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.

US Real GDP Growth Rates

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

Transition Probability Comparison

Transition Probability Comparison

Transition Probability Comparison

NBER Recession Period and Latent Factor: 1952-1984

Recession Probabilities: 1952-1984

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

Transition Probability Comparison

Transition Probability Comparison

Transition Probability Comparison

Transition Probability Comparison

Extracted Latent Factor from Volatility Model and VIX

High Volatility Probabilities: 1990-2012

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.