Model 1 with Linear Constraints

In this section we shall consider estimation of the parameters /? and a2 in Model 1 when there are certain linear constraints on the elements of /?. We shall assume that the constraints are of the form

Q’fi = c, (1.4.1)

where Q is а К X q matrix of known constants and c is a ^-vector of known constants. We shall also assume q< К and rank (Q) = q.

Equation (1.4.1) embodies many of the common constraints that occur in practice. For example, if Q’ = (1,0) where I is the identity matrix of size Kl and 0 is the Kx X K2 matrix of zeroes such that K{ + K2 = K, then the con­straints mean that the elements of a K{ – component subset of ft are specified to be equal to certain values and the remaining K2 elements are allowed to vary freely. As another example, the case in which Q’ is a row vector of ones and с = 1 implies the restriction that the sum of the regression parameters is unity.

The study of this subject is useful for its own sake in addition to providing preliminary results for the next section, where we shall discuss tests of the linear hypothesis (1.4.1). We shall define the constrained least squares estima­tor, present an alternative derivation, show it is BLUE when (1.4.1) is true, and finally consider the case where (1.4.1) is made stochastic by the addition of a random error term to the right-hand side.

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