What Drives Monetary Policy Shifts?: A New Approach to Regime - PowerPoint PPT Presentation

What Drives Monetary Policy Shifts?: A New Approach to Regime Switching in DSGE Models Yoosoon Chang Department of Economics Indiana University Macroeconomics Workshop Keio University Tokyo, Japan December 18, 2018

Main References ◮ A New Approach to Regime Switching ◮ Chang, Choi and Park (2017) A New Approach to Model Regime Switching, Journal of Econometrics , 196, 127-143. ◮ Policy Rules with Endogenous Regime Changes ◮ Chang, Kwak and Qiu (2017) U.S. Monetary-Fiscal Regime Changes in the Presence of Endogenous Feedback in Policy Rules. ◮ Endogenous Policy Shifts in a Simple DSGE Model ◮ Chang, Tan and Wei (2018) A Structural Investigation of Monetary Policy Shifts ◮ Chang, Maih and Tan (2018) State Space Models with Endogenous Regime Switching

Introduction Background: New Approach to Regime Switching Endogenous Policy Shifts in a Simple DSGE Model

Introduction Monetary policy behavior is purposeful and responds endogenously to the state of the economy. ◮ Clarida et al (2000), Lubik and Schorfheide (2004) and Sims and Zha (2006): Taylor rule displays time variation Subsequently, Markov switching processes is introduced to DSGE models to explore these empirical findings. ◮ Policy regime shifts assumed to be exogenous, inconsistent with the central tenet of Taylor rule Calls for a model that makes the policy change a purposeful response to the state of the economy. ◮ Davig and Leeper (2006) build a New Keynesian model with monetary policy rule that switches when past inflation crosses a threshold value. ◮ Is inflation true or only source of monetary policy shifts?

This Work Address why have monetary policy regimes shifted and what are the driving forces. We investigate macroeconomic sources of monetary policy shifts. We introduce the new endogenous switching by Chang, Choi and Park (2017) into state space models ◮ an autoregressive latent factor determines regimes, and generates endogenous feedback that links current monetary policy stance to historical fundamental shocks ◮ time-varying transition probabilities ◮ endogenous-switching Kalman filter ◮ application to monetary DSGE model Greater scope for understanding complex interaction between regime switching and economic behavior

Monetary Policy Shifts A. Federal Funds Rate 20% 15% 10% 5% 0% 1960 1970 1980 1990 2000 2010 B. Monetary Policy Intervention 5% 0% -5% 1960 1970 1980 1990 2000 2010 ◮ Panel A: effective federal funds rate (blue solid) vs. inertial version of Taylor rule (red dashed); Panel B: differential ◮ Loose policy in late 70s vs. tight policy in early 80s

Introduction Background: New Approach to Regime Switching Basic Switching Models Relationship with Conventional Markov Switching Model MLE and Modified Markov Switching Filter Illustrations Endogenous Policy Shifts in a Simple DSGE Model

Mean Switching Model The basic mean model with regime switching is given by ( y t − µ t ) = γ ( y t − 1 − µ t − 1 ) + u t with µ t = µ ( s t ) , where ◮ ( s t ) is a state process specifying a binary state of regime ◮ s t = 0 and 1 are referred respectively to as low and high mean regimes in the model.

Volatility Switching Model The basic volatility model with regime switching is given by y t = σ t u t with σ t = σ ( s t ) , where ◮ ( s t ) is a state process specifying a binary state of regime ◮ s t = 0 and 1 are referred respectively to as low and high volatility regimes in the model.

Conventional Regime Switching Model The state process ( s t ) is assumed to be entirely independent of other parts of the underlying model, and specified as a two state markov chain. Therefore, the two transition probabilities a = P { s t = 0 | s t − 1 = 0 } b = P { s t = 1 | s t − 1 = 1 } , completely specify the state process ( s t ) .

A New Regime Switching Model Chang, Choi and Park (2017) specify a model y t = m t + σ t u t = m ( x t , y t − 1 , . . . , y t − k , s t , . . . , s t − k ) + σ ( x t , s t , . . . , s t − k ) u t = m ( x t , y t − 1 , . . . , y t − k , w t , . . . , w t − k ) + σ ( x t , w t , . . . , w t − k ) u t where ◮ covariate ( x t ) is exogenous, ◮ state process ( s t ) is driven by s t = 1 { w t ≥ τ } , ◮ latent factor ( w t ) is specified as w t = αw t − 1 + v t , and � � �� 0 � � 1 �� u t ρ = d N , v t +1 0 ρ 1 with endogeneity parameter ρ .

New Mean Switching Model The mean model with autoregressive latent factor is given by γ ( L )( y t − µ t ) = u t where γ ( z ) = 1 − γ 1 z − · · · − γ k z k is a k − th order polynomial, µ t = µ ( s t ) , s t = 1 { w t ≥ τ } , w t = αw t − 1 + v t and � � �� 0 � � 1 �� u t ρ = d N , v t +1 0 ρ 1 Again a shock ( u t ) at time t affects the regime at time t + 1 , and the regime switching becomes endogenous. The endogeneity parameter ρ represents the reversion of mean in our mean model.

New Volatility Switching Model The volatility model with autoregressive latent factor is given by y t = σ t u t where σ t = σ ( s t ) = σ ( w t ) , s t = 1 { w t ≥ τ } , w t = αw t − 1 + v t and � � �� 0 � � 1 �� u t ρ = d N , v t +1 0 ρ 1 A shock ( u t ) at time t affects the regime at time t + 1 , and the regime switching becomes endogenous. The endogeneity parameter ρ , which is expected to be negative, represents the leverage effect in our volatility model.

Relationship with Conventional Switching Model ◮ The new model reduces to the conventional markov switching model when the underlying autoregressive latent factor is stationary with | α | < 1 , and exogenous with ρ = 0 , i.e., independent of the model innovation. ◮ Assume ρ = 0 , and obtain transition probabilities of ( s t ) . In our approach, they are given as functions of the autoregressive coefficient α of the latent factor and the level τ of threshold. ◮ Note that � � � � � � � w t − 1 � w t − 1 s t = 0 = P w t < τ = Φ( τ − αw t − 1 ) P � � � � � � � w t − 1 � w t − 1 s t = 1 = P w t ≥ τ = 1 − Φ( τ − αw t − 1 ) P from w t = αw t − 1 + v t and v t ∼ N (0 , 1) .

Transition of Stationary State Process Transition probabilities of state process ( s t ) from low state to low state a ( α, τ ) and high state to high state b ( α, τ ) are given by a ( α, τ ) = P { s t = 0 | s t − 1 = 0 } √ � τ � � 1 − α 2 αx Φ τ − √ ϕ ( x ) dx 1 − α 2 −∞ � � = √ 1 − α 2 Φ τ b ( α, τ ) = P { s t = 1 | s t − 1 = 1 } � ∞ � � αx 1 − α 2 Φ τ − √ ϕ ( x ) dx √ 1 − α 2 τ � � = 1 − √ 1 − α 2 1 − Φ τ ◮ One-to-one correspondence between ( α, τ ) and ( a, b ) . ◮ An important by-product from the new approach: regime factor w t determining regime and regime strength.

MLE and New Markov Switching Filter The new endogenous model can be estimated by ML method. n � ℓ ( y 1 , . . . , y n ) = log p ( y 1 ) + log p ( y t |F t − 1 ) t =2 � � where F t = σ x t , ( y s ) s ≤ t , for t = 1 , . . . , n . To compute the log-likelihood function and estimate the new switching model, we need to develop a new filter. Write � � p ( y t |F t − 1 ) = · · · p ( y t | s t , . . . , s t − k , F t − 1 ) p ( s t , . . . , s t − k |F t − 1 ) . s t s t − k � � m t , σ 2 Since p ( y t | s t , . . . , s t − k , y t − 1 , . . . , y t − k ) = N , it suffices to t compute p ( s t , . . . , s t − k |F t − 1 ) . This is done by repeated implementations of the prediction and updating steps, as in the usual Kalman filter.

Prediction Step For the prediction step, note p ( s t , . . . , s t − k |F t − 1 ) � = p ( s t | s t − 1 , . . . , s t − k − 1 , F t − 1 ) p ( s t − 1 , . . . , s t − k − 1 |F t − 1 ) , s t − k − 1 and p ( s t | s t − 1 , . . . , s t − k − 1 , F t − 1 ) = p ( s t | s t − 1 , . . . , s t − k − 1 , y t − 1 , . . . , y t − k − 1 ) .

Prediction Step - Transition Probability The transition probability of state is given as p ( s t | s t − 1 , . . . , s t − k − 1 , y t − 1 , . . . , y t − k − 1 ) = (1 − s t ) ω ρ ( s t − 1 , . . . , s t − k − 1 , y t − 1 , . . . , y t − k − 1 ) � � + s t 1 − ω ρ ( s t − 1 , . . . , s t − k − 1 , y t − 1 , . . . , y t − k − 1 ) , where, in turn, if | α | < 1 , ω ρ ( s t − 1 , . . . , s t − k − 1 , y t − 1 , . . . , y t − k − 1 ) � τ √ � � 1 − α 2 � ∞ � � τ − ρy t − 1 − m t − 1 αx (1 − s t − 1 ) + s t − 1 Φ ρ − √ ϕ ( x ) dx τ √ σ t − 1 1 − α 2 1 − α 2 −∞ = , � � � � 1 − α 2 ) + s t − 1 1 − α 2 ) (1 − s t − 1 )Φ( τ 1 − Φ( τ Therefore, p ( s t , . . . , s t − k |F t − 1 ) can be readily computed, once p ( s t − 1 , . . . , s t − k − 1 |F t − 1 ) obtained from the previous updating step.

Updating Step For the updating step, we have p ( s t , . . . , s t − k |F t ) = p ( s t , . . . , s t − k | y t , F t − 1 ) = p ( y t | s t , . . . , s t − k , F t − 1 ) p ( s t , . . . , s t − k |F t − 1 ) , p ( y t |F t − 1 ) where p ( y t | s t , . . . , s t − k , F t − 1 ) = p ( y t | s t , . . . , s t − k , y t − 1 , . . . , y t − k ) We may now readily obtain p ( s t , . . . , s t − k |F t ) from p ( s t , . . . , s t − k |F t − 1 ) and p ( y t |F t − 1 ) from previous prediction step.