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Financial Crises and Time-Varying Risk Premia in
a Small Open Economy: A Markov-Switching
DSGE Model for Estonia
Boris Bla...
Introduction
• Non-linearities in DSGE models
• Currency board in DSGE models
◦ Absence of money
◦ Fixed exchange rate
e =...
Introduction
1996 1998 2000 2002 2004 2006 2008 2010
0
5
10
15
20
1996 1998 2000 2002 2004 2006 2008 2010
−15
−10
−5
0
5
1...
Model Overview
• Follows Benignio (2001), Monacelli (2005), Justiniano and Preston
(2010)
• Consumers maximize utility by ...
The Model
Domestic Demand: Consumers maximize utility
E0
∞
∑
t=0
βt
ϑt
C1−σ
t
1−σ
−
N
1+ϕ
t
1+ϕ
s.t.
PtCt +Bt +etB∗
t = Bt...
The Model
Domestic Supply: Calvo pricing and hybrid inflation dynamics
(1+βδH)πH,t = βEt{πH,t+1}+δHπH,t−1 +λHmct +µH,t
(1+β...
The Model
Integration in the world economy:
Terms of Trade: st = pF,t −pH,t
Real exchange rate: qt = et +p∗
t −pt = ψt +(1...
The Model
Monetary policy:
et = 0
substituting through the above equations:
it = i∗
t −χdt −φt
Exogenous Processes
φt = ρφ...
Markov-Switching extension
The different states are characterized by time-varying stochastic volatility
of the risk-premiu...
Solving a Markov-switching DSGE Model
• 2nd Order Perturbation Method (Foerster et.al [2013], wp)
• Forward Solution Metho...
Forward Solution
xt = Et{A(st,st+1)xt+1}+B(st)xt−1 +C(st)zt
with zt following an AR(1) process. From the perspective of ti...
It may be shown that given initial values, under some regularity conditions
such as invertibility of Ξ ∀k the sequences Ωk...
Determinacy and Uniqueness
The existence of the above solution alone is necessary but not sufficient
condition for determin...
Solution and Estimation
Solution:
xt = Ω∗
(st)xt−1 +Γ∗
(st)zt
Combined with the measurement equation
Yt = Hxt +Rt
Likeliho...
Taking the Model to Data
Estonia: Real GDP growth, Consumption growth, Inflation (HICP),
3-month TALIBOR (—)
Europe: Real G...
Priors
Dist. Mean Std.Dev.
p11 Beta 0.9 0.1
p22 Beta 0.9 0.1
β PM 0.995 —
ϕ Gamma 2 0.25
θH Beta 0.75 0.1
θF Beta 0.5 0.1
...
Posterior Estimates
Distribution Prior Mean M1 M2 : St = 1 M2 : St = 2
p11 Beta 0.900 — 0.936
[0.862, 0.984] —
p22 Beta 0....
Posterior Estimates
Distribution Prior Mean M1 M2 : St = 1 M2 : St = 2
χ Gamma 0.010 0.028
[0.014, 0.043]
0.017
[0.006, 0....
Estimated Probabilities of the High State
1996 1998 2000 2002 2004 2006 2008 2010
0
5
10
15
20
1996 1998 2000 2002 2004 20...
2 4 6 8 10 12
−1
−0.5
0
0.5
Consumption
2 4 6 8 10 12
−0.5
0
0.5
Output
2 4 6 8 10 12
−0.5
0
0.5
1
Interest Rate
2 4 6 8 1...
Variance Decomposition and Moments
2 4 6 8 10 12
0
20
40
60
80
100
ε
y
*
ε
π
*
εi
*
εµ
F
εµ
H
εa
εν
ε
φ
2 4 6 8 10 12
0
20...
Robustness Checks
Differences between the specifications:
M1: No regime shifts.
M2: Switching in the volatility of the risk...
Conclusion
• The Markov-switching extension is able to capture the non-linearities
of TALIBOR
• Risk-premium shocks play n...
0.8 1
0
5
10
p11
0.5 1
0
2
4
6
8
10
p22
1 2 3
0
0.5
1
1.5
ϕ
0.8 0.9 1
0
5
10
15
20
θH
0.4 0.6 0.8
0
2
4
6
θF
0 2 4 6 8
0
0...
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Boris Blagov. Financial Crises and Time-Varying Risk Premia in a Small Open Economy: A Markov-Switching DSGE Model for Estonia

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Boris Blagov (Hamburg University)
Eesti Pank Open Seminar 04.09.2013

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Boris Blagov. Financial Crises and Time-Varying Risk Premia in a Small Open Economy: A Markov-Switching DSGE Model for Estonia

  1. 1. Financial Crises and Time-Varying Risk Premia in a Small Open Economy: A Markov-Switching DSGE Model for Estonia Boris Blagov University of Hamburg September 2013
  2. 2. Introduction • Non-linearities in DSGE models • Currency board in DSGE models ◦ Absence of money ◦ Fixed exchange rate e = 0 ◦ Identity between the interest rates (Galí 2008) i = i∗ 2
  3. 3. Introduction 1996 1998 2000 2002 2004 2006 2008 2010 0 5 10 15 20 1996 1998 2000 2002 2004 2006 2008 2010 −15 −10 −5 0 5 10 Top: Interbank interest rates in %, Estonia’s TALIBOR (—) and EURIBOR (- -). Bottom: Real GDP growth. Source: Eesti Pank. 3
  4. 4. Model Overview • Follows Benignio (2001), Monacelli (2005), Justiniano and Preston (2010) • Consumers maximize utility by choosing consumption and labour, have access to foreign markets • Consumption is a bundle of home produced goods (H) and imported goods (F) • Nominal rigidities in terms of sticky prices (hybrid inflation dynamics) • Perfect labour market • LOOP does not hold/incomplete asset markets/imperfect risk sharing • Currency board (fixed exchange rate) • Markov-Switching component 4
  5. 5. The Model Domestic Demand: Consumers maximize utility E0 ∞ ∑ t=0 βt ϑt C1−σ t 1−σ − N 1+ϕ t 1+ϕ s.t. PtCt +Bt +etB∗ t = Bt−1(1+it−1)+etB∗ t−1(1+i∗ t−1)Φ(Dt)+WtNt +Tt • Φ(Dt,φt) is a debt elastic interest rate premium with Φt = exp{−χ(Dt +φt)} • Dt is the real quantity of the consumer’s net foreign asset position in relation to steady state output ¯Y Dt = EtB∗ t−1 ¯YPt−1 • The usual Euler equation applies 5
  6. 6. The Model Domestic Supply: Calvo pricing and hybrid inflation dynamics (1+βδH)πH,t = βEt{πH,t+1}+δHπH,t−1 +λHmct +µH,t (1+βδF)πF,t = βEt{πF,t+1}+δFπF,t−1 +λFψt +µF,t with λH = (1−θH)(1−βθH) θH and λF = (1−θF)(1−βθF) θF and Ψt = EP∗ t PF,t = 1 ⇔ ψt = et +p∗ t −pF,t = 0, i.e. LOOP may not hold 6
  7. 7. The Model Integration in the world economy: Terms of Trade: st = pF,t −pH,t Real exchange rate: qt = et +p∗ t −pt = ψt +(1−α)st CPI inflation: πt = (1−α)πH,t +απF,t Nominal exchange rate dynamics: et = qt −qt−1 +πt −π∗ t UIP: (it −Et{πt+1})−(i∗ t −Et{π∗ t+1}) = Et{qt+1}−qt −χdt −φt Evolution of the net FA position: dt − 1 β dt−1 = yt −ct −α(qt +αst) 7
  8. 8. The Model Monetary policy: et = 0 substituting through the above equations: it = i∗ t −χdt −φt Exogenous Processes φt = ρφφt−1 +ε φ t with ε φ t ∼ N(0,σ2 φ(st)) at = ρaat−1 +εa t with εa t ∼ N(0,σ2 a) ϑt = ρϑϑt−1 +εϑ t with εϑ t ∼ N(0,σ2 ϑ) µH,t = ρµµH,t−1 +ε µH t with ε µH t ∼ N(0,σ2 µH ) µF,t = ρµµF,t−1 +ε µF t with ε µF t ∼ N(0,σ2 µF ) y∗ t = cy∗ y∗ t−1 +ε y∗ t with ε y∗ t ∼ N(0,σ2 y∗ ) π∗ t = cπ∗π∗ t−1 +επ∗ t with επ∗ t ∼ N(0,σ2 π∗ ) i∗ t = ci∗i∗ t−1 +εi∗ t with εi∗ t ∼ N(0,σ2 i∗ ) 8
  9. 9. Markov-Switching extension The different states are characterized by time-varying stochastic volatility of the risk-premium φt = ρφφt−1 +ε φ t with ε φ t ∼ N(0,σ2 φ(st)) which evolve according to P = p11 p12 p21 p22 The equations may then be cast in the following matrix form: B1(st)xt = Et{A1(st,st+1)xt+1}+B2(st)xt−1 +C1(st)zt zt = R(st)zt−1 +εt with εt ∼ N(0,Σ2 (st)) 9
  10. 10. Solving a Markov-switching DSGE Model • 2nd Order Perturbation Method (Foerster et.al [2013], wp) • Forward Solution Method (Cho [2011], wp) • Minimal State Variable (MSV) Solution method (Farmer et.al [2011], JEDC) • Solution under bounded shocks (Davig and Leeper [2007], AER) The first three use unbounded shocks and use the notion of a MSV solution. That is, the system has a solution of the sort: xt = Ωxt−1 +Γεt +ϒwt with xt = ΩXt−1 +Γεt being the "fundamental part" and ϒwt being a sunspot component. 10
  11. 11. Forward Solution xt = Et{A(st,st+1)xt+1}+B(st)xt−1 +C(st)zt with zt following an AR(1) process. From the perspective of time period t by forward iteration the model in period t +k may be represented by: xt = Et{Mk(st,st+1,...,st+k)xt+k}+Ωk(st)xt−1 +Γk(st)zt where Ω1(st) = B(st), Γ1(st) = C(st) and for k = 2,3... Ωk(st) = Ξk−1(st)−1 B(st) Γk(st) = Ξk−1(st)−1 C(st)+Et{Fk−1(st,st+1)Γk−1(st+1)}R Ξk−1(st) = (In −Et{A(st,st+1)Ωk−1(st+1)}) Fk−1(st,st+1) = Ξk−1(st)−1 A(st,st+1) 11
  12. 12. It may be shown that given initial values, under some regularity conditions such as invertibility of Ξ ∀k the sequences Ωk(st) and Γk(st) are well defined, unique and real-valued. The model is said to be forward convergent if the parameter matrices are convergent with k → ∞, i.e: lim k→∞ Ωk(st) = Ω∗ (st) lim k→∞ Γk(st) = Γ∗ (st) lim k→∞ Fk(st,st+1) = F∗ (st,st+1) If (no-bubble condition): lim k→∞ Et{Mk(st,st+1,...,st+k)xt+k} = 0n×1 then the (fundamental) solution is: xt = Ω∗ (st)xt−1 +Γ∗ (st)zt As k tends to infinity, the above condition should hold and all solutions where it does not should be ruled out as they are not economically relevant. Thus, if the model is forward convergent and the "no-bubble condition" is satisfied, then the above is the only relevant MSV solution. 12
  13. 13. Determinacy and Uniqueness The existence of the above solution alone is necessary but not sufficient condition for determinacy, due to the volatility induced by the regime-switching feature. There may exist a non-fundamental part that is arbitrary and there may be a multiplicity of equilibria. Assuming the non-fundamental component takes the form wt = Et{F(st,st+1)wt+1} then the concept for determinacy and indeterminacy deals with interaction of the matrices when switching between states. Defining ΨΩ∗×Ω∗ = [pijΩ∗ j ⊗Ω∗ j ] and ΨF∗×F∗ = [pijF∗ j ⊗F∗ j ] then, mean-square stability is characterized by rσ(ΨΩ∗×Ω∗ ) < 1 rσ(ΨF∗×F∗ ) ≤ 1 13
  14. 14. Solution and Estimation Solution: xt = Ω∗ (st)xt−1 +Γ∗ (st)zt Combined with the measurement equation Yt = Hxt +Rt Likelihood: Kalman filter is inoperable ⇒ Kim’s filter Maximization: Covariance Matrix Adaptation Evoltuion Strategy (CMA-ES) Posterior: Formed through Bayes’ rule, conditioning on the states. p(θ,P,S|Y) = p(Y |θ,P,S)p(S|P)p(θ,P) p(Y |θ,P,S)p(S|P)p(P,θ)d(θ,P,S) 14
  15. 15. Taking the Model to Data Estonia: Real GDP growth, Consumption growth, Inflation (HICP), 3-month TALIBOR (—) Europe: Real GDP growth, Inflation (HICP), 3-month EURIBOR (- -) 1996 1998 2000 2002 2004 2006 2008 2010 2012 −15 −10 −5 0 5 10 GDP 1996 1998 2000 2002 2004 2006 2008 2010 2012 −10 −5 0 5 10 Consumption 1996 1998 2000 2002 2004 2006 2008 2010 2012 −3 −2 −1 0 1 2 3 Inflation 1996 1998 2000 2002 2004 2006 2008 2010 2012 −1 −0.5 0 0.5 1 1.5 2 TALIBOR and EURIBOR 15
  16. 16. Priors Dist. Mean Std.Dev. p11 Beta 0.9 0.1 p22 Beta 0.9 0.1 β PM 0.995 — ϕ Gamma 2 0.25 θH Beta 0.75 0.1 θF Beta 0.5 0.1 α PM 0.5 — σ Gamma 1 1 η Gamma 2 0.25 δH Beta 0.5 0.15 δF Beta 0.5 0.15 χ Gamma 0.01 0.01 ρa Beta 0.7 0.1 ρµF Beta 0.7 0.1 Dist. Mean Std.Dev. ρµH Beta 0.7 0.1 ρν Beta 0.7 0.1 ρφ Beta 0.7 0.1 cy∗ Beta 0.85 0.1 cπ∗ Beta 0.85 0.1 ci∗ Beta 0.85 0.1 σµF IGamma 1 ∞ σµH IGamma 1 ∞ σa IGamma 1 ∞ σν IGamma 1 ∞ σφ IGamma 0.8 ∞ σy∗ IGamma 1 ∞ σπ∗ IGamma 1 ∞ σi∗ IGamma 1 ∞ 16
  17. 17. Posterior Estimates Distribution Prior Mean M1 M2 : St = 1 M2 : St = 2 p11 Beta 0.900 — 0.936 [0.862, 0.984] — p22 Beta 0.900 — 0.942 [0.852, 0.993] — β PM 0.995 0.995 0.995 — σ Gamma 1.000 2.339 [1.371, 3.694] 2.424 [1.434, 3.800] — ϕ Gamma 2.000 1.985 [1.608, 2.404] 1.982 [1.598, 2.399] — θH Beta 0.750 0.910 [0.880, 0.938] 0.912 [0.882, 0.939] — θF Beta 0.500 0.631 [0.544, 0.717] 0.645 [0.556, 0.733] — α PM 0.500 0.500 0.500 — η Gamma 2.000 2.366 [2.011, 2.760] 2.411 [2.062, 2.781] — δH Beta 0.500 0.215 [0.094, 0.371] 0.217 [0.096, 0.369] — δF Beta 0.500 0.590 [0.386, 0.786] 0.594 [0.395, 0.788] — 17
  18. 18. Posterior Estimates Distribution Prior Mean M1 M2 : St = 1 M2 : St = 2 χ Gamma 0.010 0.028 [0.014, 0.043] 0.017 [0.006, 0.029] — σµF IGamma 1.000 1.225 [0.857, 1.673] 1.159 [0.799, 1.618] — σµH IGamma 1.000 0.458 [0.326, 0.620] 0.425 [0.298, 0.586] — σa IGamma 1.000 0.855 [0.209, 2.358] 1.060 [0.205, 3.474] — σν IGamma 1.000 11.265 [7.040, 17.150] 11.438 [7.154, 17.458] — σφ IGamma 0.800 0.472 [0.406, 0.548] 0.119 [0.090, 0.156] 0.665 [0.533, 0.831] σy∗ IGamma 1.000 0.684 [0.592, 0.791] 0.685 [0.593, 0.795] — σπ∗ IGamma 1.000 0.375 [0.321, 0.438] 0.376 [0.321, 0.440] — σi∗ IGamma 1.000 0.100 [0.086, 0.116] 0.100 [0.086, 0.116] — M: -430.723 -405.5175 18
  19. 19. Estimated Probabilities of the High State 1996 1998 2000 2002 2004 2006 2008 2010 0 5 10 15 20 1996 1998 2000 2002 2004 2006 2008 2010 0 0.2 0.4 0.6 0.8 1 Figure: Top: Interbank interest rates, TALIBOR (—) and EURIBOR (- -). Bottom: Smoothed ( ) and not-smoothed (- -) probability of σφ(st) = σφ(high). 19
  20. 20. 2 4 6 8 10 12 −1 −0.5 0 0.5 Consumption 2 4 6 8 10 12 −0.5 0 0.5 Output 2 4 6 8 10 12 −0.5 0 0.5 1 Interest Rate 2 4 6 8 10 12 −0.02 0 0.02 0.04 Real Exchange Rate 2 4 6 8 10 12 −0.05 0 0.05 Terms of Trade 2 4 6 8 10 12 −0.02 −0.01 0 0.01 Inflation 2 4 6 8 10 12 −0.04 −0.02 0 0.02 Inflation Home Goods 2 4 6 8 10 12 −4 −2 0 2 Marginal Cost 2 4 6 8 10 12 0 0.5 1 Net FA Position Figure: Impulse Responses following a risk premium shock for State 1: σφ(low) (- -), State 2: σφ(high) (-.-) and the no-switching version M1 (—) . 20
  21. 21. Variance Decomposition and Moments 2 4 6 8 10 12 0 20 40 60 80 100 ε y * ε π * εi * εµ F εµ H εa εν ε φ 2 4 6 8 10 12 0 20 40 60 80 100 ε y * ε π * εi * εµ F εµ H εa εν ε φ Figure: State-conditioned variance decomposition of the interest rate. y c π i y∗ π∗ i∗ data 4.5103 4.0453 0.8323 0.4635 1.3184 0.3754 0.2169 M2 : State 1 3.4937 4.1860 0.8307 0.2480 1.2781 0.4434 0.1710 M2 : State 2 3.5142 4.2666 0.8291 0.6966 1.2705 0.4443 0.1708 Table: Standard deviation of the actual data and implied by the model based on 5000 random simulations. 21
  22. 22. Robustness Checks Differences between the specifications: M1: No regime shifts. M2: Switching in the volatility of the risk premium σ2 φ. M3: Switching in the volatility in other structural shocks: σ2 a, σ2 ϑ, σ2 µH , σ2 µF , σ2 φ. M4: Switching in σ2 φ and χ. M5: Switching in σ2 φ where the data are linearly detrended. Estimated coefficients M3 : St = 1 M3 : St = 2 M4 : St = 1 M4 : St = 2 M5 : St = 1 M5 : St = 2 χ 0.015 [0.006, 0.028] — 0.014 [0.006, 0.025] 0.024 [0.005, 0.051] 0.025 [0.021, 0.030] — σµF 0.966 [0.611, 1.442] 1.245 [0.810, 1.836] 1.160 [0.798, 1.611] — 1.203 [0.829, 1.676] — σµH 0.403 [0.275, 0.570] 0.474 [0.327, 0.662] 0.433 [0.312, 0.583] — 0.469 [0.338, 0.629] — σν 9.569 [5.430, 15.636] 10.440 [6.404, 16.143] 11.473 [7.203, 17.380] — 14.568 [9.090, 22.368] — σφ 0.121 [0.092, 0.160] 0.673 [0.535, 0.842] 0.119 [0.090, 0.156] 0.680 [0.537, 0.866] 0.145 [0.664, 1.175] 0.887 [0.097, 0.201] σa 0.769 [0.207, 2.007] 1.040 [0.212, 3.117] 0.816 [0.206, 2.346] — 0.870 [0.211, 2.472] — M: -409.2437 -405.0004 -418.835 22
  23. 23. Conclusion • The Markov-switching extension is able to capture the non-linearities of TALIBOR • Risk-premium shocks play negligible role during normal times, but have more profound effects when the system is under pressure • Successful at identifying the banking and financial crisis • Outperforms the standard model in terms of model fit • Allows for state-contingent analysis • The extension is computationally more intensive • The switching is exogenous and the number of states is ad-hoc • The framework is flexible and unexplored, presenting many opportunities for further research 23
  24. 24. 0.8 1 0 5 10 p11 0.5 1 0 2 4 6 8 10 p22 1 2 3 0 0.5 1 1.5 ϕ 0.8 0.9 1 0 5 10 15 20 θH 0.4 0.6 0.8 0 2 4 6 θF 0 2 4 6 8 0 0.2 0.4 0.6 σ 2 3 0 0.5 1 1.5 η 0 0.2 0.4 0.6 0.8 0 1 2 3 4 δH 0 0.5 1 0 1 2 3 δF 0 0.02 0.04 0.06 0 10 20 30 40 χ 0.2 0.4 0.6 0.8 1 0 1 2 3 ρa 0.2 0.4 0.6 0.8 1 0 1 2 3 4 ρµF 0.2 0.4 0.6 0.8 1 0 1 2 3 ρµH 0.2 0.4 0.6 0.8 1 0 1 2 3 ρν 0 0.5 1 0 1 2 3 4 ρφ 0.6 0.8 1 0 2 4 6 cy∗ 0 0.5 1 0 1 2 3 cπ∗ 0.6 0.8 1 0 2 4 6 8 ci∗ 0 1 2 3 0 0.5 1 1.5 σµF 0 0.5 1 0 1 2 3 4 σµH 0 5 10 1 0 0.2 0.4 0.6 0.8 σa 0 10 20 30 0 0.05 0.1 σν 0.050.10.150.20.25 0 5 10 15 20 σφ 0.5 1 0 2 4 6 σy∗ 0.2 0.4 0.6 0 2 4 6 8 10 σπ∗ 0.05 0.1 0.15 0 10 20 30 40 σi∗ 0.5 1 1.5 0 1 2 3 4 σφ Figure: Prior (dashed) and posterior (solid) distributions of M2 (no Markov-Switching). 24

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