SLIDE 1
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
SLIDE 2 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
SLIDE 3
Introduction Background: New Approach to Regime Switching Endogenous Policy Shifts in a Simple DSGE Model
SLIDE 4
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?
SLIDE 5
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
SLIDE 6 Monetary Policy Shifts
1960 1970 1980 1990 2000 2010 0% 5% 10% 15% 20%
- B. Monetary Policy Intervention
1960 1970 1980 1990 2000 2010
0% 5%
◮ 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
SLIDE 7
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
SLIDE 8
Mean Switching Model
The basic mean model with regime switching is given by (yt − µt) = γ(yt−1 − µt−1) + ut with µt = µ(st), where
◮ (st) is a state process specifying a binary state of regime ◮ st = 0 and 1 are referred respectively to as low and high mean
regimes in the model.
SLIDE 9
Volatility Switching Model
The basic volatility model with regime switching is given by yt = σtut with σt = σ(st), where
◮ (st) is a state process specifying a binary state of regime ◮ st = 0 and 1 are referred respectively to as low and high
volatility regimes in the model.
SLIDE 10 Conventional Regime Switching Model
The state process (st) is assumed to be entirely independent of
- ther parts of the underlying model, and specified as a two state
markov chain. Therefore, the two transition probabilities a = P{st = 0|st−1 = 0} b = P{st = 1|st−1 = 1}, completely specify the state process (st).
SLIDE 11 A New Regime Switching Model
Chang, Choi and Park (2017) specify a model yt = mt + σtut = m(xt, yt−1, . . . , yt−k, st, . . . , st−k) + σ(xt, st, . . . , st−k)ut = m(xt, yt−1, . . . , yt−k, wt, . . . , wt−k) + σ(xt, wt, . . . , wt−k)ut where
◮ covariate (xt) is exogenous, ◮ state process (st) is driven by st = 1{wt ≥ τ}, ◮ latent factor (wt) is specified as wt = αwt−1 + vt,
and
vt+1
1 ρ ρ 1
- with endogeneity parameter ρ.
SLIDE 12 New Mean Switching Model
The mean model with autoregressive latent factor is given by γ(L)(yt − µt) = ut where γ(z) = 1 − γ1z − · · · − γkzk is a k−th order polynomial, µt = µ(st), st = 1{wt ≥ τ}, wt = αwt−1 + vt and
vt+1
1 ρ ρ 1
- Again a shock (ut) 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.
SLIDE 13 New Volatility Switching Model
The volatility model with autoregressive latent factor is given by yt = σtut where σt = σ(st) = σ(wt), st = 1{wt ≥ τ}, wt = αwt−1 + vt and
vt+1
1 ρ ρ 1
- A shock (ut) 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.
SLIDE 14 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 (st). In
- ur approach, they are given as functions of the autoregressive
coefficient α of the latent factor and the level τ of threshold.
◮ Note that
P
- st = 0
- wt−1
- = P
- wt < τ
- wt−1
- = Φ(τ − αwt−1)
P
- st = 1
- wt−1
- = P
- wt ≥ τ
- wt−1
- = 1 − Φ(τ − αwt−1)
from wt = αwt−1 + vt and vt ∼ N(0, 1).
SLIDE 15 Transition of Stationary State Process
Transition probabilities of state process (st) 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{st = 0|st−1 = 0} = τ
√ 1−α2 −∞
Φ
αx √ 1 − α2
Φ
√ 1 − α2
- b(α, τ) = P{st = 1|st−1 = 1}
= 1 − ∞
τ √ 1−α2 Φ
αx √ 1 − α2
1 − Φ
√ 1 − α2
- ◮ One-to-one correspondence between (α, τ) and (a, b).
◮ An important by-product from the new approach: regime
factor wt determining regime and regime strength.
SLIDE 16 MLE and New Markov Switching Filter
The new endogenous model can be estimated by ML method. ℓ(y1, . . . , yn) = log p(y1) +
n
log p(yt|Ft−1) where Ft = σ
- xt, (ys)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(yt|Ft−1) =
· · ·
p(yt|st, . . . , st−k, Ft−1)p(st, . . . , st−k|Ft−1). Since p(yt|st, . . . , st−k, yt−1, . . . , yt−k) = N
t
compute p(st, . . . , st−k|Ft−1). This is done by repeated implementations of the prediction and updating steps, as in the usual Kalman filter.
SLIDE 17 Prediction Step
For the prediction step, note p(st, . . . , st−k|Ft−1) =
p(st|st−1, . . . , st−k−1, Ft−1)p(st−1, . . . , st−k−1|Ft−1), and p(st|st−1, . . . , st−k−1, Ft−1) = p(st|st−1, . . . , st−k−1, yt−1, . . . , yt−k−1).
SLIDE 18 Prediction Step - Transition Probability
The transition probability of state is given as
p(st|st−1, . . . , st−k−1, yt−1, . . . , yt−k−1) = (1 − st)ωρ(st−1, . . . , st−k−1, yt−1, . . . , yt−k−1) + st
- 1 − ωρ(st−1, . . . , st−k−1, yt−1, . . . , yt−k−1)
- ,
where, in turn, if |α| < 1,
ωρ(st−1, . . . , st−k−1, yt−1, . . . , yt−k−1) =
τ√
1−α2 −∞
+st−1 ∞
τ√ 1−α2
σt−1 − αx √ 1 − α2
(1 − st−1)Φ(τ
- 1 − α2) + st−1
- 1 − Φ(τ
- 1 − α2)
- ,
Therefore, p(st, . . . , st−k|Ft−1) can be readily computed, once p(st−1, . . . , st−k−1|Ft−1) obtained from the previous updating step.
SLIDE 19
Updating Step
For the updating step, we have p(st, . . . , st−k|Ft) = p(st, . . . , st−k|yt, Ft−1) = p(yt|st, . . . , st−k, Ft−1)p(st, . . . , st−k|Ft−1) p(yt|Ft−1) , where p(yt|st, . . . , st−k, Ft−1) = p(yt|st, . . . , st−k, yt−1, . . . , yt−k) We may now readily obtain p(st, . . . , st−k|Ft) from p(st, . . . , st−k|Ft−1) and p(yt|Ft−1) from previous prediction step.
SLIDE 20 Extraction of Latent Factor
From prediction and updating steps, we compute p(wt, st−1, ..., st−k−1, Ft−1) and p(wt, st−1, ..., st−k−1, Ft). By marginalizing obtain p (wt|Ft) =
· · ·
p (wt, st−1, ..., st−k|Ft) . which yields the inferred factor E (wt|Ft) =
We may easily extract the inferred factor, once the maximum likelihood estimates of p(wt|Ft), 1 ≤ t ≤ n, are available.
SLIDE 21
GDP Growth Rates
We use
◮ Seasonally adjusted quarterly real US GDP for two sample
periods: 1952-1984 and 1984-2012
◮ GDP growth rates are obtained as the first differences of their
logs to fit the mean model γ(L) (yt − µ(st)) = σut where γ(z) = 1 − γ1z − γ2z2 − γ3z3 − γ4z4.
SLIDE 22
US Real GDP Growth Rates
SLIDE 23 Estimation Result: GDP Growth Rate Model
Sample Periods 1952-1984 1984-2012 Endogeneity Ignored Allowed Ignored Allowed µ
(0.219) (0.161) (0.298) (0.318) µ 1.144 1.212 0.710 0.705 (0.113) (0.095) (0.092) (0.085) γ1 0.068 0.147 0.154 0.169 (0.123) (0.104) (0.105) (0.106) γ2
0.044 0.350 0.340 (0.112) (0.096) (0.105) (0.104) γ3
0.133 (0.108) (0.090) (0.106) (0.104) γ4
0.043 0.049 (0.104) (0.095) (0.103) (0.115) σ 0.794 0.784 0.455 0.453 (0.065) (0.057) (0.034) (0.032) ρ
1.000 (0.151) (0.001) log-likelihood
- 173.420
- 169.824
- 80.584
- 76.443
p-value 0.007 0.004
SLIDE 24
Transition Probability Comparison
SLIDE 25
Transition Probability Comparison
SLIDE 26
Transition Probability Comparison
SLIDE 27
NBER Recession Period and Latent Factor: 1952-1984
SLIDE 28
Recession Probabilities: 1952-1984
SLIDE 29
Stock Return Volatility
We use
◮ Monthly CRSP returns for 1926/01 - 2012/12 (1,044 obs.) ◮ One-month T-bill rates used to obtain excess returns ◮ Demeaned excess returns
to fit the volatility model yt = σ(st)ut, where σ(st) = σ(1 − st) + ¯ σst and st = 1{wt ≥ τ}.
SLIDE 30 Estimation Result: Monthly Volatility Model
Sample Periods 1926-2012 1990-2012 Endogeneity Ignored Allowed Ignored Allowed σ = σ(st) when st = 0 0.0385 0.0380 0.0223 0.0251 (0.0010) (0.0011) (0.0018) (0.0041) σ = σ(st) when st = 1 0.1154 0.1153 0.0505 0.0554 (0.0087) (0.0090) (0.0030) (0.0082) ρ
(0.0847) (0.0059) log-likelihood 1742.28 1747.98 507.70 511.28 p-value (LR test for ρ = 0) 0.001 0.007
SLIDE 31
Transition Probability Comparison
SLIDE 32
Transition Probability Comparison
SLIDE 33
Transition Probability Comparison
SLIDE 34
Transition Probability Comparison
SLIDE 35
Extracted Latent Factor from Volatility Model and VIX
SLIDE 36
High Volatility Probabilities: 1990-2012
SLIDE 37
Introduction Background: New Approach to Regime Switching Endogenous Policy Shifts in a Simple DSGE Model A Simple Fisherian Model A Prototypical New Keynesian Model A New Filtering Algorithm for Estimation
SLIDE 38 A Simple Regime Switching Fisherian Model
Fisher equation: it = Etπt+1 + Etrt+1 Real rate process: rt = ρrrt−1 + σrǫr
t
Monetary policy with endogenous feedback: it = α(st)πt + σeǫe
t
st = 1{wt ≥ τ} wt+1 = φwt + vt+1,
t
vt+1
1 ρ ρ 1
- as considered in Chang, Choi and Park (2017).
SLIDE 39 Information Structure
Agents do not observe the level of latent regime factor wt, but
- bserve whether or not it crosses the threshold, as reflected in
st = 1{wt ≥ τ}. Agents form expectations on future inflation as Etπt+1 = E(πt+1|Ft) using the information Ft = {iu, πu, ru, ǫr
u, ǫe u, su}t u=0
Monetary authority observes all information in Ft and also the history of policy regime factor (wt).
SLIDE 40 Endogenous Feedback Mechanism
To see the endogenous feedback mechanism, rewrite wt+1 = φwt + ρǫe
t +
, ηt+1 ∼ i.i.d.N(0, 1) From variance decomposition, we see that ρ2 is the contribution of past intervention to regime change
◮ ρ = 0 : fully driven by exogenous non-structural shock
wt+1 = φwt + ηt+1
◮ |ρ| = 1 : fully driven by past monetary policy shock
wt+1 = φwt + ǫe
t
SLIDE 41 Time-Varying Transition Probabilities
Agents infer transition probabilities by integrating out the latent factor wt using its invariant distribution, N(0, 1/(1 − φ2)). Transition probabilities of the state from t to t + 1 p00(ǫe
t) =
τ√
1−φ2 −∞
Φρ
φx
t
Φ(τ
p10(ǫe
t) =
∞
τ√ 1−φ2 Φρ
φx
t
1 − Φ(τ
where Φρ(x) = Φ(x/
- 1 − ρ2). Time varying and depend on ǫe
t.
SLIDE 42
Time-Varying Transition Probabilities
◮ If ρ = 0, reduces to exogenous switching model ◮ ρ governs the fluctuation of transition probabilities
SLIDE 43
Analytical Solution
We solve the system of expectational nonlinear difference equations using the guess and verify method. Davig and Leeper (2006) show that the analytical solution for the model with fixed regime monetary policy process is πt+1 = a1rt+1 + a2ǫe
t+1
with some constants a1 and a2. Motivated by this, we start with the following guess πt+1 = a1(st+1, pst+1,0(ǫe
t+1))rt+1 + a2(st+1)ǫe t+1
SLIDE 44 Analytical Solution
πt+1 = ρr α(st+1) (α1 − α0)pst+1,0(ǫe
t+1) + α1
α0 ρr − Ep00(ǫe
t+1)
t+1)
(α1 − ρr) α0 ρr − Ep00(ǫe
t+1)
t+1)
t+1))
rt+1 − σe α(st+1)
ǫe
t+1
Limiting Case 1: Exogenous switching solution (ρ = 0)
πt+1 = ρr α(st+1) (α1 − α0)¯ pst+1,0 + α1 α0 ρr − ¯ p00
p10 (α1 − ρr) α0 ρr − ¯ p00
p10
rt+1 − σe α(st+1)
ǫe
t+1
SLIDE 45 Analytical Solution
πt+1 = ρr α(st+1) (α1 − α0)pst+1,0(ǫe
t+1) + α1
α0 ρr − Ep00(ǫe
t+1)
t+1)
(α1 − ρr) α0 ρr − Ep00(ǫe
t+1)
t+1)
t+1))
rt+1 − σe α(st+1)
ǫe
t+1
Limiting Case 2: Fixed-regime solution (α1 = α0) πt+1 = ρr α − ρr
a1
rt+1 −σe α
ǫe
t+1
SLIDE 46 Macro Effects of Policy Intervention
Set a future policy intervention It = { ǫe
t+1,
ǫe
t+2, . . . ,
ǫe
t+K} and
evaluate its effect on future inflation. Consider the contractionary intervention IT = {4%, . . . , 4%
, 0, . . . , 0
8 periods
} with K = 16, sT = 0 As in Leeper and Zha (2003), define
◮ Baseline = E(πT+K|FT , st = sT , t = T + 1, . . . , T + K) ◮ Direct Effects =
E(πT+K|IT , FT , st = sT , t = T + 1, . . . , T + K) - Baseline
◮ Total Effects = E(πT+K|IT , FT ) - Baseline ◮ Expectations Formation Effects = Total Effects - Direct
Effects
SLIDE 47
Expectations Formation Effect
◮ ǫT+1 > 0 ρ>0
− − → wT+2 ↑, sT+2 ր 1 → more likely to switch
◮ price stabilized as agents adjust beliefs on future regimes ◮ black dot signifies period T + 2 total effect;
SLIDE 48
Impulse Response Function
◮ ǫT+1 > 0 ρ>0
− − → wT+2 ↑, sT+2 ր 1 → more aggressive
◮ endogenous mechanism helps explain price stabilization
SLIDE 49 Regime Switching Monetary DSGE Model
◮ Benchmark Specification
◮ A regime-switching New Keynesian DSGE model, which has
been a standard tool for monetary policy analysis Ireland (2004), An and Schorfheide (2007), Woodford (2011), and Davig and Doh (2014)
◮ Endogenous Switching in Monetary DSGE Model
◮ Links current monetary policy regime to past fundamental
shocks by a policy regime factor.
◮ Generates an endogenous feedback between monetary policy
stance and observed economic behavior.
◮ Solved by perturbation method in Maih and Waggoner (2018)
up to first-order.
◮ Implemented in RISE MATLAB toolbox developed by Junior
Maih, available at https://github.com/jmaih/RISE toolbox.
SLIDE 50
A Prototypical DSGE Model
We consider the small-scale new Keynesian DSGE model presented in An and Schorfheide (2007), whose essential elements include:
◮ a representative household ◮ a continuum of monopolistically competitive firms; each firm
produces a differentiated good and faces nominal rigidity in terms of quadratic price adjustment cost
◮ a cashless economy with one-period nominal bonds ◮ a monetary authority that controls nominal interest rate as
well as a fiscal authority that passively adjusts lump-sum taxes to ensure its budgetary solvency
◮ a labor-augmenting technology that induces a stochastic trend
in consumption and output.
SLIDE 51
Notations
◮ 0 < β < 1 : the discount factor ◮ τc > 0 : the coefficient of relative risk aversion ◮ ct : the detrended consumption ◮ Rt the nominal interest rate ◮ πt : the inflation between periods t − 1 and t ◮ Et : expectation given information available at time t ◮ 1/ν > 1 : elasticity of demand for each differentiated good ◮ φ : the degree of price stickiness ◮ π :
the steady state inflation
◮ yt :
the detrended output
◮ 0 ≤ ρR < 1 : the degree of interest rate smoothing ◮ r : the steady state real interest rate, ◮ ψπ > 0, ψy > 0 : the policy rate responsive coefficients ◮ y∗ t = (1 − ν)1/τcgt : the detrended potential output
SLIDE 52
Shocks
zt : an exogenous shock to the labor-augmenting technology gt : an exogenous government spending shock ǫR,t : an exogenous policy shock. ln gt and ln zt evolve as autoregressive processes ln gt = (1 − ρg) ln g + ρg ln gt−1 + ǫg,t and ln zt = ρz ln zt−1 + ǫz,t where 0 ≤ ρg, ρz < 1 and g is the steady state of gt. The model is driven by the three innovations ǫt = [ǫR,t, ǫg,t, ǫz,t]′ that are serially uncorrelated, independent of each other at all leads and lags, and normally distributed with means zero and standard deviations (σR, σg, σz), respectively.
SLIDE 53 A Prototypical DSGE Model
Equilibrium conditions in fixed-regime benchmark: DIS: 1 = βEt ct+1 ct −τc Rt γzt+1πt+1
1 = 1 − cτc
t
ν + φ(πt − π)
2ν
2ν
ct+1 ct −τc yt+1 yt (πt+1 − π)πt+1
Rt = R∗1−ρR
t
RρR
t−1eǫR,t,
R∗
t = rπ
πt π ψπ yt y∗
t
ψy ARC: yt = ct +
gt
2 (πt − π)2yt Regime switching process: ψπ(st) = ψ0
π(1 − st) + ψ1 πst,
0 ≤ ψ0
π < ψ1 π
SLIDE 54
A Prototypical DSGE Model
Implied time-varying transition probabilities to regime-0 are an important part of the model solution p00( ǫt) = P(st+1 = 0|st = 0, ǫt) = τ
√ 1−α2 −∞
Φρ(τ − αx/ √ 1 − α2 − ρ′ ǫt)pN(x|0, 1)dx Φ(τ √ 1 − α2) p10( ǫt) = P(st+1 = 0|st = 1, ǫt) = ∞
τ √ 1−α2 Φρ(τ − αx/
√ 1 − α2 − ρ′ ǫt)pN(x|0, 1)dx 1 − Φ(τ √ 1 − α2) where Φρ(x) = Φ(x/√1 − ρ′ρ), ǫt = [ǫR,t/σR, ǫg,t/σg, ǫz,t/σz]′, and ρ = [ρRv, ρgv, ρzv]′ = corr( ǫt, vt+1). The presence of st poses keen computational challenges to solving the model. When ǫt and vt+1 are orthogonal (i.e., ρ = 03×1), our model reduces to the conventional Markov switching model.
SLIDE 55 Regime Switching DSGE Model
Switching in state space form yt = Dst + Zstxt + Fstzt + ut, ut ∼ N(0, Ωst) xt = Cst + Gstxt−1 + Estzt + Mstǫt, ǫt ∼ N(0, Σst) New regime switching
◮ state process st driven by wt as st = 1{wt ≥ τ} ◮ latent factor wt = αwt−1 + vt,
ǫt = Σ−1/2
st
, and ǫt vt+1
0n×1
In ρ ρ′ 1
ρ′ρ < 1 Advantage
◮ st is endogenous → systematically affected by observables ◮ st can be nonstationary → allow for regime persistency ◮ wt is continuous → directly related to other variables
SLIDE 56 Endogenous Feedback Mechanism
Latent factor wt+1 = αwt + ρ′ ǫt +
where ηt ∼ N(0, 1), ρ = (ρRv, ρgv, ρzv)′ and ρ′ρ < 1. Forecast error variance decomposition: Vart(wt+h) =
3
h
ρ2
kα2(h−j)
ǫk
+
h
3
ρ2
k
◮ ρ2 k: contribution of
ǫk to regime change
◮ 1 − ρ′ρ: contribution of η to regime change
Key to quantifying macro origins of monetary policy shifts
SLIDE 57 Taking Model to Data
◮ Quarterly observations from 1954:Q3 to 2007:Q4
◮ YGR: per capita real output growth ◮ INF: annualized inflation rate ◮ INT: effective federal funds rate
◮ Measurement equations
YGRt INFt INTt = γ(Q) π(A) π(A) + r(A) + 4γ(Q) + 100 ˆ yt − ˆ yt−1 + ˆ zt 4ˆ πt 4 ˆ Rt
◮ Transition Equations
◮ perturbation solution of Maih & Waggoner (2018) ◮ RISE MATLAB toolbox developed by Maih
SLIDE 58 Endogenous-Switching Kalman Filter
Augmented state space form yt = Dst + Fstzt
+
0l×n
xt dt
+ut xt dt
= Cst + Estzt 0n×1
+ Gst 0m×n 0n×m 0n×n
xt−1 dt−1
+
st
In
◮ Key features of our filter
◮ marginalization-collapsing procedure of Kim (1994) ◮ time-varying transition probabilities ◮ filtered/smoothed regime factor as by-product
SLIDE 59 Filtering Algorithm
Step 1: Forecasting ς(i,j)
t|t−1
=
Gjςi
t−1|t−1
P (i,j)
t|t−1
=
t−1|t−1
G′
j +
Mj M′
j
p(i,j)
t|t−1
=
P(st = j|st−1 = i, λt−1)pi
t−1|t−1p(λt−1|Ft−1)dλt−1 ◮ To compute e.g., p(0,0) t|t−1
◮ since λt−1 = ρ′
ǫt−1 is latent, approximate p(λt−1|Ft−1) by pN(λt−1|ρ′ς0
d,t−1|t−1, ρ′P 0 d,t−1|t−1ρ)
◮ time-varying transition probability P(st = 0|st−1 = 0, λt−1)
τ
√ 1−α2 −∞
Φρ(τ − αx/ √ 1 − α2 − λt−1)pN(x|0, 1)dx Φ(τ √ 1 − α2)
SLIDE 60 Filtering Algorithm (Cont’d)
Step 2: Likelihood evaluation y(i,j)
t|t−1
=
Zjς(i,j)
t|t−1
F (i,j)
t|t−1
=
t|t−1
Z′
j + Ωj
p(yt|Ft−1) =
1
1
pN(yt|y(i,j)
t|t−1, F (i,j) t|t−1)p(i,j) t|t−1
SLIDE 61 Filtering Algorithm (Cont’d)
Step 3: Updating ς(i,j)
t|t
= ς(i,j)
t|t−1 + P (i,j) t|t−1
Z′
j(F (i,j) t|t−1)−1(yt − y(i,j) t|t−1)
P (i,j)
t|t
= P (i,j)
t|t−1 − P (i,j) t|t−1
Z′
j(F (i,j) t|t−1)−1
ZjP (i,j)
t|t−1
p(i,j)
t|t
= pN(yt|y(i,j)
t|t−1, F (i,j) t|t−1)p(i,j) t|t−1
p(yt|Ft−1)
◮ History truncation
◮ collapse (ς(i,j)
t|t
, P (i,j)
t|t
) into (ςj
t|t, P j t|t)
◮ further collapse (ςj
t|t, P j t|t) into (ςt|t, Pt|t)
SLIDE 62 Prior Distributions
Parameter Density Para (1) Para (2) [5%, 95%] Fixed Regime τc, coefficient of relative risk aversion G 2.00 0.50 [1.25, 2.89] κ, slope of new Keynesian Phillips curve G 0.20 0.10 [0.07, 0.39] ψπ, interest rate response to inflation G 1.50 0.25 [1.11, 1.93] ψy, interest rate response to output G 0.50 0.25 [0.17, 0.97] r(A), s.s. annualized real interest rate G 0.50 0.10 [0.35, 0.68] π(A), s.s. annualized inflation rate G 3.84 2.00 [1.24, 7.61] γ(Q), s.s. technology growth rate N 0.47 0.20 [0.14, 0.80] ρR, persistency of monetary shock B 0.50 0.10 [0.34, 0.66] ρg, persistency of spending shock B 0.50 0.10 [0.34, 0.66] ρz, persistency of technology shock B 0.50 0.10 [0.34, 0.66] 100σR, scaled s.d. of monetary shock IG-1 0.40 4.00 [0.08, 1.12] 100σg, scaled s.d. of spending shock IG-1 0.40 4.00 [0.08, 1.12] 100σz, scaled s.d. of technology shock IG-1 0.40 4.00 [0.08, 1.12] ν, inverse of demand elasticity B 0.10 0.05 [0.03, 0.19] 1/g, s.s. consumption-to-output ratio B 0.85 0.10 [0.66, 0.97] Threshold Switching ψ0
π, ψπ under regime-0
G 1.00 0.10 [0.84, 1.17] ψ1
π, ψπ under regime-1
G 2.00 0.25 [1.61, 2.43] α, persistency of latent factor B 0.90 0.05 [0.81, 0.97] τ, threshold level N −1.00 0.50 [−1.82, −0.18] ρRv, endogeneity from monetary shock U −1.00 1.00 [−0.90, 0.90] ρgv, endogeneity from spending shock U −1.00 1.00 [−0.90, 0.90] ρzv, endogeneity from technology shock U −1.00 1.00 [−0.90, 0.90]
SLIDE 63 Posterior Estimates
No Switching Model Regime Switching Model Parameter Mode Median [5%, 95%] Mode Median [5%, 95%] Fixed Regime τc 3.54 3.29 [2.50, 4.25] 2.49 2.47 [1.72, 3.34] κ 0.47 0.49 [0.36, 0.65] 0.74 0.69 [0.49, 0.94] ψπ 1.00 1.01 [1.00, 1.04] – – – ψy 0.15 0.18 [0.07, 0.38] 0.23 0.27 [0.10, 0.53] r(A) 0.47 0.47 [0.34, 0.62] 0.43 0.47 [0.35, 0.61] π(A) 2.78 2.80 [1.65, 3.97] 2.97 2.72 [2.04, 3.38] γ(Q) 0.38 0.37 [0.30, 0.45] 0.39 0.37 [0.29, 0.44] ρR 0.69 0.70 [0.66, 0.73] 0.70 0.70 [0.65, 0.74] ρg 0.95 0.95 [0.94, 0.95] 0.95 0.95 [0.94, 0.95] ρz 0.95 0.95 [0.94, 0.95] 0.95 0.95 [0.93, 0.95] 100σR 0.26 0.26 [0.23, 0.28] 0.26 0.27 [0.24, 0.30] 100σg 1.00 1.03 [0.95, 1.12] 1.03 1.04 [0.96, 1.13] 100σz 0.06 0.07 [0.05, 0.08] 0.06 0.06 [0.05, 0.08] ν 0.04 0.09 [0.03, 0.18] 0.12 0.09 [0.03, 0.18] 1/g 0.87 0.87 [0.68, 0.98] 0.89 0.87 [0.69, 0.97] Threshold Switching ψ0
π
– – – 1.02 0.99 [0.92, 1.06] ψ1
π
– – – 1.28 1.25 [1.10, 1.43] α – – – 0.97 0.95 [0.91, 0.98] τ – – – −1.20 −0.98 [−1.72, −0.23] ρRv – – – −0.08 −0.05 [−0.38, 0.27] ρgv – – – 0.34 0.11 [−0.29, 0.41] ρzv – – – −0.91 −0.71 [−0.91, −0.39] Log marginal likelihood −1099.97 −1076.14 Bayes factor vs. no switching 1.00 exp (23.83)
SLIDE 64 Evidence of Endogeneity
0.2 0.4 0.6 0.8 1 0.5 1 1.5
0.2 0.4 0.6 0.8 1 0.5 1 1.5 2
0.2 0.4 0.6 0.8 1 0.5 1 1.5 2
◮ ρgv > 0: MP is ‘leaning against the wind’ ◮ ρzv < 0: MP is promoting economic growth
SLIDE 65 Extracted Regime Factor
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
wtjt
2 4 6 8 10
p1
tjt
0.2 0.4 0.6 0.8 1
◮ Sluggish switching b/w more and less active regimes ◮ Timing and nature are consistent with narrative record
SLIDE 66 Main Findings
◮ Prima facie evidence of endogeneity in monetary policy shifts
◮ MP is ‘leaning against the wind’—expansionary gvt spending
shock increases the likelihood of more active regime.
◮ MP is promoting long-term growth—favorable tech shock
decreases the likelihood of shifting into more active regime.
◮ overall, non-policy shocks have played a predominant role in
driving regime changes during the post-World War II period.
◮ Estimated regime factor identifies MP as slowly fluctuating
between more and less active regimes, in ways consistent with conventional view and narrative record.
◮ Endogenizing regime changes in monetary DSGE models
provides a promising venue for understanding the purposeful nature of monetary policy.
