Constrained interest rates and changing dynamics at the zero lower bound

Gregor Bäurle; Daniel Kaufmann; Sylvia Kaufmann; Rodney Strachan

doi:10.1515/snde-2017-0098

Published by De Gruyter June 28, 2019

Constrained interest rates and changing dynamics at the zero lower bound

Gregor Bäurle , Daniel Kaufmann , Sylvia Kaufmann and Rodney Strachan

From the journal Studies in Nonlinear Dynamics & Econometrics

https://doi.org/10.1515/snde-2017-0098

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Abstract

The interaction of macroeconomic variables may change as nominal short-term interest rates approach zero. In this paper, we propose to capture these changing dynamics with a state-switching parameter model which explicitly takes into account that the interest rate might be constrained near the zero lower bound by using a Tobit model. The probability of state transitions is affected by the lagged level of the interest rate. The endogenous specification of the state indicator permits dynamic conditional forecasts of the state and the system variables. We use Bayesian methods to estimate the model and to derive the forecast densities. In an application to Swiss data, we evaluate state-dependent impulse-responses to a risk premium shock identified with sign-restrictions. We provide an estimate of the latent rate, i.e. the rate lower than the constraint on the interest rate level which would be state- and model-consistent. Additionally, we discuss scenario-based forecasts and evaluate the probability of exiting the ZLB region. In terms of log predictive scores and the Bayesian information criterion, the model outperforms a model substituting switching with stochastic volatility and another including intercept switching only combined with stochastic volatility.

Keywords: constrained variable; regime switching; stochastic volatility; time-varying probability; Tobit model

JEL Classification: C3; E3

Acknowledgement

We thank for constructive comments of the Editor, Associate Editor and two anonymous referees. Rodney Strachan acknowledges the support from the Australian Research Council, Discovery Project DP180102373.

A Appendix

Distributional properties of censored and uncensored variables

Given the normality assumption on ε_t, model (1) defines a joint normal distribution for the variables yt∗=[y1t∗′,y2t′]′, where y_2t gathers the uncensored variables.

(16)[y1t∗y2t]|Xt,It,θ∼N([m1Itm2It],[Σ11,ItΣ12,ItΣ21,ItΣ22,It])

where θ={βk,Σk,γr,γ|k=0,1} represents the model parameters and miIt=Xitβi,It and Σij,It are obtained by gathering the corresponding rows in (1) and by partitioning accordingly the moment matrices. This allows the expression of the joint observation density f(yt∗) as the product of a marginal and a conditional density, f(yt∗|⋅)=f(y1t∗|y2t,⋅)f(y2t|⋅), where:

(17)f(y2t|⋅)=N(m2It,Σ22,It)=N(X2tβ2It,Σ22,It)

(18)f(y1t|2∗|⋅)=f(y1t∗|y2t,⋅)=N(m1It|2,M1It|2)

with

m1It|2=m1It+Σ12,ItΣ22,It−1(y2t−m2It)M1It|2=Σ11,It−Σ12,ItΣ22,It−1Σ21,It

The factoring of f(yt∗|⋅) partitions the joint distribution into two parts and allows the implementation of a normal regression model for the unconstrained variables and a conditional censored normal regression model for the constrained variables

(19)[y~1t|2y2t]|Xt,It,θ∼N([m1It|2m2It],[M1It|200Σ22,It])1(y~1t≥b)

B Appendix

Bayesian framework

B.1 Likelihood

Define the number N_j, j = 1, 2, which indicates the number of, respectively, censored and uncensored variables.

Conditional on I and using (19), the data likelihood can be factorized

(20)f(y~|X,I,θ)=∏t=p+1Tf(y~t|Xt,βIt,ΣIt)1(y~1t≥b)

(21)=∏t=p+1Tf(y~1t|y2t,X1t,β1It,Σ11,It)1(y~1t≥b)f(y2t|X2t,β2It,Σ22,It)

From (17), the period t density contribution is multivariate normal for y_2t

(22)f(y2t|X2t,β2It,Σ22,It)=(2π)−N2/2|Σ22,It|−1/2×exp⁡{−12(y2t−X2tβ2It)′Σ22,It−1(y2t−X2tβ2It)}

and the period t contribution of censored variables is

(23)f(y~1t|y2t,X1t,β1It,Σ11,It)1(y~1t≥b)=Φ(M1It|2−1/2(b−m1It|2))1(y~1t=b)×|M1It|2|−1/2ϕ(M1It|2−1/2(y~1t−m1It|2))1(y~1t>b)

where Φ(M1It|2−1/2(b−m1It|2))1(y~1t=b) equals

∫−∞bN1…∫−∞b1|M1It|2|−1/2ϕ(M1It|2−1/2(y~1t−m1It|2))dy~11,t…dy~1N1,t,

and ϕ denotes the pdf (see (22)) of the standard (multivariate) normal distribution.

The likelihood of the complete data factorizes

(24)f(y∗|X,I,θ)=∏t=p+1Tf(y1t∗|y2t,X1t,β1It,Σ11,It)f(y2t|X2t,β2It,Σ22,It)

where the moments of the marginal and condition normal observation densities are given in, respectively, (17) and (18).

B.2 Prior distributions

To complete the Bayesian setup, we specify the prior density of the state indicator I:

(25)π(I|r,γ,γr)=∏t=p+1Tπ(It|rt−1,γ,γr)

The prior for the censored variables is assumed to be diffuse, π(y1∗)∝1(y1∗≤b). We might also work with a proper prior distribution, restricted to the latent area, π(y1∗)∼N(0,κI)1(y1∗≤b) with κ some real number.

Finally, we assume independent priors for the model parameters:

(26)π(θ)=π(γ,γr)∏k=01π(βk)π(Σk)

The prior for (γ^{^r}, γ) includes a state-identifying restriction and additional information on the threshold level, see (10).

The priors on β_k are independent normal, with variance structure implied by Minnesota priors, π(βk)=N(v_,V_k). The vector v_ is of dimension N(Np + 1), see (1). Given that we estimate a VAR in levels, we center the first own autoregressive lag at 1 and all other coefficients at zero, v_={v_l|l=1,…,N(Np+1)}, with

(v_l,V_k,ll)={1l=(j−1)(Np+1)+(j+1), j=1,…,N0otherwise

We specify the corresponding elements in V__k as

Var(Bkl,ij)={0.01/l2i=j0.25(0.01/l2)(σki2/σkj2)i≠j, i,j=1,…,N

for k = 0,1 and l = 1, …, p. The state-specific variances in the scale factor (σki2/σkj2) are equal to the variance of residuals of univariate state-specific autoregressions in which states are predefined as I_t=1 if the libor ≤ 1%. For the intercepts, we work with diffuse priors, Var(μki)=5.

For Σ_k, we assume an inverse Wishart prior distribution IW(s_,S_k) with degrees of freedom s_=N+2 and scale S__k with diagonal elements S_k,ii=σki2.

B.3 Posterior distributions

To obtain draws from the posterior

π(ϑ|y~)∝f(y∗|X,I,θ)π(I|r,θ)π(y1∗)π(θ)

we sample iteratively from the posterior of

the state indicator, π(I|y∗,X,r,θ). Given that there is no state dependence in the state probabilities, we are able to sample the states simultaneously. We update the period t prior odds P(It=1)/P(It=0)=exp⁡(γrrt−1+γ) to obtain the posterior odds
P(It=1|⋅)/P(It=0|⋅)=f(yt∗|Xt,β1,Σ1)exp⁡(γrrt−1+γ)f(yt∗|Xt,β0,Σ0), t=p+1,…,T
We sample T − p uniform random variables U_t, and set I_t = 1 if
P(It=1|⋅)/(P(It=0|⋅)+P(It=1|⋅))≥U
the censored variables, π(y1∗|y2,X,I,θ)1(y1∗≤b). Conditional on I and the observed variables, and given a diffuse prior, the moments of the posterior normal distribution π(y1∗|⋅) are given by (18). We sample from this distribution truncated to the region y1∗≤b.
the parameters of the state distribution, π(γ|r,I)1(γr<0)1(−γrγ_≤γ≤−γrγ¯). First, we introduce two layers of data augmentation, which renders the non-linear, non-normal model into a linear-normal model for the parameters (Frühwirth-Schnatter and Frühwirth, 2010):
1. We express the state distribution in relative terms, i.e. as difference between the latent state utilities
  ϖt=I1tu−I0tu=γrrt−1+γ+ϵt, ϵt∼Logistic
  where
  I1tu=γrrt−1+γ+ν1t, and I0tu=ν0twith νkt i.i.d. Type I EV
2. We approximate the Logistic distribution by a mixture of normals with M components, R=(R1,…,RT). Conditional on the latent relative state utilities ϖ and the components, we obtain a normal posterior distribution, N(g, G) with moments:
  G=(G0−1+∑t=p+1TZt′Zt/smt2)−1g=G(G0−1g0+∑t=p+1TZt′ϖt/smt2)
  where Zt=[rt−1,1]′ and smt2=sm2 is the variance of the mixture components R_t, see Table 2 in Frühwirth-Schnatter and Frühwirth (2010).
To implement the restrictions on 𝜸 according to (6), we partition the posterior appropriately:
π(γr,γ|⋅)∼N([g1g2],[G11G12G21G22])
Then we first sample γr,(mc) from N(g1,G11)1(γr<0) and then sample γ from the truncated conditional posterior (Robert, 2009; Botev, 2017):
γ|γr=γr,(mc)∼N(g2c,G2c)1(−γrγ_≤γ≤−γrγ¯)
with moments
g2c=g2−G21G11−1(γr,(mc)−g2)G2c=G22−G21G11−1G12
the rest of the parameters, π(θ−γ|X,y1∗,I). Conditional on I and the augmented data y∗, the model in (1) is linear. The posterior distribution of the regression parameters and of the error variances are then, respectively, normal and inverse Wishart, the moments of which can be derived in the usual way.

C Appendix

Addtional results

C.1 Duration of minimum 2 periods

$Figure 17: Duration of minimum 2 periods.Left-hand: Observed interest rate and model-based estimate of $\boldsymbol{y^{*}_{1}} < b$y1∗<b (red); the black line is the median, the areas decreasing in shades correspond to, respectively, the 25%, the 50% and the 80% interval of highest posterior density; mean posterior probability of state 1 (yellow). Right-hand: histogram of ${y^{*}_{1t}} < b$y1t∗<b for $t=2014.00,2014.25,2014.50$t=2014.00,2014.25,2014.50, i.e. 2014 first through third quarter; the second and third numbers under each histogram refer to, respectively, $P\left(y^{*}_{1t} < \min_{t}\{y_{1t}\}\right)$P(y1t∗<mint{y1t}) and $\textrm{median}\left(y^{*}_{1t}\right)$median(y1t∗).$

Figure 17:

Duration of minimum 2 periods.

Left-hand: Observed interest rate and model-based estimate of y1∗<b (red); the black line is the median, the areas decreasing in shades correspond to, respectively, the 25%, the 50% and the 80% interval of highest posterior density; mean posterior probability of state 1 (yellow). Right-hand: histogram of y1t∗<b for t=2014.00,2014.25,2014.50, i.e. 2014 first through third quarter; the second and third numbers under each histogram refer to, respectively, P(y1t∗<mint{y1t}) and median(y1t∗).

C.2 Additional results of the Full model

Figure 18:

Scatter plots: State I_t = 0 (x-axis) against I_t = 1 (y-axis) parameter draws along with the 45% line. (A) Intercept and lag 1. (B) lag 2.

C.3 Additional results of the SV model

Figure 19:

Switching VAR with SV. Scatter plots: State I_t = 0 (x-axis) against I_t = 1 (y-axis) parameter draws along with the 45% line. (A) Intercept and lag 1. (B) lag 2.

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Supplementary Material

The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/snde-2017-0098).

Published Online: 2019-06-28

Constrained interest rates and changing dynamics at the zero lower bound

Abstract

Acknowledgement

A Appendix

Distributional properties of censored and uncensored variables

B Appendix

Bayesian framework

B.1 Likelihood

B.2 Prior distributions

B.3 Posterior distributions

C Appendix

Addtional results

C.1 Duration of minimum 2 periods

C.2 Additional results of the Full model

C.3 Additional results of the SV model

References

Supplementary Material

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