Markov Switching in Structural Models

1. Markov Switching in Structural Models Roger E. A. Farmer (Joint with Dan Waggoner and Tao Zha) EABCN Conference (September 2007)

2. 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?

3. 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

4. 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

5. The New-Keynesian model

6. 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

7. Why the Taylor principle works • If the Taylor principle is satisfied, the eigenvalues of Gamma are outside the unit circle

8. 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

9. Davig-Leeper • Davig-Leeper study a model of the form States Transition probabilities

10. The New-Keynesian example with policy switches Policy parameters may switch

11. The New-Keynesian example

12. The Davig-Leeper question • What do we mean by an equilibrium in the MSRE model? • When is equilibrium unique?

13. 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

14. Davig-Leeper approach Define new variables

15. Using the newly defined variables they defines two new matrices, A and B

16. 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

17. 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

18. 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

19. Example Determinate Determinate Indeterminate Determinacy of each regime but indeterminacy of the MSRE model

20. 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

21. 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

22. Calibration These transition probabilities in conjunction with the LS estimates imply indeterminacy in the US economy

23. Simulation with LS Estimates (Only Fundamental Noise)

24. Simulation with LS Estimates (Only Sunspot Noise)

25. Simulation with LS Estimates (Only Fundamental Noise)

26. 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?

27. Answers • When do MSRE models have a unique equilibrium? • We don’t 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.

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

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

30. How to Proceed • Focus on Minimal State Variable Solutions (see our working paper on this topic)