Markov Switching in Structural Models

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Markov Switching in Structural Models

• Roger E. A. Farmer (Joint with Dan Waggoner and
  Tao Zha)
• EABCN Conference (September 2007)

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Main research questions

• When do MSRE models have a unique equilibrium?
• Can good policy rule out indeterminacy?
• Can a bad policy by one administration spillover
  into another?
• How should structural empirical work proceed?

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Why are these interesting questions?

• A large literature argues that good monetary
  policy has been effective in controlling
  inflation and reducing the variance of gdp
• This literature argues that before 1980 monetary
  policy induced sunspot driven fluctuations
  (indeterminacy)
• After 1980 the policy induced a unique
  determinate equilibrium

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What are MSRE models

• Sims Cooley-LeRoy-Ramon
• If policy may change then this should be
  accounted for by rational agents
• Davig-Leeper Generalized Taylor Principle
• Caution- these models are more subtle than they
  appear

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The New-Keynesian model
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The Taylor principle?

• In the model WITHOUT Markov switching
• Equilibrium is unique in the NK model if --
• -- the coefficient alpha in the Taylor rule is
  greater than 1 in absolute value

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Why the Taylor principle works

• If the Taylor principle is satisfied, the
  eigenvalues of Gamma are outside the unit circle

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The DL approach

• In the next few slides I will explain the
  Generalized Taylor Principle of Davig and
  Leeper
• I will then provide some intuition as to why this
  principle provides a necessary but not a
  sufficient condition for determinacy

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Davig-Leeper

• Davig-Leeper study a model of the form

States
Transition probabilities
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The New-Keynesian example with policy switches
Policy parameters may switch
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The New-Keynesian example
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The Davig-Leeper question

• What do we mean by an equilibrium in the MSRE
  model?
• When is equilibrium unique?

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The Davig-Leeper answer

• Just as there is a Taylor principle for the NK
  model without switching
• So there is a Generalized Taylor Principle for
  the NK model with switching
• Works by finding an equivalent linear model

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Davig-Leeper approach
Define new variables
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Using the newly defined variables they defines
two new matrices, A and B
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DL derive a Generalized Taylor principle

• A necessary and sufficient condition for the NK
  model to have a unique bounded equilibrium is
  that all the eigenvalues of (B-1A) are inside the
  unit circle
• In Fact this is necessary but not sufficient for
  equilibrium to be unique

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A pitfall

• The Davig-Leeper idea (find a generalized Taylor
  principle) is an excellent one
• There is a problem with its execution which
  arises from the fact that

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An implication

• Two policy makers may each follow determinate
  policies. But the Markov Switching RE model may
  have indeterminate equilibria
• Two policy makers may each follow indeterminate
  policies. But the Markov Switching RE model may
  have a determinate equilibria

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Example
Determinate
Determinate
Indeterminate
Determinacy of each regime but indeterminacy of
the MSRE model
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What we show
If we can find numbers ci that satisfy this
equation
v1 and v2 play the roles of eigenvectors
c1 and c2 play the roles of eigenvalues
Note the DL condition forces the ci to be equal.
Then there are multiple sunspot equilibria
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What this implies

• The DL condition is necessary for uniqueness (but
  not sufficient)
• Sensible policy cannot stop bad things from
  happening
• For indeterminacy in every regime we need only
  find one bad policy-maker

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Calibration
These transition probabilities in conjunction
with the LS estimates imply indeterminacy in the
US economy
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Simulation with LS Estimates (Only Fundamental
Noise)
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Simulation with LS Estimates (Only Sunspot Noise)
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Simulation with LS Estimates (Only Fundamental
Noise)
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Main research questions

• When do MSRE models have a unique equilibrium?
• Can good policy rule out indeterminacy?
• Can a bad policy by one administration spillover
  into another?

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Answers

• When do MSRE models have a unique equilibrium?
• We dont know. We have necessary conditions for
  determinacy. We have some sufficiency conditions.
  A full set of necessary and sufficient conditions
  is a hard problem in linear algebra that (to our
  knowledge) has not yet been solved.

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Answers

• Can good policy rule out indeterminacy?
• Probably not. But good policy can limit the
  impact of both fundamental and non-fundamental
  shocks.

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Answers

• Can a bad policy by one administration spillover
  into another?
• Yes. But the bad effects of bad administrations
  can be limited.

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How to Proceed

• Focus on Minimal State Variable Solutions (see
  our working paper on this topic)