A Solution Method for Linear Rational Expectation Models under Imperfect Information

This paper has developed a solution algorithm for linear rational expectation models under imperfect information. Imperfect information in this paper means that some decision makings are based on smaller information sets than others. The algorithm generates the solution in the form of k_t+1 = Hk_t + Jx^t,S f_t = Fk_t + Gx^t,S where k_t and f_t are column vectors of crawling and jump variables, respectively, while x^t,S is the vertical concatenation of the column vectors of past and present innovations. The technical breakthrough in this article is made by expanding the innovation vector, rather than expanding the set of crawling variables. Perhaps surprisingly, the H and F matrices are the same as those under the corresponding perfect information models. This implies that if the corresponding perfect information model is saddle path stable (sunspot, explosive), the imperfect model is also saddle-path stable (sunspot, explosive, respectively). Moreover, if the minimum information set in the model has all the information up to time t-S-1, then the direct effects on the impulse response functions last for only the first S periods after the impulse. In the subsequent dates, impulse response functions follow essentially the same process as in the perfect information counterpart. However, imperfect information can significantly alter the quantitative properties of a model, though it does not drastically change its qualitative nature. This article demonstrates, as an example, that adding imperfect information to the standard RBC models remarkably improves the correlation between labour productivity and output. Hence, a robustness check for information structure is recommended.