Models With Several Regimes and Changes in Exogeneity

In recent years, increasing attention has been devoted to models with a finite (usually small) number of regimes. Various strategies have been discussed in the literature to handle situations where each regime is characterized by a different value of a common parameter vector. See e.g. Barten and Bronsard (1970), Goldfeld and Quandt (1973), Poirier (1976),... . It appears however that no satisfactory treatment has yet been given to cases where the partitioning between " endogenous " and " exogenous " variables changes over time. Our objective is therefore to define a class of models with several regimes which is flexible enough to cover such situations. For convenience, we shall illustrate our argument by reference to an economy which is "controlled " by a policy maker shifting between instruments at some, possibly unknown, points of time. For tractability we shall mainly restrict our attention to a class of dynamic linear models although the concepts we introduce apply in a much broader framework. The possibility that the switching times could be endogenous to the model, such as in disequilibrium models will not be investigated here: work in progress indicates however that our approach can be extended in such directions. The paper is organized as follows: In Section 2 we shall discuss at length the issues to be faced by means of a simple example, taken from Goldfeld and Quandt (1973). In Section 3 we shall introduce the concepts which are needed for our analysis; linear dynamic models, LIML estimation and exogeneity. In Section 4, we shall discuss models with several regimes and concentrate in particular on imposing appropriate restrictions on the parameters characterizing different regimes. It will be shown that it is possible to preserve some of the operational features of LIML procedures.