Abstract
Partial linearity can be exploited to solve the numerical estimation problem by using the general principle of concentrating the criterion function. This well-known principle can be applied when the optimization problem has a unique global analytic solution for a subset of the parameters, given the values of the remaining parameters. The principle is usually implemented to build concentrated likelihood functions and in the case of separable least squares.
The explicit use of partial linearity has been a very important matter in the pioneering days of econometrics when nonlinear models were very expensive in terms of computation. In fact, nonlinearity was then mainly restricted to a small subset of the parameters (often containing a single element). With algorithms based on numerical derivatives, or no derivatives, it is fundamental to reduce the dimension of the optimization problem.
With the powerful computers available nowadays and the development of software using symbolic derivatives, it has become less crucial to base numerical estimation on the specific nature of the model. However, it remains true that, even if the chances of finding the solution are not improved, it is still valuable to reduce the quantity of computations when possible and therefore to win time.
In this paper, we consider the use of partial linearity, in relation to the computation of various estimators. For that matter, we consider a general (possibly) multiple equation, reduced form, partially linear regression model.
In the context of minimum distance estimators, we compute full analytical expressions for the Gauss-Newton algorithm in both the unconcentrated and the concentrated cases. The comparison shows that concentration offers the possibility to rule off at little cost unnecessary calculations concerning the "linear" parameters.
A subsequent part of the paper deals with estimation methods that simultaneously involve the parameters characterizing the first two moments (or more) of the distribution of the endogenous variables. What use of the partial linearity property can we make in that context?
To get an answer to that question, we consider the estimation of our basic model by maximum likelihood under normality, with various assumptions concerning the covariance matrix of the error terms, from classic particular cases to more general structures.
In the general case, the concentration method can be extended
straightforwardly. However, for a given covariance structure, the
related parameters may lead to an alternative concentration method. In
such a case, an iterative process can be implemented.