10
4	9.5
5	7

Example 3: Cross-covariance without FFT

Inputs:

x							y							adjusted	demean	fft
2	4	3	5	4	6	5	1	3	2	4	3	5	4	false	true	false

Excel formula:

=CCOVF({2,4,3,5,4,6,5}, {1,3,2,4,3,5,4}, FALSE, TRUE, FALSE)

Expected output:

Result
0	1.55102
1	0.0991254
2	0.565598
3	-0.457726
4	-0.134111
5	-0.586006
6	-0.262391

Example 4: Cross-covariance with unadjusted denominator

Inputs:

x								y								adjusted	demean	fft
1	3	2	4	3	5	4	6	2	1	3	2	4	3	5	4	false	true	true

Excel formula:

=CCOVF({1,3,2,4,3,5,4,6}, {2,1,3,2,4,3,5,4}, FALSE, TRUE, TRUE)

Expected output:

Result
0	0.75
1	1.3125
2	-0.0625
3	0.3125
4	-0.625
5	-0.3125
6	-0.6875
7	-0.3125

Python Code

import numpy as np
from statsmodels.tsa.stattools import ccovf as sm_ccovf

def ccovf(x, y, adjusted=True, demean=True, fft=True):
    """
    Estimate cross-covariance values between two time series across lags.

    See: https://www.statsmodels.org/stable/generated/statsmodels.tsa.stattools.ccovf.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        x (list[list]): First time series as a 2D range.
        y (list[list]): Second time series as a 2D range.
        adjusted (bool, optional): Use denominator n-k instead of n in covariance estimation. Default is True.
        demean (bool, optional): Subtract sample means from both series before covariance estimation. Default is True.
        fft (bool, optional): Use FFT-based convolution. Default is True.

    Returns:
        list[list]: 2D table with columns lag and ccovf.
    """
    try:
        def to1d(values):
            if isinstance(values, list):
                if all(isinstance(row, list) for row in values):
                    raw = [item for row in values for item in row]
                else:
                    raw = values
            else:
                raw = [values]

            out = []
            for item in raw:
                try:
                    out.append(float(item))
                except (TypeError, ValueError):
                    continue
            return out

        x_vals = to1d(x)
        y_vals = to1d(y)

        if len(x_vals) < 2 or len(y_vals) < 2:
            return "Error: x and y must each contain at least two numeric values"

        if len(x_vals) != len(y_vals):
            return "Error: x and y must have the same number of numeric values"

        ccovf_vals = sm_ccovf(
            np.asarray(x_vals, dtype=float),
            np.asarray(y_vals, dtype=float),
            adjusted=adjusted,
            demean=demean,
            fft=fft,
        )

        arr = np.asarray(ccovf_vals, dtype=float)
        return [[lag, float(arr[lag])] for lag in range(len(arr))]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

x *

First time series as a 2D range.

y *

Second time series as a 2D range.

adjusted

Use denominator n-k instead of n in covariance estimation.

demean

Subtract sample means from both series before covariance estimation.

fft

Use FFT-based convolution.