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An Estimated Dynamic Stochastic General Equilibrium Model

for Estonia

Paolo Gelain and Dmitry Kulikov∗

Bank of Estonia

Estonia pst 13

15095 Tallinn

Estonia

phone: +372 6680777

e-mail: dmitry.kulikov@epbe.ee

April 7, 2009

Abstract

This paper presents the first version of an open economy Dynamic Stochastic General

Equilibrium (EP DSGE) model for Estonia. The model is designed to highlight the main

driving forces behind Estonian business cycle and to understand how the euro area economic

shocks and monetary policy transmit into the small open economy of Estonia. EP DSGE is

a two area DSGE model incorporating New Keynesian features such as the nominal price

and wage rigidity, variable capital utilization, investment adjustment costs, as well as other

typical features like external consumption habits — both for Estonia and the euro area part

of the model. It is rich in structural shocks such as technology, preference, mark–up, etc.

The ultimate goal of the new model is to be used in simulation exercises, policy advice and

forecasting at the Bank of Estonia.

JEL classification: E4, E5

Keywords: DSGE, Monetary policy, New Keynesian models, Small open economy, Bayesian

statistical inference

∗The main work on EP DSGE model was completed during Paolo Gelain’s stay as a visiting researcher at

Eesti Pank over the period June to September 2008.

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Non-technical summary

This paper reports theoretical foundations and empirical results for the first version of an open

economy Dynamic Stochastic General Equilibrium model for Estonia developed at the Bank

of Estonia. One of the main goals of building a DSGE model of the Estonian economy is to

use it for understanding the monetary policy and export–import linkages between a small open

economy of Estonia and much larger economy of the euro area. The list of other potential tasks

for the Bank of Estonia’s Dynamic Stochastic General Equilibrium model include simulation

exercises, policy advice and forecasting of the main macroeconomic aggregates.

The Bank of Estonia’s DSGE model presented in this paper is a rich New Keynesian DSGE

model which incorporates many important features that are found to be essential for repro-

ducing the complex dynamics and persistence of the real–world macroeconomic time series.

It incorporates the key ingredients that are needed to effectively describe the functioning of

Estonian economy:

• The currency board regime, free capital mobility and resulting lack of an independent

monetary policy conducted by the national central bank. The monetary policy of Estonia

is effectively imported from the ECB and therefore depends on the euro area business cy-

cle. The spread between domestic and euro area interest rates is the key for understanding

the macroeconomic developments in Estonia over the last decade;

• Estonian economy is a textbook example of a small open economy in terms of the openness

to foreign trade as well. The impact of the euro area business cycle on the domestic

economy of Estonia via the mutual trade links is very important;

• Real and nominal convergence still features prominently in the main macroeconomic

aggregates of Estonia. However, this version of the Bank of Estonia’s DSGE model is

specified for the business cycle frequency only, and filters out the long run dynamics of

the empirical data. The future revisions of the model will address this issue with due

care.

The unique feature of the Bank of Estonia’s DSGE model presented in this paper is inclusion

of a fully specified, calibrated DSGE model for the euro area. The economy of Estonia is

considered to be a small open economy on the fringes of the euro area — its main trading partner

and de facto implement of Estonia’s monetary policy due to the currency board arrangement

and free capital mobility between the two areas. The euro area part of the model is a fully

articulated New Keynesian DSGE model of Smets and Wouters (2003), subject to its own set of

ten structural shocks, that is designed to reproduce the monetary policy conducted by the ECB,

and to act as a foreign market for Estonian exports and imports. The two area setup of the Bank

of Estonia’s DSGE model allows for meaningful simulations of the euro area monetary policy

effects on the home economy of Estonia. The forthcoming integration of Estonian economy

into the euro area makes a thorough understanding of these effects particularly important.

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Empirical part of this paper reports estimation results for model’s structural parameters,

impulse response functions and variance decomposition of the main variables. Out of 52 struc-

tural parameters in the Bank of Estonia’s DSGE model, 34 are estimated using a data sample

consisting of 14 macroeconomic series for Estonia and the euro area. Statistical estimates of

the main parameters are largely in line with previous studies for Estonia, when a direct com-

parison can be made. It is also worth mentioning that the net foreign asset position of Estonia

has been found an important and statistically significant factor in explaining the country risk

premium, but the results suggest that other explanatory factors may be warranted.

The empirical relevance of structural shocks is assessed using the variance decomposition.

It is found that the most important domestic shocks in explaining the variability of Estonian

macroeconomic series are the price mark up shock, that often dominates the other shocks

contributing 50% or more of the variability of the state variables, and the technology shock.

The euro area shocks also play a very significant role in driving the dynamics of Estonian

macroeconomic aggregates. Among the most prominent euro area shocks that affect Estonia

are the labor supply, the interest rate and the technology shock.

As mentioned previously, the first version of the Bank of Estonia’s DSGE model focuses on

the business cycle fluctuations of the main Estonian macroeconomic aggregates, leaving their

long run trends aside. The future developments of the model are likely to incorporate the long

run dynamics as well, considering that Estonia is still subject to effects of real and nominal

convergence stemming from its catch-up with the developed euro area economies. Other areas of

the future developments of the model include incorporation of the financial sector together with

relevant frictions, adding the housing sector combined with collateral-constrated households,

and expanding the government sector in the model.

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1 Introduction

This paper reports theoretical foundations and empirical results for the first version of an open

economy Dynamic Stochastic General Equilibrium model for Estonia developed at the Bank

of Estonia. One of the main goals of building a DSGE model of the Estonian economy is to

use it for understanding the monetary policy and export–import linkages between a small open

economy of Estonia and much larger economy of the euro area. The list of other potential

tasks for the Bank of Estonia’s Dynamic Stochastic General Equilibrium model, henceforth

abbreviated as EP DSGE, include simulation exercises, policy advice and forecasting of the

main macroeconomic aggregates.

For now, all these jobs are carried out at Eesti Pank by EMMA model, see Kattai (2005).

EMMA is a traditional medium scale backward–looking macroeconomic model estimated on an

equation-by-equation basis. It incorporates a number of theory–based restrictions, but unlike

a typical DSGE model is not derived from the ground up using utility and profit maximization

framework of the modern macroeconomics.

Recently, a new breed of microfounded DSGE models that incorporate a large number of

structural shocks, nominal and real rigidities and other features needed to describe persistence

of the real–world macroeconomic time series has received a lot of attention by the leading

monetary policy institutions around the world; refer to Tovar (2008) for a recent survey of

DSGE modeling at the central banks. These models became possible thanks to advances in the

macroeconomic theory, offering an advantage over the traditional backward-looking models in

terms of clear interpretations of the main relationships among the forward-looking economic

agents that are subject to the uncertainty stemming from a large number of well–motivated

structural disturbances. In addition, the newly found popularity of DSGE models in many

central banks comes from recent developments in powerful computational methods that permit

statistical inference for many structural parameters using the real-world macroeconomic data.

Likewise, the first version of EP DSGE model presented in this paper is a step toward

eventual phasing out of EMMA at the Bank of Estonia as the main tool for simulation of

different macroeconomic scenarios and policy advice. However, a substantial amount of work

remains to be done before the new model is sufficiently refined and ready to be used by the

policy makers.

A DSGE approach to modeling Estonian economy has been previously attempted in Colan-

toni (2007) and Lendvai and Roeger (2008). Colantoni (2007) estimates a two area DSGE

model using Estonian macroeconomic data with a goal of studying the interest rate channel

of the monetary policy transmission between Estonia and the euro area. Compared to Colan-

toni (2007), the EP DSGE model has similar objectives, but its structure has been refined to

reflect the existing monetary policy regime between the two areas, and the statistical inference

has been substantially improved. The second paper by Lendvai and Roeger (2008) calibrates

an open economy DSGE model with several types of households, a housing sector and sepa-

rate tradable and non-tradable production sectors in order to assess the relative importance of

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productivity growth and credit expansion in driving the long run trends of the main Estonian

macroeconomic aggregates over the last decade. In contrast to Lendvai and Roeger (2008),

where a specific simulation exercise is carried out to understand the long run trends, EP DSGE

model is focused on the effects of the euro area monetary policy and export–import links on

the economy of Estonia at the business cycle frequency.

The first version of EP DSGE model presented in this paper is a rich New Keynesian DSGE

model which incorporates many important features that are found to be essential for describing

the complex dynamics and persistence of the real–world macroeconomic time series. The key

references for the model are papers by Smets and Wouters (2003), Christiano, Eichenbaum and

Evans (2005) and Adolfson et al. (2005). Specifically, EP DSGE model incorporates external

habit formation in consumption, investment adjustment costs, price and wage rigidities and

indexation to the past inflation, and variable capital utilization.

In addition to these frictions, there are eleven structural shocks that drive the dynamics

of Estonian economy in the model. Among the fundamental shocks are production technology

and investment–specific technology innovations, labour supply and preference shocks, an eq-

uity premium shock, and the government consumption innovation. The domestic “cost push”

disturbances include a stochastic price mark–up in the production sector and a wage mark–up

in the labour demand function. The interactions between the euro area and the economy of

Estonia are driven by stochastic mark–ups in the export and import sectors, as well as an

idiosyncratic risk premium shock in the equation linking domestic and euro area interest rates.

The open economy aspect of the EP DSGE model is based around the paper by Adolfson

et al. (2005). In particular, export and import firms in the model operate by selling differen-

tiated consumption goods on foreign and domestic markets subject to the local currency price

stickiness and indexation to the past inflation. In contrast to Adolfson et al. (2005), trade

between the economies of Estonia and the euro area in EP DSGE model takes place the final

consumption good only. This simplification is due to unavailability of suitably disaggregated

export and import price indices in the Estonian foreign trade statistics. Other differences from

Adolfson et al. (2005) include omission of the unit root technology in favor of the stationary

one, missing working capital channel of monetary policy, much less articulated modeling of the

government sector, as well as inclusion of a fully specified, calibrated DSGE model for the euro

area.

The latter feature of EP DSGE model is particularly important considering the design goals

and prospective use of the model at the Bank of Estonia. The economy of Estonia is considered

to be a small open economy on the fringes of the euro area — its main trading partner and de

facto implement of Estonia’s monetary policy due to the currency board arrangement and free

capital mobility between the two areas. The euro area part of EP DSGE is a fully articulated

New Keynesian DSGE model of Smets and Wouters (2003), subject to its own set of ten

structural shocks, that is designed to reproduce the monetary policy conducted by the ECB,

and to act as a foreign market for Estonian exports and imports. The two area setup of EP

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DSGE model allows for meaningful simulations of the euro area monetary policy effects on the

home economy of Estonia. The forthcoming integration of Estonian economy into the euro

area makes a thorough understanding of these effects particularly important.

The empirical results obtained and reported in this paper can be considered satisfactory for

the fist version of the model. Statistical estimates of the main structural parameters are largely

in line with previous studies for Estonia, when a direct comparison can be made. However,

there are a few areas that await an improvement in the future versions of the model. The

external sector is of particular concern, where both the dynamics of trade links with the euro

area, as well as the role of net foreign assets in picking up the spread between the domestic

and euro area interest rates need further examination.

The paper is structured as follows. Section 2 provides a short summary of the main building

blocks of EP DSGE model avoiding excessive technical details. Section 3 and section 4 describe

the key equations of the model pertaining to the economies of Estonia and euro area respectively.

The log–linearized versions of these equations are reported in Appendix 8.2 and 8.3.

overview of the estimation methodology, data series and calibrated parameters is given in

section 5. Main empirical results are discussed in detail in section 6.

summarizes the main findings of the paper.

An

Finally, conclusion

2 A short summary of EP DSGE model

The EP DSGE model presented in this paper takes into account the following key features of

the Estonian economy:

• The currency board regime, free capital mobility and resulting lack of an independent

monetary policy conducted by the national central bank. The monetary policy of Estonia

is effectively imported from the ECB and therefore depends on the euro area business

cycle.1The spread between domestic and euro area interest rates is the key for under-

standing the macroeconomic developments in Estonia over the last decade;

• Estonian economy is a textbook example of a small open economy in terms of the openness

to foreign trade as well. The impact of the euro area business cycle on the domestic

economy of Estonia via the mutual trade links is very important;

• Real and nominal convergence still features prominently in the main macroeconomic

aggregates of Estonia. However, the first version of EP DSGE model in this paper is

1Prior to re-pegging Estonian Kroon to Euro in 1999 it was fixed to Deutsche Mark at the rate 1 DM =

8 EEK. During the second half of 1990-s Estonian banking system was still not completely integrated with the

European and Scandinavian ones. The Asian financial crisis of 1997 and the subsequent Russian financial crisis

of 1998 have changed the landscape of Estonian banking sector, effectively bringing all major Estonian banks

into the hands of Scandinavian owners. Since then the spreads between domestic and euro area interest rates

has narrowed dramatically.

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HOUSEHOLDS

• Consumption, labor and investment-specific shocks

• Own capital, supply labor, set wages a la Calvo, invest into

home and foreign bonds and into productive capital

INTERMEDIATE GOODS PRODUCERS

• Technology shock

•Use capital and labor in production

• Maximize profits and set prices a la Calvo

• Labor

• Capital

• Wage

• Capital income

FINAL GOODS PRODUCERS

• Price markup shock

• Aggregate multiple intermediate goods into one final good

Intermediate goods

demand

Intermediate goods

supply

Private consumption

EXPORTERS

• Price markup shock

• Set export prices a la Calvo

IMPORTERS

• Price markup shock

• Set import prices a la Calvo

EURO AREA

• Euro area shocks (10)

• Optimal monetary policy

• Export, import

Export

Import

INTEREST RATE

• Shock to the risk premium

• Interest rate setting equation

Euro area monetary policy

Figure 1: An overview of EP DSGE model

specified for the business cycle frequency only, and filters out the long run dynamics of

the empirical data.2

Figure 1 provides an overview of the main building blocks and links of the EP DSGE model.

It is a two area DSGE model, consisting of a small open economy DSGE model for Estonia

and a large closed economy DSGE model for euro area. The two parts are linked through the

monetary policy channel — one way from the euro area to Estonia — and by export–import

trading links, where the euro area economy serves as a source of imports to the home economy

of Estonia and generates demand for Estonian exports.3Foreign trade with the euro are is

assumed to be in terms of the composite final consumption good only.

The Estonian part of EP DSGE is a fairy typical small open economy DSGE model that

is similar to Adolfson et al. (2005). It has 24 state variables and 11 structural shocks, and

consists of the following main sections:

• Households own labor and capital, optimize their consumption and supply of working

hours across time, set wages in Calvo (1983) manner subject to labour demand from the

2The future versions of the model are likely to address this issue directly, by incorporating the unit root

technology and suitable steady state inflation dynamics.

3The breakdown of Estonian trade statistics in 2008 reveals that 70% of the foreign trade takes place with

the EU countries. However, the share of euro area countries in the foreign trade is around 25% because many

of Estonia’s major trading partners in the Baltic sea region, such as Latvia, Lithuania, Sweden, Denmark

and Poland, are not euro area members. Since these countries are themselves highly open to the euro area

trade, the assumption of EP DSGE model about export–import trading links with the euro area is a reasonable

approximation.

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labour aggregator, and invest into domestic and foreign bonds as well as the productive

capital;

• Firms are of four types: final good producers operating in perfectly competitive mar-

ket, monopolistically competitive domestic intermediate good producers that set prices

in Calvo (1983) manner, and export and import firms that set prices of differentiated

consumption goods in Calvo (1983) manner;

• Government sector is assumed to follow a balanced budged fiscal policy driven by an

exogenous government consumption shock;

• Domestic nominal interest rates are linked to the euro area ones via the uncovered interest

rate parity condition Rn

t

)Rn,EA

t

in the nominal exchange rate is set to zero because of the currency board.4Instead, the

idiosyncratic part of the interest rate spread is picked up by ?risk

t= Ω(FAt,?risk

, where the next period expected change

t

.

The euro area part of EP DSGE is a calibrated version of Smets and Wouters (2003) closed

economy DSGE model with 13 state variables and 10 structural shocks. Calibrated values of

all structural parameters are taken directly from Smets and Wouters (2003) study.5

3Key equations: Estonian economy

3.1 Households

Household i ∈ [0,1] maximizes its intertemporal utility function by choosing how much to

consume {Ci

in production tomorrow {Ii

rate of capital {zi

domestic {Bi

∞

?

where logεβ

and logεL

t: t ≥ 0}, how much to invest today in order to build the capital that will be used

t: t ≥ 0}, the hours it wants to work {Li

t: t ≥ 0}, how much capital to rent to the firms {Ki

t: t ≥ 0} and euro area {Bi,EA

?

t: t ≥ 0}, the utilization

t: t ≥ 0}, and how many

t

: t ≥ 0} bonds to hold:6

?Ci

t∼ N(0,σ2

t, uL

max

t,zi

{Ci

t,Ii

t,Li

t,Ki

t,Bi

t,Bi,EA

t

}

E0

t=0

βtεβ

t

1

1 − σc

t− hCt−1

?(1−σc)−

εL

t

1 + σL

?Li

t

?(1+σL)?

t= ρβlogεβ

t= ρLlogεL

t−1+uβ

t−1+ uL

t, uβ

β) is the discount factor shock (or preference shock)

t∼ N(0,σ2

L) is the labor supply shock. Households’ behavior

4In fact, the institutional arrangement of the 17 years old currency board system in Estonia rules out possi-

bility of an unilateral Euro peg rate changes by the central bank of Estonia. All such changes must be enacted

by the national parliament and therefore are likely to take some time before coming into effect.

5Future upgrades of the EP DSGE model are likely to move to the part-estimated – part-calibrated euro area

part due to differences in the sample period between Smets and Wouters (2003) and the present paper.

6Household’s domestic bond holdings Bi

tin the Estonian economy part of EP DSGE model can be thought

of as a proxy for per capita net short-term saving/borrowing by the residents in Estonian banks; ditto for the

euro area bonds Bi,EA

t

in foreign banks. There is no market for short-term government obligations in Estonia,

and almost all financing needs of Estonian households and firms are met by the banking sector. The first version

of EP DSGE presented in this paper does not explicitly model the banking sector, an omission that is likely to

be addressed in the future versions of the model.

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is characterized by external habit formation, whose degree is governed by parameter h. A

household has a positive utility in period t only if it able to consume more than a fraction h of

the economy-wide average per-household consumption at t−1. The inverse of the intertemporal

elasticity of substitution in consumption (or equivalently the coefficient of relative risk aversion)

and the inverse of the elasticity of work effort with respect to the real wage are denoted by σc

and σLrespectively.

The maximization is constrained. Firstly, at every time period t ≥ 0 each household faces

the following budget constraint in real terms:7

Ci

t+ Ii

t+ Bi

t+ eBi,EA

t

=

Rn

t−1

Bi

t−1

πc

t

+ Ω(FAt−1,?risk

t−1)eRn,EA

t−1

Bi,EA

t−1

πc

t

+Wi

Pc

t

t

Li

t+ Rk

tzi

tKi

t−1− Ψ?zi

t

?Ki

t−1+ Ti

t+ Divi

t

where Rn

firms (owned by the households), πc

with Pc

Rk

the country specific risk premium function:

t= (1 + it), Ti

tare the net transfers, Divi

tis the gross inflation rate (1+Pc

tthe CPI), e is the fix nominal exchange rate, Wi

tis the return on capital, Ψ(zi

tare dividends from the final good sector

t−Pc

Pc

t−1

tis the wage earned by the household,

t) is the cost of capital utilization function (with Ψ(1) = 0) and

t−1

or equivalently

Pc

Pc

t

t−1,

Ω(FAt,?risk

t

) = exp?− φfaFAt+ log?risk

eBEA,n

t

PD

t

the domestic consumption price index, and log?risk

an idiosyncratic component of the country specific risk.9The idea behind this formulation is to

capture the imperfect integration in the international financial markets. The higher the debt

a country has with the rest of the world, the higher the risk of a default and then the higher

the risk premium it has to pay. Moreover, the introduction of this risk premium is needed to

ensure a well defined steady state in the model (See Schmitt-Groh` e and Uribe (2003)).

Secondly, each household faces the capital accumulation equation at every time period t ≥ 0:

?

where δ is the depreciation rate of capital, logxt = ρxlogxt−1+ ux

stationary investment–specific technology shock, and S(It/It−1) is the investment adjustment

7In nominal terms the budget constraint is

t

?, (1)

where FAt≡

is the net foreign asset position of the economy of Estonia,8where PD

= ρrisklog?risk

t

is

t

t−1+ urisk

t

, urisk

t

∼ N(0,σ2

risk) is

Kt= (1 − δ)Kt−1+1 − S

?

It

It−1

??

Itxt, (2)

t, ux

t ∼ N(0,σ2

x) is a

Pc

tCi

t+Ii,n

t

+Bi,n

t

+Bi.EA,n

t

= Rn

t−1Bi,n

t−1+Ω(FAt−1,?risk

t−1)eRn,EA

t−1Bi,EA,n

t−1

+WtLi

t+Rk

tKi,n

t−1+Ti,n

t

+Divi,n

t

Dividing by Pc

t

Ci

t+ Ii

t+ Bi

t+ Bi.EA

t

= Rn

t−1

Bi,n

t−1

Pc

t

+ Ω(FAt−1,?risk

t−1)eRn,EA

t−1

Bi,EA,n

t−1

Pc

t

+Wt

Pc

t

Li

t+ Rk

tKi

t−1+ Ti

t+ Divi

t

Then, multiplying

Bi,n

t−1

Pc

t

and

Bi,EA,n

t−1

Pc

t

by

Pc

Pc

t−1

t−1, those two terms become

t

) function.

Bi

t−1

πc

t

and

Bi,EA

t−1

πc

t

8See Appendix 8.4 for details on the Ω(FAt,?risk

9See Lundvik (1992) and Benigno (2001).

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costs function. It has the same properties assumed in many previous papers, see Christiano,

Eichenbaum and Evans (2005), namely S(1) = S?(1) = 0 and S??(1) > 0.

The Lagrangian equation is as follows:

max

t,zi

{Ci

t,Ii

t,Li

t,Ki

t,Bi

t,Bi,EA

t

}

E0

∞

?

t=0

βt[?t]

?t

=

Ui

t

?

εβ

t

1 − σc

?Ci

t− hCt−1

?

?(1−σc)−

t−1

πc

t

−Ψ?zi

t−1+

εL

t

1 + σL

?Li

t

?(1+σL)

?

(3)

+λt

Rn

(1 − δ)Ki

t−1

Bi

+ Ω(FAt−1,?risk

?Ki

1 − S

t−1)eRn,EA

t+ Divi

?

t−1

t− Ci

??

Bi,EA

t−1

πc

t− Ii

t

+Wi

t−Bi

?

t

Pc

tLi

t−Bi.EA

t+ Rk

tzi

tKi

t−1

t

t−1+ Ti

t

Pc

t

Pc

t

+Qt

?

Ii

t

Ii

t−1

Ii

txt− Ki

t

The first order conditions are10

∂?t

∂Ct

∂?t

∂It

= 0 :

βtεβ

t(Ct− hCt−1)−σc− βtλt= 0

?

?

(4)

= 0 :

−βtλt+ βtQtxt

1 − S

?

Ii

t

Ii

t−1

?It+1

?

− S?

??It+1

?

Ii

t

Ii

t−1

?

Ii

t

Ii

t−1

?

+βt+1Et

Qt+1xt+1S?

It

It

?2?

= 0 (5)

∂?t

∂zt

∂?t

∂Kt

= 0 :

Rk

t= Ψ?(zt)

= 0 :

βt+1Et

?

λt+1

?

zt+1Rk

t+1− Ψ(zt+1)

??

− βtQt

+βt+1Et{Qt+1(1 − δ)} = 0

βtλt− βt+1Et

?

(6)

∂?t

∂Bt

∂?t

∂BEA

= 0 :

?

λt+1Rn

t

1

πc

t+1

?

= 0(7)

t

= 0 :

βteλt− βt+1Et

λt+1Ω(FAt,?risk

t

)eRn,EA

t

1

πc

t+1

?

= 0(8)

The first order condition for the labour supply is derived in the next section because house-

holds are assumed to be able to supply labour monopolistically. We report here the derivative in

the case in which households offer labour in a competitive way, underlying that this is also the

equation which the next section one reduces to when their non-competitive nature disappears

∂?t

∂Lt

= 0 :

−βtεβ

tεL

t(Lt)σL+ βtλtWi

t

Pc

t

= 0(9)

From the first order condition (4) it is possible to derive the consumption Euler equation:

εβ

t(Ct− hCt−1)−σc= λt,

10The index i is skipped because the decentralized solution is the same as the centralized one, hence the first

order conditions are the same.

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which implies

Etεβ

t+1(Ct+1− hCt)−σc= Etλt+1.

Combining the two previous equations:

Et

εβ

εβ

t(Ct− hCt−1)−σc

t+1(Ct+1− hCt)−σc= Et

Using equation (7):

λt

λt+1. (10)

λt= βEtλt+1Rn

t

1

πc

t+1

,

Et

λt

λt+1

= βEtRn

t

1

πc

t+1

.

Equations (5) and (6) may be re-written defining the marginal Tobin Q as qt=Qt

of the two lagrangian multipliers, or more loosely the value of installed capital in terms of its

replacement cost). They become respectively11

λt(the ratio

1 = qtxt

?

1 − S

?

Ii

t

Ii

t−1

?

?

− S?

?

Ii

t

Ii

t−1

?

Ii

t

Ii

t−1

?

+βEt

?

qt+1λt+1

λt

xt+1S?

?It+1

It

??It+1

It

?2?

(11)

qt= βEt

?λt+1

λt

qt+1(1 − δ) + zt+1Rk

t+1− Ψ(zt+1)

??

(12)

Equation (11) is nothing more then an investment Euler equation which describes the

optimal path for investment. Equation (12) establishes the optimal way to determine the price

of capital, taking into account its future return and its depreciation rate.

Finally, combining equations (7) and (8) a modified UIP condition is obtained, taking into

account the country specific risk:

?

βtλt− βt+1Et

λt+1Rn

t

1

πc

t+1

?

= βteλt− βt+1Et

?

λt+1Ω(FAt,?risk

t

)eRnEA

t

1

πc

t+1

?

Rn

t= Ω(FAt,?risk

t

)Rn,EA

t

.(13)

Aggregate consumption is assumed to be given by a CES index of domestically produced

and imported goods according to

Ct=

?

and CF

(1 − αc)

1

ηc?CD

t are consumption of the domestic and imported goods, αcis the share of

import in consumption and ηcis the elasticity of substitution between domestic and foreign

consumption goods.

t

?ηc−1

ηc + (αc)

1

ηc?CF

t

?ηc−1

ηc

?

ηc

ηc−1

where CD

t

11Note that when there are not investment adjustment costs, i.e. S(Ii

t/Ii

t−1) = 0, the investment dynamics

equation implies that

qt =

1

xt

namely the Tobin’s Q is equal to the replacement cost of capital (the relative price of capital). Furthermore, if

xt = 1, as in the standard neoclassical growth model, qt = 1.

11

Page 12

Household maximizes Ctsubject to the two following expenditure constraint

PD

tCD

t+ PF,c

t

CF

t= Pc

tCt

where

Pc

t=

?

(1 − αc)?PD

and PF

t

?1−ηc+ αc

?

PF,c

t

?1−ηc?

1

1−ηc

(14)

Both PD

imported consumption goods index respectively.

From the maximization we obtain the following two conditions

tt are expressed in domestic currency and are the domestic price index and the

CD

t = (1 − αc)

?PD

?−ηc

t

Pc

t

?−ηc

Ct

(15)

CF

t= αc

?

PF,c

t

Pc

t

Ct

(16)

3.1.1 Labour Supply

Each household is a monopoly supplier of a differentiated labour service required by domestic

intermediate goods producers.12This implies that households in the economy can determine

their wages subject to substitutability between different labour services governed by the pa-

rameter λw

inelastically at the going wage rate.

The framework is similar to the one used to derive the New Keynesian Phillips Curve in

the next section. A labour aggregator is assumed to hire differentiated labour services form

the households and transform them into a homogenous input good Lt using the following

technology:

t. After setting their wages, households supply the required amount of working hours

Lt=

??1

0

?Li

t

?

1

1+λw

t

di

?1+λw

t

,

where Li

varying wage mark-up, governed by logλw

steady state value of λw

tis the i-th household’s labour supply, Ltis the aggregated labour, and λw

t = logλw+ uw

t.13

tis the time

t, uw

t ∼ N(0,σ2

w), where λwis the

12The main references are Kollmann (2001), Erceg et al. (2000), Christiano, Eichenbaum and Evans (2005).

Most recent references are Adolfson et al. (2005) and Fernandez-Villaverde and Rubio-Ramirez (2007). The

latter has a good mathematical appendix with detailed derivations of all relevant formulas.

13See Chari et al. (2008) for a discussion and criticism of the wage mark-up shock and other shocks as well.

12

Page 13

The implied solution for Li

tis14

Li

t=

?Wi

t

Wt

?1+λw

t

λw

t

Lt,

where

Wt=

??1

0

?Wi

t

?−1

λw

t

di

?−λw

t

It is also assumed that not all households can optimally re-set their wages each time. Using

Calvo (1983) assumptions, only a fraction 1−θwof all households can optimally set their wages

each time period. Those households who cannot re-set their wages are assumed to be able to

index them to the past inflation according to the following formula:

Wi

t+1= (πc

t)τwWi

t. (17)

Given this set up, households optimize their wages taking into account the probability of

being unable to re-set their wages for some number of time periods in the future. The resulting

wage equation in log–linear form is given by:15

? wt=

β

1 + βEt? wt+1+

1

1 + β? wt−1+

1 + β

1 +(1+λw)σl

β

1 + βEt? πc

?

t+1−1 + βτw

1 + β

? πc

t+

τw

1 + β? πc

t−1

−

1

(1 − βθw)(1 − θw)

?

λw

θw

?

? wt− σl?lt−

σc

1 − h(? ct− h? ct−1) + ? εL

t

?

+ uw

t.

(18)

If wages are completely flexible, that is when θw= 0, this equation reduces to (9).

14The maximization problem for the labour aggregator is:

Z1

s.t.

2

0

max

{Li

t}WtLt−

0

Wi

tLi

tdi

Lt =

4

Z1

“

Li

t

”

1

1+λw

t

di

3

5

1+λw

t

.

15It is derived by solving the following maximization problem, which is a part of the lagrangian equation (3):

"

1 + σL

max

{Wi

t}E0

∞

X

k=0

(βθw)k

−εβ

tεL

t

“

Li

t+k

”(1+σL)

+ λt+k

k

Y

s=1

`πD

t+s−1

πD

t+s

´τw

Wi

Pc

t

t

Li

t+k

#

s.t.

Li

t+k=

"

k

Y

s=1

`πD

t+s−1

πD

t+s

´τw

Wi

Wt+k

t

#−1+λp

t

λp

t

Lt.

The first order condition from this maximization problems needs to be combined with the law of motion of the

aggregate wage index:

W

−1

t

λp

t

= θw

h

Wt−1

“

πD

t−1

”τwi−1

λp

t+ (1 − θw)(W∗

t)

−1

λp

t ,

where W∗

t is the wage set by the optimizing households.

13

Page 14

3.2Firms

3.2.1Final Good Producers

The final good is produced using the intermediate goods j-s by the following technology16

Yt=

??1

t= logλp+ uλp

0

?

Yj

t

?

1

1+λp

tdj

?1+λp

t, uλp

t

where logλp

steady state value. This is interpreted as a cost push shock to the inflation equation.

The cost minimization condition17in the final goods sector can be written as

t

∼ N(0,σ2

λp) is the time varying price mark-up and λpis its

Yj

t=

?

Pj

PD

t

t

?−

1+λp

λp

t

t

Yt

(19)

where Pj

be written as

tis the price of the intermediate good j and PD

t is the domestic price index which can

PD

t =

??1

0

?

Pj

t

?−1

λp

tdj

?−λp

t

3.2.2Intermediate Goods Producers

Firms producing intermediate goods operate in a competitive market. They hire labour form

households, paying the salary Wt, and they rent the capital they need paying a return Rk

Firm j produces output Yj

?? Kj

»Z1

where ε is the price elasticity of demand for good j. It is known the the gross mark up (1 + λp) is equal to

t.

ton the basis of the following Cobb-Douglas production function

?1−α

16In a standard set up, this technology is reported as follows

Yj

t= At

t−1

?α?

Lj

t

− Φ

Yt =

0

“

Yj

t

”ε−1

ε

dj

–

ε

ε−1

ε

ε − 1

In the text we have just substituted ε with its expression in terms of the mark-up and assumed that it is time

varying:

εt =1 + λp

t

λp

t

17This condition is obtained minimizing the following cost function

Z1

min

{Yj

t}

0

Pj

tYj

t

subject to the quantity constraint

»Z1

0

“

Yj

t

”

1

1+λp

tdj

–1+λp

t

≥ Yt

14

Page 15

where? Kj

stationary technology shock.

Firms minimize costs under the production function constraint. The objective function is

?Wt

The lagrangian function is:

?Wt

The first order conditions are:

?? Kj

∂?t

∂Lj

t

PD

t

where the lagrangian multiplier ζtrepresents the real marginal cost.

Solving (21) for the lagrangian multiplier and substituting the result into (20) gives:

t−1is the effective capital stock given by? Kj

t−1= ztKj

t−1, Φ are fixed costs to assure

t, ua

that profits are zero in the steady state, and logAt = ρalogAt−1+ ua

t∼ N(0,σ2

a) is a

min

t−1,Lj

{e Kj

t}

PD

t

?

Lj

t+ Rk

t? Kj

t−1

min

t−1,Lj

{e Kj

t}

?t=

PD

t

?

Lj

t+ Rk

t? Kj

t−1+ ζt

?

Yj

t− At

?? Kj

t−1

?α?

Lj

t

?1−α?

∂?t

∂? Kj

t−1

= 0 :

Rk

t− ζtAtα

t−1

?α−1?

?? Kj

Lj

t

?1−α

?α?

= 0 (20)

= 0 :

Wt

− ζtAt(1 − α)

t−1

Lj

t

?−α

= 0(21)

Rk

t=

α

1 − α

Wt

PD

t

Li

t

? Kj

?−1

t−1

which implies that

Li

t

? Kj

t−1

=

?Wt

PD

t

Rk

t

1 − α

α

.

Then, using this result to substitute out

cost:

?

Intermediate goods producers face another type of problem. Each period, only a fraction

1−θ of them, randomly chosen, can optimally adjust their prices (see Calvo (1983)). For those

that cannot re-optimize, prices are indexed to the past inflation as follows:

Li

t−1in (21) we have an expression for the real marginal

t

e Kj

MCt=

1

At

1

1 − α

?1−α?1

α

?α?Wt

PD

t

?1−α ?

Rk

t

?α

PD

t+1=?πD

t

?τπPD

t, (22)

where τπis the parameter governing the degree of price indexation.

Maximizing the expected discounted profits:18

??

s=1

max

{Pj

t}

E0

∞

?

i=0

(βθ)iλt+i

λt

i?

?πD

t+s−1

?τπ Pj

t

PD

t+i

− MCt+i

?

Yj

t+i

?

18In order to maintain the paper self-contained we do not report the derivation of the New Keynesian Phillips

curve. Moreover, it has been derived in many papers and books, so we refer to them. See Walsh (2003), Adolfson

et al. (2005), Fernandez-Villaverde (2007) among others.

15

Page 16

subject to the intermediate good demand function by the final good producers, see (19):

Yj

t+i=

?

i?

s=1

?πD

t+s−1

?τπ Pj

t

PD

t+i

?−

1+λp

λp

t

t

Yt+i,

it is possible to derive the first order condition for the optimal price P∗

t:19

E0

∞

?

i=0

(βθ)iλt+i

???

−1

λp

t

??

i?

??

s=1

?

πD

t+s−1

πD

t+s

?τπ?−1

λp

t P∗

PD

t

t

+

?1 + λp

t

λp

t

i?

s=1

?

πD

t+s−1

πD

t+s

?τπ?−

1+λp

λp

t

t

MCt+i

?

Yj

t+i

?

= 0.

(23)

Given that in each time period a fraction of firms can re-set their prices optimally, and

the rest index their prices using the previous period’s inflation rate, the aggregate price index

evolves according to the following weighted average formula:

PD

t =

??θ

0

??πD

t−1

?τπPD

t−1

?−1

λp

t+

?1

θ

(P∗

t)

−1

λp

t

?−λp

t

,

equivalently:

?PD

t

?−1

λp

t = θ

??πD

t−1

?τπPD

t−1

?−1

λp

t+ (1 − θ)(P∗

t)

−1

λp

t . (24)

Combining the log–linearized equation (24) with the first order condition for the optimal

price in (23) leads to an equation for the domestic inflation rate. It is given by the hybrid New

Keynesian Phillips Curve:

? πD

t=

β

1 + βτπ

Et? πD

t+1+

τπ

1 + βτπ? πD

t−1+

1

1 + βτπ

(1 − βθ)(1 − θ)

θ

?

mct+ uλp

t .(25)

When prices are completely flexible, that is when θ = 0, and the price mark-up is zero, the

previous equation reduces to the usual flexible price condition whereby the real marginal cost

are equal to one.

3.3 Importers

The import and export sectors in EP DSGE model are based on Adolfson et al. (2005). The

import sector consists of a large number of firms that buy a homogenous good in the euro area

19Since all firms face the same technology shock and the resulting optimal capital–output ratio is similar

across all intermediate producers, the optimal price P∗

t is the same for all firms. Solving this equation for P∗

t

and assuming flexible prices (θ = 0) gives the standard monopolistic competition outcome that firms set their

price a mark up over their nominal marginal cost:

P∗

t = (1 + λp

t)MCt

16

Page 17

market and turn it into differentiated consumption goods20using a brand naming technology,

i.e. without costs. These differentiated consumption goods are then sold to domestic households

subject to price stickiness in the local currency.

Firms buy the homogenous good at a price PEA

The framework in which these firms operate is identical to the one of the intermediate goods

producers in terms of price setting behaviour. A fraction of importers 1 − θc,F are allow to

optimize in each period. For those which do not optimize, prices evolve as follows:

?

??1

uλc,F

t

, uλc,F

t

λc,F

of price indexation. The final imported good is a composite of continuum of j differentiated

imported goods, each supplied by a different firm and with price Pj,F,c

function:

t

, which is the CPI of the Euro Area.

PF,c

t+1=

? πc,F

t

?τc,FPF,c

?

t

, (26)

where PF,c

t

=

0

Pj,F,c

t

?−

?is the imported goods time varying mark-up21, and τc,Fis the degree

1

λc,F

t

dj

?−λc,F

t

is the imported good price index, logλc,F

t

= logλc,F+

t

∼ N?0,σ2

t

, which follows the CES

CF

t=

??1

0

Cj,F

t

1

1+λc,F

t

dj

?1+λc,F

t

.

This implies the following demand for each imported good:

Cj,F

t

=

?

Pj,F,c

t

P,cF

t

?−

1+λc,F

λc,F

t

t

CF

t.

Importing firms maximize their profits subject to the Calvo (1983) price stickiness restric-

20In Adolfson et al. (2005) there is a distinction between imported consumption and investment goods.

This version of EP DSGE model does not make this distinction because the statistical data related to prices of

imported investment goods for Estonia is not readily available. This would make estimation of the corresponding

Phillips Curve difficult. It is assumed that only consumption goods and services are imported. The same applies

to the export sector in subsection 3.4.

21Note that the steady state value of this mark up does not enter in any of the equation we are going to

estimate. It is only relevant for the steady state values of some variables.

17

Page 18

tion.22The resulting inflation equation in log-linearized form is given by:

? πc,F

t

=

β

1 + τc,FβEt? πc,F

Exporters

t+1+

τc,F

1 + τc,Fβ? πc,F

t−1+

1

1 + τc,Fβ

(1 − θc,F)(1 − βθc,F)

θc,F

?

?

mcc,F

t

+uλc,F

t

t

?

.

3.4

The exporting firms buy final domestic good and differentiate it by brand naming. They sell

the continuum of differentiated consumption goods to the households in the euro area. The

nominal marginal cost is thus the price of the domestic good PD

goods are exported, each exporting firm j faces the following demand function for its product:

t. Since only consumption

Cj,x

t

=

?

Pj,x,c

t

Px,c

t

?−

1+λc,x

λc,x

t

t

Cx

t,

where Px,c

and logλc,x

export goods. Once again, the problem that exporting firms are faced with is determined by

the Calvo (1983) framework. A fraction 1−θc,xof the firms can set optimal prices each period.

For the remaining share of firms the evolution of prices is following:

t

is the export price index expressed in the local currency of the export market,

= logλc,x+ uλc,x

t

, uλc,x

t

∼ N(0,σ2

t

λc,x) is the stochastic mark-up on differentiated

Px,c

t+1= (πc,x

t)τc,xPx,c

t

. (27)

Exporting firms maximize their profits subject to the price stickiness restriction.23The

22They maximize discounted stream of profits:

max

{Pj,F,c

t

}

E0

∞

X

i=0

(βθc,F)iλt+i

λt

("

iY

s=1

“

πc,F

t+s−1

”τc,F Pj,F,c

t

PF,c

t+i

− MCc,F

t+i

#

Cj,F

t+i

)

s.t.

Cj,F

t+i=

"

iY

s=1

“

πc,F

t+s−1

”τc,F Pj,F,c

t

PF,c

t+i

#−1+λc,F

t

λc,F

t

CF

t+i,

where MCc,F

t+i=

PEA

t

ePF,c

t

.

In addition, the usual aggregate price equation needs to be combined with the first order condition obtained

from the above maximization problem:

“

PF,c

t

”−

1

λc,F

t

= θc,F

h“

πc,F

t−1

”τc,FPF,c

t−1

i−

1

λc,F

t

+ (1 − θc,F)

“

P∗F,c

t

”−

1

λc,F

t

.

23They maximize discounted stream of profits:

max

{Pj,x,c

t

}E0

∞

X

i=0

(βθc,x)iλt+i

λt

("

iY

s=1

`πc,x

t+s−1

´τc,xPj,x,c

t

Px,c

t+i

− MCc,x

t+i

#

Cj,x

t+i

)

s.t.

Cj,x

t+i=

"

iY

s=1

`πc,x

PD

t

ePx,c

t

t+s−1

´τc,xPj,x,c

t

Px,c

t+i

#−1+λc,x

t

λc,x

t

Cx

t+i,

where MCc,x

t+i=.

18

Page 19

resulting inflation equation in log-linearied form is given by:

? πc,x

In addition, Estonian economy is assumed to be small relative to the euro area economy in

EP DSGE model, and hence it plays just a negligible part in aggregate foreign consumption.

Assuming that the aggregate foreign consumption follows a CES function, the euro area demand

for the aggregate domestic consumption good (and in this case also for the total export) is given

by:

?Px,c

where ηEAis the elasticity of substitution between the foreign and home consumption goods,

and CEA

t

is the euro area aggregate consumption. Here it is assumed that YEA

?Px,c

t

=

β

1 + τc,xβEt? πc,x

t+1+

τc,x

1 + τc,xβ? πc,x

t−1+

1

1 + τc,xβ

(1 − θc,x)(1 − βθc,x)

θc,x

?

?

mcc,x

t

+ uλc,x

t

?

.

Expt= Cx

t=

t

PEA

t

?−ηEA

CEA

t

,

t

= CEA

t

, hence:24

Expt=

t

PEA

t

?−ηEA

YEA

t

. (28)

3.5Policies

3.5.1Fiscal Policy

Fiscal policy is exogenous and assumed to behaves as follows:

logGt= ρglogGt−1+ ug

t, (29)

where ug

t∼ N(0,σ2

g). In addition, the balanced budget condition implies that Gt= −Tt.

3.5.2Monetary Policy

The monetary policy of Estonia is subject to the currency board arrangement and free capital

mobility between the domestic and euro area markets. The UIP condition derived previously

in (13) implies that the domestic nominal interest rate is given by:

Rn

t= Ω(FAt,?risk

t

)Rn,EA

t

.

In other words, it is determined by the monetary policy in the euro area and by the country

specific risk premium.

In addition, the usual aggregate price equation needs to be combined with the first order condition obtained

from the above maximization problem:

(Px,c

t

)

−

1

λc,x

t

= θc,x

ˆ`πc,x

t−1

´τc,xPx,c

t−1

˜−

1

λc,x

t

+ (1 − θc,x)(P∗x,c

t

)

−

1

λc,x

t

24This assumption is not properly correct. In fact, as we will see in one of the following sections, YEA

CEA

t

+ INVEA

t

+ GEA

t

+ Ψ(zt)KEA

capital utilization. This assumption does not affect the estimation results.

t

=

t−1, where Ψ(zt)KEA

t−1is the cost associated with variations in the degree of

19

Page 20

3.6Aggregate Resource Constraint

The aggregate resource constraint is:

Yt≡ CD

t+ CF

t+ It+ Gt− Ψ(zt)Kt−1+ Expt− Mt.

Imports are defined as follows:

(30)

Mt= CF

t. (31)

Now we can substitute out all the components of the domestic output into (30) using

equations (15), (31) and (28):

?PD

Yt= (1 − αc)

t

Pc

t

?−ηc

Ct+ CF

t+ It+ Gt− Ψ(zt)Kt−1+

?Px

c,t

PEA

t

?−ηEA

YEA

t

− CF

t,

Yt= (1 − αc)

?PD

t

Pc

t

?−ηc

Ct+ It+ Gt− Ψ(zt)Kt−1+

?Px

c,t

PEA

t

?−ηEA

YEA

t

.

3.7The Net Foreign Assets

The net foreign assets evolve in the following manner:

eBEA

t

= ePx,c

t

Expt− ePEA

t

Mt+ Ω(FAt,?risk

t

)eRn,EA

t−1BEA

t−1. (32)

Dividing both sides of this equation by PD

CF

?Px,c

t

and using the definitions of Expt, Mt, MCc,x

t

and

t from equations (28), (31), (16), and footnote 23, equation (32) can be written as:

?−ηEA

FAt=ePx,c

t

PD

t

t

PEA

t

YEA

t

−ePEA

PD

t

t

CF

t+ Ω(FAt,?risk

t

)Rn,EA

t−1

FAt−1

πD

t

,

FAt= (MCc,x

t )−1

?Px,c

t

PEA

t

?−ηEA

YEA

t

−ePEA

PD

t

t

αc

?

PF,c

t

PD

t

?−ηc

Ct+Ω(FAt,?risk

t

)Rn,EA

t−1

FAt−1

πD

t

,

where πD

t=

PD

t

PD

t−1= 1 +

PD

t−PD

PD

t−1

t−1

.

4 Key equations: Euro area economy

The euro area part of EP DSGE model is based on Smets and Wouters (2003) paper.25In

contrast to Adolfson et al. (2005), where the rest of the world is described by a low dimensional

25Originally, the idea was to model euro area by a basic three equations NK DSGE model, and estimate it

either separately or jointly with the Estonian economy part. This is more in line with the spirit of small open

economy DSGE models, where the rest of the world is often just a three equation VAR system. However, some

estimation difficulties led to a change in the approach, and the model by Smets and Wouters (2003) was chosen

to describe the monetary policy and business cycles in the euro area. An interesting future extension would be

to consider an open economy model for the euro area as well, allowing to study the effects of the rest of the

world on the Estonian economy through the corresponding impact on the euro area. In this respect, Adolfson

et al. (2005) is an excellent reference.

20

Page 21

VAR system, this papers specifies a full fledged DSGE model as a counterpart to the Estonian

economy part described in Section 3. Since one of the main interests of this paper is to examine

various euro area disturbances that hit the economy of Estonia, it is necessary to have a model

which incorporates a range of structural shocks with clear economic interpretations attached

to them. For this reason, and due to increased dimensions of the euro area part of EP DSGE

model, values of the structural parameters for the euro area economy are calibrated according

to Smets and Wouters (2003).

In terms of equations, the model is similar to the Estonian economy part in Section 3, but

with a few substantial differences.26The differences are due to the fact that it is a closed

economy model having an independent monetary policy described by a monetary policy rule.

The aggregate resource constraint is given by:

YEA

t

= CEA

t

+ INVEA

t

+ GEA

t

+ Ψ?zEA

t

?KEA

t−1.

The monetary policy rule is as follows:

? rn,EA

t

=

φm? rn,EA

t−1

+ (1 − φm)

?? πEA

?

πt+ rπEA?? πEA

t

t−1− πt

− ? yP,EA

?+ ryEA

−

?

? yEA

t−1

t−1− ? yP,EA

t−1

+ urn,EA

t

??

+

+r∆π

t

− ? πEA

t−1

?+ r∆y

t ∼ N(0,σ2

?

? yEA

t

?

? yEA

t−1− ? yP,EA

is the potential output. In

??

,

where logπt = ρπlogπt−1+ uπ

N(0,σ2

DSGE literature, the potential output is defined as the level of output that would prevail under

flexible prices and wages in the absence of the cost push shocks (uw,EA

Smets and Wouters (2003) the model reported in Appendix 8.3 is expanded with flexible prices

and wages version (θEA= θEA

w

= uw,EA

t

model-consistent output gap.

t, uπ

π) is the inflation objective shock, urn,EA

t

∼

rn,EA) is a idiosyncratic monetary policy shock, and YP,EA

t

t

,uλp,EA

t

,uq,EA

t

). As in

= uλp,EA

t

= uq,EA

t

= 0) in order to calculate the

5 Data and estimation

5.1Bayesian estimation methodology

Statistical inference for the structural parameters of the EP DSGE model introduced in sec-

tions 3 and 4 is obtained by Bayesian methods, the corresponding empirical results can be

found in section 6 of this paper. Bayesian statistical methods have recently gained popularity

in applied macro-economic modeling, for a recent overview of main literature and methods of

Bayesian analysis of DSGE models refer to An and Schorfheide (2007). This subsection gives

an overview of the main stages of the Bayesian statistical inference for DSGE models.

In contrast to the traditional approach to statistical estimation and testing known under

the banner of “frequentist statistics” where the inference based on repeated sampling plays a

pivotal role and a “true” model with an unknown but constant set of parameters is assumed

to exist, Bayesian statistics adopts a view that parameters are just “mental constructs that

26The set of log-linearized equations is reported in Appendix 8.3 as a reference.

21

Page 22

exist only in the mind of the researcher”, see Poirier (1995).

on a fusion of priors about the model parameters with the likelihood function based on the

real–world data, where “the latter represents a “window” for viewing the observable world

shared by a group of researches who agree to disagree in terms of possibly different prior

distributions”, see Poirier (1995). Bayesian statistics knowingly departs from the assumptions

of repeated sampling experiments and underlying data generating process based on unknown

and constant set of parameters — the two assumptions that are crucial to the traditional

frequentist approach.

Bayesian statistics can be characterized as a learning process, where observed data collected

in Y is used to learn about the posterior distribution f(θ|Y) of a k-dimensional vector of

model parameters θ, given the likelihood function L(θ;Y) and the prior distribution f(θ).

This learning process is based on a version of the Bayes’ Theorem:

Bayesian statistics is based

f(θ|Y) =f(θ)L(θ;Y)

?

distribution function f(θ|Y), which summarizes information available in the data Y about

the vector of parameters θ. The posterior distribution function may further be combined with

a statistical loss function in order to arrive to point and interval inference about θ as well as

other forms of statistical decisions involving the vector of parameters.

It follows from expression (33) that Bayesian statistical inference requires both the likelihood

function and the prior distribution. Remaining part of this section provides a general overview

of the steps involved in construction of the likelihood function for a typical DSGE model. This

discussion is applicable not only to the EP DSGE model introduced in sections 3 and 4, but

also to other DSGE and real business cycle models found in the literature and estimated in

the form of first-order linear or log-linear approximations around the steady state. The issues

related to the particular choice of priors for the EP DSGE model are deferred to section 5.2.

In general, the likelihood function of a typical DSGE model cannot be written in closed

form as a function of data Y and model parameters θ. However, given Y and θ, the procedure

to evaluate the likelihood function at these values in an implicit form involves three stages.

They are described below in detail.

The first stage involves writing a theoretical macro-economic model as a system of linear

expectational and non-expectational equations, including exogenous stochastic processes. Ap-

pendices 8.2 and 8.3 list the corresponding system of log-linearized equations for the EP DSGE

model. Let xtdenote a m × 1 vector of endogenous model variables, ytdenote n × 1 vector of

other endogenous model variables and ztbe k × 1 vector of exogenous stochastic processes. A

f(Y)

∝ f(θ)L(θ;Y),(33)

where f(Y) =

of posterior inference.

Rkf(θ)L(θ;Y)dθ can be treated as a normalizing constant for the purpose

The main object of interest for Bayesian inference is the posterior

22

Page 23

DSGE model can be written in linearized form as follows:

0 = Axt+ Bxt−1+ Cyt+ Dzt

0 = Et

zt+1= Nzt+ ?t,

?Fxt+1+ Gxt+ Hxt−1+ Jyt+1+ Kyt+ Lzt+1+ Mzt

?

(34)

where a k × 1 vector ?tof stochastic shocks has mean zero and variance–covariance matrix Σ.

Vector of model parameters θ is mapped into the matrices A to Σ of this system according to

the theoretical model.

Linear system of expectational equations (34) is solved in the second stage of the likelihood

function evaluation. The method of undetermined coefficients stipulates the following solution

of the system (34):

xt= Pxt−1+ Qzt

yt= Rxt−1+ Szt,

(35)

where matrices P to S are mappings of matrices A to M defined in (34) and therefore also

functions of the vector of model parameters θ. Uhlig (1999) gives a comprehensive overview of

the method of undetermined coefficients for linear systems of expectational equations like (34),

including conditions on dimensions and ranks of matrices A to Σ that are necessary to obtain

the solution (35). In particular, stability or lack of thereof of the system of linear expectational

equations (34) depends on the vector of model parameters θ through the matrices A to M and

is reflected by the eigenvalues of P matrix in (35).

Having obtained the system of stochastic difference equations (35) for endogenous variables

xtand ytand using the law of motion of the exogenous stochastic processes zt, the likelihood

function L(θ;Y) of a DSGE model is evaluated using the Kalman filter in the third stage of the

procedure. The Kalman filter is needed because endogenous variables of the model in vectors xt

and ytusually involve some quantities for which no empirical counterparts can be observed in

macro-economic statistics. Let m+k×1 dimensional vector ˜ xtbe defined as ˜ xT

and let ˜ ytbe a vector of observed variables at time period 1 ≤ t ≤ T s.t. the data matrix

is given by Y := (˜ yT

using the solution (35) as follows:

?

0N

?t

t:= (xT

t−1,zT

t),

1, ˜ yT

2,..., ˜ yT

T). A DSGE model can be written in the Kalman filter form

˜ xt+1=

PQ

?

˜ xt+

?

0

?

˜ yt= Γ˜ xt,

(36)

where matrix Γ maps a subset of endogenous model variables into the observed data, and may

include elements of the matrices R and S from the system (35). In some cases the measurement

equation in (36) may additionally involve measurement errors, if observed data is deemed to

be imperfect counterpart of endogenous model variables. In the estimation of the EP DSGE

model no measurement errors are included in the Kalman filter equations (36). The value of

23

Page 24

the likelihood L(θ;Y) at the vector of model parameters θ is computed using standard Kalman

filter recursions as detailed in Hamilton (1994) pp. 372–408.

It is necessary to note that the mapping of parameters θ into the likelihood function L(θ;Y)

of a typical DSGE model is highly complicated, involving nonlinear transformations at the

solution stage (35). This might give rise to identifiability issues which are difficult to deal with

because of the lack of developed diagnostic methods. Some of the issues related to identification

in DSGE models are discussed in Canova (2008) and Iskrev (2008).

Apart from the likelihood and the priors, Bayesian statistical inference based on (33) re-

quires a set of techniques to evaluate the posterior distribution f(θ|Y). Specifically, one is

usually interested in at least first few moments of the posterior distribution of θ, but accepted

current practice requires reporting of the kernel posterior density estimates of the model param-

eters. Since for a typical DSGE model f(θ|Y) is not available in closed form, computationally

intensive Monte Carlo sampling methods are needed to generate draws from the posterior dis-

tribution f(θ|Y). For a good survey of Monte Carlo methods in Bayesian statistics refer to

Robert and Casella (2004).

Metropolis–Hastings Markov chain Monte Carlo algorithm offers a general and easy-to-

implement way to draw random numbers from probability distributions for which no procedural

random number generators are available. A particularly simple implementation of the algorithm

is called random walk Metropolis–Hastings and involves the following four steps:

1. Given the previous draw θi−1from f(θ|Y), generate a candidate draw as follows:

θ∗

i= θi−1+ vi, where vi∼ N(0,S);

2. Compute:

αi:= min

?

1,

f(θ∗

f(θi−1|Y)

i|Y)

?

;

3. Assign the new draw θifrom f(θ|Y) as:

?

θi−1

θi=

θ∗

i

if αi≥ ui

if αi< ui

, where ui∼ U[0,1];

4. Repeat steps 1 to 3 until enough random draws are generated from the distribution

f(θ|Y).

Given N draws {θ1,...,θN} from the posterior distribution f(θ|Y) supplied by the Metropolis–

Hastings or other sampling method, empirical moments, kernel density estimates and other

posterior statistics can be computed in the usual fashion.

The practical implementation of the steps associated with evaluation of the likelihood func-

tion and drawing from the posterior distribution varies from low-level programming of all

necessary steps in one of the mathematical programming languages, such as MATLAB, to us-

ing higher-level packages specifically written for the analysis and estimation of DSGE models,

24

Page 25

such as Dynare.27The latter is used for simulation and estimation of the EP DSGE model in

this paper.

Posterior distributions of the model parameters reported in Section 6 of this paper have

been obtained using MATLAB/Dynare toolbox as follows. The inference is based on 2 parallel

chains of 800000 draws each, where the last 400000 draws are used for statistics and diagnostics.

The algorithm is started by simulation–based maximization of the posterior kernel,28followed

by evaluation of the Hessian matrix at the posterior kernel maximum, which it then used as an

input parameter for the main run of Metropolis–Hastings algorithm to compute the posterior

inference for the model parameters.

5.2 Data and priors

Open economy DSGE model for Estonia introduced in sections 3 and 4 is estimated in the form

of log-deviations from the long-run steady-state. In other words, the model is not designed to

explain long-run trends and seasonal fluctuations in macro-economic variables, but rather is

focused on the business cycle frequency features of the main macro aggregates. Empirical data

series are therefore required to undergo a certain treatment before being used in evaluation of

the model’s likelihood function. This section also includes a discussion of the priors choice for

the model’s structural parameters.

The likelihood function of the EP DSGE model, which includes 24 domestic and 13 euro

area endogenous variables, is based on 14 data series, including 3 series that describe most

important euro area macro-economic indicators. The data series used for model estimation are

shown on Figures 2 to 7. All empirical variables are quarterly, covering the time interval from

1995Q1 to 2007Q4, thus giving 52 observations per data series. All series pertaining to the

domestic economy are sourced from Eurostat’s database29, euro area series are taken from the

AWM database, refer to Fagan et. al. (2005).

As mentioned previously, since the theoretical model is not designed to pick up seasonal

fluctuations in the macro-economic data, all seasonal features of the series are removed using

27See Dynare’s homepage at www.cepremap.cnrs.fr/dynare and Juillard (2004).

28In Dynare this procedure is implemented using the option mode compute=6. It takes 350000 iterations in

order to obtain the reasonable results. In fact, the complexity of the model prevented the widely used Sims’

optimizer to find acceptable starting points for the Metropolis–Hastings sampling. Because of the random nature

of Sims’ optimizer, it stopped at different modes each time, which in most cases where not the actual modes.

Using the simulation–based procedure to compute the modes helped to improve the situation, although some

minor problems remained. This is also clear from a visual inspection of the posterior kernel surface graphs

(available upon request) for each parameter, given that all other parameters are fixed at their modes. Those

graphs show that for some parameters the mode computed is not at the minimum of the objective function,

and for some parameters the surface is so flat that it is difficult to make out if the picked mode is really at the

minimum. Another advantage of using mode compute=6 is that it automatically tunes mh jscale option (the

option which sets the scale parameters of the covariance matrix of the proposal distribution) in order to have

the acceptance ratio of the Metropolis–Hastings algorithm at around 30%.

29Refer to ec.europa.eu/eurostat page.

25

Page 26

either TRAMO/SEATS software package30or, as in the case of inflation series, year-on-year

changes in the associated nominal price variables.

Given below is a detailed list of individual data series used in evaluation of the model’s

likelihood function:

• Upper left part of Figure 2 shows linearly de-trended real per capita output variable ? yt,

real per capita government consumption and real per capita trade balance;

constructed as the sum of real per capita private consumption, real per capita investment,

• Lower left part of Figure 2 shows linearly de-trended real per capita private consumption

variable ? ctbased on the national accounts statistics;

as employment share of the total working age population, defined as 15 to 74 years old,

because no statistics on the actual hours of work is available;

• Upper left part of Figure 3 displays de-meaned labour supply variable?ltwhich is computed

• Lower left part of Figure 3 depicts linearly de-trended per capita real wage variable ?

deflator;

wpt

calculated using statistics on nominal wage net of social security contributions and GDP

• Upper left part of Figure 4 shows linearly de-trended real per capita investment series?it

• Lower left part of Figure 4 depicts linearly de-trended real per capita export series ? xt

• Upper left part of Figure 5 displays linearly de-trended real per capita import series ? mt

• Lower left part of Figure 5 shows linearly de-trended euro area real output variable ? yEA

• Four panels of Figure 6 show de-meaned quarterly inflation rate variables: ? πd

in export deflator series, ? πm

three are sourced from Eurostat’s database, the last is from the AWM database;

based on the national accounts statistics;

based on the national accounts statistics;

based on the national accounts statistics;

t

sourced from the AWM database;

tcalculated

using year-on-year change in GDP deflator series, ? πx

and ? πEA

• Finally, two panels of Figure 7 display de-meaned nominal quarterly interest rates: ? rn

interest rates on government obligations is available, ? rEA

tcalculated using year-on-year change

tcalculated using year-on-year change in import deflator series

calculated using year-on-year change in euro area GDP deflator series. First

t

t

is given by 3 months average deposit rate in Estonia, since no statistics on short-term

t

variable sourced from the AWM

database.

30Developed by Victor G´ omez and Agust´ ın Maravall at the Bank of Spain, refer to www.bde.es for details.

26

Page 27

9.6

9.8

10.0

10.2

10.4

10.6

10.8

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Log per capita consumption

Log per capita output

-.06

-.04

-.02

.00

.02

.04

.06

.08

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Detrended consumption

Detrended output

9.0

9.2

9.4

9.6

9.8

10.0

10.2

95 96 97 98 99 00 01 02 03 04 05 06 07 08

-.10

-.05

.00

.05

.10

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Figure 2: Real per capita output and consumption in Estonia, 1995Q1 to 2007Q4

.52

.54

.56

.58

.60

.62

.64

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Employment share of working age population (15-74)

-.06

-.04

-.02

.00

.02

.04

.06

.08

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Labour supply

-.10

-.05

.00

.05

.10

.15

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Detrended real wage

8.6

8.8

9.0

9.2

9.4

9.6

9.8

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Log per capita real wage

Figure 3: Labour supply and real per capita wage in Estonia, 1995Q1 to 2007Q4

27

Page 28

8.0

8.4

8.8

9.2

9.6

10.0

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Log per capita export

Log per capita investment

-.2

-.1

.0

.1

.2

.3

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Detrended export

Detrended investment

9.0

9.2

9.4

9.6

9.8

10.0

10.2

10.4

10.6

95 96 97 98 99 00 01 02 03 04 05 06 07 08

-.16

-.12

-.08

-.04

.00

.04

.08

.12

.16

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Figure 4: Real per capita investment and export in Estonia, 1995Q1 to 2007Q4

9.2

9.4

9.6

9.8

10.0

10.2

10.4

10.6

10.8

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Log per EA output

Log per capita import

-.15

-.10

-.05

.00

.05

.10

.15

.20

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Detrended EA output

Detrended import

2.28

2.30

2.32

2.34

2.36

2.38

2.40

2.42

2.44

95 96 97 98 99 00 01 02 03 04 05 06 07 08

-.02

-.01

.00

.01

.02

.03

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Figure 5: Real per capita import in Estonia and real per capita output in the EA, 1995Q1 to

2007Q4

28

Page 29

-.04

-.02

.00

.02

.04

.06

.08

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Demeaned import inflation, quarterly rate

Demeaned domestic inflation, quarterly rate

-.03

-.02

-.01

.00

.01

.02

.03

.04

.05

.06

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Demeaned EA inflation, quarterly rate

Demeaned export inflation, quarterly rate

-.02

-.01

.00

.01

.02

.03

.04

.05

95 96 97 98 99 00 01 02 03 04 05 06 07 08

-.003

-.002

-.001

.000

.001

.002

.003

.004

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Figure 6: Quarterly domestic, export and import inflation rates in Estonia, and the EA do-

mestic quarterly inflation rate, 1995Q1 to 2007Q4

-.010

-.005

.000

.005

.010

.015

.020

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Demeaned nominal 3 month interest rate

-.010

-.005

.000

.005

.010

.015

.020

95 96 97 98 99 00 01 02 03 04 05 06 07 08

Demeaned EA nominal 3 month interest rate

Figure 7: Nominal quarterly interest rates in Estonia and in the EA, 1995Q1 to 2007Q4

29

Page 30

The prior distributions and associated hyper-parameters are selected according to Adolfson

et. al. (2005) for the Estonian part of the model; refer to the first part of Table 2 for a detailed

list of the prior distributions and their associated hyper-parameters. Recall that the euro area

part is not estimated in this version of EP DSGE model, and all corresponding parameters

are fixed at the values reported in Smets and Wouters (2003). Effectively, degenerate priors

have been imposed on the euro area parameters of the model. In addition, a number of steady-

state ratios and deep structural parameters for which a good reference value is available on the

theoretical grounds are fixed during the estimation. These parameters are listed in Table 1.

Among values in the table, steady-state ratios are calibrated to sample averages, the capital–

output ratio α is taken from Ratto et. al. (2008), the discount rate β and the wage mark up

parameter λwfrom Smets and Wouters (2003), and the remaining parameters are selected to

have empirically plausible steady-state values for import–output ratio.31

Table 1: Values of fixed parameters

Steady state ratios

C/Y

INV/Y

G/Y

IMP/Y

EXP/Y

C/IMP

INV/IMP

G/IMP

EXP/IMP

YEA/Y

YEA/EXP

Other parameters

Capital output ratio

Capital depreciation rate

Intertemp. discount factor

Wage mark up

Return on capital

Risk free interest rate

Relative price exp./EA

Share import in consump.

El. substit. goods consump/EA

Price mark up imp.

Relative price CPI/dom.

0.55

0.29

0.24

0.85

0.77

0.65

0.34

0.28

0.92

2

2.6

α

0.46

0.025

0.9875

0.5

0.0378

0.0126

1

0.5

5

1.25

1.11

δ

β

λw

Rk= (1 − β + βδ)/β

R = 1/β

γc,x

αc

ηF,c

γF,c= ηF,c/(ηF,c− 1)

γc= [1 − αc(γF,c)1−ηc]1/(1−ηc)

6Empirical results

6.1Posterior distributions of the parameters

Figure 8 depicts three diagnostic graphs for the Metropolis–Hastings sampling algorithm runs

that are used to compute posterior inference in Table 2, refer to Brooks and Gelman (1998).

Given the recursive nature of the algorithm, it is required that the runs display as stable

31We tried to estimate the model changing the gross mark up on imported goods up to a value of 6 (which

means a mark up of 500 per cent), and this does not affect results. We decided for 1.25 because 25 per cent

seems a reasonable mark up and because it is usually found in other estimation (See Adolfson et al. (2005) and

(2007)) that the mark up is higher in the importing sector, because the price elasticity of demand for good j is

lower and firms can exploit this lower willingness to switch from a good to another one and fix a higher mark

up. We also fixed the mark up for the domestic producers at 20 per cent, implied by an elasticity of 6 (See

Dabuˇ sinskas and Kulikov (2007)). We did not estimate neither that mark up (or elasticity), nor any other mark

up, because they created a lot of problem in terms of convergence. This is left for future development.

30

Page 31

behavior as possible. Moreover, it is important that the convergence measures computed within

and between the chains converge. Figure 8 shows that an overall converges is reached after

about 200000 draws. As for individual parameters, results are in general satisfactory, although

in some cases the diagnostic measures do not appear to be sufficiently stable, especially the

ones based on the third moment. But in all cases converges is achieved after about 200000

draws.32

Table 2 reports summary statistics of prior and posterior distributions for all estimated

parameters of EP DSGE model. The last two columns of the table show 90% confidence

intervals; most of the estimated parameters appear to be statistically different from zero based

on this measure.33In the rest of this subsection the parameters in Table 2 are discussed in an

order corresponding to their perceived economic significance.

The parameter φfa that enters the country specific risk premium function Ω(FAt,?risk

in (1) has the mean of estimated posterior distribution equal to 0.0088. This value appears to

be relatively low in comparison to some previously reported estimates in the literature. For

example, in Adolfson et al. (2005) the mode of this parameter is 0.14 in their benchmark model

for the euro area.34Adolfson et al. (2007) report even higher value for Finland at around 0.3,

unaffected by various assumptions about structural shocks and when the UIP equation in their

model is modified to include dependence on the exchange rate. The relatively low estimate

of φfain Table 2 may indicate that the net foreign asset position of Estonia is not a good

explanatory factor for the observed interest rate spread. This is corroborated by the fact

that the idiosyncratic component of Ω(FAt,?risk

t

posterior mean of ρriskequal to 0.9562, suggesting that ?risk

premium variation in the data. This can be partly explained by data issues: Figure 7 clearly

indicates presence of two pronounced interest rate spikes in Estonian interest rates in the second

half of 1990-s induced by the Asian and Russian financial crises. These events have coincided

with substantial structural shifts in Estonian banking sector, and a dramatic reduction in the

interest rate spread in the following years. The specification of Ω(FAt,?risk

might be too simple to pick up these changes.

Empirical Calvo parameters reported in Table 2 carry information about the timing of price

and wage setting decisions by the domestic firms and households. The posterior mean of price

stickiness parameter θ of the domestic intermediate goods producers is given by 0.7080, implying

t

)

) is estimated to be highly persistent, with the

captures a high share of the risk

t

t

) function in (1)

32Diagnostic graphs for the individual parameters are available separately on request.

33Note that the standard deviations shown in Table 2 are computed prior to running the Metropolis–Hastings

posterior density sampling algorithm. They are based on the quadratic approximation using Hessian evaluated

at the posterior kernel mode, and therefore are not fully reliable for two reasons. Firstly, the mode of the final

posterior distribution may be different from the one obtained by maximizing the posterior kernel, although it

should not be the case in a good estimation. Secondly, even if the two modes coincide, the standard deviations

are based on the normality assumption, which is not necessary satisfied by the posterior distribution.

34Their estimations reveal that φfa is not robust to different specification of the model. In particular, one of

their models estimated assuming i.i.d. mark-up shocks similar to the specification in this paper, has φfa equal

to 0.035. This parameter needs additional robustness checks in the future versions of EP DSGE model.

31

Page 32

a price duration around 3.5 quarters. This is in line with the recent finings by Dabuˇ sinskas and

Kulikov (2007), who report price duration around 4 quarters (Calvo parameter of 0.75) from

a similarly specified New Keynesian Phillips Curve for Estonia. On the other hand, estimated

values of Calvo parameters are usually higher for the euro area, see Smets and Wouters (2003),

Adolfson et al. (2005) and Gal` ı and Gerlter (1999).

Turning to the export–import sector, it is worth noting that the corresponding price sticki-

ness parameters are higher in both cases than θ for the domestic intermediate goods producers.

The implied price duration ranges from over 8 quarters in the import sector to over 5 quarters

in the export sector. On the other hand, Adolfson et al. (2005) report lower price stickiness in

the export–import sector relative to the domestic one in their euro area model.

The posterior mean of the wage stickiness parameter θwis remarkably low in comparison

with the price stickiness coefficients. Implied average duration of a wage contract is below 3

quarters. Both Smets and Wouters (2003) and Adolfson et al. (2005) also find lower degree

of wage stickiness relative to the price stickiness for the euro area, although their wage Calvo

parameters are somewhat higher than the one obtained for Estonia.

The next set of parameters in Table 2 is related to price and wage indexation, see equa-

tions (17), (22), (26), and (27). These parameters are directly linked to the weights of forward

and backward looking components in the corresponding Phillips curves and the real wage equa-

tion. The posterior mean of the domestic price indexation coefficient τπis estimated at 0.8985,

giving the weight of the forward looking component in (25) at 0.52 versus 0.48 for the backward

looking one. This result is in line with Dabuˇ sinskas and Kulikov (2007). Empirical indexa-

tion parameters in the export–import sector are slightly lower, with estimated posterior means

around 0.81 for both parameters. There is a high degree of wage indexation to the past inflation

as indicated by the mean of estimated posterior distribution of τw. The implied weights of the

present and past inflation rates in (18) are given by 0.92 and 0.42 respectively.

Another parameter of interest that can be calculated using empirical results in Table 2 links

the real marginal costs to the domestic inflation rate in the Phillips curve equation (25). The

implied value of this parameter computed at the posterior means of θ and τπis equal to 0.066.

This is notably higher than the previously reported coefficient by Dabuˇ sinskas and Kulikov

(2007), but a precise statistical comparison of the two results is infeasible due to differences in

the estimation methodologies.

The remaining parameters in Table 2 are as follows. The posterior distribution of σc, which

is the inverse of intertemporal elasticity of substitution of consumption, is centered around 1.85.

This is higher than the value 1.39 reported by Smets and Wouters (2003) for the euro area,

and implies that Estonian households are less responsive to the variation in the real interest

rate than their european counterparts. At the same time, the posterior mean of external

consumption habit parameter h is higher in Estonia than in the euro area, the respective

values are 0.83 and 0.60. This result may be attributed to the “catching up with Joneses”

effect that can characterize a country with high GDP growth rate. The inverse elasticity of

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work effort with respect to the real wage is governed by the parameter σL, posterior mean

of which is estimated at 1.59, with the value of corresponding elasticity equal to 0.63. This

result is close to the one obtained in Staehr (2008), where he finds that ”1 per cent increase in

after-tax hourly income would lead to 0.6 percentage point more individuals being employed”.

Two parameters linked to the export–import sector in EP DSGE model are the elasticity

of substitution between domestic and imported consumption goods ηc and the elasticity of

substitution between exported and domestic consumption goods in the euro area ηEA. Their

posterior means reported in Table 2 are respectively 2.05 and 1.29, both in line with empirical

results in the literature, see for instance Ratto et al. (2008).35The inverse elasticity of the

investment adjustment cost function parameter ϕ is centered at 7.71. The corresponding elas-

ticity is 0.13, and according to the interpretation of this parameter in Christiano, Eichenbaum

and Evans (2005) this point estimate implies that a 1 per cert permanent change in the price

of capital induces about a 10 per cent change in investment.

Among estimated autoregressive parameters of the structural shocks reported in Table 2,

ρβand ρxstand out as low relative to the benchmark studies for the euro area in Smets and

Wouters (2003) and Adolfson et al. (2005). All other estimated autoregressive coefficients are

in line with the literature, with ρgemerging as the largest among them. Recall that the fiscal

policy in EP DSGE model in this paper is effectively exogenous w.r.t. all other sectors, refer

to (29). A relatively large estimated ρg parameter may indicate that the persistence coming

from other structural disturbances is insufficient to describe the observed dynamics of Estonian

macroeconomic time series.

6.2Fit to the data

The fit of EP DSGE model to the macroeconomic data series for Estonia and euro area described

in section 5 is shown on Figure 9.36Specifically, the real-world data are plotted against the

one-sided Kalman filter predicted values of the corresponding series. The procedure to compute

one step ahead forecast of ˜ ytbased on the partially unobserved state vector ˜ xttogether with

data up to the period t in the Kalman filter representation of DSGE model (36) is described

in detail in Hamilton (1994).

Figure 9 shows that the fit of EP DSGE model to Estonian macroeconomic series related to

the domestic sector is good, with an exception of the nominal interest rate variable. However,

the model can be considered inadequate in reproducing the essential features of the main foreign

sector variables, including the euro area variables. Recall that the first version of EP DSGE

35The typical estimates for the elasticity of substitution between home and foreign goods are around 5 to 20

using the micro data, see the references in Obstfeld and Rogoff (2000). However, the macro data–based estimates

are usually a lot lower, in the range of 1.5 to 2, see e.g. Collard and Dellas (2002).

36There are several other methods that can be used to assess the goodness of fit of DSGE models, refer to

An and Schorfheide (2007). Among them is posterior odds comparison of DSGE model with a bayesian VAR

model, and comparison of the autocorrelation and cross-correlation structure of the real-world and DSGE model-

generated time series. A more thorough model checking exercise is left for the future version of the EP DSGE

model.

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Table 2: Priors and posteriors of the EP DSGE model parameters

ParametersPriors Posteriors

distribution means.d.mean s.d.90% interval

Shocks’ standard deviations

Preference (σβ) Inv. Gamma0.2

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

0.07330.0133 0.04230.1062

Labour supply (σL) Inv. Gamma0.2 0.09850.0455 0.04580.1611

Investment specific (σx) Inv. Gamma0.1 0.02840.0041 0.0195 0.0359

Technology (σa) Inv. Gamma0.4 0.05080.0032 0.04700.0549

Equity premium (σq) Inv. Gamma 0.40.2792 0.08890.1072 0.4681

Government spending (σg)Inv. Gamma 0.30.18550.0337 0.15210.2184

Risk premium (σrisk)Inv. Gamma 0.080.0134 0.00230.01070.0157

Wage (σw)Inv. Gamma 0.250.03380.00260.02940.0383

Price mark-up domestic (σλp)Inv. Gamma0.150.0215 0.00360.01760.0238

Price mark-up imports (σλc,F )

Price mark-up exports (σλc,x)

Inv. Gamma0.40.23820.04560.08940.3585

Inv. Gamma0.40.15940.0222 0.0745 0.1763

Auto-regressive coefficients

Fiscal policy (ρg) Beta0.850.1 0.96950.01250.9504 0.9895

Preference (ρβ) Beta0.85 0.10.5482 0.0561 0.39820.6976

Technology (ρa) Beta 0.850.10.7999 0.02210.6538 0.9449

Investment specific (ρx) Beta0.85 0.1 0.44400.0964 0.28580.6177

Labour supply (ρL)Beta0.85 0.10.8078 0.0372 0.64440.9778

Risk premium (ρrisk)Beta 0.650.10.95620.0225 0.93180.9817

Calvo parameters

Inflation domestic (θ) Beta 0.750.05 0.70800.0360 0.65580.7606

Inflation imports (θc,F) Beta0.50.1 0.87950.03800.83230.9284

Inflation exports (θc,x) Beta0.50.10.81610.0637 0.7574 0.8735

Wage (θw) Beta 0.7 0.050.63110.0184 0.5611 0.6998

Indexation parameters

Inflation domestic (τπ) Beta 0.750.15 0.8985 0.11760.80230.9940

Inflation imports (τc,F)Beta0.50.150.82400.1281 0.71620.9364

Inflation exports (τc,x)Beta 0.5 0.150.80600.10480.7021 0.9152

Wage (τw)Beta 0.750.15 0.82870.10630.66510.9927

Elasticities

Inverse el. of intertemporal subst. (σc) Normal10.375 1.84880.16041.42642.2506

Inverse el. of labour supply (σL)Normal20.751.59170.74270.44622.6772

El. of substitution dom/imp goods (ηc)Inv. Gamma20.12.05080.18331.75402.3576

El. of substitution exp/EA goods (ηEA)Inv. Gamma20.11.28950.14911.19911.3715

Inverse el. of capital utilization (ψ) Normal0.20.0750.16620.0386 0.04980.2799

Inverse el. of investment adj. cost (ϕ)Normal41.57.70531.3986 5.81089.6105

Other parameters

1+share of fix costs (φ) Normal1.450.2 1.19860.31910.74251.6277

Habit (h)Beta0.70.050.82970.02080.78670.8723

Risk premium (φfa)Inv. Gamma0.5

∞

0.00880.00080.00700.0104

Notes: The EP DSGE model parameter name and symbol are shown in the first column. The following three

columns describe the corresponding prior distribution: its type, and the first two moments. The last four columns

summarize the posterior distribution: its mean, and 5% and 95% percentiles are computed using the Metropolis–

Hastings sampler, its standard deviation is a quadratic approximation at the posterior mode.

34

Page 35

model presented in this paper uses calibration for the euro area part of the model based on

Smets and Wouters (2003). It is based on the sample period 1980Q2 to 1999Q4 which only

partially overlaps with the newer data sample used for the EP DSGE model in this paper.

Foreign sector variables in EP DSGE model, such as export, import and the corresponding

prices, depend on the evolution of the main euro area variables, and therefore their fit is likely

to be substantially improved by switching from calibration to estimation of the euro area part

of EP DSGE model in the future.

6.3Impulse Response Functions

The impulse response functions for most important model variables are shown on Figures 14

to 22. These are orthogonalized responses to one standard deviation of all model shocks.37

I want to comment first the responses to a risk premium shock (i.e. an increase in the risk

premium), depicted on Figure 20. This would require to talk about the real exchange rate,

because of the effects on imports and exports. Nevertheless, we do not have any real explicit

exchange rate in the mode, but we have several relative prices which are indirectly related to

it. We derived that relationship in appendix 8.1 and we refer to that, freely referring the the

real exchange rate.

A positive risk premium shock generate a depreciation of the real exchange rate. This

leads to an increase of export (Estonian goods are more competitive abroad) and to a decrease

in import (Euro Area goods are more expensive).38This positive effect on exports together

with the switch of consumption from foreign goods to presumably domestic goods should in-

crease domestic output. Nevertheless, there is another effect of the risk premium shock which

counteracts the fist one, i.e. the implied increase of the domestic interest rate. It depresses

investment and consumption (the latter because of the intertemporal substitution effect) and

the total effect on output is higher that the one coming from exports and imports, generating a

downturn. Employment decreases as a consequence (implying a decrease of wages as well). In

the end, domestic inflation decreases because the standard mechanism governing it at the basis

of the NK models (incorporate in the NK Phillips Curve). If output drops above the potential,

it means that also the marginal cost are below their natural level. Given that mark up is the

inverse of the marginal costs, it is above the potential (or desired) level. Firms wants then to

reduce prices (and hence inflation) to drive back their mark up to their desired level.

The technology shock is presented on Figure 14. The responses are in line with the all

other estimations. A rise in productivity leads to an increase in output, consumption and

investment. In line with what found by Gal` ı 1999 (and confirmed by Smets and Wouters (2005)

and Adolfson et al. (2005) among others), hours worked decrease after a positive technology

37We have chosen to report only those responses to save space. All the other responses are available on request.

38A one standard deviation risk premium shock leads on impact to a response of γEA

of γc

t equal to 0.0005501. The negative response is higher and, given the equation for the real exchange rate,

equation (A.3), the latter increases (i.e. there is a depreciation). This generates the responses of imports and

t

equal -0.001201 and

exports described in the text.

35

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shock. In a model where there is a rule that describe monetary policy, the nominal interest

rate in this situation would decrease (as in Smets and Wouters 2003) and in part compensate

the reduction in marginal costs. Moreover, marginal costs are not recovered enough and then

inflation decreases as well. In our model, interest rate does not react because of domestic

changes but as a consequence of changes in foreign (Euro Area) related variables caused by

domestic development. In fact, the technology shock causes a real exchange rate depreciation39,

and this has the standard consequences on import and export. Interest rate falls because the

net foreign assets increase, due to the excess of exports on imports.

Another interesting shock is the labour supply one, it is shown on Figure 16. Its implications

are very similar to the technology shock ones. Nevertheless, it is worth noting that now hours

worked increase and as a consequence wages drop, forcing domestic inflation down (via the effect

on marginal costs). The fall in inflation affects consumption which in turns stimulates output

and investments. The increase in domestic output together with the reduction in inflation

stimulate exports and reduce imports, causing the net freing assets to increase. Although an

initial increase in imports, the effect on the interest rate is always negative and very low.

A positive wage mark up is supposed to have completely reverse effects. In fact after

such a shock wages increase.40This reduction transmit to both labour and domestic inflation,

decreasing the former and increasing the latter. Both those consequences generate a fall in

consumption and then on output and in turns to investments. In addition, there is a real

exchange appreciation (due the increase in domestic inflation) which causes exports to fall and

imports to goes up. The positive effect on the interest rate due to the negative net foreign

asset position is observed.

The most part of the reaming shocks are quite standard and their effects are by now

extensively described in many paper. Moreover, the description above for the other shocks

make clear which are the mechanism underlying the model and make the interpretation easy.

Hence we leave it to the reader.

We want instead to highlight the effects of the two Euro Area monetary shocks.

The interest rate in Estonia is related one to one to the Euro Area interest rate as described

by the UIP equation, with the net foreign asset positions and with the stochastic part. A

positive interest rate shock in the Euro Area has the well know effects in that area on output

and inflation. They are the same affect that can be found in Smets and Wouters (2003).

But what happens in Estonia? There are two effects which have a direct negative impact on

the Estonian output. On the one hand the domestic interest rate increases. On the other

39Again, the depreciation of the real exchange rate id due to the contemporaries variation of γEA

move in the opposite direction but the negative movement of γEA

of γc

40This increases in wages may be explained in different ways. If the mark up shock is seen as an increased

t

and γc

t. They

t

(-0.007599) dominates the positive movement

t(0.00348), hence ret increases

power of the Trade Unions, wages are higher because they have more bargaining power and then they can

contract a higher wage. As a result, and as prescribed by the theories about the relationship between wages

and Trade Unions power, there is a negative effect on employment. Contrary, if we consider that shock as an

increased workers’ preference for leisure. In this case wages drop because there is a drop in the labour supply.

36

Page 37

hand exports fall, because of the decrease of the Euro Area output. The domestic downturn

depress investments and has a negative effect on employment, which in turns negatively affects

consumption and wages. The net negative effect of exports and imports (the latter decrease as

a consequence of the domestic drop in consumption, but less then exports) on the net foreign

asset position justifies the extra positive response of the interest rate.41

As for the inflation objective shock, there are many mechanisms in place. An increase in the

inflation target has the first effects on inflation, which increases as well. A direct consequence

is the increase of the Euro Area interest rate. Nevertheless the Euro Area output does not

decrease, but increases. Why? Let’s see what happens in Estonia. Interest rate goes up, but

again that does not cause any reduction in output investment and consumption. The reason

lays in the fact that the effect on domestic inflation is higher that the effects on the inflation

of the imported goods, and this shifts the domestic consumption towards the imported goods.

This is the explanation of the increase in the output in the Euro Area. In addition, the negative

effect of the increased interest rate in Estonia on consumption is mode the compensated by

the increase in Estonia Exports due to the increase in the Euro Area change in output. The

positive effect on consumption is then the cause of the expansion in the estonia economy, with

the consequences of increasing hours worked and wages (there is more demand for workers form

the firms which are growing)

6.4 Variance Decomposition

The relative importance of structural shocks for the dynamics of state variables in EP DSGE is

measured by the share of total variation that a particular shock helps to explain for each given

state variable of the model. The variance decomposition methodology is standard in time series

analysis, refer to Hamilton (1994) pp. 323–324, and can be applied to DSGE models written

in the vector autoregressive form (36).

In this subsection the variance decomposition of EP DSGE model is carried for the full

two–area model as well as for the Estonian economy part only. The separate decomposition

for Estonia is needed because the euro area structural shocks dominate the variability of the

model’s state variables, making it difficult to assess the relative importance of Estonian shocks

for the Estonian economy.42The variance decomposition for Estonia is computed by setting the

standard deviations of the euro area structural shocks to zero. In addition to 10 standard euro

area shocks, the risk premium innovation ?risk

t

is also shut down in the variance decomposition

for Estonia, although this shock cannot be considered as bona fide outside shock for the Estonian

economy.43

41Without this effect the domestic interest rate would have increased one by one with the Euro Area interest

rate.

42Recall that the euro area part of the EP DSGE model is calibrated, including the standard errors of the

corresponding structural shocks. The latter are notably higher than the estimated standard errors of the struc-

tural shocks for the Estonian economy part of the model, leading to an imbalance in the variance decomposition

results for the full model.

43The variance decomposition in Table3 is not substantially affected by the choice to shout down the

37

Page 38

Table 3 presents the variance decomposition results for the Estonian economy part of EP

DSGE model.44It is possible to see that the most important shock is the domestic price mark

up shock, which in some cases explains almost 50% of state variables dynamics. This result

is somewhat unexpected, as this shock is usually found to be of a secondary importance in

explaining the variance of endogenous variables, apart inflation and wages, even at a very short

time horizon. The second most important shock, as is normal in the literature, is the technology

shock. It explains about the 30% of the variance of many state variables. The remaining shocks

have a marginal contribution to the overall variability of the model, apart from direct effects

on the variables that they are linked to. Notably, the wage mark shock, although not very

dominant, appears to be more relevant that usually found in literature, see Smets and Wouters

(2003).

Table 4 reports the variance decomposition for the full model that includes Estonian and

euro area part with a full complement of 21 structural shocks. Ideally, one can divide the table

into four quadrants corresponding to the two parts of the model in the vertical dimension, and

two sets of structural shocks in the horizontal one. Contribution of the Estonian shocks to

the Estonian part of the model appears in the upper left quadrant. It is easy to see, that the

importance of these shocks is considerably smaller in the full model than in Table 3, although

the structure of their relative contribution to the variation of domestic endogenous variables

remains the same. The lower left quadrant of the table shows that Estonian shocks have no

effects on the euro area state variables. The lower right quadrant is almost entirely similar to

the variance decomposition reported in Smets and Wouters (2003). Finally, the upper right

quadrant of Table 4 shows that the most important euro area shocks for the Estonian part of

the model are the labour supply shock, the interest rate shock, and the technology shock.

idiosyncratic risk premium component. When this shock is present, it serves to decrease the effect of the

technology and the domestic mark up shocks in explaining the variability of the states, without dominating the

overall picture.

44All results in this subsection are based on 20 lags approximation to the unconditional variance–covariance

matrix of the vector of state variables.

38

Page 39

Table 3: Variance decomposition for the Estonian economy part of EP DSGE (in per cent)

States Shocks

ua

ug

uβ

ux

ul

uw

uλp

uλc,F

uλx

uq

y39.055.630.49 3.171.78 7.8740.04 0.051.630.28

c33.863.17 16.742.031.466.6835.65 0.060.26 0.1

inv 35.613.25 3.8727.32 2.366.16 19.010.05 0.342.04

r 30.59 0.561.680.76 0.827.05 58.020.1 0.38 0.04

rn

πD

πc

31.210.553.06 0.850.89 7.02 55.960.150.27 0.04

21.26 0.390.35 0.42 0.475.09 71.660.030.310.02

21.150.380.35 0.42 0.475.0671.27 0.570.3 0.02

k40.85 5.215.01 21.193.196.61 16.26 0.070.39 1.21

l 31.847.640.674.36 2.8113.1337.010.072.07 0.38

q10.33 0.230.98 0.53 0.35 2.1715.080.02 0.0770.23

mc52.97 0.410.40.34 0.4214.64 30.560.010.220.03

w

rk

33.450.220.670.15 0.81 26.6537.87 0.02 0.150.01

34.86 3.560.63.261.28 7.49 47.470.05 1.180.26

fa 31.210.553.06 0.85 0.897.0255.96 0.150.27 0.04

πc,F

mcc,F

γc,x

γc,F

πc,x

mcc,x

γc

γEA

0000000 10000

0000000 10000

35.171.22 1.26 1.41.257.5945.89 0.175.980.07

28.48 0.620.65 0.740.766.56 60.80.86 0.510.04

30.220.890.68 0.96 0.936.7146.810.0912.650.05

23.350.410.320.44 0.55.56 64.88 0.024.49 0.03

28.540.620.650.74 0.766.5760.940.620.51 0.04

28.7 0.620.65 0.740.776.61 61.29 0.07 0.51 0.04

Exp35.171.221.26 1.41.257.5945.89 0.17 5.980.07

m24.480.56 8.010.53 0.58 5.72 58.93 0.750.4 0.03

39

Page 40

Table 4: Variance decomposition for the full EP DSGE model (in per cent)

States

Shocks

ua

ug

uβ

ux

ul

uw

uλp

urisk

uλc,F

uλx

uq

uq,EA

uλp,EA

uw,EA

urn,EA

uβ,EA

ua,EA

ux,EA

ul,EA

uπ,EA

ug,EA

y

0.14

0.02

0

0.01

0.01

0.03

0.14

0

0

0.01

0

0.01

7.58

0.36

11.36

16.13

7.01

0.53

56.52

0.03

0.12

c

0.35

0.03

0.17

0.02

0.02

0.07

0.37

0.11

0

0

0

0

1.58

0.21

13.32

1

12.9

0.61

69.2

0.03

0.01

inv

1.11

0.1

0.12

0.85

0.07

0.19

0.59

0.16

0

0.01

0.06

0

1.86

0.39

12.02

1.32

10.16

2.7

68.25

0.03

0.01

r

0.1

0

0.01

0

0

0.02

0.19

0.1

0

0

0

0

2.55

0.22

14.26

1.44

13.79

0.91

66.35

0.02

0.02

rn

0.12

0

0.01

0

0

0.03

0.21

0.27

0

0

0

0

1.53

0.09

11.82

5.27

29.64

1.26

49.7

0.01

0.02

πD

0.11

0

0

0

0

0.03

0.38

0.01

0

0

0

0

3.35

0.45

13.5

10.18

5.94

0.66

65.31

0.04

0.02

πc

0.09

0

0

0

0

0.02

0.3

0.01

0

0

0

0

4.99

0.7

14.16

9.54

4.39

0.54

65.19

0.05

0.02

k

1.71

0.22

0.21

0.89

0.13

0.28

0.68

0.24

0

0.02

0.05

0

1.73

0.56

11.76

1.84

8.22

5.45

65.96

0.03

0.03

l

0.1

0.02

0

0.01

0.01

0.04

0.11

0

0

0.01

0

0.01

8.16

0.36

11.16

17.21

6.88

0.53

55.21

0.03

0.14

q

0.18

0

0.02

0.01

0.01

0.04

0.27

0.1

0

0

1.24

0

1.8

0.2

13.4

1.02

13.14

0.59

67.94

0.02

0.01

mc

0.63

0

0

0

0

0.17

0.36

0.01

0

0

0

0.01

4.76

0.34

12.21

10.92

7.8

0.49

62.2

0.03

0.05

w

0.42

0

0.01

0

0.01

0.34

0.48

0.01

0

0

0

0

3.91

0.33

12.52

9.27

8.01

0.5

64.11

0.03

0.04

rk

0.14

0.01

0

0.01

0.01

0.03

0.19

0.01

0

0

0

0.01

6.84

0.34

11.57

14.9

7.4

0.46

57.94

0.03

0.1

fa

0.27

0

0.03

0.01

0.01

0.06

0.49

0.11

0

0

0

0.02

4.29

0.28

10.9

21.7

6.49

0.89

54.29

0.03

0.13

πc,F

0

0

0

0

0

0

0

0

0.05

0

0

0

12.2

2.09

15.21

6.33

1.37

0.04

62.65

0.07

0

mcc,F

0

0

0

0

0

0

0

0

0.04

0

0

0

59.44

2.21

4.23

3.33

2.61

0.01

28.13

0.02

0

γx,c

0.06

0

0

0

0

0.01

0.08

0.01

0

0.01

0

0

13.86

0.28

11.7

8.62

15.68

2.33

47.27

0.03

0.03

γF,c

0.08

0

0

0

0

0.02

0.18

0.01

0

0

0

0

1.68

0.3

12.5

10.81

7.42

0.97

65.96

0.03

0.02

πc,x

0.04

0

0

0

0

0.01

0.07

0.01

0

0.02

0

0

2.63

0.49

15.46

9.74

4.25

0.82

66.38

0.05

0.02

mcc,x

0.15

0

0

0

0

0.04

0.42

0.01

0

0.03

0

0

3.97

0.4

12.04

10.62

7.21

0.52

64.52

0.03

0.03

γc

0.08

0

0

0

0

0.02

0.18

0.01

0

0

0

0

1.7

0.28

12.64

10.82

7.25

0.96

65.98

0.04

0.02

γEA

0.11

0

0

0

0

0.03

0.24

0.01

0

0

0

0

4.16

0.09

11.72

10.26

12.96

1.34

59

0.03

0.03

Exp

0.05

0

0

0

0

0.01

0.07

0.01

0

0.01

0

0.02

9.27

0.42

10.3

20.78

5.97

0.71

52.21

0.02

0.16

m

0.05

0

0.02

0

0

0.01

0.11

0.02

0

0

0

0

1.76

0.29

12.37

8.55

8.61

0.63

67.52

0.03

0.02

cEA

0

0

0

0

0

0

0

0

0

0

0

0

0.39

0.06

5.59

48.13

7.23

8.2

30.36

0.01

0.03

invEA

0

0

0

0

0

0

0

0

0

0

0

0.05

1

0.12

9.93

2.33

9.03

31.12

46.33

0.03

0.05

qEA

0

0

0

0

0

0

0

0

0

0

0

6.42

0.51

0.19

9.24

4.09

11.59

8.69

59.22

0.01

0.03

rk,EA

0

0

0

0

0

0

0

0

0

0

0

0.01

1.85

5.21

9.04

9.87

7.68

2.18

64.12

0.02

0.03

rn,EA

0

0

0

0

0

0

0

0

0

0

0

0.01

2.01

0.26

7.71

37.86

35.21

4.36

12.55

0.01

0.02

πEA

0

0

0

0

0

0

0

0

0

0

0

0

53.04

1.59

6.96

3.55

1.59

0.02

33.22

0.03

0

kEA

0

0

0

0

0

0

0

0

0

0

0

0.02

1.09

0.18

10.71

2.24

8.24

32.43

45

0.03

0.05

wpEA

0

0

0

0

0

0

0

0

0

0

0

0

3.49

24.05

3.99

8.23

0.91

0.32

58.99

0.01

0

lEA

0

0

0

0

0

0

0

0

0

0

0

0.01

0.88

1.71

9.63

15.11

29.38

3.19

39.93

0.02

0.13

yEA

0

0

0

0

0

0

0

0

0

0

0

0.01

0.98

0.13

10.68

17.13

12.14

2.29

56.5

0.03

0.11

yP,EA

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1.46

35.18

1.29

62.03

0

0.04

cf,EA

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

6.48

34.52

6.6

52.35

0

0.05

invf,EA

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

13.74

22.86

13.21

50.14

0

0.06

qf,EA

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

30.45

29.47

2.92

37.14

0

0.03

rk,f,EA

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

3.84

32.23

3.84

60.07

0

0.02

rn,f,EA

0

0

0

0

0

0

0

0

0

0

0

0

0

0

3.14

33.5

28.02

1.48

33.81

0.03

0.03

πf,EA

0

0

0

0

0

0

0

0

0

0

0

0

0

0

10.48

33.85

25.7

0.54

29.39

0.03

0.02

kf,EA

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

11.99

21.37

13.27

53.29

0

0.07

wf,EA

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.07

98.85

0.07

1.02

0

0

lf,EA

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1.96

19.29

1.76

76.95

0

0.04

40

Page 41

7Conclusion

This paper presents a detailed theoretical structure and the set of empirical results for the

first version of EP DSGE model. The latter is a two area DSGE model specifically designed to

match the essential characteristics of Estonian economy: the currency board regime, free capital

mobility, and dependence on the outside economic environment via foreign trade. These are

typical features of a small open economy in the vicinity of a much bigger economic zone. The EP

DSGE model consists of two interlinked parts, the domestic economy part describing Estonian

economy, and the euro area part acting as a large outside closed economy with monetary policy

and trade links to the first part. The Estonian economy part has 24 state variables and 11

structural shocks, while the euro area part consists of 13 state variables and 10 structural

shocks.

The first version of EP DSGE model focuses on the business cycle frequency fluctuations

of the main Estonian macroeconomic aggregates, leaving their long run trends aside. The

future developments of the model are likely to incorporate the long run dynamics as well,

considering that Estonia is still subject to effects of real and nominal convergence stemming

from its catch-up with the developed euro area economies. Other areas of the future theoretical

developments of the model include incorporation of the financial sector together with relevant

frictions, adding the housing sector combined with collateral-constrated type of households,

and expanding the government sector part of the model.

Empirical part of this paper reports Bayesian estimation results for model’s structural

parameters, impulse response functions and variance decomposition of the state variables. Out

of 52 structural parameters in EP DSGE, 34 are estimated using a data sample consisting of 14

macroeconomic series for Estonia and the euro area. Statistical estimates of the main structural

parameters are largely in line with previous studies for Estonia, when a direct comparison can

be made. It is also worth mentioning that the net foreign asset position of Estonia has been

found an important and statistically significant factor in explaining the country risk premium

in UIP equation for the interest rates, but the results suggest that other explanatory factors

need to be considered as well.

The empirical relevance of structural shocks is assessed using the variance decomposition.

It has been found that the most important domestic shocks in explaining the variability of

Estonian macroeconomic series are the price mark up shock, that often dominates the other

shocks contributing 50% or more of the variability of the state variables, and the technology

shock. The euro area shocks also play a very significant role in driving the dynamics of Estonian

macroeconomic aggregates. Among the most prominent euro area shocks that affect Estonia

are the labor supply, the interest rate and the technology shock.

41

Page 42

12345678

5

x 10

4

6

8

10

12

Interval

12345678

5

x 10

0

5

10

15

20

m2

12345678

5

x 10

0

50

100

150

m3

Figure 8: The red and blue lines represent specific measures of the parameter vector both

within and between chains. For the results to be sensible, they should be relatively constant

(although there will always be some variation) and should converge. Dynare reports three

measures: “interval”, being constructed from an 80% confidence interval around the parameter

mean, “m2”, being a measure of the variance and “m3” based on third moment. In each case,

Dynare reports both within and between chains measures. The overall convergence picture

presents results of the same nature, except that they reflect an aggregate measure based on the

eigenvalues of the variance-covariance matrix of each parameter

42

Page 43

0 50

!0.2

0

0.2

con()*p,ion

0 50

!0.05

0

0.05

no*in.l in,. r.,2

0 50

!0.5

0

0.5

inv2(,*2n,(

0 50

!0.5

0

0.5

l.4o)r

050

!0.1

0

0.1

do*2(,ic in7l.,ion

0 50

!0.2

0

0.2

o),p),

0 50

!0.2

0

0.2

r2.l w.92

0 50

!0.05

0

0.05

2xpor, in7l.,ion

0 50

!0.05

0

0.05

i*por, in7l.,ion

0 50

!1

0

1

2xpor,(

0 50

!0.2

0

0.2

i*por,(

0 50

!0.1

0

0.1

E< no*. in,. r.,2

0 50

!0.2

0

0.2

E< in7l.,ion

0 50

!0.2

0

0.2

E< o),p),

Figure 9: Sample data (blue solid line) and one-sided predicted values (black dotted line)

43

Page 44

0 0.51

0

10

20

SE_ub

0 0.51

0

5

10

15

SE_ul

0 0.2 0.4

0

20

40

60

80

SE_ux

0.51 1.52

0

50

100

150

SE_ua

012

0

1

2

3

4

SE_uq

0.51 1.5

0

5

10

15

20

SE_ug

0.1 0.20.3 0.4

0

100

200

SE_urisk

0.2 0.4 0.6 0.8 1 1.2 1.4

0

50

100

150

SE_uw

0.2 0.4 0.60.8

0

50

100

150

SE_ulambdapi

Figure 10: The prior (light colored line) and posterior (dark colored line) distributions; green

vertical line is the posterior mode obtained by the posterior kernel maximization

44

Page 45

012

0

2

4

6

SE_ulambdacf

012

0

5

10

SE_ulambdax

0510 15

0

0.1

0.2

0.3

psi

012

0

0.5

1

1.5

phi

0 0.20.4

0

2

4

FI

0123

0

0.5

1

1.5

sigmac

0246

0

0.2

0.4

0.6

sigmal

0.50.6 0.7 0.80.9

0

5

10

theta

0.60.81

0

10

20

30

rhog

Figure 11: The prior (light colored line) and posterior (dark colored line) distributions; green

vertical line is the posterior mode obtained by the posterior kernel maximization

45

Page 46

0 0.51

0

2

4

rhob

0.40.6 0.81

0

2

4

rhoa

0 0.51

0

2

4

rhox

0.2 0.4 0.6 0.81 1.2

0

2

4

rhol

0.40.6 0.81

0

10

20

rhorisk

0.6 0.70.80.9

0

5

10

15

h

0 0.51

0

1

2

3

4

gammaw

0.40.6 0.81

0

2

4

6

gammapi

0.4 0.60.8

0

5

10

cw

Figure 12: The prior (light colored line) and posterior (dark colored line) distributions; green

vertical line is the posterior mode obtained by the posterior kernel maximization

46

Page 47

0.2 0.4 0.6 0.81

0

2

4

6

taucf

0.40.6 0.81

0

5

10

thetacf

0.2 0.4 0.6 0.81

0

2

4

6

taux

0.40.60.81

0

5

10

thetax

0.05 0.1 0.15 0.2 0.25

0

100

200

300

400

phifa

1.52 2.5

0

2

4

6

etaEA

123

0

0.5

1

1.5

2

etac

Figure 13: The prior (light colored line) and posterior (dark colored line) distributions; green

vertical line is the posterior mode obtained by the posterior kernel maximization

47

Page 48

5 10 15 20

0

0.01

0.02

0.03

c

5 101520

!8

!6

!4

!2

0

2x 10

!3

rn

5 10 1520

0

0.05

0.1

0.15

0.2

inv

5 1015 20

!0.04

!0.02

0

0.02

0.04

l

5 1015 20

!10

!5

0

5

x 10

!3

pid

5 10 1520

0

0.05

0.1

y

5 1015 20

0

0.02

0.04

0.06

wp

5 10 1520

!4

!2

0

2

4

x 10

!3

pix

5 10 15 20

!0.02

0

0.02

0.04

Expo

5 1015 20

!0.01

0

0.01

0.02

Imp

Figure 14: Model variables response to one standard deviation technology shock measured in

percentage deviation from the steady state

48

Page 49

5 1015 20

0

0.005

0.01

0.015

0.02

c

5 1015 20

0

1

2

3

4

x 10

!3

rn

5 101520

!0.08

!0.06

!0.04

!0.02

inv

5 101520

!15

!10

!5

0

x 10

!3

l

5 1015 20

!2

!1

0

1

x 10

!3

pid

5 101520

!15

!10

!5

0

x 10

!3

y

510 1520

!4

!2

0

2

4

6

8

x 10

!3

wp

51015 20

!10

!5

0

5

x 10

!4

pix

5101520

0

5

10

15

20

x 10

!3

Expo

5101520

0

5

10

15

%&10

!'

Imp

Figure 15: Model variables response to one standard deviation preference shock measured in

percentage deviation from the steady state

49

Page 50

5 1015 20

0

2

4

6

8

x 10

!3

c

5 10 15 20

!20

!15

!10

!5

0

x 10

!4

rn

5 1015 20

0.02

0.04

0.06

inv

5 101520

0

0.01

0.02

0.03

l

510 15 20

!1

0

1

x 10

!3

pid

5 1015 20

0

0.01

0.02

0.03

y

5 10 15 20

!8

!6

!4

!2

0

2

x 10

!3

wp

510 1520

!5

0

5

x 10

!4

pix

51015 20

!5

0

5

10

x 10

!3

Expo

510 1520

0

2

4

x 10

!3

Imp

Figure 16: Model variables response to one standard deviation labour supply shock measured

in percentage deviation from the steady state

50