RDP 2007-12: Dynamic Pricing and Imperfect Common Knowledge Appendix B: The Law of Motion of the Expectations Hierarchy

Define a new state vector X_t as

For a given W, firm j's estimate of X_t will evolve according to the updating equation

where K is the Kalman gain matrix and S_t(j) is firm j's observation vector. L is a matrix that maps an expected state into an expected observation vector. They are given by

Take averages of Equation (B2) across firms and use the definitions of the observables in S_t to write the average updating equation as

Finally, use the definition of the hierarchy :

and the fact that the exogenous state x_t|t follows the known process

to identify M and N:

where [·]₁₁ is the upper left submatrix of the appropriately partitioned matrix [·].