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Constrained estimation

Suppose it is known that the coefficients in the regression satisfy a system of linear equations

where Q is a p × q matrix of full rank, and c is a q ×1 vector of known constants, where q < p. In this case least squares estimation is equivalent to minimizing the sum of squared residuals of the model subject to the constraint A. The constrained least squares (CLS) estimator can be given by an explicit formula:

This expression for the constrained estimator is valid as long as the matrix X X is invertible. It was assumed from the beginning of this article that this matrix is of full rank, and it was noted that when the rank condition fails, β will not be identifiable. However it may happen that adding the restriction A makes β identifiable, in which case one would like to find the formula for the estimator. The estimator is equal to

where R is a p ×(p − q) matrix such that the matrix is non-singular, and R Q = 0. Such a matrix can always be found, although generally it is not unique. The second formula coincides with the first in case when X X is invertible.