Procedure of Calculating Policy Functions 1 Motivation Previous Works 2 Advantages and Summary 3 Model NK Model with MS Taylor Rule under ZLB Expectations Function Static One-Period Problem of a MS-DSGE subject to ZLB Stochastic, Rational-Expectations Equilibrium (SREE) 4 5 6 Calibration Methods Procedure of Calculating Policy Functions Calbration Parameters Calibration Results Policy and Impulse Response Functions Policy Implications Conclusion 26
27 Procedure of Calculating Policy Functions Global Numerical Procedure (Billi, 2011, AEJ Macro) A fixed point in the space of policy functions is found with an iterative update rule ŷ k+1 = ŷ k + ι k (ŷ,k+1 ŷ k ), from step k to k + 1 Step 1. Assign interpolation nodes and make an initial guess ŷ 0. Step 2. Updata the state, evaluate the expectations function, and apply update rule above to derive a new guess ŷ +1.. Step 3. Stop if max n=1,,n ŷ k+1 ŷ k < τ where τ > 0 is convergence tolerance. Otherwise, repeat step 2.
Calbration Parameters 1 Motivation Previous Works 2 Advantages and Summary 3 Model NK Model with MS Taylor Rule under ZLB Expectations Function Static One-Period Problem of a MS-DSGE subject to ZLB Stochastic, Rational-Expectations Equilibrium (SREE) 4 5 6 Calibration Methods Procedure of Calculating Policy Functions Calbration Parameters Calibration Results Policy and Impulse Response Functions Policy Implications Conclusion 28
Calbration Parameters Table : Calibration Parameters parameter Economic interpretation assigned value β quarterly discount factor 0.9913 = (1+ 3.5% ) 1 4 σ real rate elasticity of output 6.25 κ slope of the Phillips curve 0.024 φ π1 reaction cofficient of inflation under Aggressive regime 2.2 φ y 1 reaction cofficient of output under Aggressive regime 0.5 φ π2 reaction cofficient of inflation under Passive regime 0.8 φ y 2 reaction cofficient of output under Passive regime 0.15 ρ u AR-coefficient Agg Supply shocks 0.0 ρ g AR-coefficient Agg Demand shocks 0.8 σ u S.d. Agg Supply shock innovations (quarterly %) 0.154 σ g S.d. Agg Demand shock innovations (quarterly %) 3.048 (=1.524*2) p 11 transition probability from Aggresive to Aggresive 0.7 p 22 transition probability from Passive to Passive 0.7 29
Policy and Impulse Response Functions 1 2 3 4 5 Motivation Previous Works Advantages and Summary Model NK Model with MS Taylor Rule under ZLB Expectations Function Static One-Period Problem of a MS-DSGE subject to ZLB Stochastic, Rational-Expectations Equilibrium (SREE) Calibration Methods Procedure of Calculating Policy Functions Calbration Parameters Calibration Results Policy and Impulse Response Functions Policy Implications 6 Conclusion 30
Policy and Impulse Response Functions Figure : Policy Functions w.r.t. Agg Demand Shock g t ; non-zlb vs. ZLB (a) Under Aggressive Policy Regime (b) Under Passive Policy Regime non-zlb -> linear, ZLB -> non-linear. 31
Policy and Impulse Response Functions Figure : Policy Functions w.r.t. Agg Demand Shock g t ; Stochastic Expectations vs. Perfect Foresight (a) Under Aggressive Policy Regime (b) Under Passive Policy Regime drop of output and inflation under stochastic rational expectation is bigger than under perfect foresight 32
Policy and Impulse Response Functions Figure : Policy Functions w.r.t. Agg Demand Shock g t ; Aggressive Regime vs. Passive Regime (a) Under Perfect Foresight (b) Under Stochastic Rational Expectations slope under aggressive regime is more moderate than under passive regime the larger size of negative shock is, the closer difference between both regimes 33
Policy and Impulse Response Functions Figure : Impulse Response Functions of Agg Demand Shock g t under Stochastic Expectations; Aggressive vs. Passive (a) Response of Positive Shock (b) Response of Negative Shock In positive shock, big difference between both regimes In negative shock hitting zero interest rate, similar impulse between both regimes 34
Policy and Impulse Response Functions Figure : Impulse Response Functions of Agg Demand Shock g t under Aggressive Regime; Stoc. Expec vs. Perfect Foresight (a) Response of Positive Shock (b) Response of Negative Shock In positive shock, almost same impulse response In negative shock, size of decline of output and inflation in stochastic > in perfect 35
1 Motivation Previous Works 2 Advantages and Summary 3 Model NK Model with MS Taylor Rule under ZLB Expectations Function Static One-Period Problem of a MS-DSGE subject to ZLB Stochastic, Rational-Expectations Equilibrium (SREE) 4 5 6 Calibration Methods Procedure of Calculating Policy Functions Calbration Parameters Calibration Results Policy and Impulse Response Functions Policy Implications Conclusion 36
Figure : Simulations conditional on Specified Regime (100 periods); Aggressive Regime vs. Passive Regime (a) the Case in absence of the ZLB (b) the Case in presence of the ZLB Without ZLB, the effects under both regime are symmetry between postive and negative areas. With ZLB, size of declines of output and inflation is similar between both 37
Table : Simulation conditional on Specified Regime (100,000 samples) (a) the Case in absence of the ZLB constraint Regime Stoc. Expect. Perfect Fore. variables under the non-zlb Aggressive. Passive diffe. ( A - P ) mean Std Dev mean Std Dev mean Std Dev output 0.00 1.27 0.00 2.87 0.00-1.60 inflation 0.00 0.13 0.00 0.16 0.00-0.03 interest rate 0.00 0.87 0.00 0.62 0.00 +0.25 output 0.00 1.29 0.01 2.92-0.01-1.63 inflation 0.00 0.13 0.00 0.16 0.00-0.03 interest rate 0.00 0.89 0.00 0.64 0.00 +0.25 means = steady states (=0.0) means under Aggressive = means under Passive St.D. under Aggressive < St.D. under Passive 38
39 Figure : Simulation conditional on Specified Regime; under Non-ZLB
Table : Simulation conditional on Specified Regime (100,000 samples) (b) the Case in presence of the ZLB constraint Regime Stoc. Expect. Perfect Fore. Aggressive. Passive diffe. ( A - P ) variables mean Std Dev mean Std Dev mean Std Dev output -0.93 3.01-0.54 4.00-0.39-0.99 inflation -0.02 0.20-0.01 0.22-0.01-0.02 interest rate 0.13 0.80 0.07 0.63 +0.06 +0.17 output -0.57 2.85-0.12 3.90-0.45-1.05 inflation -0.01 0.19 0.00 0.21-0.01-0.02 interest rate 0.12 0.78 0.06 0.62 +0.06 +0.16 under the ZLB means < steady states (=0.0) means of y t and π t under Aggressive < means under Passive St.D. of y t and π t under Aggressive < St.D. under Passive 40
41 Figure : Simulation conditional on Specified Regime; under ZLB
Figure : Simulations of Regime Switching Policy and Fixed Policy (100 periods) ; Fixed Policy = one regime fixed under aggressive policy. (a) the Case in absence of the ZLB (b) the Case in presence of the ZLB Without ZLB, the effects under both policies are symmetry between postive and negative areas. With ZLB, size of declines of output and inflation is similar between both 42
Table : Simulations of Regime Switching and Fixed Policies (100,000 samples) (a) the Case in absence of the ZLB constraint Policy variables R.S. Policy Fixed Policy diffe ( RS FP ) mean Std Dev mean Std Dev mean Std Dev output -0.01 2.22 0.00 1.21-0.01 +1.01 Stoc. Expect. inflation 0.00 0.15 0.00 0.12 0.00 +0.03 interest rate 0.00 0.76 0.00 0.79 0.00-0.03 output -0.01 2.21 0.00 1.21-0.01 +1.01 Perfect Fore. inflation 0.00 0.14 0.00 0.12 0.00 +0.02 interest rate 0.00 0.75 0.00 0.79 0.00-0.04 Note: Fixed Policy denotes one regime fixed under aggressive policy. under the non-zlb means = steady states (=0.0) means under R.S. policy = means under Fixed policy St.D. under R.S. policy > St.D. under Fixed policy 43
Table : Simulations of Regime Switching and Fixed Policies (100,000 samples) (b)the Case in presence of the ZLB constraint Policy Stoc. Expect. Perfect Fore. R.S. Policy Fixed Policy diffe ( RS - FP ) variables mean Std Dev mean Std Dev mean Std Dev output -0.71 3.60-0.91 2.91 +0.20 +0.69 inflation -0.01 0.21-0.02 0.18 +0.01 +0.03 interest rate 0.12 0.73 0.12 0.74 0.00-0.01 output -0.38 3.40-0.51 2.64 +0.13 +0.76 inflation -0.01 0.20-0.01 0.17 0.00 +0.03 interest rate 0.08 0.70 0.09 0.69-0.01 +0.01 under the ZLB means < steady states (=0.0) means of y t and π t under RS policy > means under Fixed policy St.D. of y t and π t under RS policy > St.D. under Fixed policy 44
Policy Implications 1 Motivation Previous Works 2 Advantages and Summary 3 Model NK Model with MS Taylor Rule under ZLB Expectations Function Static One-Period Problem of a MS-DSGE subject to ZLB Stochastic, Rational-Expectations Equilibrium (SREE) 4 Calibration Methods Procedure of Calculating Policy Functions Calbration Parameters 5 Calibration Results Policy and Impulse Response Functions Policy Implications 6 Conclusion 45
Policy Implications Under the ZLB, small difference in dopped sizes of output and inflation between Active Policy regime and Passive Policy Regime. Next, we consider what policy dose work in this sitiation? Policy Imprication The effect of 20% Reduction of St.D (or Uncertainty) of Agg Demand Shock Figure : Distributions of Stochastic Rational Expectations at -2% Agg Demand Shock 46
Policy Implications Table : Effects of 20% Reduction of St.D. (or Uncertainty) of Agg Demand Shock (a) the Case in absence of the ZLB constraint Policy RS Policy Fixed Policy 20% reduction Original difference variables mean Std Dev mean Std Dev mean Std Dev output 0.00 1.84-0.01 2.22 0.01-0.38 inflation 0.00 0.14 0.00 0.15 0.00-0.01 interest rate 0.00 0.63 0.00 0.76 0.00-0.23 output 0.00 1.00 0.00 1.21 0.00-0.21 inflation 0.00 0.12 0.00 0.12 0.00 0.00 interest rate 0.00 0.64 0.00 0.79 0.00-0.15 under the non-zlb means = steady states (=0.0) just 20 % down for St.D under RS policy and Fixed policy 47
48 Policy Implications Table : Effects of 20% Reduction of St.D. (or Uncertainty) of Agg Demand Shock Policy RS Policy Fixed Policy (b) the Case in presence of the ZLB constraint under the ZLB variables 20% reduction Original differences mean Std Dev mean Std Dev mean Std Dev output -0.41 2.63-0.71 3.60 +0.30-0.97 inflation -0.01 0.18-0.01 0.21 0.00-0.03 interest rate 0.08 0.62 0.12 0.73-0.04-0.11 output -0.57 1.84-0.91 2.91 +0.34-1.07 inflation -0.02 0.15-0.02 0.18 0.00-0.03 interest rate 0.09 0.61 0.12 0.74-0.03-0.13 Both of means and St.D of 20% reduction are around 2/3 of Original
Conclusion Under the ZLB 1 Small difference in dropped level of output and inflation between Active (or Aggressive) and Passive policy regimes. 2 Non-negligible gap between Stochastic Expectations and Perfect Foresight Perfect Foresight make output and inflation biased upward. 3 Intensifying uncertainty ( bigger variance of shocks) would deepen further declines of output and inflation even for the same exogenous shocks, regardless of monetary policy regimes. a policy forming expectations would play an important role of recovering an economy by mitigating uncertainty of aggregate demand shock, rather than monetary policy regime should remain aggressively. the means of Output and Inflation are biased downward from their steady state. 49