# Cross-covariance matrix

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

## Definition

This is a formal definition.

Definition Let be a random vector and be a random vector. The cross-covariance matrix between and is a matrix, denoted by and defined as follows:

Note that in the formula above is a column vector and is a row vector.

## Example

Define two random vectors and as follows:

The cross-covariance matrix between and is

## Special case

When , then the cross-covariance matrix coincides with the covariance matrix of :

## Autocovariance matrix

Let be a sequence of random vectors, where is a time-index.

Let be a time lag. Then, the cross-covariance matrixis called autocovariance matrix.

## Lack of symmetry

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

In fact,and, in general,

For example, if is and is , is and is .

## Other properties

We have just demonstrated that

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

• formula in terms of cross-moments:

• homogeneity:where is a constant scalar;

• formula for linear transformations:where and are constant vectors and and are constant conformable matrices.

## 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 is said to be covariance stationary if and only ifwhere and are integers.

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

Property (2) means that the cross-covariance between a term of the sequence () and the term that is located positions before it () must always be the same, irrespective of how has been chosen. In other words, depends only on and not on .

## Sum of covariance stationary vectors

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

Proof

This is demonstrated as follows:where: in step we have used the fact that the cross-covariance of a vector with itself is equal to its covariance matrix; in step we have used the additivity property; in step we have used the stationarity of the sequence; in step we have re-grouped the summands.

## Use in asymptotic theory and statistical inference

Consider a sequence of covariance stationary random vectors .

The sample mean of the first terms of the sequence is

Under some regularity conditions, the sample mean is asymptotically normal, with mean and asymptotic covariance matrixwhere the cross-covariance matrices are defined as in equation (2) above.

In other words, if a multivariate central limit theorem for dependent sequences applies, thenwhere denotes convergence in distribution and denotes a multivariate normal distribution with zero mean and long-run covariance matrix equal to .

## Cross-covariance function

Let be a sequence of random vectors.

When the cross-covarianceis viewed as a function of the two indices and , then it is called cross-covariance function.

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