Abstract
Under a currency board, the central bank relinquishes control over its monetary policy and domestic interest rates converge towards the foreign rates. Nevertheless, a spread between both usually remains. This spread can be persistently positive due to elevated risk in the economy. This paper models that feature by building a DSGE model with a currency board, where the domestic interest rate is endogenously derived as a function of the foreign rate, the external debt position and an exogenous risk premium component. Time variation in the volatility of the risk premium component is then modelled via a Markov-switching component. Estimating the model with Bayesian methods and Estonian data shows that the economy does not react much to shocks to domestic interest rates in quiet times but is much more sensitive during crises, and matches the financial and banking crises, which cannot be captured by the standard DSGE model.
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Notes
The model presents an alternative method to Justiniano and Primiceri (2008) in dealing with time variation in the volatility of macroeconomic shocks.
Following the convention of the open-economy literature, the foreign variables are denoted by an asterisk (*) and logs of the variables—with lower-case letters.
A detailed list of the equations is found in “Log-linearized system of equations” of Appendix.
For example, Cho and White (2007) demonstrate that because of the unusually complicated nature of the null space, the appropriate measure for test of multiple regimes is a quasi-likelihood-ratio (QLR) statistic, for which an asymptotic null distribution and critical values for a small class of models may be computed. Unfortunately, Carter and Steigerwald (2012) show that the QLR-likelihood statistic is inconsistent if the covariates include lagged dependent variables.
The estimation of the posterior for a three-state MS-DSGE model is highly computationally intensive and, even more so, the computation of the predictive likelihood. Therefore, the calculation of p(Y|k) is based on the Laplace approximation of the predictive marginal likelihood around the posterior mode. The technical details are documented in Warne (2012, pp. 188–196).
A technical discussion of the solution method is found in “Solving a MS-DSGE model” of Appendix.
Due to having the switching component in the volatility, the steady state is the same for each regime. This avoids issues that are yet to be resolved in the literature, such as transitions between different steady states.
Both TALIBOR and EURIBOR exhibit non-stationary behaviour. Nevertheless, detrending of the interest rate is not a standard practice in the literature. Therefore, several models are estimated, with and without detrending, and the results remain qualitatively the same. In the main section, the model is estimated with an HP-filtered series. The robustness Sect. 5 discusses a model without detrending of the interest rate. The data span from the first quarter of 1996 to the last quarter of 2012.
Even though the use of Kim’s filter avoids a sample split, the overall sample size is relatively short. The model is constrained by the length of the interest rate series to 64 observations, and therefore, the estimates should be taken with a grain of salt.
The prior and posterior distributions for the parameters of \(\mathcal {M}_1\) are plotted in the “\(\mathcal {M}_1\): Convergence diagnostics—figures and tables” of Appendix.
Distribution plots and convergence diagnostics for this specification may be found in “\(\mathcal {M}_2\): Convergence diagnostics—figures and tables” section of Appendix.
For a detailed timeline of the events, see Adahl (2002).
The variance decomposition of output, consumption, inflation, and interest rate can be found in “\(\mathcal {M}_2\): Variance decomposition tables” of Appendix.
Two further robustness checks have been carried out—a linear detrending and a model with a VAR representation. The results are similar across all specifications.
To illustrate this point, assume a two-state model. Drawing \(p_{11} \in [0, 1]\) ensures \((1-p_{11})\in [0, 1]\), hence allowing for an unconstrained maximization procedure given a suitable transformation \(g(p)\in (-\infty , \infty )\). However, adding an additional regime with \(p_{11} \in [0, 1]\) and \(p_{12} \in [0, 1]\) does not ensure \((1-p_{11}-p_{12}) \in [0, 1]\) even under fairly tight priors.
The only notable difference between Figs. 3 and 6 regarding the second state is the period following the global financial crisis including 2011, when Estonia had already adopted the Euro and TALIBOR was equivalent to EURIBOR. This result might be an artefact of the smoothing of the regime probabilities.
Note that \( A(s_t,s_{t+1}) = B_1(s_t)^{-1} A_1(s_t,s_{t+1})\) in (53). \(B(s_t)\) and \(C(s_t)\) are similarly defined.
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Acknowledgements
Parts of this work have been completed, while the author was a Ph.D. student at the University of Hamburg. I would like to thank Prof. Robert Kunst, an anonymous referee, and Prof. Michael Funke for the invaluable comments and remarks.
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Appendix
Appendix
1.1 Log-linearized system of equations
Endogenous equations:
Euler equation:
Domestic price inflation:
Import price inflation:
Market clearing:
Law of one price:
Terms of trade:
Nominal exchange rate:
Interest rate parity:
Marginal cost:
CPI:
Foreign asset budget constraint:
Interest rate reaction function:
Exogenous processes:
Domestic shocks:
World variables:
1.2 Solving a MS-DSGE model
This section will sketch the solution method employed in the paper. For details and proofs, see Cho (2016). The model is cast in the following state-space form:
with \(Z_t\) following an AR(1) process.Footnote 19 From the perspective of time point t by forward iteration the model at time \(t+n\) may be represented by
where \({\varOmega }_1(s_t) = B(s_t)\), \({\varGamma }_1(s_t) = C(s_t)\) and for \(n = 2,3,{\ldots }:\)
It may be shown that given initial values, under some regularity conditions such as invertibility of \({\varXi }_n\) \(\forall n\), the sequence \(E_t \{M_n (s_t, s_{t+1},{\ldots } s_{t+n}) X_{t+n}\} \) is well defined, unique and real-valued. In the limit as \(n \rightarrow \infty \), the model (54) is said to be forward convergent if the parameter matrices are convergent, i.e. \(\lim \nolimits _{n \rightarrow \infty }{{\varOmega }_n(s_t)} = {\varOmega }^*(s_t)\); \(\lim \nolimits _{n \rightarrow \infty }{{\varGamma }_n(s_t)} = {\varGamma }^*(s_t)\) and \(\lim \nolimits _{n \rightarrow \infty }{F_n(s_t,s_{t+1})} = F^*(s_t,s_{t+1}) \). If
then the solution is
Equation (59) is called the non-bubble condition and, if satisfied, implies the existence of a unique solution to the model. As n tends to infinity, this condition should hold and all solutions, for which it does not should be ruled out as they are not economically relevant. Thus, if the model is forward convergent and Eq. (59) is satisfied, then Eq. (60) is the only relevant MSV solution to the model cast in the form of (53).
The existence of (60) alone is a necessary but not sufficient condition for determinacy, due to the volatility induced by the regime-switching feature. The MSV solution is only the fundamental part of the solution, but there may exist a non-fundamental part that is arbitrary, which leads to a multiplicity of equilibria. Assuming the non-fundamental component takes the form
the concept for determinacy and indeterminacy deals with interaction of the matrices \({\varOmega }_j^*\) and \(F_j^*\) when switching between states. Defining
\(j = \{1,2\}\), mean-square stability is characterized by
where \(r_\sigma (\cdot )\) represents the maximum absolute eigenvalue of the argument matrix. The intuition behind these conditions is straightforward. The first one concerns the transition between the matrices \({\varOmega }^*(s_t)\) of the fundamental part of the solution (60). As long as the biggest absolute eigenvalue is smaller than one, the system would be stable under regime-switching. The \(F^*_j\) matrix governs the non-fundamental switching part and as long the biggest eigenvalue lies on or within the unit circle, the forward solution is the determinate equilibrium.
1.3 \(\mathcal {M}_1\): Convergence diagnostics—figures and tables
See Figs. 7, 8, 9 and Table 6.
1.4 \(\mathcal {M}_2\): Convergence diagnostics—figures and tables
See Figs. 10, 11, 12 and Table 7.
1.5 \(\mathcal {M}_3\): Additional figures
1.6 \(\mathcal {M}_4\): Additional figures
1.7 \(\mathcal {M}_5\): Additional figures
1.8 \(\mathcal {M}_6\): Additional figures
See Fig. 19.
1.9 \(\mathcal {M}_2\): Variance decomposition tables
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Blagov, B. Financial crises and time-varying risk premia in a small open economy: a Markov-switching DSGE model for Estonia. Empir Econ 54, 1017–1060 (2018). https://doi.org/10.1007/s00181-017-1256-z
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DOI: https://doi.org/10.1007/s00181-017-1256-z