What Drives Monetary Policy Shifts?: A NewApproach to Regime Switching in DSGE Models
Yoosoon Chang
Department of EconomicsIndiana University
Macroeconomics WorkshopKeio University
Tokyo, JapanDecember 18, 2018
Main References
I A New Approach to Regime SwitchingI Chang, Choi and Park (2017) A New Approach to Model
Regime Switching, Journal of Econometrics, 196, 127-143.
I Policy Rules with Endogenous Regime ChangesI Chang, Kwak and Qiu (2017) U.S. Monetary-Fiscal Regime
Changes in the Presence of Endogenous Feedback in PolicyRules.
I Endogenous Policy Shifts in a Simple DSGE ModelI Chang, Tan and Wei (2018) A Structural Investigation of
Monetary Policy Shifts
I Chang, Maih and Tan (2018) State Space Models withEndogenous 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 endogenouslyto the state of the economy.
I Clarida et al (2000), Lubik and Schorfheide (2004) and Simsand Zha (2006): Taylor rule displays time variation
Subsequently, Markov switching processes is introduced to DSGEmodels to explore these empirical findings.
I Policy regime shifts assumed to be exogenous, inconsistentwith the central tenet of Taylor rule
Calls for a model that makes the policy change a purposefulresponse to the state of the economy.
I Davig and Leeper (2006) build a New Keynesian model withmonetary policy rule that switches when past inflation crossesa threshold value.
I Is inflation true or only source of monetary policy shifts?
This Work
Address why have monetary policy regimes shifted and what arethe driving forces. We investigate macroeconomic sources ofmonetary policy shifts.
We introduce the new endogenous switching by Chang, Choi andPark (2017) into state space models
I an autoregressive latent factor determines regimes, andgenerates endogenous feedback that links current monetarypolicy stance to historical fundamental shocks
I time-varying transition probabilities
I endogenous-switching Kalman filter
I application to monetary DSGE model
Greater scope for understanding complex interaction betweenregime switching and economic behavior
Monetary Policy Shifts
A. Federal Funds Rate
1960 1970 1980 1990 2000 20100%
5%
10%
15%
20%
B. Monetary Policy Intervention
1960 1970 1980 1990 2000 2010
-5%
0%
5%
I Panel A: effective federal funds rate (blue solid) vs. inertialversion of Taylor rule (red dashed); Panel B: differential
I Loose policy in late 70s vs. tight policy in early 80s
Introduction
Background: New Approach to Regime SwitchingBasic Switching ModelsRelationship with Conventional Markov Switching ModelMLE and Modified Markov Switching FilterIllustrations
Endogenous Policy Shifts in a Simple DSGE Model
Mean Switching Model
The basic mean model with regime switching is given by
(yt − µt) = γ(yt−1 − µt−1) + ut
withµt = µ(st),
where
I (st) is a state process specifying a binary state of regime
I st = 0 and 1 are referred respectively to as low and high meanregimes
in the model.
Volatility Switching Model
The basic volatility model with regime switching is given by
yt = σtut
withσt = σ(st),
where
I (st) is a state process specifying a binary state of regime
I st = 0 and 1 are referred respectively to as low and highvolatility regimes
in the model.
Conventional Regime Switching Model
The state process (st) is assumed to be entirely independent ofother parts of the underlying model, and specified as a two statemarkov chain.
Therefore, the two transition probabilities
a = Pst = 0|st−1 = 0b = Pst = 1|st−1 = 1,
completely specify the state process (st).
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
I covariate (xt) is exogenous,
I state process (st) is driven by st = 1wt ≥ τ,I latent factor (wt) is specified as wt = αwt−1 + vt,
and (utvt+1
)=d N
((00
),
(1 ρρ 1
))with endogeneity parameter ρ.
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 = 1wt ≥ τ, wt = αwt−1 + vt and(
utvt+1
)=d N
((00
),
(1 ρρ 1
))
Again a shock (ut) at time t affects the regime at time t+ 1, andthe regime switching becomes endogenous. The endogeneityparameter ρ represents the reversion of mean in our mean model.
New Volatility Switching Model
The volatility model with autoregressive latent factor is given by
yt = σtut
where σt = σ(st) = σ(wt), st = 1wt ≥ τ, wt = αwt−1 + vt and(utvt+1
)=d N
((00
),
(1 ρρ 1
))
A shock (ut) at time t affects the regime at time t+ 1, and theregime switching becomes endogenous. The endogeneityparameter ρ, which is expected to be negative, represents theleverage effect in our volatility model.
Relationship with Conventional Switching Model
I The new model reduces to the conventional markov switchingmodel when the underlying autoregressive latent factor isstationary with |α| < 1, and exogenous with ρ = 0, i.e.,independent of the model innovation.
I Assume ρ = 0, and obtain transition probabilities of (st). Inour approach, they are given as functions of the autoregressivecoefficient α of the latent factor and the level τ of threshold.
I Note that
Pst = 0
∣∣wt−1
= P
wt < τ
∣∣wt−1
= Φ(τ − αwt−1)
Pst = 1
∣∣wt−1
= P
wt ≥ τ
∣∣wt−1
= 1− Φ(τ − αwt−1)
from wt = αwt−1 + vt and vt ∼ N(0, 1).
Transition of Stationary State Process
Transition probabilities of state process (st) from low state to lowstate a(α, τ) and high state to high state b(α, τ) are given by
a(α, τ) = Pst = 0|st−1 = 0
=
∫ τ√
1−α2
−∞Φ
(τ − αx√
1− α2
)ϕ(x)dx
Φ(τ√
1− α2)
b(α, τ) = Pst = 1|st−1 = 1
= 1−
∫ ∞τ√
1−α2
Φ
(τ − αx√
1− α2
)ϕ(x)dx
1− Φ(τ√
1− α2)
I One-to-one correspondence between (α, τ) and (a, b).
I An important by-product from the new approach: regimefactor wt determining regime and regime strength.
MLE and New Markov Switching Filter
The new endogenous model can be estimated by ML method.
`(y1, . . . , yn) = log p(y1) +
n∑t=2
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 newswitching model, we need to develop a new filter. Write
p(yt|Ft−1) =∑st
· · ·∑st−k
p(yt|st, . . . , st−k,Ft−1)p(st, . . . , st−k|Ft−1).
Since p(yt|st, . . . , st−k, yt−1, . . . , yt−k) = N(mt, σ
2t
), it suffices to
compute p(st, . . . , st−k|Ft−1). This is done by repeatedimplementations of the prediction and updating steps, as in theusual Kalman filter.
Prediction Step
For the prediction step, note
p(st, . . . , st−k|Ft−1)
=∑st−k−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).
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)
=
[(1−st−1)
∫ τ√
1−α2
−∞+st−1
∫ ∞τ√
1−α2
]Φρ
(τ−ρyt−1−mt−1
σt−1− αx√
1− α2
)ϕ(x)dx
(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, oncep(st−1, . . . , st−k−1|Ft−1) obtained from the previous updating step.
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) fromp(st, . . . , st−k|Ft−1) and p(yt|Ft−1) from previous prediction step.
Extraction of Latent Factor
From prediction and updating steps, we computep(wt, st−1, ..., st−k−1,Ft−1) and p(wt, st−1, ..., st−k−1,Ft).
By marginalizing obtain
p (wt|Ft) =∑st−1
· · ·∑st−k
p (wt, st−1, ..., st−k|Ft) .
which yields the inferred factor
E (wt|Ft) =
∫wtp (wt|Ft)dwt.
We may easily extract the inferred factor, once the maximumlikelihood estimates of p(wt|Ft), 1 ≤ t ≤ n, are available.
GDP Growth Rates
We use
I Seasonally adjusted quarterly real US GDP for two sampleperiods: 1952-1984 and 1984-2012
I GDP growth rates are obtained as the first differences of theirlogs
to fit the mean model
γ(L) (yt − µ(st)) = σut
where γ(z) = 1− γ1z − γ2z2 − γ3z
3 − γ4z4.
Estimation Result: GDP Growth Rate Model
Sample Periods 1952-1984 1984-2012
Endogeneity Ignored Allowed Ignored Allowed
µ -0.165 -0.083 -0.854 -0.756(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 -0.015 0.044 0.350 0.340(0.112) (0.096) (0.105) (0.104)
γ3 -0.175 -0.260 -0.077 0.133(0.108) (0.090) (0.106) (0.104)
γ4 -0.097 -0.067 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 -0.923 1.000(0.151) (0.001)
log-likelihood -173.420 -169.824 -80.584 -76.443p-value 0.007 0.004
Stock Return Volatility
We use
I Monthly CRSP returns for 1926/01 - 2012/12 (1,044 obs.)
I One-month T-bill rates used to obtain excess returns
I Demeaned excess returns
to fit the volatility model
yt = σ(st)ut,
whereσ(st) = σ(1− st) + σst
andst = 1wt ≥ τ.
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 -0.9698 -1.0000(0.0847) (0.0059)
log-likelihood 1742.28 1747.98 507.70 511.28p-value (LR test for ρ = 0) 0.001 0.007
Introduction
Background: New Approach to Regime Switching
Endogenous Policy Shifts in a Simple DSGE ModelA Simple Fisherian ModelA Prototypical New Keynesian ModelA New Filtering Algorithm for Estimation
A Simple Regime Switching Fisherian Model
Fisher equation:it = Etπt+1 + Etrt+1
Real rate process:rt = ρrrt−1 + σrε
rt
Monetary policy with endogenous feedback:
it = α(st)πt + σeεet
st = 1wt ≥ τ
wt+1 = φwt + vt+1,(εetvt+1
)=d iid N
(0,
(1 ρρ 1
))
as considered in Chang, Choi and Park (2017).
Information Structure
Agents do not observe the level of latent regime factor wt, butobserve whether or not it crosses the threshold, as reflected inst = 1wt ≥ τ.
Agents form expectations on future inflation as
Etπt+1 = E(πt+1|Ft)
using the information
Ft = iu, πu, ru, εru, εeu, sutu=0
Monetary authority observes all information in Ft and also thehistory of policy regime factor (wt).
Endogenous Feedback Mechanism
To see the endogenous feedback mechanism, rewrite
wt+1 = φwt + ρεet +√
1− ρ2ηt+1︸ ︷︷ ︸vt+1
, ηt+1 ∼ i.i.d.N(0, 1)
From variance decomposition, we see that ρ2 is the contribution ofpast intervention to regime change
I ρ = 0 : fully driven by exogenous non-structural shock
wt+1 = φwt + ηt+1
I |ρ| = 1 : fully driven by past monetary policy shock
wt+1 = φwt + εet
Time-Varying Transition Probabilities
Agents infer transition probabilities by integrating out the latentfactor wt using its invariant distribution, N(0, 1/(1− φ2)).
Transition probabilities of the state from t to t+ 1
p00(εet ) =
∫ τ√
1−φ2
−∞Φρ
(τ − φx√
1− φ2− ρεet
)ϕ(x)dx
Φ(τ√
1− φ2)
p10(εet ) =
∫ ∞τ√
1−φ2Φρ
(τ − φx√
1− φ2− ρεet
)ϕ(x)dx
1− Φ(τ√
1− φ2)
where Φρ(x) = Φ(x/√
1− ρ2). Time varying and depend on εet .
Time-Varying Transition Probabilities
I If ρ = 0, reduces to exogenous switching model
I ρ governs the fluctuation of transition probabilities
Analytical Solution
We solve the system of expectational nonlinear differenceequations using the guess and verify method.
Davig and Leeper (2006) show that the analytical solution for themodel with fixed regime monetary policy process is
πt+1 = a1rt+1 + a2εet+1
with some constants a1 and a2.
Motivated by this, we start with the following guess
πt+1 = a1(st+1, pst+1,0(εet+1))rt+1 + a2(st+1)εet+1
Analytical Solution
πt+1 =ρr
α(st+1)
(α1 − α0)pst+1,0(εet+1) + α1
(α0
ρr− Ep00(εet+1)
)+ α0Ep10(εet+1)
(α1 − ρr)(α0
ρr− Ep00(εet+1)
)+ (α0 − ρr)Ep10(εet+1)︸ ︷︷ ︸
a1(st+1,pst+1,0(εet+1))
rt+1
−σe
α(st+1)︸ ︷︷ ︸a2(st+1)
εet+1
Limiting Case 1: Exogenous switching solution (ρ = 0)
πt+1 =ρr
α(st+1)
(α1 − α0)pst+1,0 + α1
(α0
ρr− p00
)+ α0p10
(α1 − ρr)(α0
ρr− p00
)+ (α0 − ρr)p10︸ ︷︷ ︸
a1(st+1)
rt+1−σe
α(st+1)︸ ︷︷ ︸a2(st+1)
εet+1
Analytical Solution
πt+1 =ρr
α(st+1)
(α1 − α0)pst+1,0(εet+1) + α1
(α0
ρr− Ep00(εet+1)
)+ α0Ep10(εet+1)
(α1 − ρr)(α0
ρr− Ep00(εet+1)
)+ (α0 − ρr)Ep10(εet+1)︸ ︷︷ ︸
a1(st+1,pst+1,0(εet+1))
rt+1
−σe
α(st+1)︸ ︷︷ ︸a2(st+1)
εet+1
Limiting Case 2: Fixed-regime solution (α1 = α0)
πt+1 =ρr
α− ρr︸ ︷︷ ︸a1
rt+1−σeα︸︷︷︸a2
εet+1
Macro Effects of Policy Intervention
Set a future policy intervention It = εet+1, εet+2, . . . , ε
et+K and
evaluate its effect on future inflation.
Consider the contractionary intervention
IT = 4%, . . . , 4%︸ ︷︷ ︸8 periods
, 0, . . . , 0︸ ︷︷ ︸8 periods
with K = 16, sT = 0
As in Leeper and Zha (2003), define
I Baseline = E(πT+K |FT , st = sT , t = T + 1, . . . , T +K)
I Direct Effects =E(πT+K |IT ,FT , st = sT , t = T + 1, . . . , T +K) - Baseline
I Total Effects = E(πT+K |IT ,FT ) - Baseline
I Expectations Formation Effects = Total Effects - DirectEffects
Expectations Formation Effect
I εT+1 > 0ρ>0−−→ wT+2 ↑, sT+2 1 → more likely to switch
I price stabilized as agents adjust beliefs on future regimesI black dot signifies period T + 2 total effect;
Impulse Response Function
I εT+1 > 0ρ>0−−→ wT+2 ↑, sT+2 1 → more aggressive
I endogenous mechanism helps explain price stabilization
Regime Switching Monetary DSGE Model
I Benchmark Specification
I A regime-switching New Keynesian DSGE model, which hasbeen a standard tool for monetary policy analysis
Ireland (2004), An and Schorfheide (2007), Woodford (2011),and Davig and Doh (2014)
I Endogenous Switching in Monetary DSGE Model
I Links current monetary policy regime to past fundamentalshocks by a policy regime factor.
I Generates an endogenous feedback between monetary policystance and observed economic behavior.
I Solved by perturbation method in Maih and Waggoner (2018)up to first-order.
I Implemented in RISE MATLAB toolbox developed by JuniorMaih, available at https://github.com/jmaih/RISE toolbox.
A Prototypical DSGE Model
We consider the small-scale new Keynesian DSGE model presentedin An and Schorfheide (2007), whose essential elements include:
I a representative household
I a continuum of monopolistically competitive firms; each firmproduces a differentiated good and faces nominal rigidity interms of quadratic price adjustment cost
I a cashless economy with one-period nominal bonds
I a monetary authority that controls nominal interest rate aswell as a fiscal authority that passively adjusts lump-sum taxesto ensure its budgetary solvency
I a labor-augmenting technology that induces a stochastic trendin consumption and output.
Notations
I 0 < β < 1 : the discount factor
I τc > 0 : the coefficient of relative risk aversion
I ct : the detrended consumption
I Rt the nominal interest rate
I πt : the inflation between periods t− 1 and t
I Et : expectation given information available at time t
I 1/ν > 1 : elasticity of demand for each differentiated good
I φ : the degree of price stickiness
I π : the steady state inflation
I yt : the detrended output
I 0 ≤ ρR < 1 : the degree of interest rate smoothing
I r : the steady state real interest rate,
I ψπ > 0, ψy > 0 : the policy rate responsive coefficients
I y∗t = (1− ν)1/τcgt : the detrended potential output
Shocks
zt : an exogenous shock to the labor-augmenting technologygt : an exogenous government spending shockεR,t : an exogenous policy shock.
ln gt and ln zt evolve as autoregressive processes
ln gt = (1− ρg) ln g + ρg ln gt−1 + εg,t
andln zt = ρz ln zt−1 + εz,t
where 0 ≤ ρg, ρz < 1 and g is the steady state of gt.
The model is driven by the three innovations εt = [εR,t, εg,t, εz,t]′
that are serially uncorrelated, independent of each other at allleads and lags, and normally distributed with means zero andstandard deviations (σR, σg, σz), respectively.
A Prototypical DSGE Model
Equilibrium conditions in fixed-regime benchmark:
DIS: 1 = βEt
[(ct+1
ct
)−τc Rtγzt+1πt+1
]
NKPC: 1 =1− cτctν
+ φ(πt − π)
[(1− 1
2ν
)πt +
π
2ν
]− φβEt
[(ct+1
ct
)−τc yt+1
yt(πt+1 − π)πt+1
]
MP: Rt = R∗1−ρRt RρRt−1eεR,t , R∗t = rπ
(πtπ
)ψπ ( yty∗t
)ψyARC: yt = ct +
(1− 1
gt
)yt +
φ
2(πt − π)2yt
Regime switching process:
ψπ(st) = ψ0π(1− st) + ψ1
πst, 0 ≤ ψ0π < ψ1
π
A Prototypical DSGE Model
Implied time-varying transition probabilities to regime-0 are animportant part of the model solution
p00(εt) = P(st+1 = 0|st = 0, εt)
=
∫ τ√1−α2
−∞ Φρ(τ − αx/√
1− α2 − ρ′εt)pN(x|0, 1)dx
Φ(τ√
1− α2)
p10(εt) = P(st+1 = 0|st = 1, εt)
=
∫∞τ√
1−α2 Φρ(τ − αx/√
1− α2 − ρ′εt)pN(x|0, 1)dx
1− Φ(τ√
1− α2)
where Φρ(x) = Φ(x/√
1− ρ′ρ), εt = [εR,t/σR, εg,t/σg, εz,t/σz]′,
and ρ = [ρRv, ρgv, ρzv]′ = corr(εt, vt+1).
The presence of st poses keen computational challenges to solvingthe model. When εt and vt+1 are orthogonal (i.e., ρ = 03×1), ourmodel reduces to the conventional Markov switching model.
Regime Switching DSGE Model
Switching in state space form
yt = Dst + Zstxt + Fstzt + ut, ut ∼ N(0,Ωst)
xt = Cst +Gstxt−1 + Estzt +Mstεt, εt ∼ N(0,Σst)
New regime switching
I state process st driven by wt as st = 1wt ≥ τI latent factor wt = αwt−1 + vt, εt = Σ
−1/2st , and
(εtvt+1
)∼ N
((0n×1
0
),
(In ρρ′ 1
)), ρ′ρ < 1
Advantage
I st is endogenous → systematically affected by observables
I st can be nonstationary → allow for regime persistency
I wt is continuous → directly related to other variables
Endogenous Feedback Mechanism
Latent factor
wt+1 = αwt + ρ′εt +√
1− ρ′ρ ηt+1,
where ηt ∼ N(0, 1), ρ = (ρRv, ρgv, ρzv)′ and ρ′ρ < 1.
Forecast error variance decomposition:
Vart(wt+h) =
3∑k=1
h∑j=1
ρ2kα
2(h−j)
︸ ︷︷ ︸k-th internal, εk
+
h∑j=1
(1−
3∑k=1
ρ2k
)α2(h−j)
︸ ︷︷ ︸external, η
I ρ2k: contribution of εk to regime change
I 1− ρ′ρ: contribution of η to regime change
Key to quantifying macro origins of monetary policy shifts
Taking Model to Data
I Quarterly observations from 1954:Q3 to 2007:Q4
I YGR: per capita real output growth
I INF: annualized inflation rate
I INT: effective federal funds rate
I Measurement equationsYGRtINFtINTt
=
γ(Q)
π(A)
π(A) + r(A) + 4γ(Q)
+ 100
yt − yt−1 + zt4πt4Rt
I Transition Equations
I perturbation solution of Maih & Waggoner (2018)I RISE MATLAB toolbox developed by Maih
Endogenous-Switching Kalman Filter
Augmented state space form
yt = Dst + Fstzt︸ ︷︷ ︸Dst
+(Zst 0l×n
)︸ ︷︷ ︸Zst
(xtdt
)︸ ︷︷ ︸ςt
+ut
(xtdt
)︸ ︷︷ ︸ςt
=
(Cst + Estzt
0n×1
)︸ ︷︷ ︸
Cst
+
(Gst 0m×n
0n×m 0n×n
)︸ ︷︷ ︸
Gst
(xt−1
dt−1
)︸ ︷︷ ︸ςt−1
+
(MstΣ
1/2st
In
)︸ ︷︷ ︸
Mst
εt
I Key features of our filter
I marginalization-collapsing procedure of Kim (1994)
I time-varying transition probabilities
I filtered/smoothed regime factor as by-product
Filtering Algorithm
Step 1: Forecasting
ς(i,j)t|t−1 = Cj + Gjς
it−1|t−1
P(i,j)t|t−1 = GjP
it−1|t−1G
′j + MjM
′j
p(i,j)t|t−1 =
∫RP(st = j|st−1 = i, λt−1)pit−1|t−1p(λt−1|Ft−1)dλt−1
I To compute e.g., p(0,0)t|t−1
I since λt−1 = ρ′εt−1 is latent, approximate p(λt−1|Ft−1) by
pN(λt−1|ρ′ς0d,t−1|t−1, ρ′P 0d,t−1|t−1ρ)
I 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)
Filtering Algorithm (Cont’d)
Step 2: Likelihood evaluation
y(i,j)t|t−1 = Dj + Zjς
(i,j)t|t−1
F(i,j)t|t−1 = ZjP
(i,j)t|t−1Z
′j + Ωj
p(yt|Ft−1) =
1∑j=0
1∑i=0
pN(yt|y(i,j)t|t−1, F
(i,j)t|t−1)p
(i,j)t|t−1
Filtering Algorithm (Cont’d)
Step 3: Updating
ς(i,j)t|t = ς
(i,j)t|t−1 + P
(i,j)t|t−1Z
′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−1Z
′j(F
(i,j)t|t−1)−1ZjP
(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)
I History truncation
I collapse (ς(i,j)t|t , P
(i,j)t|t ) into (ςjt|t, P
jt|t)
I further collapse (ςjt|t, Pjt|t) into (ςt|t, Pt|t)
Prior Distributions
Parameter Density Para (1) Para (2) [5%, 95%]
Fixed Regimeτc, 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
ψ0π , ψπ under regime-0 G 1.00 0.10 [0.84, 1.17]
ψ1π , 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]
Posterior Estimates
No Switching Model Regime Switching Model
Parameter Mode Median [5%, 95%] Mode Median [5%, 95%]
Fixed Regimeτc 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
ψ0π – – – 1.02 0.99 [0.92, 1.06]
ψ1π – – – 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 −1076.14Bayes factor vs. no switching 1.00 exp (23.83)
Evidence of Endogeneity
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0.5
1
1.5
B. ;gv
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0.5
1
1.5
2A. ;Rv
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
C. ;zv
I ρgv > 0: MP is ‘leaning against the wind’
I ρzv < 0: MP is promoting economic growth
Extracted Regime Factor
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
wtjt
-10
-8
-6
-4
-2
0
2
4
6
8
10
p1tjt
0
0.2
0.4
0.6
0.8
1
I Sluggish switching b/w more and less active regimes
I Timing and nature are consistent with narrative record
Main Findings
I Prima facie evidence of endogeneity in monetary policy shifts
I MP is ‘leaning against the wind’—expansionary gvt spendingshock increases the likelihood of more active regime.
I MP is promoting long-term growth—favorable tech shockdecreases the likelihood of shifting into more active regime.
I overall, non-policy shocks have played a predominant role indriving regime changes during the post-World War II period.
I Estimated regime factor identifies MP as slowly fluctuatingbetween more and less active regimes, in ways consistent withconventional view and narrative record.
I Endogenizing regime changes in monetary DSGE modelsprovides a promising venue for understanding the purposefulnature of monetary policy.
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