Exiting from QE by Fumio Hayashi and Junko Koeda Some comments - PowerPoint PPT Presentation

Exiting from QE by Fumio Hayashi and Junko Koeda Some comments from Mark Watson ¡ 1 ¡

Questions: • How can we estimate effects of monetary policy when the policy instrument changes endogenously from the short term interest rate to QE? • How can we estimate the effect of changing instruments, i.e., exiting from QE? Answer: • Use standard recursive SVAR that allows for discrete breaks associated with changes in policy instruments. o Model breaks in terms of observables o Do some careful data work o Estimate parameters using appropriate methods (breaks, truncation associated with bounds). o Make some sensible empirical choices o Compute IRFs and counterfactuals using nonlinear methods. ¡ 2 ¡

Models and VARs ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ X t Output gap Bank Rate ⎢ ⎥ , X t = t = Notation: ⎢ ⎥ and P ⎥ ⎢ ⎢ P ⎥ ⎢ ⎥ Excess Reserves ⎣ ⎦ Inflation ⎣ ⎦ ⎣ ⎦ t ( ) X ), η X , t ⎧ X t = f X X t − 1 , P t + 1 , … | Ω t t − 1 , E ( P t , X t + 1 , P ⎪ ⎨ Model: ( ) t = f P X t − 1 , P t + 1 , … | Ω t P ), η P , t ⎪ P t − 1 , E ( P t , X t + 1 , P ⎩ Linearize model and solve ⎧ X t = Φ X t − 1 + Β P t − 1 + Γ XX η X , t + Γ XP η P , t ⎪ ⎨ SVAR: t = Λ X t − 1 + Ψ P t − 1 + Γ PX η X , t + G PP η P , t P ⎪ ⎩ ¡ 3 ¡

2 Models ( ) X ), η X , t ⎧ X t = f X X t − 1 , P t + 1 , … | Ω t t − 1 , E ( P t , X t + 1 , P ⎪ ⎨ Model 0: ( ) t = f 0, P X t − 1 , P t + 1 , … | Ω t P ), η P , t ⎪ P t − 1 , E ( P t , X t + 1 , P ⎩ ( ) X ), η X , t ⎧ X t = f X X t − 1 , P t + 1 , … | Ω t t − 1 , E ( P t , X t + 1 , P ⎪ ⎨ Model 1: ( ) t = f 1, P X t − 1 , P t + 1 , … | Ω t P ), η P , t ⎪ P t − 1 , E ( P t , X t + 1 , P ⎩ ⎧ X t = Φ 0 X t − 1 + Β 0 P t − 1 + Γ 0, XX η X , t + Γ 0, XP η P , t ⎪ ⎨ SVAR 0: t = Λ 0 X t − 1 + Ψ 0 P t − 1 + Γ 0, PX η X , t + G 0, PP η P , t P ⎪ ⎩ ⎧ X t = Φ 1 X t − 1 + Β 1 P t − 1 + Γ 1, XX η X , t + Γ 1, XP η P , t ⎪ ⎨ SVAR 1: t = Λ 1 X t − 1 + Ψ 1 P t − 1 + Γ 1, PX η X , t + G 1, PP η P , t P ⎪ ⎩ ¡ 4 ¡

1 Model, 2 Regimes ( s t = 0 or s t = 1) ( ) X ), η X , t ⎧ X t = f X X t − 1 , P t + 1 , … | Ω t t − 1 , E ( P t , X t + 1 , P ⎪ Model( s t = 0): ⎨ ( ) t = f 0, P X t − 1 , P t + 1 , … | Ω t P ), η P , t ⎪ P t − 1 , E ( P t , X t + 1 , P ⎩ ( ) X ), η X , t ⎧ X t = f X X t − 1 , P t + 1 , … | Ω t t − 1 , E ( P t , X t + 1 , P ⎪ Model( s t = 1): ⎨ ( ) t = f 1, P X t − 1 , P t + 1 , … | Ω t P ), η P , t ⎪ P t − 1 , E ( P t , X t + 1 , P ⎩ ⎧ X t = Φ 0 X t − 1 + Β 0 P t − 1 + Γ 0, XX η X , t + Γ 0, XP η P , t ⎪ SVAR( s t = 0): ?? ⎨ t = Λ 0 X t − 1 + Ψ 0 P t − 1 + Γ 0, PX η X , t + G 0, PP η P , t P ⎪ ⎩ ⎧ X t = Φ 1 X t − 1 + Β 1 P t − 1 + Γ 1, XX η X , t + Γ 1, XP η P , t ⎪ SVAR( s t = 1): ?? ⎨ t = Λ 1 X t − 1 + Ψ 1 P t − 1 + Γ 1, PX η X , t + G 1, PP η P , t P ⎪ ⎩ ¡ 5 ¡

t + 1 , … | Ω t ) What matters: E ( P t , X t + 1 , P which depends on forecasts of s t , s t +1 , s t +2 , s t +3 , … ( ) = P s t + k s t { } j ≥ 1 ( ) A special case: P s t + k s t , X t , P t , s t − j , X t − j , P t − j That is, s t is an exogenous first-order Markov process ¡ 6 ¡

( ) = P s t + k s t { } j ≥ 1 ( ) Special Case: P s t + k s t , X t , P t , s t − j , X t − j , P t − j ( ) ⎧ X ), η X , t X t = f X X t − 1 , P t + 1 , … | Ω t t − 1 , E ( P t , X t + 1 , P ⎪ ⎪ ( ) t = f ( s t = 0), P X t − 1 , P t + 1 , … | Ω t P ), η P , t ⎨ Model: P t − 1 , E ( P t , X t + 1 , P ⎪ ( ) t = f ( s t = 1), P X t − 1 , P t + 1 , … | Ω t P ), η P , t ⎪ P t − 1 , E ( P t , X t + 1 , P ⎩ X = Ω t Suppose Ω t P (and both include s t ). Things are simple ⎧ X t = Φ 0 X t − 1 + Β 0 P t − 1 + Γ 0, XX η X , t + Γ 0, XP η P , t ⎪ SVAR( s t = 0): ⎨ t = Λ 0 X t − 1 + Ψ 0 P t − 1 + Γ 0, PX η X , t + G 0, PP η P , t P ⎪ ⎩ ⎧ X t = Φ 1 X t − 1 + Β 1 P t − 1 + Γ 1, XX η X , t + Γ 1, XP η P , t ⎪ SVAR( s t = 1): ⎨ t = Λ 1 X t − 1 + Ψ 1 P t − 1 + Γ 1, PX η X , t + G 1, PP η P , t P ⎪ ⎩ ¡ 7 ¡

Suppose Ω t X includes s t − 1 but not s t Ω t P include s t ⎧ X t = Φ 0,0 X t − 1 + Β 0,0 P t − 1 + Γ 0,0, XX η X , t + Γ 0,0, XP η P , t ⎪ SVAR( s t = 0, s t − 1 = 0): ⎨ t = Λ 0,0 X t − 1 + Ψ 0,0 P t − 1 + Γ 0,0, PX η X , t + G 0,0, PP η P , t P ⎪ ⎩ ⎧ X t = Φ 1,0 X t − 1 + Β 1,0 P t − 1 + Γ 1,0, XX η X , t + Γ 1,0, XP η P , t ⎪ SVAR( s t = 1, s t − 1 = 0): ⎨ t = Λ 1,0 X t − 1 + Ψ 1,0 P t − 1 + Γ 1,0, PX η X , t + G 1,0, PP η P , t P ⎪ ⎩ ⎧ X t = Φ 0,1 X t − 1 + Β 0,1 P t − 1 + Γ 0,1, XX η X , t + Γ 0,1, XP η P , t ⎪ SVAR( s t = 0, s t − 1 = 1): ⎨ t = Λ 0,1 X t − 1 + Ψ 0,1 P t − 1 + Γ 0,1, PX η X , t + G 0,1, PP η P , t P ⎪ ⎩ ⎧ X t = Φ 1,1 X t − 1 + Β 1,1 P t − 1 + Γ 1,1, XX η X , t + Γ 1,1, XP η P , t ⎪ SVAR( s t = 1, s t − 1 = 1): ⎨ t = Λ 1,1 X t − 1 + Ψ 1,1 P t − 1 + Γ 1,1, PX η X , t + G 1,1, PP η P , t P ⎪ ⎩ ¡ 8 ¡

Truth: SVAR( s t = 0, s t − 1 = 0), SVAR( s t = 0, s t − 1 = 1), SVAR( s t = 1, s t − 1 = 0),SVAR( s t = 1, s t − 1 = 1) Estimate: SVAR( s t = 0) , SVAR( s t = 1) model. What will I find ? SV AR( s t = 0) will be a mixture of SVAR( s t = 0, s t − 1 = 0), SVAR( s t = 0, s t − 1 = 1) SVAR( s t = 1) will be a mixture of SVAR( s t = 1, s t − 1 = 0), SVAR( s t = 1, s t − 1 = 1) but P ( s t = 0, s t − 1 = 0) ≫ P ( s t = 0, s t − 1 = 1), so SVAR( s t = 0) ≈ SVAR( s t = 0, s t − 1 = 0) P ( s t = 1, s t − 1 = 1) ≫ P ( s t = 1, s t − 1 = 0), so SVAR( s t = 1) ≈ SVAR( s t = 1, s t − 1 = 1) ¡ 9 ¡

Some questions for this misspecified model (1) IRFs of Policy variables under s = 1 ( R – shocks) or s = 0 ( m – shocks) SVAR( s t = 1) ≈ SVAR( s t = 1, s t − 1 = 1) SVAR( s t = 0) ≈ SVAR( s t = 0, s t − 1 = 0) Approximately correct “continuing regime” answers. ¡ 10 ¡

(2) Counterfactual “Exit from Regime” s t s t +1 s t +2 s t − 1 Factual 0 1 1 1 Counterfactual 0 0 1 1 Correct Answers from t − 1 t t + 1 t + 2 Factual SVAR( s t = 1, s t − 1 = 0) SVAR( s t = 1, s t − 1 = 1) Counterfactual SVAR( s t = 0, s t − 1 = 0) SVAR( s t = 0, s t − 1 = 1) Misspecfied Model Answers from t − 1 t t + 1 t + 2 Factual SVAR( s t = 1, s t − 1 = 1) SVAR( s t = 1, s t − 1 = 1) Counterfactual SVAR( s t = 0, s t − 1 = 0) SVAR( s t = 1, s t − 1 = 1) ¡ 11 ¡

( ) = P s t + k s t { } j ≥ 1 ( ) Special Case: P s t + k s t , X t , P t , s t − j , X t − j , P t − j Hyashi-Koeda: ( ) + (1 − ρ ) r s t − 1 = 0, ρ r * X t , … , X t − 11 ⎧ ⎧ t − 1 > r t > 0 and π t > q t ∼ N ( π , σ 2 ) ⎪ ⎪ ⎨ s t = 1 if ( ) + (1 − ρ ) r s t − 1 = 1, ρ r * X t , … , X t − 11 t − 1 > r t > 0 ⎨ ⎪ ⎩ ⎪ ⎩ 0 otherwise ( ) ⎧ X ), η X , t X t = f X X t − 1 , P t + 1 , … | Ω t t − 1 , E ( P t , X t + 1 , P ⎪ ⎪ ( ) t = f ( s t = 0), P X t − 1 , P t + 1 , … | Ω t P ), η P , t ⎨ Model: P t − 1 , E ( P t , X t + 1 , P ⎪ ( ) t = f ( s t = 1), P X t − 1 , P t + 1 , … | Ω t P ), η P , t ⎪ P t − 1 , E ( P t , X t + 1 , P ⎩ SVAR … more complicated (!) Approximations: SVAR( s t = 0) and SVAR( s t = 1) are averages over these regimes ¡ 12 ¡

Data: 30000 20000 10000 0 -10000 -20000 -30000 -40000 -50000 -60000 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Output Gap ¡ 13 ¡

4 3 2 1 0 -1 -2 -3 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Inflation ¡ 14 ¡

10 8 6 4 2 0 -2 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 R-RBar ¡ 15 ¡

200 175 150 125 100 75 50 25 0 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Excess Reserves ¡ 16 ¡

Approximations: SVAR( s t = 0) and SVAR( s t = 1) are averages over these regimes (1) IRFs of Policy variables under s = 1 ( R – shocks) or s = 0 ( m – shocks) (2) Counterfactual “Exit from Regime” s t s t +1 s t +2 s t − 1 Factual 0 1 1 1 Counterfactual 0 0 1 1 ¡ 17 ¡