The Linear Model with Restricted Explanatory Variables
研究了解释变量受限时线性模型的预测问题，讨论了奇异解释矩阵下递归广义逆估计的唯一性条件，并证明了似然函数最大值唯一的充要条件。
Linear models with restrictions on the explanatory variables apply in many situations. For prediction, when there are not available observations on the whole range of conceivable combinations of these variables, quasimulticolinearity is frequently accepted, with care about not applying the prediction model on unusual cases. Nevertheless, detecting possible specification changes in the rare regions may help correcting the model fit and improving prediction in the regions where information is not sparse. This can be done even when there is not only a rare region but also an empty region, that is, a subspace on which the projection of every row of the explanatory matrix is null. Tools for checking specification changes under the singular explanatory matrix condition are developed by McGilchrist et al. (1983) and include the analysis of the series of recursive generalized inverse estimates. We discuss here conditions for uniqueness of such estimates in a general framework and for the preservation of chosen uniqueness restrictions throughout the recursive algorithm iterations. In ? 2 we consider the different presentations that the restricted model may have and set equivalences between identification through constraints on the linear coefficients and on the covariance matrix. Formulating in terms of singular explanatory matrices helps clarifying the relation between estimability of the coefficients and testability of linear hypotheses about them. This is shown in ? 3, where we prove necessary and sufficient conditions for unicity of the maximum of the likelihood. In ? 4 we consider the case of nonuniqueness and the new constraints that then may be imposed. The simpler choice is that of null projection on the null space of the explanatory matrix. We show that this restriction, that implies the choice of the Penrose inverse in the generalized recursive algorithm, may be automatically retained throughout the iterations. In the final section, we comment on application of the previous results to prediction.