Financial crises and time-varying risk premia in a small open economy: a Markov-switching DSGE model for Estonia

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.

Appendix

1.1 Log-linearized system of equations

Endogenous equations:

Euler equation:

$$\begin{aligned} c_t = E_{t}\{c_{t+1} \} + \frac{1}{\sigma } (E_{t}\{ \pi _{t+1} \} - i_t ) + \frac{1}{\sigma } (1-\rho _{\vartheta }) \vartheta _t. \end{aligned}$$

(33)

Domestic price inflation:

$$\begin{aligned} (1 + \beta \delta _H) \pi _{H,t} = \beta E_t \{\pi _{H,t+1}\} +\delta _H \pi _{H,t-1} +\lambda _H mc_t + \mu _{H,t}. \end{aligned}$$

(34)

Import price inflation:

$$\begin{aligned} (1 + \beta \delta _F) \pi _{F,t} = \beta E_t \{\pi _{F,t+1}\} +\delta _F \pi _{F,t-1} +\lambda _F \psi _t + \mu _{F,t}. \end{aligned}$$

(35)

Market clearing:

$$\begin{aligned} y_t - (1-\alpha )c_t - \alpha \eta (s_t + q_t) = \alpha y^*_t. \end{aligned}$$

(36)

Law of one price:

$$\begin{aligned} \psi _t = q_t - (1-\alpha ) s_t. \end{aligned}$$

(37)

Terms of trade:

$$\begin{aligned} \triangle s_t = \pi _{F,t} - \pi _{H,t}. \end{aligned}$$

(38)

Nominal exchange rate:

$$\begin{aligned} \triangle e_t = 0. \end{aligned}$$

(39)

Interest rate parity:

$$\begin{aligned} \pi ^*_t - \pi _t = \triangle q_t. \end{aligned}$$

(40)

Marginal cost:

$$\begin{aligned} mc_t = \sigma c_t + \varphi y_t + \alpha s_t - (1+\varphi ) a_t . \end{aligned}$$

(41)

CPI:

$$\begin{aligned} \pi = (1-\alpha )\pi _{H,t} + \alpha \pi _{F,t}. \end{aligned}$$

(42)

Foreign asset budget constraint:

$$\begin{aligned} c_t + d_t + \alpha (q_t + \alpha s_t) - \frac{1}{\beta }d_{t-1} = y_t. \end{aligned}$$

(43)

Interest rate reaction function:

$$\begin{aligned} i_t = i^*_t - \chi d_t - \phi _t. \end{aligned}$$

(44)

Exogenous processes:

Domestic shocks:

$$\begin{aligned} a_t&= \rho _a a_{t-1} + \varepsilon ^a_t \quad \text {with} \quad \varepsilon ^a_t \sim N(0,\sigma ^2_a), \end{aligned}$$

(45)

$$\begin{aligned} \vartheta _{t}&= \rho _{\vartheta } \vartheta _{t-1} + \varepsilon ^\vartheta _t \quad \text {with} \quad \varepsilon ^{\vartheta }_t \sim N(0,\sigma ^2_{\vartheta }), \end{aligned}$$

(46)

$$\begin{aligned} \mu _{H,t}&= \rho _\mu \mu _{H,t-1} + \varepsilon ^{\mu _H}_t \quad \text {with} \quad \varepsilon ^{\mu _H}_t \sim N(0,\sigma ^2_{\mu _H}), \end{aligned}$$

(47)

$$\begin{aligned} \mu _{F,t}&= \rho _\mu \mu _{F,t-1} + \varepsilon ^{\mu _F}_t \quad \text {with} \quad \varepsilon ^{\mu _F}_t \sim N(0,\sigma ^2_{\mu _F}), \end{aligned}$$

(48)

$$\begin{aligned} \phi _t&= \rho _{\phi } \phi _{t-1} + \varepsilon ^{\phi }_t \quad \text {with} \quad \varepsilon ^{\phi }_t \sim N(0,\sigma ^2_\phi ). \end{aligned}$$

(49)

World variables:

$$\begin{aligned} y^*_t&= c_{y^*} y^*_{t-1} + \varepsilon ^{y{^*}}_{t} \quad \text {with} \quad \varepsilon ^{y{^*}}_{t} \sim N(0,\sigma ^2_{y^*}), \end{aligned}$$

(50)

$$\begin{aligned} \pi ^*_t&= c_{\pi *} \pi ^*_{t-1} + \varepsilon ^{\pi {^*}}_{t} \quad \text {with} \quad \varepsilon ^{\pi {^*}}_{t} \sim N(0,\sigma ^2_{\pi ^*}), \end{aligned}$$

(51)

$$\begin{aligned} i^*_t&= c_{i*} i^*_{t-1} + \varepsilon ^{i{^*}}_{t} \quad \text {with} \quad \varepsilon ^{i{^*}}_{t} \sim N(0,\sigma ^2_{i^*}). \end{aligned}$$

(52)

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:

$$\begin{aligned} X_t = E_t\{ A(s_t,s_{t+1}) X_{t+1}\} + B(s_t)X_{t-1} + C(s_t)Z_t, \end{aligned}$$

(53)

with $Z_t$ following an AR(1) process.¹⁹ From the perspective of time point t by forward iteration the model at time $t+n$ may be represented by

$$\begin{aligned} X_t = E_t \{M_n (s_t, s_{t+1},{\ldots }, s_{t+n}) X_{t+n}\} + {\varOmega }_n (s_t) X_{t-1} + {\varGamma }_n (s_t) Z_t, \end{aligned}$$

(54)

where ${\varOmega }_1(s_t) = B(s_t)$, ${\varGamma }_1(s_t) = C(s_t)$ and for $n = 2,3,{\ldots }:$

$$\begin{aligned} {\varOmega }_n(s_t)&={\varXi }_{n-1}^{-1}(s_t) B(s_t), \end{aligned}$$

(55)

$$\begin{aligned} {\varGamma }_n(s_t)&={\varXi }_{n-1}^{-1}(s_t) C(s_t) + E_t \{ F_{n-1} (s_t, s_{t+1}) {\varGamma }_{n-1}(s_{t+1}) \} R, \end{aligned}$$

(56)

$$\begin{aligned} {\varXi }_{n-1}(s_t)&=(I_n - E_t \{ A (s_t, s_{t+1}) {\varOmega }_{n-1(s_{t+1})} \}), \end{aligned}$$

(57)

$$\begin{aligned} F_{n-1}(s_t,s_{t+1})&={\varXi }_{n-1}^{-1}(s_t)^{-1} A (s_t, s_{t+1}) . \end{aligned}$$

(58)

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

$$\begin{aligned} \lim \limits _{n \rightarrow \infty } E_t \{M_n (s_t, s_{t+1},{\ldots }, s_{t+n}) X_{t+n}\} = \mathbf 0 , \end{aligned}$$

(59)

then the solution is

$$\begin{aligned} X_t = {\varOmega }^* (s_t) X_{t-1} + {\varGamma }^* (s_t) Z_t. \end{aligned}$$

(60)

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

$$\begin{aligned} W_t = E_t \{ F(s_t, s_{t+1}) W_{t+1} \}, \end{aligned}$$

(61)

the concept for determinacy and indeterminacy deals with interaction of the matrices ${\varOmega }_j^*$ and $F_j^*$ when switching between states. Defining

$$\begin{aligned} {\varPsi }_{{\varOmega }^* \times {\varOmega }^*} = [p_{ij} {\varOmega }^*_j \otimes {\varOmega }^*_j], \quad {\varPsi }_{F^* \times F^*} = [p_{ij} F^*_j \otimes F^*_j], \end{aligned}$$

$j = \{1,2\}$, mean-square stability is characterized by

$$\begin{aligned} r_\sigma ({\varPsi }_{{\varOmega }^* \times {\varOmega }^*} )< 1,\quad r_\sigma ({\varPsi }_{F^* \times F^*} ) \le 1, \end{aligned}$$

(62)

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.

Fig. 7

$\mathcal {M}_1$: Prior (dashed curve) and posterior (long dashed curve) distributions of the parameters

Fig. 8

$\mathcal {M}_1$: Recursive means of the parameters calculated over the draws from the posterior distribution

Fig. 9

$\mathcal {M}_1$: Trace plots of the parameters

Table 6 Left: Autocorrelation among the draws, based on a sample of 10,000; right: Raftery–Lewis convergence diagnostics with $q=0.025, r=0.1, s=0.95$

1.4 $\mathcal {M}_2$: Convergence diagnostics—figures and tables

See Figs. 10, 11, 12 and Table 7.

Fig. 10

$\mathcal {M}_2$: Prior (dashed curve) and posterior (long dashed curve) distributions of the parameters

Fig. 11

$\mathcal {M}_2$: Recursive means of the parameters calculated over the draws from the posterior distribution

Fig. 12

$\mathcal {M}_2$: Trace plots of the parameters

Table 7 Left: Autocorrelation among the draws, based on a sample of 10,000; right: Raftery–Lewis convergence diagnostics with $q=0.025, r=0.1, s=0.95$

1.5 $\mathcal {M}_3$: Additional figures

See Figs. 13 and 14.

Fig. 13

Impulse responses following a risk premium shock for $\mathcal {M}_3(1)$ (blue long dashed curve) and $\mathcal {M}_3(2)$ (red dotted curve) compared to the baseline MS-DSGE model $\mathcal {M}_2(1)$ (green solid curve) and $\mathcal {M}_2(2)$ (orange dashed curve). (Color figure online)

Fig. 14

Regime probabilities and interest rates. Top panel estimated probability of the high risk premium volatility regime for $\mathcal {M}_3$. Values below 0.5 indicate a realization of the first regime and values above 0.5—a realization of the second regime. Bottom panel annualized 3-month interbank interest rates

1.6 $\mathcal {M}_4$: Additional figures

See Figs. 15 and 16.

Fig. 15

Impulse responses following a risk premium shock for $\mathcal {M}_4(1)$ (blue long dashed curve) and $\mathcal {M}_4(2)$ (red dotted curve) compared to the baseline MS-DSGE model $\mathcal {M}_2(1)$ (green solid curve) and $\mathcal {M}_2(2)$ (orange dashed curve). (Color figure online)

Fig. 16

Regime probabilities and interest rates. Top panel estimated probability of the high risk premium volatility regime for $\mathcal {M}_4$. Values below 0.5 indicate a realization of the first regime and values above 0.5—a realization of the second regime. Bottom panel annualized 3-month interbank interest rates

1.7 $\mathcal {M}_5$: Additional figures

See Figs. 17 and 18.

Fig. 17

Impulse responses following a risk premium shock for $\mathcal {M}_5(1)$ (blue long dashed curve) and $\mathcal {M}_5(2)$ (red dotted curve) compared to the baseline MS-DSGE model $\mathcal {M}_2(1)$ (green solid curve) and $\mathcal {M}_2(2)$ (orange dashed curve). (Color figure online)

Fig. 18

Regime probabilities and interest rates. Top panel estimated probability of the high risk premium volatility regime for $\mathcal {M}_5$. Values below 0.5 indicate a realization of the first regime and values above 0.5—a realization of the second regime. Bottom panel annualized 3-month interbank interest rates

1.8 $\mathcal {M}_6$: Additional figures

See Fig. 19.

Fig. 19

Impulse responses following a risk premium shock for $\mathcal {M}_6(1)$ (blue long dashed curve), $\mathcal {M}_6(2)$ (red dotted curve), and $\mathcal {M}_6(3)$ (orange long and short dashed curve) compared to the baseline MS-DSGE model $\mathcal {M}_1$ (green solid curve). (Color figure online)

1.9 $\mathcal {M}_2$: Variance decomposition tables

See Tables 8, 9, 10, 11, 12, 13, 14 and 15.

Table 8 Forecast error variance decomposition of consumption for horizon $h = \{1,\ldots ,\infty \}$

Table 9 Forecast error variance decomposition of consumption for horizon $h = \{1,{\ldots },\infty \}$

Table 10 Forecast error variance decomposition of inflation for horizon $h = \{1,{\ldots },\infty \}$

Table 11 Forecast error variance decomposition of inflation for horizon $h = \{1,{\ldots },\infty \}$

Table 12 Forecast error variance decomposition of output for horizon $h = \{1,{\ldots },\infty \}$

Table 13 Forecast error variance decomposition of output for horizon $h = \{1,{\ldots },\infty \}$

Table 14 Forecast error variance decomposition of the interest rate for horizon $h = \{1,{\ldots },\infty \}$

Table 15 Forecast error variance decomposition of the interest rate for horizon $h = \{1,{\ldots },\infty \}$