Orthogonality in Econometrics

In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle.

In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns (or rows) are orthogonal unit vectors (i.e., orthonormal). Because the columns are unit vectors in addition to being orthogonal, some people use the term orthonormal to describe such matrices.
Equivalently, a matrix Q is orthogonal if its transpose is equal to its inverse:

    alternatively,   

The concept of orthogonality tends to be very important in econometrics, since we have been building almost all of the methods and rules based on the matrix platform. For example, if it happens that a relevant independent variable is omitted, in general, the OLS estimator of the coefficients of the remaining variables is biased. If the omitted variable is orthogonal to the included variables, the slope coefficient estimator will be unbiased; the intercept estimator will retain its bias unless the mean of the observations on the omitted variable is zero.
In the case of inclusion of an irrelevant variable, unless the irrelevant variable is orthogonal to the other independent variables, the variance-covariance matrix βOLS becomes larger; the OLS estimator is not as efficient. Thus in this case the MSE of the estimator is unequivocally raised.