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.
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.
Distributional properties of censored and uncensored variables
Given the normality assumption on εt, model (1) defines a joint normal distribution for the variables , where y2t gathers the uncensored variables.
where represents the model parameters and and 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 as the product of a marginal and a conditional density, , where:
The factoring of 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
Define the number Nj, 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
From (17), the period t density contribution is multivariate normal for y2t
and the period t contribution of censored variables is
and ϕ denotes the pdf (see (22)) of the standard (multivariate) normal distribution.
The likelihood of the complete data factorizes
B.2 Prior distributions
To complete the Bayesian setup, we specify the prior density of the state indicator I:
The prior for the censored variables is assumed to be diffuse, . We might also work with a proper prior distribution, restricted to the latent area, with κ some real number.
Finally, we assume independent priors for the model parameters:
The prior for (γr, γ) includes a state-identifying restriction and additional information on the threshold level, see (10).
The priors on βk are independent normal, with variance structure implied by Minnesota priors, . The vector 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, , with
We specify the corresponding elements in k as
for k = 0,1 and l = 1, …, p. The state-specific variances in the scale factor are equal to the variance of residuals of univariate state-specific autoregressions in which states are predefined as if the libor ≤ 1%. For the intercepts, we work with diffuse priors, .
For Σk, we assume an inverse Wishart prior distribution with degrees of freedom and scale k with diagonal elements .
B.3 Posterior distributions
To obtain draws from the posterior
we sample iteratively from the posterior of
the state indicator, . 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 to obtain the posterior odds
We sample T − p uniform random variables Ut, and set It = 1 if
the censored variables, . Conditional on I and the observed variables, and given a diffuse prior, the moments of the posterior normal distribution are given by (18). We sample from this distribution truncated to the region .
the parameters of the state distribution, . 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):
We express the state distribution in relative terms, i.e. as difference between the latent state utilities
We approximate the Logistic distribution by a mixture of normals with M components, . Conditional on the latent relative state utilities ϖ and the components, we obtain a normal posterior distribution, N(g, G) with moments:
To implement the restrictions on 𝜸 according to (6), we partition the posterior appropriately:
the rest of the parameters, . Conditional on I and the augmented data , 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.1 Duration of minimum 2 periods
C.2 Additional results of the Full model
C.3 Additional results of the SV model
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The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/snde-2017-0098).
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