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Cross-covariance matrix

by , PhD

The cross-covariance matrix between two random vectors X and Y is a matrix containing the covariances between all the possible couples of random variables formed by one entry of X and one entry of Y.

Table of Contents


This is a formal definition.

Definition Let X be a Kx1 random vector and Y be a $L	imes 1$ random vector. The cross-covariance matrix between X and Y is a $K	imes L$ matrix, denoted by [eq1] and defined as follows:[eq2]

Note that in the formula above [eq3] is a column vector and [eq4] is a row vector.


Define two random vectors X and Y as follows:[eq5]

The cross-covariance matrix between X and Y is[eq6]

Special case

When $Y=X$, then the cross-covariance matrix coincides with the covariance matrix of X:[eq7]

Autocovariance matrix

Let [eq8] be a sequence of random vectors, where n is a time-index.

Let $j$ be a time lag. Then, the cross-covariance matrix[eq9]is called autocovariance matrix.

Lack of symmetry

Note that, in general, the cross-covariance is not symmetric.

In fact,[eq10]and, in general,[eq11]

For example, if X is Kx1 and Y is $L	imes 1$, [eq12] is $K	imes L$ and [eq13] is $L	imes K$.

Other properties

We have just demonstrated that [eq14]

Other useful properties that can be easily derived from the definition (exercise!) are:

Use in time-series analysis

The cross-covariance matrix is often used in time-series analysis and in the theory of stochastic processes.

For example, it is used to define the concept of covariance stationarity for random vectors.

A sequence of random vectors [eq8] is said to be covariance stationary if and only if[eq20]where n and $j$ are integers.

Property (1) means that all the random vectors belonging to the sequence [eq21] must have the same mean.

Property (2) means that the cross-covariance between a term of the sequence (X_n) and the term that is located $j$ positions before it ($X_{n-j}$) must always be the same, irrespective of how n has been chosen. In other words, [eq22] depends only on $j$ and not on n.

Sum of covariance stationary vectors

If [eq8] is a covariance stationary sequence of vectors, we can use the following important formula for the covariance matrix of a sum:[eq24]


This is demonstrated as follows:[eq25]where: in step $rame{A}$ we have used the fact that the cross-covariance of a vector with itself is equal to its covariance matrix; in step $rame{B} $ we have used the additivity property; in step $rame{C}$ we have used the stationarity of the sequence; in step $rame{D}$ we have re-grouped the summands.

Use in asymptotic theory and statistical inference

Consider a sequence of covariance stationary random vectors [eq26].

The sample mean of the first n terms of the sequence is[eq27]

Under some regularity conditions, the sample mean is asymptotically normal, with mean [eq28] and asymptotic covariance matrix[eq29]where the cross-covariance matrices $Gamma _{j}$ are defined as in equation (2) above.

In other words, if a multivariate central limit theorem for dependent sequences applies, then[eq30]where [eq31] denotes convergence in distribution and [eq32] denotes a multivariate normal distribution with zero mean and long-run covariance matrix equal to V.

Cross-covariance function

Let [eq8] be a sequence of random vectors.

When the cross-covariance[eq34]is viewed as a function of the two indices n and $m$, then it is called cross-covariance function.

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How to cite

Please cite as:

Taboga, Marco (2021). "Cross-covariance matrix", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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