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

11 tttt CBuAss

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:

tt

t

t

t uB

ss

IAA

ss

00 1

211

tttt BusAsAs 1211

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)

tt

tet

t uBB

xz

AAAA

xz

2

11

2221

1211

1

What happens with RE? (2)

• What if we have a feedback rule for u?

• The model ‘under control’ is:

t

tet

t

xz

AFBAAFBA

xz 1

22221

12111

1

1 tt Fzu

Generalized BK

• Now modify the model to:

• May be that E21 = 0 and E22 is singular• Can be solved using Klein (1997),

Soderlind (1999)

tt

tet

t uBB

xz

AAAA

xz

EEEE

2

11

2221

1211

12221

1211

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?

tettt BuEsAsDs 11

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

tet

et

t

t

t

t uBB

xz

Ixz

AA

xz

AAI

2

1

1

1

1

1

21

11

22

12

000

00

0

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