The effects of monetary policy regime shifts on the term structure of interest rates

Appendix C.1 State Space Form

We begin by providing the details for the state space form, which comprises the transition and measurement equations and is the basis for model estimation. The transition equation of the state space form is given by equation (3). To derive the measurement equation, we follow Dai, Singleton, and Yang (2007) and assume that one yield, in particular the 12 quarter maturity yield (R_12,t), is priced without error. This yield is entitled basis yield. We choose the 12 quarter maturity yield to be priced without error based on the finding in Chib and Kang (2013) that the yields in the middle of the yield curve have the lowest variance of the measurement errors. As a result, the pricing equation for this yield has the form:

(21)R12,t=a12st+b12st′ft=a12st+bu,12stut+bm¯,12st′m¯t, (21)

where

b12st=(bu,12stbm¯,12st)

and m¯t denotes the vector of macro factors (π_t, g_t)′. This assumption allows the latent factor to be expressed in terms of observable yields and macro variables:

(22)ut=(bu,12st)−1(R12t−a12st−bm¯,12st′m¯t). (22)

Thus,

(23)ft=(utm¯t)=((bu,12st)−1(R12t−a12st−bM¯,12st′m¯t)m¯t). (23)

By denoting the vector of all yields other than R_12t by R_t and y_t(R_t, f_t)′, the measurement equation can be expressed as

(24)yt=(a¯st0)︸A¯st+(b¯stI3)︸B¯stft+(I703×7)ε˜t,ε˜t~iidN(0,Σ), (24)

where Σ is the variance-covariance matrix for the measurement errors, which is assumed to be a diagonal and regime independent, and α¯st and b¯stdenote the vector and matrix of all stacked aτst and bτst′ excluding α12st and βu,12st.

Appendix C.2 Prior Distribution

We set the prior distributions of the model parameters based on the general observation that, on average, the yield curve is upward sloping. Following Chib and Ergashev (2009), we simulate parameters and model-implied yield curves from the prior distributions to ensure that our prior produces, on average, a reasonably shaped yield curve. At the same time, we set the variances of key parameter distributions to be relatively large so that the distributions cover economically reasonable values of parameters. The prior for the diagonal elements of G is based on the fact that interest rates, inflation, and the output gap are all persistent time series. Since λ0st and Ωst are key parameters determining the term premium, their means are set based on the simulation outcomes of the model-implied yield curve. To see the prior implied outcomes, we sample the parameters 25,000 times from the prior distributions and simulate factor dynamics and yield curves. This simulation exercise produces, on average, a slightly upward-sloping yield curves with substantial variation between –3% and 15%. Our specific prior distributions are as follows.

First, we describe the approach for estimating the transition probabilities. We estimate the transition probabilities separately for each regime process as functions of normally distributed parameters

(25)prgjk=11+exp(ηrgjk),j≠k, (25)

which truncates the transition probability values to be within 0 and 1 bounds.

We assume that all parameters, denoted as θ, are distributed independently from each other. Table 3 provides detail for the prior distributions of the parameters. We set the prior for all variances to be defuse to ensure that the prior implied yield curve and the factor processes have considerable variations. Parameters Ω¹, Ω², Σ are reparameterized using coefficients

(26)dΩ=(5×1055×1057×104) (26)

and

(27)dΣ=(7×1054×1063×1076×107107107107). (27)

Appendix C.3 Posterior Distribution and MCMC Sampling

The posterior distributions of parameters are simulated by Markov Chain Monte Carlo (MCMC) methods. The joint posterior distribution to be simulated is described by

(28)π(θ,ST|y)∝f(y|θ,ST)f(ST|θ)π(θ), (28)

where f(y∣θ, S_T) is the likelihood function for data, denoted by y comprising time series of all yields and macro factors, given all parameters of interest θ and time series of regimes S_T={s_t}_t=0,1,…,T; f(S_T∣θ) is the density function for regime-indicators given the parameters; π(θ) is the prior density of the parameters.

The MCMC procedure is summarized as follows:

• Step 1: Initialize (θ, u_T, S_T); where u_T={u_t}_t=0…T is the time series of the latent factor and S_T={s_t}_t=0…T is the time series of regimes;

• Step 2: Sample θ conditional on (S_T, F_T, R_T), where F_T={f_t}_t=0…T is the time series of factors and R_T={R_t}_t=0…T is the time series of yields;

• Step 3: Sample S_T conditional on (θ, F_T, R_T);

• Step 4: Compute u_T conditional on (θ,ST,m¯T,R12,T) using equation (22), where m¯T={m¯t}t=0...T is the time series of macro factors and R_12,T={R_12,t}_t=0…T is the time series of basis yield;

• Step 5: Repeat Steps 2–4 (n₀+n) times, then disregard the first n₀ iterations, which are burn-in iterations, and save n draws of the parameters.

The following provides details of the MCMC algorithm and the construction of the likelihood function.

Step 2: Samplingθ

Parameters θ conditional on (S_T, F_T, R_T) are sampled using the Metropolis-Hastings (MH) algorithm. Because it is difficult to find an optimal parameter blocking scheme due to the high dimension of parameter space of the model, we use the tailored randomized block M-H (TaRB-MH) method developed by Chib and Ramamurthy (2010). The general idea of this method is in setting a number and composition of blocks randomly in each sampling iteration. We let the proposal density q(θ_i∣θ_–i, y) for parameters θ_i in the ith block, conditional on the value of parameters in the remaining blocks θ_–i to take the form of a multivariate student t distribution with 15 degrees of freedom

q(θi|θ−i,y)=St(θi|θ^i,Vθ^i,15),

where

θ^i=argmaxθiln{f(y|θi,θ−i,ST)π(θi)}andVθ^i=(−∂2ln{f(y|θi,θ−i,ST)π(θi)∂θi∂θ′i)|θi=θ^i−1.

Following Chib and Kang (2013) and Chib and Ergashev (2009), we solve numerical optimization problem using the simulated annealing algorithm, which has better performance in this problem than deterministic optimization routines due to high irregularity of the likelihood surface.

Next, we draw a proposal value θi† from the multivariate student t distribution with 15 degrees of freedom, mean θ^i and variance Vθ^i. If the proposed value does not satisfy the model imposed constrains, then it is immediately rejected. The proposed value, satisfying the constraints, is accepted as the next value in the Markov chain with probability

α(θi(g−1),θi†|θ−i,y)=min{f(y|θi†,θ−i,ST)π(θi†)f(y|θi(g−1),θ−i,ST)π(θi(g−1))St(θi(g−1)|θ^i,Vθ^i,15)St(θi†|θ^i,Vθ^i,15),1},

where g is an index for the current iteration. The completed simulation of θ in the gth iteration with h_g blocks produces sequentially updated parameters in all blocks:

π(θ1|θ−1,y,ST),π(θ2|θ−2,y,ST),…,π(θhg|θ−hg,y,ST).

Now we derive the log-likelihood function conditional on θ and S_T, which has the form:

logf(y|θ,ST)=∑t=1Tlogf(yt|It−1,θ,ST),

where It−1={yn}n=0t−1 denotes the information set available for the econometricians at time t–1. Given the model specification, y_t conditional on s_t–1=j, s_t=k, I_t–1, and θ is distributed normally with the mean and variance defined as

yt|t−1jk≡E[yt|st−1=j,st=k,It−1,θ]=A¯k+B¯kμt−1j,kVt|t−1jk≡Var[yt|st−1=j,st=k,It−1,θ]=B¯kLkLk′B¯k'+(Σ000)︸W¯.

Thus, the conditional density of y_t becomes

(29)f(yt|st−1=j,st=k,It−1,θ)=1(2π)10/2|Vt|t−1jk|1/2(−12(yt−yt|t−1jk)′[Vt|t−1jk]−1(yt−yt|t−1jk)). (29)

Step 3: Sampling regimesS_T

Regimes S_T are sampled from f(S_T∣I_T, θ) in a single block in backward order. First, the regime probabilities conditional on I_t and θ are obtained by applying the filtering procedure developed by Hamilton (1989) as follows:

• Step 1: Probabilities of regime s₀ conditional on available information at time t=0 and parameters are initialized at unconditional probabilities of regimes denoted by p_{steady–state}:

  Pr(s0|I0,θ)=psteady−state.

• Step 2: The joint density of s_t–1 and s_t conditional on information at time t–1 and parameters is given by

  (30)Pr(st−1=j,st=k|It−1,θ)=pjkPr(st−1=j|It−1,θ). (30)

• Step 3: Then, the density of y_t conditional on information at time t–1 and parameters is given by

  (31)f(yt|It−1,θ)=∑j,kf(yt|st−1=j,st=k,It−1,θ)Pr(st−1=j,st=k|It−1,θ), (31)

  where the first and second terms are given by equations (29) and (30), respectively.

• Step 4: The joint density of s_t–1 and s_t conditional on information at time t and parameters is obtained by using the Bayes rule:

  Pr(st−1=j,st=k|It,θ)=f(yt,st−1=j,st=k|It−1,θ)f(yt|It−1,θ)=f(yt|st−1=j,st=k,It−1,θ)Pr(st−1=j,st=k|It−1,θ)f(yt|It−1,θ),

  where the first and second terms of the nominator are given by equations (29) and (30) and the denominator is given by equation (31).

• Step 5: By integrating out regime s_t–1 we obtain the probabilities of regime s_t conditional of information at time t and parameters:

  Pr(st=k|It,θ)=∑jPr(st−1=j,st=k|It,θ).

Next, the regimes are drawn backward based on regime probabilities. In particular, regime s_T is sampled from Pr(s_T∣I_T, θ) and then for t from T–1 to 1 regimes are sampled from probabilities computed sequentially backward as

Pr(st=j|It,st+1=k,θ)=Pr(st+1=k|st=j)Pr(st=j|It,θ)∑j=1nPr(st+1=k|st=j)Pr(st=j|It,θ),

where n is the total number of regimes.