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Threshold models with time-varying threshold values and their application in estimating regime-sensitive Taylor rules

  • Yanli Zhu , Haiqiang Chen EMAIL logo and Ming Lin
Published/Copyright: September 29, 2018
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Studies in Nonlinear Dynamics & Econometrics
From the journal Studies in Nonlinear Dynamics & Econometrics Volume 23 Issue 5

Abstract

The literature of time series models with threshold effects makes the assumption of a constant threshold value over different periods. However, this time-homogeneity assumption tends to be too restrictive owing to the fact that the threshold value that triggers regime switching could possibly be time-varying. This study herein proposes a threshold model in which the threshold value is assumed to be a latent variable following an autoregressive (AR) process. The newly proposed model was estimated using a Markov Chain Monte Carlo (MCMC) algorithm under a Bayesian framework. The Monte Carlo simulations are presented to assess the effectiveness of the Bayesian approaches. An illustration of the model was made through an application to a regime-sensitive Taylor rule employing U.S. data.

Keywords: bayesian inference; Markov chain Monte Carlo; model selection; threshold model; time-varying threshold value

Funding source: National Nature Science Foundation of China

Award Identifier / Grant number: 71703030

Award Identifier / Grant number: 71571152

Award Identifier / Grant number: 11101341

Award Identifier / Grant number: 71131008

Award Identifier / Grant number: 71631004

Funding source: Fundamental Research Funds for the Central Universities

Award Identifier / Grant number: 2018B20314

Award Identifier / Grant number: 20720171002

Award Identifier / Grant number: 20720181004

Funding source: Fujian Provincial Key Laboratory of Statistics (Xiamen University)

Award Identifier / Grant number: 2016006

Funding source: Fok Ying-Tong Education Foundation

Award Identifier / Grant number: 151084

Funding statement: Yanli Zhu acknowledges the financial supports from the National Nature Science Foundation of China #71703030, the Fundamental Research Funds for the Central Universities #2018B20314, and funds provided by Fujian Provincial Key Laboratory of Statistics (Xiamen University) #2016006. Haiqiang Chen acknowledges the financial supports from the National Nature Science Foundation of China #71571152, the Fundamental Research Funds for the Central Universities #20720171002 as well as # 20720181004, and the Fok Ying-Tong Education Foundation #151084. Ming Lin acknowledges the financial supports from the National Nature Science Foundation of China #11101341, #71131008 and #71631004.

Acknowledgement

We would like to thank Yongmiao Hong, Zongwu Cai, Ying Fang and Yingxing Li for their helpful comments.

Appendix A

Estimation via MCMC

We use the Metropolis-within-Gibbs approach (Tierney 1994) to draw samples from the posterior distribution,

p(γ1:T,Θ∣y1:T,x1:T,z1:T)∝p(Θ)∏t=1Tp(γt∣γt−1,Θ)p(yt∣xt,zt,γt,Θ),

where Θ=(β1,β2,σ12,σ22,ν,ϕ,σu2).

We briefly outline the features of our MCMC sampler. With conjugate priors, β1, β2, σ12, σ22, ν and σu2 can be sampled directly from their conditional posterior distributions, which is normal or inverse Gamma distributions, through Gibbs sampling steps (Casella and George, 1992). However, a Metropolis-Hastings step (Hastings 1970) is needed to sample ϕ and γ1:T, because the conditional posterior distributions of ϕ and γ1:T do not take a known form. Based on our approach, the Markov chain (γ1:T(s),Θ(s)), s = 1, ⋯, S, can be constructed.

Under the Bayesian framework, the Metropolis-within-Gibbs algorithm we have adopted here prescribes a transition rule (detailed balance) with respect to which the target distribution is invariant. By the standard Markov chain theory, if the chain is irreducible, aperiodic, which is almost surely for the Metropolis-within-Gibbs algorithm, and possesses an invariant distribution, then the Markov chain will attain stationarity and converge to the target distribution ideally. Therefore, if the Markov Chain is long enough, the distribution of the MCMC samples can be considered as the target distribution approximately (Liu 2001).

In the following, we provide the detailed steps that update (γ1:T(s),Θ(s)) from (γ1:T(s−1),Θ(s−1)). To save notation, we use rest to denote the rest of the most updated sample and the observations {(yt,xt,zt),t=1,⋯,T}. We also suppress the superscripts of the samples except for the component to be updated.

  1. Sequentially update γt(s), t = 1, ⋯, T, as follows.

    1. Draw γt∗ from distribution

      q(γt∣rest)∼N(gγ,t,hγ,t2)∝p(γt∣γt−1,Θ)p(γt+1∣γt,Θ),

      where

      gγ,t={ν+ϕ(γ2−ν),if t=1;ν+ϕ((γt−1−ν)+(γt+1−ν))1+ϕ2,if t=2,…,T−1;ν+ϕ(γT−1−ν),if {t} =\text{T},

      and

      hγ,t2={σu2,if t=1;σu21+ϕ2,if t=2,…,T−1;σu2,if t=T.

    2. Accept γt∗ as γt(s) with probability

      min{1,p(yt∣xt,zt,γt∗,Θ)p(yt∣xt,zt,γt(s−1),Θ)}.

      If rejected, let γt(s)=γt(s−1).

  2. Draw ν(s) from p(ν∣rest)∼N(gν,hν2), where

    gν=γ1(1−ϕ2)+(1−ϕ)∑t=2T(γt−ϕγt−1)1−ϕ2+(1−ϕ)2(T−1)σν2+σu21−ϕ2+(1−ϕ)2(T−1)μνσν2+σu21−ϕ2+(1−ϕ)2(T−1), andhν2=σν2σu21−ϕ2+(1−ϕ)2(T−1)σν2+σu21−ϕ2+(1−ϕ)2(T−1).

  3. Update ϕ(s) as follows.

    1. Draw ϕ* from the truncated normal distribution q(ϕ∣rest)∼N(gϕ,hϕ2)I(−1<ϕ<1),where

      gϕ=∑t=2T(γt−ν)(γt−1−ν)∑t=2T−1(γt−ν)2,hϕ2=σu2∑t=2T−1(γt−ν)2,

      and I(⋅) is the indicator function.

    2. Accept ϕ* as ϕ(s) with probability

      min{1,1−[ϕ∗]21−[ϕ(s−1)]2}.

  4. Draw [σu2](s) from p(σu2∣rest)∼IG(au/2,bu/2), where

    au=Au+T,bu=Bu+(1−ϕ2)(γ1−ν)2+∑t=2T[(γt−ν)−ϕ(γt−1−ν)]2.

  5. Draw β1(s) from p(β1∣rest)∼N(gβ,1,hβ,12) and draw β2(s) from p(β2∣rest)∼N(gβ,2,hβ,22), where

    gβ,1=(∑t,zt≤γtxtxt′σ12+Σβ−1)−1(∑t,zt≤γtxtytσ12+Σβ−1μβ),hβ,12=(∑t,zt≤γtxtxt′σ12+Σβ−1)−1,gβ,2=(∑t,zt>γtxtxt′σ22+Σβ−1)−1(∑t,zt>γtxtytσ22+Σβ−1μβ),hβ,22=(∑t,zt>γtxtxt′σ22+Σβ−1)−1.

  6. Draw [σ12](s) from p(σ12∣rest)∼IG(a1/2,b1/2) and draw [σ22](s) from p(σ22∣rest)∼IG(a2/2,b2/2), where

    a1=Aε+∑t=1TI(zt≤γt),b1=Bε+∑t:zt≤γt(yt−xt′β1)2,a2=Aε+∑t=1TI(zt>γt),b2=Bε+∑t:zt>γt(yt−xt′β2)2.

Appendix B

Calculating likelihood function via PF

For Model 1, we have

p(y1:T∣x1:T,z1:T,Θ)=∫∏t=1Tp(γt∣γt−1,Θ)p(yt∣xt,zt,γt,Θ) dγ1:T,

which does not have an analytic form. We can use a particle filter to calculate p(y1:T∣x1:T,z1:T,Θ) as follows.

  1. At t = 1, draw samples γ~1, j = 1, ⋯, m, from p(γ1∣Θ). Let weight be w1(j)=p(y1∣x1,z1,γ~1(j),Θ).

  2. At t = 2, ⋯, T,

    1. Draw γ~t(j) from p(γt∣γ~t−1(j),Θ) for j = 1, ⋯, m.

    2. Update weights by letting wt(j)=wt−1(j)p(yt∣xt,zt,γ~t(j),Θ).

    3. Resampling: When t < T,

      1. draw γ~tnew(j), j = 1, ⋯, m, from distribution

        ∑j=1mwt(j)∑k=1mwt(k) δ(γt−γ~t(j)),

        and set wtnew(j)=1m∑j=1mwt(j); then

      2. let γ~t(j)=γ~tnew(j) and wt(j)=wtnew(j) for j = 1, ⋯, m.

  3. The likelihood function is estimated by 1m∑i=1mwT(i).

Doucet and Johansen (2011) show that 1m∑i=1mwT(i)→pp(y1:T∣x1:T,z1:T,Θ) as the Monte Carlo sample size, m, tends to infinity.

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Supplementary Material

The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/snde-2017-0114).

Published Online: 2018-09-29

©2019 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/snde-2017-0114
Keywords for this article
bayesian inference; Markov chain Monte Carlo; model selection; threshold model; time-varying threshold value

Articles in the same Issue

  1. Research Articles
  2. Variance reduction estimation for return models with jumps using gamma asymmetric kernels
  3. Threshold models with time-varying threshold values and their application in estimating regime-sensitive Taylor rules
  4. Smart or stupid depends on who is your counterpart: a cobweb model with heterogeneous expectations
  5. An explicit formula for the smoother weights of the Hodrick–Prescott filter
  6. An intuitive skewness-based symmetry test applicable to stationary time series data
  7. An elementary business cycle mechanism: learning from Harrod and Kaldor
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