Rational Sunspots Guido Ascari a , Paolo Bonomolo b , HedibertF.Lopes c a University of Oxford, b Sveriges Riksbank 1 , c INSPER Nonlinear Models in Macroeconomics and Finance for an Unstable World Norges Bank, Oslo, 2018 1 The views expressed are solely the responsibility of the authors and should not to be interpreted as reflecting the views of Sveriges Riksbank. Ascari, Bonomolo and Lopes Oslo, January 2018 1 / 35
This Paper Propose a method to consider a broader class of solutions to 1 stochastic linear models. Two generalizations: A novel way to introduce sunspots that yields drifting parameters and stochastic volatility Include temporary unstable solutions: we allow for determinacy, indeterminacy and instability Develop an econometric strategy to verify if unstable paths are 2 empirically relevant Application: 3 Example of U.S. Great inflation (Lubik and Schorfheide, 2004, model and data) U.S. inflation dynamics in the 70’s are better described by unstable equilibrium paths. Ascari, Bonomolo and Lopes Oslo, January 2018 2 / 35
Motivation RE generally implies multiple equilibria Explosive Stable How can we get uniqueness? (Sargent and Wallace ,1973; Phelps and Taylor, 1977; Taylor, 1977; Blanchard, 1979) Stability Criterion : Transversality conditions In saddle paths dynamics only one solution is stable This became the standard in Macroeconomics (Blanchard and Kahn, 1980) Ascari, Bonomolo and Lopes Oslo, January 2018 3 / 35
Example: U.S. Great Inflation period Is it appropriate to rule out unstable paths from the empirical analysis? 16 US Inflation 14 12 10 8 6 4 2 0 -2 1960 1965 1970 1975 1980 1985 1990 1995 2000 Figure: CPI inflation, quarterly data. Sample: 1960Q1 - 1997Q4 Is there any evidence that inflation is described (at least for a while) by unstable equilibria? Ascari, Bonomolo and Lopes Oslo, January 2018 4 / 35
A simple example: multiple RE solutions Consider the following simple one equation model: y t = 1 ε t ∼ N ( 0 , σ 2 θ E t y t + 1 + ε t , ε ) (1) Equation (1) has an infinite number of solutions: y t + 1 = E t y t + 1 + η t + 1 y t + 1 = θ y t − θε t + η t + 1 (2) where E t η t + 1 = 0 . Assume: η t + 1 = ( 1 + M ) ε t + 1 + ζ t + 1 where ζ t + 1 = sunspot or non-fundamental error. Two sources of multiplicity: This paper considers the FIRST term: intrinsic multiplicity of RE solutions Ascari, Bonomolo and Lopes Oslo, January 2018 5 / 35
A simple example: multiple RE solutions Assume ζ t = 0 ∀ t . All the solutions for y t are described by y t = θ y t − 1 − θε t − 1 + ( 1 + M ) ε t (3) Degree of freedom: the solution is parameterized by M ∈ ( − ∞ , + ∞ ) Two famous particular cases: Ascari, Bonomolo and Lopes Oslo, January 2018 6 / 35
A simple example: multiple RE solutions Assume ζ t = 0 ∀ t . All the solutions for y t are described by y t = θ y t − 1 − θε t − 1 + ( 1 + M ) ε t (3) Degree of freedom: the solution is parameterized by M ∈ ( − ∞ , + ∞ ) Two famous particular cases: "pure" forward-looking solution: M = 0 ( η t = ε t ) y F t − θ y F = ε t − θε t − 1 t − 1 y F = ε t ∀ t (4) t Ascari, Bonomolo and Lopes Oslo, January 2018 6 / 35
A simple example: multiple RE solutions Assume ζ t = 0 ∀ t . All the solutions for y t are described by y t = θ y t − 1 − θε t − 1 + ( 1 + M ) ε t (3) Degree of freedom: the solution is parameterized by M ∈ ( − ∞ , + ∞ ) Two famous particular cases: "pure" forward-looking solution: M = 0 ( η t = ε t ) y F t − θ y F = ε t − θε t − 1 t − 1 y F = ε t ∀ t (4) t "pure" backward-looking solution ( M = − 1 ) y B t = θ y B t − 1 − θε t − 1 (5) Ascari, Bonomolo and Lopes Oslo, January 2018 6 / 35
The interpretation for M For M � = − 1 , the expected value = an exponentially weighted average of the past observations (Muth, 1961) � � i t θ ∑ E t y t + 1 = M y t + 1 − i 1 + M i = 0 Natural interpretation for M: the way agents form expectations Ascari, Bonomolo and Lopes Oslo, January 2018 7 / 35
The interpretation for M For M � = − 1 , the expected value = an exponentially weighted average of the past observations (Muth, 1961) � � i t θ ∑ E t y t + 1 = M y t + 1 − i 1 + M i = 0 Natural interpretation for M: the way agents form expectations M defines the importance the agents give to the past data, both in absolute terms ( M vs 0), and in relative terms. Ascari, Bonomolo and Lopes Oslo, January 2018 7 / 35
The interpretation for M For M � = − 1 , the expected value = an exponentially weighted average of the past observations (Muth, 1961) � � i t θ ∑ E t y t + 1 = M y t + 1 − i 1 + M i = 0 Natural interpretation for M: the way agents form expectations M defines the importance the agents give to the past data, both in absolute terms ( M vs 0), and in relative terms. Infinite solutions = infinite way we can set that weights = > how to choose? Ascari, Bonomolo and Lopes Oslo, January 2018 7 / 35
The stability criterion (e.g., Blanchard and Kahn, 1980) y t = θ y t − 1 − θε t − 1 + ( 1 + M ) ε t Is the stability criterion sufficient to identify a unique path? If | θ | > 1 YES determinacy, by imposing M = 0 = f.l. 1 solution y F t = ε t (MSV solution) If | θ | < 1 NO indeterminacy 2 = > "Sunspot equilibria can often be constructed by randomizing over multiple equilibria of a general equilibrium model" Benhabib and Farmer (1999, p.390) Ascari, Bonomolo and Lopes Oslo, January 2018 8 / 35
Introducing sunspot equilibria We have infinite equilibria because: - there is an infinite number of ways of forming expectations - parametrized by M hence we introduce sunspots randomizing over M : M t = M t ( ζ t ) (6) where ζ t i.i.d., orthogonal to the fundamental shocks ε s ( s = 1 , 2 , ... ), and E t ζ t = 0 ∀ t . Ascari, Bonomolo and Lopes Oslo, January 2018 9 / 35
Introducing sunspot equilibria: drifting parameters and unstable paths If M t is a stochastic process with E t M t + 1 = M t then y t = α t y t − 1 − α t ε t − 1 + ( 1 + M t ) ε t (7) with α t = θ M t (with M t − 1 � = 0 otherwise FL solution ) . M t − 1 Same form as y t = θ y t − 1 − θε t − 1 + ( 1 + M ) ε t Ascari, Bonomolo and Lopes Oslo, January 2018 10 / 35
Introducing sunspot equilibria: drifting parameters and unstable paths If M t is a stochastic process with E t M t + 1 = M t then y t = α t y t − 1 − α t ε t − 1 + ( 1 + M t ) ε t (7) with α t = θ M t (with M t − 1 � = 0 otherwise FL solution ) . M t − 1 Same form as y t = θ y t − 1 − θε t − 1 + ( 1 + M ) ε t Drifting parameters and stochastic volatility within the rational expectations framework. Cogley and Sargent (2005), Primiceri (2005). Ascari, Bonomolo and Lopes Oslo, January 2018 10 / 35
Introducing sunspot equilibria: drifting parameters and unstable paths If M t is a stochastic process with E t M t + 1 = M t then y t = α t y t − 1 − α t ε t − 1 + ( 1 + M t ) ε t (7) with α t = θ M t (with M t − 1 � = 0 otherwise FL solution ) . M t − 1 Same form as y t = θ y t − 1 − θε t − 1 + ( 1 + M ) ε t Drifting parameters and stochastic volatility within the rational expectations framework. Cogley and Sargent (2005), Primiceri (2005). Intuition:agents can modify in every period the expectation formation process Ascari, Bonomolo and Lopes Oslo, January 2018 10 / 35
Introducing sunspot equilibria: drifting parameters and unstable paths If M t is a stochastic process with E t M t + 1 = M t then y t = α t y t − 1 − α t ε t − 1 + ( 1 + M t ) ε t (7) with α t = θ M t (with M t − 1 � = 0 otherwise FL solution ) . M t − 1 Same form as y t = θ y t − 1 − θε t − 1 + ( 1 + M ) ε t Drifting parameters and stochastic volatility within the rational expectations framework. Cogley and Sargent (2005), Primiceri (2005). Intuition:agents can modify in every period the expectation formation process Reconsidering unstable paths: | θ | > 1 and M t temporarily different from zero Ascari, Bonomolo and Lopes Oslo, January 2018 10 / 35
Which process for M? RE Solutions - With M t random variable the forecast error becomes: � � t − 1 θ i ε t − i ∑ η t = ( 1 + M t ) ε t + ( M t − M t − 1 ) (8) i = 1 - Under RE E t − 1 η t = 0, then: E t − 1 ( M t ) = M t − 1 ( M t martingale) 1 E t − 1 [( 1 + M t ) ε t ] = 0 , ( M t must be uncorrelated with ε t ) 2 if | θ | < 1 : Use conditions 1 and 2 if | θ | > 1 : Unstable paths. Consider the role of transversality condition Ascari, Bonomolo and Lopes Oslo, January 2018 11 / 35
Temporarily explosive paths All the paths corresponding to expected temporary deviations of M t from 0 will not violate the transversality condition RE requires M t martingale, but if | θ | > 1 and M t martingale, when M t � = 0 the economy is expected to remain on the unstable path, so transversality condition would be violated = > To allow for temporary unstable paths relax the martingale assumption (and RE) Deviation can be minimal without practical implications Mprocess Ascari, Bonomolo and Lopes Oslo, January 2018 12 / 35
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