What are State Space Models? Andrew P. Blake CCBS/HKMA May 2004

Model form Introduced into the rational expectations literature by Blanchard and Kahn (1980) Develops a model form used in much of the optimal control literature Later generalised to handle slightly more complicated economic models

Linear stochastic model We consider a model in state space form: u is a vector of control instruments, s a vector of endogenous variables, ε is a shock vector The model coefficients are in A, B and C

Properties of the model Linear, time invariant System of first order equations –Is this restrictive? May be of any size, and may be quite sparse Superficially like a first order VAR

Properties of the model (2) Companion form What if the model is (ignoring stochastics): We can write in first order form as:

What happens with RE? Modify the model to: Now we have z as predetermined variables and x as jump variables Model has a saddlepath structure Solved using Blanchard & Kahn (1980)

What happens with RE? (2) What if we have a feedback rule for u? The model ‘under control’ is:

Generalized BK Now modify the model to: May be that E 21 = 0 and E 22 is singular Can be solved using Klein (1997), Soderlind (1999)

How does this compare with other forms? Binder & Pesaran, Sims, Dennis have used RE models of the form: Sometimes called semi-structural form How do they compare?

How does a BK model fit into this? We can rewrite our BK form model as: Larger, sparser, redundant future terms Can be more compact Still requires ‘companion form’ type transform for further lags

Why use state-space? Trivially: Most of the literature does Our Ox programs/WinSolve use SS SS requires us to identify the variables as predetermined or not This is advantageous for understanding models –Understand role of states and co-states