Which statement about the variance-covariance matrix is NOT correct?

Which statement about the variance-covariance matrix is NOT correct?

• A. The diagonal elements are variances.
• B. The off-diagonal elements are covariances.
• C. The matrix is square.
• D. The diagonals include covariances.

Answer: (D)

The variance-covariance matrix encodes how much each variable varies and how pairs of variables covary. For n variables, it is an n by n square matrix where each entry is Cov(X_i, X_j). When the two indices are the same, Cov(X_i, X_i) equals Var(X_i), so the diagonal entries are variances. Covariances between different variables appear in the off-diagonal positions. For example, with two variables X and Y, the matrix is: [ [Var(X), Cov(X,Y)], [Cov(X,Y), Var(Y)] ] Thus, diagonals contain variances, not covariances between different variables. The statement that diagonals include covariances is not correct.

The variance-covariance matrix encodes how much each variable varies and how pairs of variables covary. For n variables, it is an n by n square matrix where each entry is Cov(X_i, X_j). When the two indices are the same, Cov(X_i, X_i) equals Var(X_i), so the diagonal entries are variances. Covariances between different variables appear in the off-diagonal positions.

For example, with two variables X and Y, the matrix is:

[ [Var(X), Cov(X,Y)],

[Cov(X,Y), Var(Y)] ]

Thus, diagonals contain variances, not covariances between different variables. The statement that diagonals include covariances is not correct.