Testing exact rational expectations in cointegrated vector autoregressive models
Rational expectations are an important concept in theoretical as well as empirical economics. Typically, the hypothesis of rational expectations (REH) imposes nonlinear cross-equation restrictions on the coefficients of the equations of the model which are used to evaluate the model by testing the validity of the restrictions. Unfortunately, rational expectations cannot be observed directly so that in most cases some proxies have to be taken into account. To overcome the problem, one possibility, when the models contain stochastic elements, is to use conditional expectations in the probabilistic sense given some previous information. Yet there is another source of trouble. Usually, dynamic models containing rational expectations of future values have a multitude of solutions. \textit{R.T. Baillie} [Econ. Rev. 8, No. 2, 151-216 (1989; Zbl 0718.62284)] advocated a procedure for testing the restrictions implied by rational expectations on a set of variables by assuming that the solutions could be described by a VAR-model. This paper follows this line of research, but the authors put more emphasis on the proper specification of cointegration relationships. Especially, if the cointegration rank in the VAR model is larger than what is specified by the REH, then the remaining cointegration vectors that are used to transform to stationarity must then be estimated before the transformation is carried out. With respect to testing, this two-step procedure leads to a sort of Wald test to test the validity of the restrictions imposed. The authors present a sequence of likelihood ratio tests when the restrictions implied by the rational expectations hypothesis have a particular form. Starting with a VAR model, the authors assume that the variables are integrated of order one. They show that the restrictions on the expectations entail restrictions on the cointegrating relationships. Furthermore, some restrictions on the short-run parameters of the model must be satisfied. This approach allows the authors to define three nested models: the VAR, the cointegrated VAR and the cointegrated VAR with the restrictions implied by rational expectations. For each model the ML-estimator has to be worked out, which may then be used to construct the likelihood ratio test. If the rational expectations restrictions are of a particular form, their procedure allows an explicit solution in terms of reduced rank regression. The particular form of the rational expectations hypothesis requires that they are formulated as linear restrictions involving the conditional expected value of the observations one-step-ahead and the present and lagged observed values. In the case where the restrictions can be expressed as linear combinations with known coefficients, an explicit expression for the maximum-likelihood estimators and the maximal value of the likelihood can be found. The authors demonstrate their approach with an application to a present value model.
scientific article; zbMATH DE number 1369434