Variance-covariance matrix

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The variance – covariance matrix is ​​a square matrix of dimension nxm that collects the variances in the main diagonal and the covariances in the elements outside the main diagonal.

In other words, the variance-covariance matrix is ​​a matrix that has the same number of rows and columns and has the variances distributed on the main diagonal and the covariances on the elements outside the main diagonal.

Covariance

Matrix representation

The variance-covariance matrix is ​​usually expressed as

Sigma

Although it seems that it is the symbol of the summation and that it has no relation to the variance-covariance matrix, this Greek letter perfectly represents the content of this matrix.

To understand it, let's first look at its expression:

Variance-covariance matrix nxm

Knowing that there are m columns, the ellipses indicate that the columns between the second and the last column have not been represented. In the same way, knowing that there are n rows, the ellipses indicate that the rows between the second and the last row have not been represented.

In this case, we use sigma to represent the covariances and sigma squared for the variances. As an example:

Two elements of the variance-covariance matrix

What Greek letter appears in all the elements of the matrix? The sigma.

So, it is logical that, to define the variance-covariance matrix, a sigma is also used.

Greek letter

Sigma

is the capital form of

Lowercase sigma

So if we remember that the variance-covariance matrix is ​​expressed as the uppercase of sigma, it will be easier to remember its definition.

Requirements for it to be a variance-covariance matrix

The requirements for a matrix to be variance-covariance are the following:

  • Square matrix: same number of rows (n) as columns (m), therefore, n = m, and therefore, the dimension of this matrix can be expressed both nxm and nxn.
  • On the main diagonal are the variances:

Variances of the variance-covariance matrix
  • Outside the main diagonal are the covariances:

Covariances of the variance-covariance matrix

App

The variance-covariance matrix is ​​very popular in econometrics since it is used mainly in the matrix calculation of the coefficients of linear regression using Ordinary Least Squares, among other uses.

In finance, it is used to get a general picture of the volatility of financial assets.

Mathematical expression of variance and covariance

Mathematics is expressed as follows:

  • Covariance of the element n = 1 and m = 2

Covariance calculation for element n = 1 and m = 2
  • Variance of the element n = 1 and m = 1

Variance calculation for element n = m = 1

Both variance and covariance can be corrected. That is, the denominator is n-1 instead of n. This is due to the degrees of freedom and depends on whether we are talking about population or sample variances and covariances.

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