Olm Not of Full Rank

Abstract

Consider the Ordinary Linear Model (OLM)

$$ y = Ax + \varepsilon , $$

((2.1))

where A ∈ ℜ^mxn (m > n) is the exogenous data matrix, y ∈ ℜ^m is the response vector and ε ∈ ℜ^m is the noise vector with zero mean and dispersion matrix σ²I _m }. The least squares estimator of the parameter vector x ℜ^m

$$ \arg \min \varepsilon ^T \varepsilon = \mathop {\arg \min }\limits_x \left\| {Ax - \left. y \right\|} \right.^2 , $$

((2.2))

has an infinite number of solutions when A does not have full rank. However, a unique minimum 2-norm estimator of x say, x, can be computed.