Download Can Open Economy Business Cycle Models Explain Business Cycle Facts?

Can Open Economy Business Cycle Models
Explain Business Cycle Facts?∗
Arnab Bhattacharjee†
Jagjit S. Chadha‡
Sun Qi§
June 2007
Abstract
We assess the capability of a flex-price open economy DSGE model to
explain two main puzzles in open-economy macro: an absence of complete
international risk sharing and the existence of exchange rate disconnect.
Within the context of a suite of econometric tests for model fit, we
examine the role of (i) traded and non-traded sectors; (ii) financial market
incompleteness; (iii) preference shocks as well as traded and non-traded
productivity shocks; (iv) expectational errors in exchange rates; and (v)
∗
Acknowledgements: We are grateful for constructive comments from colleagues and seminar participants, particularly at the Centre for Dynamic Macroeconomic Analysis, the ESRC
Evolving Macroeconomics Seminar, the Society of Computational Economics in Amsterdam,
2004, the Money, Macro, Finance Conference, 2004 and ‘Exchange Rates: Choice and Consequences’, Cambridge 2007. In particular we thank Michale Bordo, Ehsan Choudhri, John
Driffill, Chris Meissner, Marcus Miller, Charles Nolan, Joe Pearlman, Alan Sutherland, Lucio
Sarno, Mark Taylor, Christoph Thoenissen and Simon Wren-Lewis. A version of this paper
has been presented at St Andrews University, Aberdeen University, Brunel University, London
Metropolitan University, the Cass Business School, University College, Oxford, Kent University and at the Bank of Iceland and the Norges Bank. Finally, we are particularly grateful to
Christoph Thoenissen for sharing the early results of his open economy work with us.
†
School of Economics and Finance, Castlecliffe, The Scores, St Andrews University, UK.
E-mail: [email protected]
‡
BNP Paribas, 10 Harewood Avenue, Marylebone, London NW1 6AA. Professor of
Economics at Kent University at Canterbury. Professorial Fellow of the Centre for International
Macroeconomics and Finance, Cambridge University. E-mail: [email protected].
§
Centre for Dynamic Macroeconomic Analysis, School of Economics and Finance, Castlecliffe,
The Scores, St Andrews University, UK. E-mail: [email protected]
creditor status in net foreign assets. We find that there is a case both
for traded and non-traded productivity shocks and alongside expectational
errors in exchange rates to explain these twin puzzles.
JEL Classification: E32; F32; F41.
Keywords: Current account dynamics, real exchange rates, incomplete
markets, financial frictions.
Preliminary and incomplete: comments very welcome.
1. Introduction
In this paper we try to understand the twin puzzles of why it appears that
international risk sharing and the real exchange rate seem to divert so far the levels
that would be associated with their complete market allocations. Many authors,
originating with Backus and Smith (1993), have pointed to a lack of aggregate risk
sharing across open economies and as an analogue many have also commented on
the disconnect between the relative price of goods and their relative consumption,
see, for example, Obstfeld and Rogoff (2000). We concentrate on a flexible price
solution to the problem in the vein on Baxter and Crucini (1995), Backus, Kehoe
and Kydland (1995) and Stockman and Tesar (1995) but also allow for financial
market imperfections in the spirit of Devereux and Engel (2002). We find, with
the context of a new methodology for testing the fit of calibrated models, that a
two-sector open economy replete with financial market imperfections and driven
by productivity, preference and expectational errors may provide a reasonably
satisfactory contribution to the solution of these puzzles.
To understand the puzzle we use a two-sector version of Chari, Kehoe and
McGrattan (2002), developed by Benigno and Thoenissen (2004), in which there
are optimizing infinitely-lived representative households, a two-sector production
sector for traded and non-traded goods, in which the law of one price holds but in
which there are incomplete financial markets. As is well known under a complete
markets environment, there exists a strong motivation for cross-country holdings
2
of equity and it is this strong motivation that makes the Backus-Smith correlation
such a puzzle.1 As data suggest that international portfolios are home-biased
(Tesar and Werner, 1995) and imply that an important channel for risk sharing
is impeded, a popular treatment is to introduce incomplete markets by assuming
that portfolio diversification relies only on non-state contingent bonds, as in Kehoe
and Perri (2002), and accordingly we adopt this feature.
The model we adopt is a flexible device and we are also able to allow for
costly capital accumulation and the possibility of a country being a net creditor
(or debtor).2 Full price flexibility is maintained in all versions of the model. The
model is driven by three types of shocks: to both traded and non-traded sector
productivity; to preferences in the allocation of time between work and leisure
of the representative household, and by expectational errors in the relationship
between relative interest rates and the expected change in the exchange rate (see,
Frankel, 1996, and Sarno and Taylor, 2002).
A further innovation of this paper is the development of summary statistics
on the closeness of each model simulation to the data in the sense of Geweke’s
(1999) ‘weak’ interface with the data. We define a model as a structural set of
equations, which are parameterized and simulated with forcing variables defined
over a given variance-covariance matrix of shocks. The model then produces
an artificial economy which can be thought of as lying some distance from our
systematic observations on real-world economies (Watson, 1993). In this sense,
the open-economy puzzles drive a large wedge between theory and observation and
so we construct a number of empirical measures of this wedge across models and
choice of forcing variables to understand which models provide a more satisfactory
1
Baxter and Jermann (1997) conclude, with a wealth holding model with a production sector,
that domestic individuals should hold only foreign shares against loss caused for labour income
by a domestic negative shock.
2
The importance of these creditor or debtor positions have been explored comprehensively
by Lane and Milesi-Ferretti (2002).
3
resolution of the puzzles (see, for example, Ledoit and Wolf, 2002, for related
work).
Our results suggest that some form of financial market incompleteness will
probably be required to solve the open-economy puzzles (Engel, 2000). A key
result is that price stickiness can be shown not to necessarily contribute to the
resolution to the puzzles. It turns out that reasonable answers can be found
with reference to traded and non-traded forcing processes and to expectational
errors in exchange rates.3 In the former case, with a dominant role for traded
over non-traded productivity shocks, in an incomplete financial market, domestic
households raise consumption for traded and non-traded goods compared to
overseas but the real exchange rate depreciates if the terms of trade effect
outweighs the Harrod-Belassa-Samuleson effect (Corsetti et al, 2004). In the
case of preference (for work over leisure) shocks, the labour supply curve shifts
out and hence demand for goods increases (Hall, 1997). But an elastic supply
of investment, and hence output, means that there is little response in the real
exchange rate. Expectation errors in exchange rates operate to drive the exchange
rate to appreciate even if domestic interest rates fall. Consumption increases in
response to the fall in real rates and investment also increases, with wage growth
attenuated by the exchange rate appreciation and this results in a reduction in
net foreign assets. Finally, it can also be shown that a combination of all shocks
seems to explain the puzzles best.
3
We seem to have located a statistical analogue of the Sonnennschein-Mantel-Mas-Colell
(1977) theorem, which states that for any set of prices and allocations there will exist an economy
for which consumers will be at an equilibrium. Informally, we locate shock structures that take
us some distance to the actual data, which may imply that for any given distance there will
always be some shock process we can find to close the gap.
4
1.1. Some Stylized Facts
We examine open economy data from ten OECD countries. Table 1 gives the
descriptive statistics of HP filtered cyclical data and Figure 1 illustrates some
clues that the behaviour of the current account over the cycle is likely to help
explain the puzzles. Figure 1 is set over four panels. Figure 1 (a) shows the extent
to which the real exchange rate seems noisy and significantly more volatile than
its fundamentals would imply. The observed relative volatility of real exchange
rate to GDP is between 2-8, with an average, over this dataset, larger than 3.
Researchers have explained this high volatility from many dimensions in the
literature.4 Certainly, we find that compared to relative consumption, which
varies from 0.5 to 2, the real exchange rate does look ‘disconnected’. Figure 1
(b) plots the correlation of consumption with US consumption for our OECD
economies against the correlation of output with US consumption and suggests
in general that output is more closely related across countries than consumption,
which implies somewhat less than perfect risk sharing.
Figure 1(c) shows the close correspondence between the current account and
trade balance over the business cycle across these economies - suggesting a strong
role for intertemporal trade over the business cycle and some deviation from
complete markets. The finding that the current account is likely to play an
important role in the resolution of puzzles has two implications for our work,
we will want to adopt a model where current account dynamics play an important
role and assess the fit of any models we develop with, inter alia, their match to
current account data.
Finally, Figure 1(d) shows that the current account is more countercyclical
(deficit when output is high) when real exchange rates tend to appreciate over
4
These explanations include price stickiness and exchange rate overshooting (Dornbusch,
1976).
5
the economic cycle (that is negatively correlated with output). Higher demand
for foreign assets seems to be associated with an expansion but that when the
real exchange rate appreciates less over the cycle (i.e. the bottom right-hand
quadrant), some of the higher demand can be choked off.
A second modelling question concerns whether price stickiness is required for
the resolution of the puzzles. Figure 2 shows the forecast error correlation of up
to 25 quarters of US and UK current account and real exchange rate and relative
consumption and the real exchange rate (den Haan, 2000). The panels show that
over long run, these quantities are countercyclical but over the short term, all
three measures somewhat less so. As price stickiness can be expected to play a
less important role in long run dynamics, than in short run, there is some initial
motivation for excluding this feature from our model.
The rest of the paper is organized as follows. Section 2 describes the model,
section 3 outlines the solution technique and model calibration, section 4 offers
the model results, section 5 compared the model to the data VCM and section 6
concludes. Appendices A and B offer more detail on model and shock selection
and testing methodology.
2. The Model
This section describes our baseline model. Essentially, we take the flexible price
two-country, two sector model derived by Benigno and Thoenissen (2004) and
emphasize the specification of driving forces as in Chadha, Janssen and Nolan
(2001). The model is driven variously by forcing variables in domestic and overseas
traded and non-traded productivity shocks, domestic and overseas preference
shocks and by expectational errors in the exchange rate.
6
2.1. Consumer behavior
We adopt a two-country model with infinitely lived consumers.
The world
economy is populated by a continuum of agents on the interval [0, 1].
The
population on the segment [0, n) belongs to the country H (Home), while the
population on segment [n, 1] belongs to F (Foreign). Preferences for a generic
Home consumer are described by the following utility function:
∞
X
¤
£
Ut = Et
β s−t U(Csj , ξ C,s )V (lsj ) ,
(2.1)
s=t
where Et denotes the expectation conditional on the information set at date t,
and β is the intertemporal discount factor, with 0 < β < 1. The Home consumer
obtains utility from consumption, C j , and receives disutility from supplying labor,
lj . ξ C,s is a stochastic disturbance affecting the utility the agent receives from a
unit of consumption. The preference is non-separable in leisure.
The asset market structure in the model is relatively standard and is described
in detail in Benigno (2001) and Benigno and Thoenissen (2004). We assume that
Home individuals are able to trade two nominal risk-less bonds denominated in
the domestic and foreign currency. These bonds are issued by residents in both
countries in order to finance their consumption expenditure. On the other hand,
foreign residents can allocate their wealth only in bonds denominated in the foreign
currency. Home households face a cost (i.e. transaction cost) when they take a
position in the foreign bond market. This cost depends on the net foreign asset
position of the home economy as in Benigno (2001). Alternative ways of closing
open economy models are discussed in Schmitt-Grohe and Uribe (2003).
The generic Home consumer maximizes utility subject to the following budget
constraint.
Pt Ctj
j
j
St BF,t
BH,t
j
j
´ = BH,t−1
³
+
+
+ St BF,t−1
+ Pt wt ltj + Πjt
S
B
t
F,t
(1 + it ) (1 + i∗ )Θ
t
Pt
7
(2.2)
where Pt is the price index corresponding the basket of final goods C, w is the
real wage earned by agent in return for supplying labour and Π are dividends
received by the agent from holing an equal share of the economy’s intermediate
goods producing firms.
The asset market is incomplete in the sense that Home agents can only hold
j
two types of nominal, non-state contingent bonds. BH
denotes agent j’s holdings
of Home-currency denominated bonds. The one-period return from these bonds
is denoted by (1 + it ) . S denotes the nominal exchange rate, defined as Home
currency price of a unit of foreign currency.
BFj denotes agent j’s holdings
of Foreign-currency denominated bonds. The one-period return from foreign´
³
S B
currency denominated bonds is (1 + i∗t )Θ tPtF,t , where (1 + i∗t ) is the gross rate
³
´
St BF,t
of return and Θ
is a factor of proportionality (cost associated with foreign
Pt
currency-denominated bond holding) that depends on the economy-wide holdings
of foreign-currency denominated bonds.
The market structure ensures that the steady state of our model is well defined.
³
´
St BF,t
The factor of proportionality Θ
is equal to unity only when economy-wide
Pt
bond holdings are at their initial steady state level, thus ensuring that in the longrun the economy returns to its initial steady state level of bond holdings.
The first order condition of the representative consumer can be summarized
as follows:
Uc,t+1
∙
¸
Pt
Uc,t = (1 + it )βEt Uc,t+1
Pt+1
∙
¸
Pt∗
∗
Uc∗,t = (1 + it )βEt Uc∗,t+1 ∗
Pt+1
¶
∙
¸
µ
St+1 Pt
St BF,t
∗
βEt Uc,t+1
.
= (1 + it )Θ
Pt
St Pt+1
Uc,s wt = Vl (ls )
8
(2.3)
(2.4)
(2.5)
(2.6)
where Uc,t ≡ Uc (Ct , ξ C,t , 1 − lt ) and Uc∗,t ≡ Uc (Ct∗ , ξ ∗C,t , 1 − lt∗ ). As in Benigno
(2001), we assume that all individual belonging to the same country have the same
level of initial wealth. This assumption, along with the fact that all individuals
face the same labor demand and own an equal share of all firms, implies that
within the same country all individuals face the same budget constraint. Thus
they will choose identical paths for consumption. As a result, we are able to drop
the j superscript and focus on a representative individual for each country.
2.2. The supply side
There are three layers of production in our model economy. Final goods are
produced by a competitive final goods producing sector using Home traded and
non-traded intermediate goods as well as foreign-produced traded intermediategoods. Final goods are non-traded and are either consumed or used as investment
goods to augment the domestic capital stock. Intermediate goods producers
combine labor and capital according to a constant returns to scale production
technology to produce intermediate goods. Each country produces two types of
goods, a differentiated traded good and a non-traded good.
2.2.1. Final good producers
Let Y be the output of final good produced in the home country. Final goods
producers combine domestic and foreign-produced intermediate goods which they
must obtain from the distributor to produced Y in the a two-step process. The
final good Y is made up of traded, yT , and non-traded inputs, yNT , combined in
the following manner:
κ
h 1 κ−1
κ−1 i κ−1
1
κ
κ
κ
κ
,
Y = ω yT + (1 − ω) yN
(2.7)
where ω is the share of traded goods in the final good, and κ is the intra-temporal
elasticity of substitution between traded and non-traded intermediate goods. The
9
traded component, yT , is, in turn, produced using home, yH , and foreign-produced
traded goods, yF , in the following manner:
θ
h 1 θ−1
θ−1 i θ−1
1
θ
θ
,
yT = v θ yH + (1 − v) θ yF
(2.8)
where v is the domestic share of home produced traded intermediate goods in total
traded intermediate goods and θ is the elasticity of substitution between home
and foreign-produced traded goods. Final goods producers are competitive and
maximize profits, where P is the aggregate or sectoral price index and Y aggregate
output.
max P Y − PT yT − PN yN
(2.9)
max PT yT − PH yH − PF yF
(2.10)
yN, yT
yH, yF
subject to (2.8).
This maximization yields the following input demand functions:
µ ¶−κ
PN
Y
yN = (1 − ω)
P
µ ¶−θ µ ¶−κ
PT
PH
Y
yH = ωv
PT
P
µ ¶−θ µ ¶−κ
PT
PF
Y.
yF = ω(1 − v)
PT
P
(2.11)
The foreign final goods firm carries out a similar maximization exercise which
10
yields:
∗
yN
∗
yH
yF∗
¶−κ
PN∗
= (1 − ω )
Y∗
P∗
µ ∗ ¶−θ µ ∗ ¶−κ
PH
PT
∗ ∗
= ω v
Y∗
∗
∗
PT
P
µ ∗ ¶−θ µ ∗ ¶−κ
PT
PF
= ω∗ (1 − v∗ )
Y ∗.
∗
PT
P∗
∗
µ
(2.12)
The price index that corresponds to the above maximization problem is:
PT1−θ = [vPH1−θ + (1 − v)PF1−θ ]
(2.13)
P 1−κ = [ωPT1−κ + (1 − ω) PN1−κ ]
and
PT∗1−θ = [v∗ PH∗1−θ + (1 − v ∗ )PF∗1−θ ]
(2.14)
P ∗1−κ = [ωPT∗1−κ + (1 − ω ∗ ) PN∗1−κ ].
The goods produced in the final goods sector are only used domestically, either
for consumption or investment, xt :
Yt = Ct + xt .
(2.15)
2.2.2. Traded-intermediate goods sector
Firms in the traded intermediate goods sector produce goods using capital and
labor services. The typical firm maximizes the following profit function:
∗
max PHt yHt + St PH∗ t yH
− Pt wt lH,t − Pt xH,t ,
11
(2.16)
or because the law of one price holds at the wholesale level,
∗
) − Pt wt lH,t − Pt xH,t .
max PHt (yHt + yH
Ht
This maximisation is subject to:
∗
1−α
= F (kH,t−1 , lH,t ) = (At lH,t )α kH,t−1
yHt + yH
t
¶
µ
xHt
kHt−1 .
kH,t = (1 − δ)kH,t−1 + xH,t − φ
kHt−1
(2.17)
The stochastic maximization problem of the domestic intermediate goods firm
is given by:
L = Et
∞
X
t=0
β
t Uc,t
Pt
(
)
¤
£
1−α
α
− ³Pt wt lH,t
hPH,t (At lt ) (kH,t−1 )
´ − Pt xH,t
i .
xH,t
+λt (1 − δ)kH,t−1 + xH,t − φ kHt−1
kH,t−1 − kH,t
(2.18)
The first order condition with respect to labor input is:
Pt wt = αPH,t (At )α (
kH,t−1 1−α
) .
lH,t
The first order condition with respect to investment:
0
Pt = λt − φ
µ
xH,t
kH,t−1
¶
λt .
The first order condition with respect to capital:
λt = Et β
⎧
⎨
³
At+1 lH,t+1
kHt
´α
⎫
⎬
PHt+1 (1 − α)
+
Uc,t+1 Pt
h
³
´
³
´
i , (2.19)
Uc,t Pt+1 ⎩ λt+1 (1 − δ) − φ xHt+1 + φ0 xH,t+1 xH,t+1 ⎭
kH,t
kH,t
kH,t
12
using the expression for PH,t from the wage equation yields:
⎫
⎧
³ ´
(1−α) lt+1
⎬
⎨
Pt+1 wt+1 +
Uc,t+1 Pt
α
h
³ kt ´
³
´
i .
λt = Et β
Uc,t Pt+1 ⎩ λt+1 (1 − δ) − φ xH,t+1 + φ0 xH,t+1 xH,t+1 ⎭
kH,t
kH,t
kH,t
Next, substitute in the expression for λ
Uc,t
µ
¶¸
∙
fk
xt
0
EβUc,t+1 wt+1 t+1 +
= 1−φ
(2.20)
kt−1
flt+1
³
´
0
xt
∙
µ
¶
µ
¶
¸
1 − φ kt−1
xH,t+1
xH,t+1 xH,t+1
0
³
´ Uc,t+1 (1 − δ) − φ
Et β
+φ
,
kH,t
kH,t
kH,t
1 − φ0 xt+1
kt
where fkt is the marginal product of capital and flt+1 the marginal product of
labor and wt+1 is the real wage, Uc,t ≡ Uc (Ct , ξ C,t , 1 − lt ).
2.2.3. Non-traded-intermediate goods sector
Non-traded intermediate goods producer has the similar maximization problem:
max PNt yNt − Pt wt lN,t − Pt xN,t ,
(2.21)
which is subject to
yNt = F (kt−1, lN,t )
kN,t = (1 − δ)kN,t−1 + xt − φ
µ
xN,t
kN,t−1
¶
(2.22)
kN,t−1 ,
and where ψyH,t + ψyF,t are demands for non-traded goods coming from the
distribution sector. If we now set up the stochastic maximization problem of
the domestic intermediate goods firm:
13
L = Et
∞
X
t=0
Uc,t
βt
Pt
( £
¤ )
PN,t (AN,t lN,th)α (kN,t−1 )1−α + PN,t (ψyH,t
³ + ψy´F,t ) − Pt wt lNt i− Pt xN,t
.
xN,t
+λt (1 − δ)kN,t−1 + xN,t − φ kN,t−1
kN,t−1 − kN,t
(2.23)
The first order condition with respect to labor input is then given by:
kN,t−1 1−α
) .
lN,t
The first order condition with respect to investment is:
Pt wt = αPN,t (AN,t )α (
¶
xN,t
λt .
Pt = λt − φ
kN,t−1
The first order condition with respect to capital is:
0
λt = Et β
Uc,t+1 Pt
Uc,t Pt+1
µ
⎧
⎨
⎫
´α
³
A
lN,t+1
⎬
PNt+1 (1 − α) t+1kN,t
+
h
³
´
³
´
i ,
⎩ λt+1 (1 − δ) − φ xN,t+1 + φ0 xN,t+1 xN,t+1 ⎭
kN,t
kN,t
kN,t
(2.24)
and using the expression for PN from the wage equation yields:
⎫
⎧
³
´
(1−α) lNt+1
⎬
⎨
P
w
+
t+1 t+1
Uc,t+1 Pt
α
h
³ kNt ´
³
´
i .
λt = Et β
Uc,t Pt+1 ⎩ λt+1 (1 − δ) − φ xN,t+1 + φ0 xN,t+1 xN,t+1 ⎭
kN,t
kN,t
kN,t
We next substitute in the expression for λ
Uc,t
µ
¶¸
∙
fk
xN,t
0
EβUc,t+1 wt+1 t+1 +
= 1−φ
(2.25)
kN t−1
flt+1
³
´
xN,t
0
∙
µ
¶
µ
¶
¸
1 − φ kN,t−1
xN.t+1
xN,t+1 xN,t+1
0
³
´ Uc,t+1 (1 − δ) − φ
+φ
,
Et β
x
kN,t
kN,t
kN,t
1 − φ0 N,t+1
kN,t
where fkt is the marginal product of capital and flt+1 the marginal product of
labour and wt+1 is the real wage, Uc,t ≡ Uc (Ct , ξ C,t , 1 − lt ).
14
2.3. The real exchange rate
In this model, the real exchange rate is defined as:
RSt =
St Pt∗
Pt
(2.26)
and can deviate from purchasing power parity (PPP) as a result of three channels.
As in Benigno and Thoenissen (2004) allowing for the possibility of home bias in
consumption (v 6= v ∗ ), via the terms of trade channel (because of home bias) and
via the internal real exchange rate channel (because of non-traded goods). We
can see this by expanding (2.26) to give:
∗
∗
St PH,t
St Pt∗
PH,t PT,t
PT,t Pt∗
=
,
∗
∗
Pt
PH,t PT,t PH,t
Pt PT,t
SP ∗
P
equals unity, we
c t = (v − v ∗ )T̂t + (ω − 1) R̂t + (1 − ω ∗ ) R̂t∗ .
RS
(2.27)
which if we then linearise around the steady state, where
find:
The deviation of the real exchange rate around its steady state depends on
deviations of of the home and foreign retail to wholesale price ratios, the terms of
trade, T , defined as
PF
,
PH
and the relative price of non-traded to traded goods, R.
2.4. The current account
The current account is defined as changes in foreign asset holding, within the
incomplete financial market. Home and foreign agents trade intermediate goods
and the trade balance is used to buy foreign bonds. Buying foreign bonds is the
only way of arbitraging, therefore the flow budget constraint shows the current
account dynamics below. The left hand side is the changes in foreign asset
holding. The right hand side shows the total production (first two terms) minus
consumption and investment, yielding adjustment of bond wealth:
15
F
St Bt−1
St BtF
1
PHt
PN
∗
´
³
=
(yHt + yH
) + t yNt − Ct − xt .
−
F
∗
Pt (1 + it ) Φ St Bt
Pt
Pt
Pt
(2.28)
Pt
2.5. Forcing Variables
We adopt the specification of Stockman and Tesar (1995) and Chadha, Janssen
and Nolan (2001) by investigating role of both productivity and preference
shocks for an open economy. We use both traded sector and non-traded sector
productivity, which drive the input and hence product price, shocks to the
allocation of time spent in work over leisure, which affect labour supply, a random
shock in the UIP condition, which directly affects the terms of trade, and interpret
as expectational errors to the exchange rate. Each shock originates from a different
sector but allows us to attribute exchange rate volatility attributable to more
than one exogenous factor. In total, we enable seven shocks (two sectoral and a
preference shock in each of two countries, plus UIP shock) and try to locate the
importance in explaining open economy business cycles. The construction of each
shock process is explained in Appendix A.
3. Solution and Model Calibration
3.1. Solution method
Before solving the model, it is log-linearized around the steady state to obtain a
set of equations describing the equilibrium fluctuations of the model. The loglinearization yields a system of linear difference equations which we list in an
appendix and can be expressed as a singular dynamic system of the following
form:
AEt y(t + 1 | t) = By(t) + Cx(t)
16
where y(t) is ordered so that the non-predetermined variables appear first and the
predetermined variables appear last, and x(t) is a martingale difference sequence.
There are up to seven shocks in C. The variance-covariance as well as the
autocorrelation matrices associated with these shocks are described in Table 2.
Given an initial parametrization of the model, which we describe in the next
section, we solve this system using the King and Watson (1998) solution algorithm.
3.2. Data and Estimation
Table 2 in the summarizes the calibration parameters. We collect both quarterly
and annual data and calibrate an annual (1980-2000) model for UK-US. Values
of parameters are either estimated from US or UK data or taken from extant
literature. An annual risk free rate of 4% and depreciation of 10% is assumed.
Labor share is 0.6 and 0.577 for UK and US annual data. We take the consumption
and leisure curvature of 2 (Corsetti et al, 2004) and 4 (Chadha et al, 2001). The
elasticity of substitution between home and foreign goods in UK is 1.5 as in Chari
et al (2002). For the trade-off between traded and non-traded goods we adopt
the elasticity suggested by Corsetti et al (2004) of 0.74. We measure share in
aggregation as percentage share in quantities. UK and US trade data reveals the
shares of UK produced goods in UK and US production to be 0.73 and 0.0157.
Estimate on traded goods weights in all household consumption are estimated
to be 0.3 and 0.24, smaller than that of Corsetti et al (2004), 0.45 to 0.5. Cost
of financial intermediation is 4bp as in Benigno (2001). The cost of investment,
b=2, is chosen to match the relative volatility of investment.5 Steady state of net
foreign asset is set to be 0 or 0.2. Each means UK has a balanced current account
or a is creditor.
5
However we run experiments with also b=5, which are available on request and covered in
the robustness exercise of Figure 7.
17
We have at most seven exogenous shocks in our experiments. The vector of
shocks Πt are assumed to follow a VAR(1) process:
Πt+1 = AΠt+1 + Ut ,
Ut ∼ N (0, Σ) .
We run an OLS regression to estimate the parameters of the A matrix.
Following Stockman and Tesar (1995), we allow three type of evolution of shocks:
(1) no spillovers; (2) estimated spillovers; (3) symmetric spillover.
4. Model Results
We now turn to the evaluation of the structural linear model by its simulation and
comparison to our observations on the economy. As well as standard matching of
moments, we develop a new approach for model evaluation and model selection.
4.1. Methodology
Conventional tools such as the impulse response function and variance
decomposition help us understand the dynamics of an artificial economy. The
standard practice is also to assess models against the second moments of the
data. But in this paper we also use the variance covariance matrix (VCM)
of endogenous variables to evaluate goodness of fit of the model. We define a
better model, as one that can render a better match between data VCM and
unconditional VCM. In general, we believe any constructed model cannot be
thought of as the whole truth and may wish to find a better approximation
subject to the class of economic investigation we adopt. In order to pin down
18
some parameter value or decide on certain features of a model, we work on a class
of candidate models (or calibrations). We locate criteria to see which candidate
models deliver the closest match to the criteria. By examining the corresponding
match for candidate models, we call any improvement towards the criterion a gain
in marginal information. We can also evaluate the gain on a particular parameter,
by which we can signal the importance of any one feature of the model. Strictly
speaking, we cannot guarantee the marginal information gain is reliable, or nearer
to the ‘true’ model, unless we are quite certain about the rest of the model. The
proposition of a marginal information gain we make is therefore a ‘weak-form’
model selection (see Geweke, 1999).
The criteria we use is the statistical divergence of the two VCMs. We develop
formal and also trivial distance measures elsewhere but some details are available
in Appendix B. These measures all have a one-side empirical distribution. A higher
value of distance denotes a model that is further from our measure on ‘true’ data
process.6 The data required to test the open economy model is of high dimension
and a relatively short sample, which tends to make the model evaluation and
selection very difficult. We calculate for each candidate model a distance and
compare across each measure. We are cautious in making a proposition of model
selection, especially for a particular parameter constellation, but feel able to make
some statements on the validity of the joint choices on model and shock processes.
4.2. Impulse Responses
The impulse response functions are based on the seven-shock model with a
symmetric calibration. In this calibration, the foreign country has the same
6
In developing this approach, we use Monte Carlo simulations on some artificial models.
We find: (1) this approach works very well for models close to ‘true model’, as long as the
multivariate normality is tenable; (2) our approach helps overcome small-sample bias, and (3)
experiments on a sub-block of the full VCM may be inconclusive.
19
properties as home, such as shares of traded and home goods on market. We
change v=0.85 and v*=0.15 in order to highlight the effect of foreign sector.
4.2.1. Traded productivity shocks
Figure 3 plots the response of quatities and relative prices to a traded productivity
shock in the home country. The response of real exchange rate depends on
two effects: the terms of trade and the Harrod-Balassa-Samuelson (HBS) effect.
The former requires an adjustment in relative traded prices, which requires a
depreciation in the real exchange rate in the long run. But the latter effect drives
up wages in both the traded and non-traded sector but with no productivity
improvement in the non-traded sector, non-traded prices will rise and hence so
will the real exchange rate. This effect is especially strong, see equation 2.26, when
there is a home bias in consumption, which acts to accentuate the real exchange
rate change. Finally, the lack of complete risk sharing means that consumption
is more elastic to a productivity shock than under a complete markets allocation.
The combination of forward-looking domestic consumption responding to higher
productivity (income) but an attenuated overall investment response, i.e. traded
sector investment rises but non-traded sector investment falls, leads to and an
elastic response of investment leads to the accumulation of foreign debt to finance
current demand.
4.2.2. Non-traded productivity shocks
Following a non-traded productivity shock (Figure 4), investment and labor
increase. Households at home enjoy somewhat higher consumption and more
so than in the case of traded sector productivity shock. In this case, the terms
of trade effect and HBS effect is the same, causing the real exchange rate to
depreciate. Although relative consumption is positive, it is not large enough to
20
bring about a current account deficit, because there is a larger response from the
labour input, and hence there is net lending overseas. In general the impulse
responses suggest that strong traded-sector productivity shocks can lead to the
matching of some elements of the open economy. A lack of complete risk sharing
raises consumption at home compared to abroad and a strong preference for home
goods consumption also amplifies the extent to which output increases.
4.2.3. Preference Shocks
Preference shocks determine the household trade-off between leisure and
consumption. Following Hall (1997) such shocks simply suggest that the household
decides to allocate more time to work, which finances consumption, rather than
leisure. As one would expect preference shocks help increase the volatility of
the labor input by introducing exogenous shifts in work and may act to solve
the puzzle of the Backus-Smith correlation (Figure 5). A home preference shock
drives up labour input and consumption and reduces relative prices, as the supply
response is elastic. If home agents remain inelastic in the substitution of leisure
across periods, increased consumption is also met by an increase in investment.
The current account remains acyclical.
In principle, preference shocks can break the Backus-Smith puzzles simply as
marginal utility is now, inter alia, a function of the preference shocks rather than
just consumption growth:
UC∗
RS =
UC
but by themselves are problematic as they imply relatively acyclical current
account dynamics and a reduction of real exchange rates along with higher
domestic supply.
21
4.2.4. UIP Shocks
Following the suggestion of Devereux and Engel (2002), we explore the implication
of adopting errors in the uncovered interest rate parity (UIP) determination of
exchange rate changes. These shocks, motivated by expectational errors or the
poor empirical performance of UIP equations, imply that the exchange rate does
not move equiproportionally with interest rate differentials. In fact it moves in the
opposire direction. A shock that brings about an initial exchange rate appreciation
is equivalent to a demand shock as it depresses traded and non-traded wages via
competition with overseas traded-sector wages. To deal with the temporary fall
in wages, consumption - which is tilted up by the fall in domestic interest rates
- is maintained by overseas borrowing and investment is stimulated by the fall in
wages.
4.3. Variance Decomposition
Table 3 shows the decomposition of unconditional variances for relative
consumption, the real exchange rate and the current account. The top panel c(ii)
and c(iii) shows the results for a model with all seven shocks, with persistent UIP
shocks for shocks with and without spillover, respectively. The panel d(ii) d(iii) are
for realistic (transitory) UIP shocks. The Table suggests that productivity shocks
play an important role in explaining fluctuations in relative consumption, the real
exchange rate and the current account but that UIP persistent shocks can also play
an important role in explaining real exchange rate and current account behavior.
Preference shocks increase in importance when we select symmetric shocks suggesting that they may be able to substitute to some extent for idiosyncratic
home or foreign responses to symmetric shocks. The two lower panels show the
variance decomposition when we exclude productivity shocks and we find that
preference and UIP shocks play an equivalent role in explaining these variables.
22
These findings are similar to other empirical studies such as Straub and Tchakarov
(2004).
4.4. Moments result
We present selected second moment properties of the artificial simulated model in
Tables 4 and 5, for each of the model with no-spillover and symmetric spillovers.7
The four sets of results: a-d, correspond to the case of no UIP shock, the case
when the UK is a net creditor, and when there are persistent then temporary UIP
shocks, respectively.
We find that varying the spill-over of productivity and preference shocks does
not change the moments qualitatively. The models with productivity shocks
capture well the main moments of the data. If we compare the model with or
without UIP shocks, we find that exploring the possibility of UIP deviations can
help explain the puzzles. Note finally that in the model with only productivity
shocks, labor input is countercyclical and adding preference shocks can solve this
problem. It also helps to weaken the strong correlation between wages and output.
We also find that a model with preference shocks alone cannot generate sufficient
exchange rate volatility without the addition of UIP shocks.
The overall performance of baseline calibrated model is reasonable.
To
conclude the model performance in explaining the puzzles, we have (1) the model
enables different shocks to interact and solve Backus-Smith puzzle with the help of
a non-traded sector and incomplete financial markets; (2) this model stresses the
HBS effect and therefore generates volatile real exchange rates; (3) countercyclical
current account is a robust result, as the current account move together with real
exchange rate.
7
See Annex A.2 for a definition of spillover and non-spillover shocks.
23
5. Model-data comparison
If we choose to define the best model as that with the least deviation from the
data, we can choose a number of metrics to locate the best model. Our model
selection from a class of candidate models is based on the comparison of the VCM
of endogenous variables simulated by our model to the actual data, see Appendix
B for some further details. We consider a sub-set of the variables that comprises
10 variables, including other domestic business cycle components.
As well as RMSE and MAE, two likelihood ratio methods can be used to
determine how different the two matrices are: (a) the Box M-test; (b) Kullback
Leibler Information Criterion (KLIC). A singular variable decomposition method
is adopted to find a revised KLIC distance for a singular VCM. Finally, the third
criterion we use is W-test for matrix equality, which is modified form of the Nagao
test for covariance matrix equality. The W-test test constructs a measure mainly
relevant to diagonal elements of VCMs. Ledoit and Wolf (2004) argue that this
test can solve problem of singularity.
For each case, we obtain six statistical measures of distance and the results
are given in Table 6. The smaller statistics indicate a better fit of data to model
and we find for the main model selection criterion the models with persistent UIP
shocks are closest to the observed data.
5.1. Model selection with VCM
Table 6 shows the main results of measuring distance in model-data matching on
the variance covariance matrices. The measures of distance are shown in lower
panel in each table. We first try to find in each table which model is the best
according to each of the six criteria and second we compare among the three tables
and the best model or models is marked with an asterisk.
Table 6 shows the best model is the model c(iii) with productivity, preference
24
and expectational shocks. The preference shock helps explain dynamics of wages
and labor input, therefore more welcome for the whole variables in the model.
From our initial examination of impulse responses and cursory examination of
moments in Tables 5-7, we had suggested there may be a case for all three classes
of shocks to explain the open economy data and based on our statistical tests of
the distance from our model and the 10 key variables we argue that indeed there
is.
5.2. Sensitivity Analysis
The sensitivity analysis is shown in Figures 7-18 and is based on the sevenshock model with a base-case calibration of Table 2. We simulate the model
and allow parameters to changes and check the sensitivity with respect to several
main statistical measures: the Backus-Smith correlation, the extent of exchange
rate disconnect, the correlation between the trade and current account and the
cyclicality of the current account. The vertical solid line(s) denotes the initial
calibration for UK and US.
First, we consider frictions in the model: costly investment and costly foreign
asset holding. In Figure 7, although higher cost of investment alters volatility of
open economy variables, it does not change the basic correlation structure. In
Figure 8, costly foreign asset holding make the channel of risk sharing smaller,
therefore the Backus-Smith correlation tend to zero. However, this will happen
when the cost is extremely high. As the model has very simple assumption for
financial markets, we emphasize its qualitative implication instead of its value
denoted by basis points.
Secondly, we discuss the characteristics of the market and production. Steady
state NFA does not alter real exchange rate dynamics significantly but it is crucial
for current account dynamics. For a net debtor, a positive traded TFP shock leads
25
to current account deficit. For example, upon a positive traded productivity shock,
output increases, the real exchange rate appreciates, Home country borrows and
a current account deficit results. But as a debtor there is requirement for paying
interest, making the borrowing incentive lower and thus the extent to which the
current account is countercyclical is mitigated, as shown in Figure 11.
Thirdly, we consider varying source of dynamics, the exogenous shocks.
The UIP shock in baseline calibration is highly transitory and by examining
a more persistent UIP shock as in Figure 10, we find there are real effects
only in the case of highly persistent shocks.
Adding UIP shocks reinforces
the pattern of correlation we find in the data.
When we vary the relative
magnitude of non-traded productivity shocks in Figure 11, it leads to changes
in the key correlations. A combination of relatively strong traded compared to
non-traded productivity shocks contribute to negative Backus-Smith correlation
and countercyclical current account. Turning to Figure 12, as preference shocks
are strengthened, negative correlation is weakened.
5.3. From model selection to parameter estimation
Figures 13-18 replicate the sensitivity analysis for each of these key parameters
but in terms of the distance measures. We can use the diagnostics to obtain
estimates of the parameters which provides the best match to the data and we
can run the sensitivity analysis.
It is clear that the minimum distance is achieved when treating the cost of
investment is in the neighbourhood of 2 (Figure 13). That the costs of financial
intermediation do seem especially high (Figure 14) i.e. that liquidity does not seem
to be an important cause of the puzzles. We examine the sensitivity analysis on
cost of financial intermediation, i.e., the spread between return on foreign and
domestic bond. This parameter affects the trade-off between home and foreign
26
bonds. The four criterion all suggest model will improve when this spread is big.
We attribute this to strong home bias in asset holding. The distance of model
from data reduces as we increase the interest spread, or the holding cost of foreign
bond, to 100%, but in absolute terms the interest spread remains relatively small.
The conclusion we draw here is, we may not need to model the cost of financial
intermediation as being especially large as in Benigno (2001).
The adoption of the assumption that the small open economy is a net debtor
seems to help model fit. And although the UK as a have a steady-state level of
debt near zero. Lane and Milesi-Ferretti (2002) document the mean net foreign
asset over GDP at 0.06, i.e., UK is a small net creditor. Our approach seems to
locate the correct approximate region for the level of steady-state debt.
Finally we examine the correct level of persistence for the shock processes.
More persistent UIP shocks are preferred (Figure 16) and certainly increasing the
relative volatility of traded to non-traded shocks seems to help the fit. But as
with earlier results increasing the volatility of preference shocks does not seem to
improve the fit of the model markedly.
6. Conclusion
Open-economy general equilibrium models offer an attractive laboratory in which
to examine the insolubility or otherwise of data puzzles.
We examine the
properties of a two-sector real business cycle model with incomplete financial
markets. The model is driven by a number of driving forces to both domestic and
overseas traded and non-traded productivity, to the work-leisure margin at home
and overseas and to errors observed between UIP equations and the evolution of
the exchange rate. We find some evidence to support that when all these shocks
perturbate the economy there is some move towards resolution of the puzzles. The
model we use is flexible enough to allow examination of deep parameters for small
27
open economies.
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31
A. Appendix A - Shocks and Data
A.1. Forcing Variables
A.1.1. Productivity Measurement
Sectoral productivities are calculated as Total Factor Productivity in traded or
non-traded sector. In following formula the k is the number of industry in that
sector.
i
is the corresponding weights in sectoral output.
T /N
T F Pt
=
k
X
i,t
i=1
⎛
⎜
log ⎝ ³
T /N
Kt,i
T /N
Yi,t
´1−αt,i ³
T /N
Nt,i
⎞
⎟
´αt,i ⎠
(A.1)
A.1.2. Measuring the Allocation of Time
Preference measures, ξ t , are calculated from the Euler equation for leisure-labor
output and solved for ξ t with output, labor input or wage, consumption. In
equilibrium the real wage equals
UL
,
UC
the marginal rate of substitution between
leisure and consumption (L denotes leisure while l denotes labor input).
Wt =
ULt
UCt
Yt = At F (Kt−1 , lt )
(A.2)
(A.3)
Wt = At Fl (Kt−1 , lt )
ULt
= At Fl (Kt−1 , lt )
UCt
32
(A.4)
In this equation, the left hand side contains preference shocks if we have defined
and therefore it can be expressed explicitly by simple observables. Hall (1997)
constructs a measure of preference shocks:
xt = ct − yt + (1 + φ) lt − ln (α) .
(A.5)
This residual is obtained from the equilibrium condition for consumptionleisure substitution.
He analyzed US business cycles at both low and high
frequencies and found a strong link between this measure and hours worked. In
this equation lower case letters denote taking logarithm. α is labor share in CobbDouglas production function. 1/φ is the compensated elasticity of labor supply
(marginal disutility of work). We constructed UK and US preference measure in
a similar manner, instead using data of real wages. The non-separable-in-leisure
utility function is:
U=
1
t
Ct1−ρ · Lη·X
t
1−ρ
(A.6)
The residual measure for preference shock xt from equilibrium solution is:
µ
(1 − ρ) Wt · (1 − lt )
·
xt = ln
η
Ct
¶
(A.7)
A.1.3. Expectational Errors in Exchange Rates
Furthermore, we allow a random shock in the UIP condition, making exchange
rate volatility attributable to more factors. Our version of UIP shocks is a simple
treatment allowing noise trader bias exchange rate in the short run. The nominal
exchange rate adjustment is according to UIP and a shock xu,t by:
bt + xu,t
Et ∆st+1 = it − i∗t + εB
33
(A.8)
These three types of shocks enter the dynamics of the economy in different way.
Productivity shocks enter the optimal allocation of input factors and drive up or
down the input prices therefore the product prices. Trade among two countries
adjusts mostly as a consequence of term of trade effect and Balassa-Samuelson
effect. Preference shocks drive fluctuations of labor demand. Allocation of laborleisure is affected by new equilibrium on labor market and it determines the
new quantity supplied. Relative prices and trade patterns adjust accordingly.
Exchange rate bubble lead to changes in terms of trade and thus on trade flows
and relative prices. Fluctuations on domestic markets follow.
A.2. Data and calibration
We show some stylized open economy business cycle facts in Table - 1. The
data set contains ten OECD countries and the post-war quarterly data is taken
from IMF IFS data base. We find in all the countries, real exchange rate is very
volatile and persistent. There is no clear correlation between real GDP and real
exchange rate. Current account is contra-cyclical and quite smooth, although the
magnitude varies much among all samples.
According to lack of quarterly sectoral data, we calibrate the UK-US model
to be annual, 1980-2000. The construction of total and sectoral TFP series are
following routine Solow Residuals. Sectoral data is taken from OECD STAN.
However, the net capital stock is not available. US capital stock on 2000 is
obtained in BEA (Bureau of Economic Analysis) fixed asset table, where sectoral
fixed asset data is combined according to conventional industry classification.
Previous capital stock is derived with data of gross fixed capital formation and
consumption of fixed capital. UK capital stock data is replaced with net capital
stock from ONS (series IZSF, IZSJ, IZSN, JABD, GSQV, GSRG, GSTN, GSTQ,
GSTU, GSTY, GSUC, GSUG, GSUK, GSUO, GSUS, GSUW, GSVA). Other
34
business cycles data is taken from IMF IFS database.
We have either four, five, six or seven exogenous shocks in the model,
depending assumptions of sources of the fluctuations. Preference measures are
calculated as the residual of equilibrium real wages (A.7). The measures we
construct for UK and US are quite persistent and positively correlated with each
other. UIP disturbance is the residuals of an OLS estimation of UIP condition:
∆st+1 = α + β (it − i∗t ) + xut
(A.9)
UIP shocks are transitory and volatile in our calibration. Four productivity
shocks are calculated as TFP with sectoral annual series for (A.1). Univariate
time series of seven shocks are assumed to be AR(1). Statistical properties are
summarized in following table.
Table A1 - Univariate properties of seven shocks
Shocks
xA
xAN
xA∗
xAN∗ xP
xP ∗
xU IP
Correlation with output 0.39
0.32
0.48
0.19
-0.57 -0.12 -0.25
Persistence
0.81
0.77
0.68
0.73
0.75
0.47
0.23
Standard Deviation
4.96% 2.77% 5.56% 4.37% 3.36% 1.84% 5.00%
Variances
0.25% 0.08% 0.31% 0.19% 0.11% 0.03% 0.25%
Note: A, AN, P and UIP denote shocks to traded productivity, non-traded
productivity, preference and UIP conditions. ”*” denotes foreign country (US).
According to Stockman and Tesar (1995), we allow three type of evolution
of shocks: (1) No spillover; (2) Estimated spillover; (3) Symmetric spillover.
Transmission in technology, either international or intersectoral, could be
important for models explaining international risk-sharing (Corsetti et al, 2004).
We model a vector of n (n = {4, 5, 6, 7}) shocks Πt as a VAR(1) process.
Πt+1 = AΠt+1 + Ut
Ut ∼ N (0, Σ)
35
(A.10)
No spillover refers to the case that A matrix is diagonal with each shocks’
persistence along the diagonal. Spillover introduces nonzero off-diagonal elements.
However, spillover of shocks can be incorporated in the covariance matrix of
shocks Σ as well. Stockman and Tesar (1995) stress a positive spillover of these
productivity shocks characterized by Σ. We assume spillover of technology is
attributable to both A and Σ. The estimated spillover (case (2)) scenario takes
an unrestricted OLS regression for (A.10). The symmetric spillover imposes a
restriction of symmetry in A so that the shocks in home and foreign economy
are affected by each other in the same way. Nevertheless, we do not assume
volatility of shocks are symmetric in home and overseas, as in Stockman and
Tesar (1995). As we have a small open economy model, it is not imperative for
us to model symmetricity. We also allow spillovers between preference shocks and
productivity shocks. We further assume UIP shocks are orthogonal with all other
shocks.
For example, calibration for a productivity shock-only model is:
(1) No spillover (Corr denotes correlation matrix of shock residuals):
⎡
⎤
0.25%
0
0
0
0.81 0
0
0
⎢ 0
⎢ 0 0.77 0
⎥
0.08%
0
0
0
b=⎢
b=⎢
⎥;Σ
A
⎣ 0
⎣ 0
0
0.31%
0
0 0.68 0 ⎦
0
0
0
0.19%
0
0
0 0.73
⎡
⎡
1
³ ´ ⎢ 0
b =⎢
Corr Σ
⎣ 0
0
(2) Estimated spillover:
36
0
1
0
0
0
0
1
0
⎤
0
0 ⎥
⎥;
0 ⎦
1
⎤
⎥
⎥;
⎦
⎡
0.58 −0.23 −0.09
⎢ −0.34 0.57
0.23
b=⎢
A
⎣ 0.26
0.24
0.08
0.26
0.17 −0.34
⎡
⎤
0.83
⎢
0.21 ⎥
b=⎢
⎥;Σ
⎣
0.69 ⎦
0.74
⎡
1.00
³ ´ ⎢ 0.24
b =⎢
Corr Σ
⎣ 0.61
0.22
0.24
1.00
0.71
0.43
0.12%
0.02%
0.09%
0.03%
0.02%
0.06%
0.07%
0.04%
0.61
0.71
1.00
0.29
⎤
0.22
0.43 ⎥
⎥;
0.29 ⎦
1.00
(3) Symmetric spillover:
⎡
0.45
⎢ −0.29
b=⎢
A
⎣ 0.03
0.20
0.23 0.03
0.68 0.20
0.44 0.45
0.16 −0.29
⎤
⎡
0.44
0.18% 0.03%
⎢ 0.03% 0.06%
0.16 ⎥
b=⎢
⎥;Σ
⎣ 0.15% 0.08%
0.23 ⎦
0.68
0.04% 0.04%
⎡
1.00
³ ´ ⎢ 0.26
b =⎢
Corr Σ
⎣ 0.71
0.21
0.26
1.00
0.71
0.42
0.71
0.71
1.00
0.29
0.09%
0.07%
0.18%
0.05%
0.15%
0.08%
0.25%
0.06%
⎤
0.03%
0.04% ⎥
⎥;
0.05% ⎦
0.17%
⎤
0.04%
0.04% ⎥
⎥;
0.06% ⎦
0.17%
⎤
0.21
0.42 ⎥
⎥;
0.29 ⎦
1.00
The short period of observations (21 annual) leads to significant sample bias
problem. An unrestricted VAR(1) regression implies a very low autoregressive
coefficient (0.08) for foreign traded productivity shocks, which is problematic.
Our calibration is in line with that of Corsetti et al (2004) and Stockman and
Tesar (1995). An broadly accepted finding is, home traded production is the most
highly correlated with foreign traded productivity shocks, then with home nontraded productivity shocks, and they are least correlated with foreign non-traded
productivity shocks, as shown in above correlation matrix of shock residuals.
37
B. Appendix B - Testing Model Fit
B.1. Naive approach - RMSE and MAE
The root mean squared error (RMSE) and mean absolute error (MAE) are two
measures one can use for evaluating model accuracy. In the first instance we
can evaluate the accuracy of the model VCM, ΣM , by calculating the RMSE and
MAE of the difference ΣM − Σ0 , where Σ0 is the VCM of the data. This informal
approach provides an impression of the distance between two DGPs. The RMSE
and MAE are defined as:
RMSE =
MAE =
1 XX e 2
Σ ,
p2 i j i,j
1 XX ¯¯ e ¯¯
¯Σi,j ¯ ,
p2 i j
where the VCM forecasting error is:
e = ΣM − Σ0 .
Σ
B.2. Testing approach - Box’s M Test
The canonical hypothesis testing on equality of several covariance matrices is
developed by Bartlett (1937) and improved by Box (1949). For our problem, we
assume the filtered cyclical series (both data and model simulation) of certain
endogenous variables follow a multivariate normal distribution with zero mean.
Ybt ∼ N (0p×1 , Σ0 ) , t = 1, ..., T
¡
¢
YcS t ∼ N 0p×1 , ΣSM , t = 1, ..., K
38
(B.1)
where ΣSM means covariance matrix obtained from model simulation. The
covariance matrices are the unbiased estimate of sample covariance matrices.
The null and alternative hypotheses we want to test are:
H0 : Σ0 = ΣSM
(B.2)
H1 : Σ0 6= ΣSM
(B.3)
The philosophy of hypothesis testing is to obtain the likelihood ratio of two
scenarios: (1) Two sets of series are jointly distributed as a common multivariate
normal vector process. (2) Two sets of series are distributed with respect to
different process specified by Σ0 and ΣSM . The first scenario is the restricted
model as we will obtain the sample covariance matrix for a pooled sample by
combining observation of two sets of series. The second is the unrestricted case.
We multiply individual likelihood of two single observations to get the overall
likelihood. We firstly consider the unrestricted case. The likelihood function of
actual data on estimated data covariance matrix is:
LD =
1
1
1
(2π) 2 pT |ΣD | 2 T
#
T
´0
´
³
1X ³b
Yt − Yb Σ−1
Ybt − Yb
exp −
D
2 t=1
"
(B.4)
In the likelihood function ΣD denotes the estimator of population covariance
matrix. Estimate of population mean is just the sample mean Yb . The maximum
likelihood estimate of ΣD is given by:
´³
´0
1 X ³b
b
b
b
b
Yt − Y
ΣD =
Yt − Y
T t=1
T
eD .
When we plug in this estimate in LD , it achieves the maximum L
39
(B.5)
eD
L
"
#
T
³
´0
´
1
1X ³b
b −1 Ybt − Yb
Yt − Yb Σ
=
¯ ¯ 1 T exp − 2
D
1
2
¯
¯
bD¯
t=1
(2π) 2 pT ¯Σ
Ã
"
!#
T ³
´0 ³
´
X
1
1
b −1
Ybt − Yb
Ybt − Yb
=
¯ ¯ 1 T exp − 2 tr Σ
D
1
2
¯
¯
pT
bD¯
t=1
(2π) 2 ¯Σ
∙
¸
1
1
=
¯ ¯ 1 T exp − 2 tr (Ip×p T )
1
pT ¯ b ¯ 2
2
(2π)
¯ΣD ¯
¸
∙
1
1
(B.6)
=
¯ ¯ 1 T exp − 2 pT
1
2
¯
¯
pT
bD¯
(2π) 2 ¯Σ
For the rest group of the data, the likelihood function of simulated data on
estimated model covariance matrix is:
LS =
1
1
1
(2π) 2 pK |ΣS | 2 K
¶0
µ
¶#
K µ
1 X cS
exp −
Y t − YcS Σ−1
YcS t − YcS
S
2 t=1
"
(B.7)
In the likelihood function ΣS denotes the estimator of population covariance
matrix. Estimate of population mean is just the sample mean YcS . The maximum
likelihood estimator of ΣS is given by:
¶µ
¶0
K µ
X
1
c
c
c
c
S
S
S
S
bS =
Y t−Y
Y t−Y
Σ
K t=1
(B.8)
Similarly, we may obtain the maximized likelihood for simulated value.
¸
∙
1
1
eS =
L
¯ ¯ 1 K exp − 2 pK
1
pK ¯ b ¯ 2
2
(2π)
¯ΣS ¯
(B.9)
The likelihood function of two separate multivariate normal vector processes
are just the product of two likelihood function.
40
Lunrestricted = LD LS
eunrestricted = L
eS
eD L
L
(B.10)
For the restricted case, we pool all the observation together the assume they
have the same covariance matrix. The likelihood function is:
Lrestricted =
1
1
1
(2π) 2 p(T +K) |ΣP | 2 (T +K)
"
#
T +K
1X
exp −
(yt − y)0 Σ−1
(B.11)
P (yt − y)
2 t=1
where yt and y are observations and sample mean for the pooled series. We
obtain the maximum likelihood estimate for sample covariance matrix of it:
1 X
(yt − y)0 Σ−1
P (yt − y)
T + K t=1
T +K
bP =
Σ
(B.12)
This estimator make the likelihood for the pooled observation achieve a
maximum.
erestricted =
L
∙
¸
1
1
¯ ¯ 1 (T +K) exp − 2 p (T + K)
1
p(T +K) ¯ b ¯ 2
2
(2π)
¯ΣP ¯
(B.13)
The likelihood ratio is then obtained by:
1
λ=
erestricted
L
=
eunrestricted
L
|Σe P |
1
|Σe D |
1T
2
1 (T +K)
2
·
1
|Σe S |
1K
2
¯ ¯1T ¯ ¯1K
¯b ¯2 ¯b ¯2
¯ΣD ¯ ¯ΣS ¯
= ¯ ¯ 1 (T +K)
¯b ¯ 2
¯ΣP ¯
(B.14)
The log-likelihood ratio times -2 should follow a scaled Chi-square distribution
of degree of freedom
1
p (p
2
+ 1) since we can only consider upper triangle for
restriction of homogeneity in two covariance matrices.
41
µ
¶
¯ ¯
¯ ¯
¯ ¯
1
¯b ¯
¯b ¯
¯b ¯
2
−2 ln λ = (T + K) ln ¯ΣP ¯ − T ln ¯ΣD ¯ − K ln ¯ΣS ¯ ∼ Gχ
p (p + 1)
(B.15)
2
This test is very similar to Bartlett test for equality of several variances
in univariate case.
In fact, the determinant of sample covariance matrix is
the generalized variances and is the multivariate counterpart of Bartlett test
(Morrison, 1967). Because maximum likelihood estimate of covariance matrices
are biased, Bartlett (1937) worked on the univariate example and suggested using
the degrees of freedom of sum of square matrices. Then we replace T and K with
T − 1 and K − 1. Moreover, after evaluating the distribution of likelihood ratio,
Box (1949) use the scale factor C −1 to construct the Chi-square test statistics.
Revised for our two matrices case we have:
MC
−1
C −1
µ
¶
1
p (p + 1)
∼ χ
2
µ
¶
2p2 + 3p − 1
1
1
1
= 1−
+
−
6 (p + 1)
T −1 K −1 T +K −2
2
(B.16)
B.3. Distribution approach - KLIC
As argued by Watson (1993), Kullback-Leibler Information Criterion can be
used to measure similarity of two multinormal distributions in DSGE models.
For a statistical model, KLIC distance is a measure of distance between model
distribution f (z) and data distribution g (z). It is the expected log likelihood
ratio of two densities.
Z ∞
g (Y )
g (y)
I (g; f ) = EY ln
ln
=
g (y) dy
(B.17)
f (Y )
f (y)
−∞
When model represent the data precisely, the KLIC will be very close to zero.
Because model is always a simplified representation of the true data,
42
g(Y )
f (Y )
is
expected to be larger than one, therefore KLIC is larger than zero. KLIC can
become a model selection criterion and the model with lower KLIC is preferred.
Assume we have a DSGE model with data yt , which is an n × T vector, n
is the dimension of multivariate system and T is toal observation t = 1, ..., T .
Accordingly the model estimate the data series xt . Log PDF of these series:
1
1
n
ln g (z) = − ln 2π − ln |Sy | − (z − y)0 Sy−1 (z − y)
2
2
2
(B.18)
1
1
n
(B.19)
ln f (z) = − ln 2π − ln |Sx | − (z − x)0 Sx−1 (z − x)
2
2
2
Sx and Sy are variance covariance matrices of model and data. Equation (1)
is difficult to solve analytically when n is large. To obtain KLIC, two numerical
methods can be applied: Monte Carlo study and Bootstrapping method.
With Monte Carlo I draw k = 10, 000 observations with distribution of the
data. They are {zi } i = 1, ..., 10000. An estimated KLIC between model and data
is then:
1 X g (zi )
ln
IbMonte Carlo (g; f ) =
k i=1 f (zi )
k
(B.20)
With bootstrapping the true data itself is used to check the difference of pdf
of two set of data. Now we have {zi } i = 1, ..., T . The KLIC is now:
1 X g (zi )
ln
IbBootstrap (g; f ) =
k i=1 f (zi )
T
(B.21)
The Kullback-Leibler Information Criterion is very similar to a likelihood ratio
test. Note we can rewrite B.21:
k
k
1X
1X
b
ln g (zi ) −
ln f (zi )
IMonte Carlo (g; f ) =
k i=1
k i=1
43
(B.22)
=
´
1 ³b
bModel ({zMonte Carlo })
LData ({zMonte Carlo }) − L
k
(B.23)
This two methods are actually using simulated or realistic series to find out
how different are there between model distribution and data distribution. Or
equivalently, we look the log likelihood ratio of two alternatives: the data (without
restriction) and the model (with restriction). In this sense, KLIC is similar to
Box’s M test. But KLIC is only a measure for the distance of two distribution
instead a test. However, both Box’s M test and KLIC can be model selection
criterion.
B.4. Principal Component analysis for rank deficit problem
Likelihood ratio based approach in earlier section requires the VCM to be full rank.
But we constantly encounter rank deficit problem, i.e., the VCM is singular. The
reason might be: (1) Dynamic model have fewer driving forces than number of
endougenous variables; (2) The model implies linear relationship among some of
variables. One might choose a list of variables that have same dimension as driving
forces, or increase more shocks and measurement errors to the model, but they
are not preferable. The formal solution we present is to run principal component
analysis to decrease the dimension of endogenous variables.
The leading factor of a multivariate system can be represented by leading
eigenvalues and eigenvectors. Identification for driving forces can be achieved
by finding how many eigenvalues are significant. Significant eigenvalues and
eigenvectors convey information on relevant importance of each dimension of this
multivariate system. The philosophy here is to compare the data and model only
on these leading dimensions.
The procedure of PCA solution is as following. From solution of eigenvalues
and eigenvectors, we usually decompose a positive definite (or semidefinite) matrix
44
in the following way:
Γ = λ1 C1 C10 + λ2 C2 C20 + ... + λp Cp Cp0
(B.24)
(λ1 ...λp ) and (C1 ...Cp ) are eigenvalues and corresponding eigenvectors. λi ≥ 0,
i = 1...p. For positive semidefinite matrix some of eigenvalues are zero. By picking
out some information of "insignificant dimensions" we can construct a new matrix
by summing up first n principal components.
e = λ1 C1 C10 + λ2 C2 C20 + ... + λn Cn Cn0
Γ
(B.25)
−1
−1
0
0
0
e−1 = λ−1
Γ
1 C1 C1 + λ2 C2 C2 + ... + λn Cn Cn
¯ ¯
¯e¯
¯Γ¯ = λ1 λ2 ...λn
(B.26)
We could construct the inverse and determinant by:
(B.27)
From previous section we know that rank deficit problem make many likelihood
ratio based testing and measuring approach infeasible. PCA can help this out in
two possible ways: (1) By principal components decomposition, we can compare
the density of the multivariate normal distribution (MVN) of data with the density
of the multivariate singular normal distribution (MVSN) implied by the model;
(2) We can transform the original data and simulation by summarizing first n
principal components, where n is the number of shocks implied by the model.
In first approach, we adopt KLIC to measure distance between the two
distributions. The density of MVSN distribution with sample covariance matrix
Γ is:
1
g (z; z, Γ) =
(2π)
1
pT
2
¯ ¯1T
¯e¯ 2
¯Γ¯
∙
¸
1
0 e −1
exp − (z − z) Γ (z − z)
2
45
(B.28)
We use the generalized inverse and determinant according to equation (B.26)
and (B.27). For the data, we still measure the density as before. This new density
enable us to calculated the approximate KLIC.
Alternatively, we could reduce the dimension of the data and simulation with
the first n principal components. The implied assumption for doing so is there are
n driving factors for the multivariate process, represented by n exogenous shocks
we specify. The transformed series:
Yet =
YfS t =
h
h
C10 Ybt C20 Ybt . . . Cn0 Ybt
i
C10 YcS t C20 YcS t . . . Cn0 YcS t
(B.29)
i
Ybt denotes data and Yet denotes principal component from data. Superscript
s denotes simulation from calibrated model.
These new series both have a
nonsingular covariance matrices, therefore we can come back to usual KLIC
approximation and hypothesis test such as Box’s M-test and Nagao’s test. For
KLIC we are now calculating the density of original data and simulation in a
subspace of reduced dimensions.
B.5. Ledoit and Wolf’s W Test
Ledoit and Wolf (2004) construct following statistic to test whether a variance
covariance matrix S is identity matrix I.
∙
¸2
1 £
p
p 1
2¤
W = tr (S − I) −
tr (S) +
(B.30)
p
n p
n
The test statistics is a Chi-square distributed (asymptotically) number when
the dimension is very big.
npW
∼ χ2
2
µ
¶
1
p (p + 1)
2
46
(B.31)
We construct the S matrix with Cholesky Decomposition for positive definite
matrix.
Σ0 = P 0 P
(B.32)
S = P 0−1 ΣM P −1
I = P 0−1 Σ0 P −1
It is equivalent to test the hypothesis of ΣM = Σ0 and S = I. This approach
is proved to be superiors for large dimension system. Another advantage of this
approach is that it work for rank deficit problem.
47
1.80
1.60
Dispersion of
Relative Consumption and
the Real Exchange Rate
1.40
sd(cc*)/sd(y)
1.20
1.00
0.80
0.60
0.40
0.20
0.00
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
sd(rer)/sd(y)
1.00
International Output and
Consumption Correlations with US
0.80
corr(y, y_US)
0.60
0.40
0.20
-0.50
-0.40
-0.30
-0.20
-0.10
0.00
0.00
-0.20
-0.40
corr(c,c_US)
0.10
0.20
0.30
0.40
0.50
-0.60
-0.50
-0.40
-0.30
-0.20
-0.10
0.00
0.00
0.10
0.20
-0.10
Current Account and Trade Balance
Business Cycle Dynamics
-0.20
corr(TB,y)
-0.30
-0.40
-0.50
-0.60
-0.70
-0.80
corr(CA,y)
0.10
-0.60
-0.40
-0.20
corr(CA,y)
Real Exchange Rate and Current
Account Business Cycle Dynamics
0.00
0.00
0.20
0.40
-0.10
-0.20
-0.30
-0.40
-0.50
-0.60
corr(REER,y)
Figure 1: OECD International Economy Stylised Facts
0.60
0.80
UK Current Account and Output
0.2
0
-0.2
-0.4
-0.6
0
5
10
15
US Current Account and Output
20
25
5
10
15
20
UK Relativ e Consumption to US and Bilateral Real Exchange Rate
25
0.2
0
-0.2
-0.4
-0.6
0
0.2
0
-0.2
-0.4
-0.6
0
5
10
15
20
25
Quarters
Note: Forecast error correlation calculated with Den Haan (1996) code. Solid lines denote correlations
and the dotted lines represent a 90% conf idence interv al of point estimate.
Figure 2: Price Stickiness
Output
0.4
HomeHome
Ov erseas
Ov erseas
0.3
0.1
0.1
0.05
0
10
20
Labour input
0.15
0
30
HomeHome
Ov erseas
Ov erseas
0.1
C
C
C*
C*
CC*
CC*
0.15
0.2
0
Deviations in percentage
Consumption
0.2
0
0
10
20
Home Inv estment
4
2
-0.05
30
0
0
10
20
Net Foreign Asset
30
-2
-0.5
0
10
20
30
Current Account and Trade Balance
0.15
Traded
Traded
CACA
Nontraded
Nontraded
TBTB
0.1
Total
Total
0.05
0.05
0
Real Exchange Rate and Terms of Trade
1
RER
RER
TOTTOT
0.5
0
0
10
20
30
UIP
-0.05
0
10
20
30
Relativ e price of nontraded goods
0.8
HomeHome
Ov erseas
Ov erseas
0.6
0.2
0.15
0.15
0.1
0.1
0.05
0.4
0.05
0
0.2
0
0
10
20
30
-0.05
ds
ds
i
i
i*
i*
0
10
20
Years af ter shock
30
0
0
Figure 3: Response to Traded Productivity Shock
10
20
30
Output
Consumption
1.5
Real Exchange Rate and Terms of Trade
1
RER
TOT
0.5
1
Home
Ov erseas
1
C
C*
CC*
0.5
0.5
0
0
Deviations in percentage
-0.5
0
10
20
Labour input
30
0.6
-0.5
0
0
10
20
Home Inv estment
30
4
Home
Ov erseas
0.4
Traded
Nontraded
Total
2
0.2
0
10
20
30
Current Account and Trade Balance
0.1
CA
TB
0.05
0
0
0
-0.2
-0.5
0
10
20
Net Foreign Asset
30
-2
-0.05
0
10
20
30
UIP
0.4
1
0.3
ds
i
i*
0.5
-0.1
0
10
20
30
Relativ e price of nontraded goods
0
Home
Ov erseas
-0.5
0.2
0
0.1
0
0
10
20
30
-0.5
-1
0
10
20
Years af ter shock
30
-1.5
0
10
Figure 4: Response to Non-Traded Productivity Shock
20
30
Output
0.8
HomeHome
Ov erseas
Ov erseas
0.6
C
C
C*
C*
CC*
CC*
0.6
Real Exchange Rate and Terms of Trade
0.6
RER
RER
TOTTOT
0.4
0.4
0.4
0.2
0.2
0.2
0
0
Deviations in percentage
Consumption
0.8
0
10
20
Labour input
1.5
30
HomeHome
Ov erseas
Ov erseas
1
0
0
10
20
Home Inv estment
-0.2
0
10
20
30
Current Account and Trade Balance
0.04
Traded
Traded
CACA
Nontraded
Nontraded
TBTB
Total
0.02
Total
2
1
30
0.5
0
0
-0.5
0
10
20
Net Foreign Asset
30
-1
0.6
0.3
0.4
0.2
0.2
0.1
0
0
10
20
0
10
20
30
UIP
0.4
0
0
30
-0.2
ds
ds
i
i
i*
i*
-0.02
0
10
20
30
Relativ e price of nontraded goods
0.4
HomeHome
Ov erseas
Ov erseas
0.2
0
-0.2
0
10
20
Years af ter shock
30
-0.4
0
Figure 5: Response to Preference Shock
10
20
30
Output
Consumption
0.06
Home
Ov erseas
0.04
C
C*
CC*
0.1
0
0.02
0.05
-0.5
0
0
-1
-0.02
Deviations in percentage
Real Exchange Rate and Terms of Trade
0.5
0.15
0
10
20
Labour input
30
-0.05
0.01
0.6
0
0.4
10
20
Home Inv estment
30
Traded
Nontraded
Total
-0.02
0
10
20
Net Foreign Asset
30
-0.2
0
-0.25
-1
CA
TB
-0.1
0
10
20
30
1
-0.2
0
10
20
30
Current Account and Trade Balance
0.2
0
UIP
-0.15
-1.5
0.1
Home
0.2
Ov erseas
0
-0.01
-0.03
0
RER
TOT
ds
i
i*
-0.2
0
10
20
30
Relativ e price of nontraded goods
0.6
Home
Ov erseas
0.4
0.2
-0.3
0
10
20
30
-2
0
0
10
20
Years af ter shock
30
Figure 6: Response to UIP Shock
-0.2
0
10
20
30
1
2.8
SD(RER)/SD(CC*)
Correlation (CC*,RER)
2.6
0.8
0.6
0.4
2.4
2.2
2
1.8
1.6
0.2
0
2
4
Cost of inv estment
1.4
6
0
2
4
Cost of inv estment
6
0
2
4
Cost of inv estment
6
0.4
Correlation (CA,Y)
Corr(CA, TB)
1.2
1
0.8
0.6
0
2
4
Cost of inv estment
6
0.2
0
-0.2
-0.4
Figure 7: Sensitivity - Investment Cost (b)
2.8
0.9
2.6
SD(RER)/SD(CC*)
Correlation (CC*,RER)
1
0.8
0.7
0.6
0.5
0.4
0.3
2.4
2.2
2
1.8
1.6
0
0.01
0.02
0.03
0.04
Cost of f inancial intermediation
1.4
0.05
0
0.01
0.02
0.03
0.04
Cost of f inancial intermediation
0.05
0
0.01
0.02
0.03
0.04
Cost of f inancial intermediation
0.05
0.3
0.2
Correlation (CA,Y)
Corr(CA, TB)
1.2
1
0.8
0.1
0
-0.1
-0.2
-0.3
0.6
0
0.01
0.02
0.03
0.04
Cost of f inancial intermediation
0.05
-0.4
Figure 8: Sensitivity - Financial Intermediation Costs ( )
2.6
SD(RER)/SD(CC*)
0.8
0.6
0.4
2.4
2.2
2
1.8
1.6
0.2
-0.4
Corr(CA, TB)
2.8
1.4
-0.4
-0.2
0
0.2
0.4
Steady state NFA to Consumption ratio
1.5
0.4
1
0.2
Correlation (CA,Y)
Correlation (CC*,RER)
1
0.5
0
-0.5
-0.4
-0.2
0
0.2
0.4
Steady state NFA to Consumption ratio
-0.2
0
0.2
0.4
Steady state NFA to Consumption ratio
0
-0.2
-0.4
-0.4
-0.2
0
0.2
0.4
Steady state NFA to Consumption ratio
Figure 9: Sensitivity - NFA ratio (a)
10
0.5
SD(RER)/SD(CC*)
Correlation (CC*,RER)
1
0
-0.5
-1
0
0.2
0.4
0.6
0.8
Persistence of UIP Shock
4
2
0
0.2
0.4
0.6
0.8
Persistence of UIP Shock
1
0
0.2
0.4
0.6
0.8
Persistence of UIP Shock
1
0.4
0.2
Correlation (CA,Y)
1.2
Corr(CA, TB)
6
0
1
1.4
1
0.8
0.6
8
0
-0.2
-0.4
0
0.2
0.4
0.6
0.8
Persistence of UIP Shock
1
-0.6
Figure 10: Sensitivity - UIP Shock (
U IP )
3
1.4
SD(RER)/SD(CC*)
Correlation (CC*,RER)
1.6
1.2
1
0.8
0.6
0
1.5
0.5
2
4
6
Relativ e v olatility of traded productiv ity
to non-traded sector
1.6
0.6
1.4
0.4
Correlation (CA,Y)
Corr(CA, TB)
2
1
0.4
0.2
2.5
1.2
1
0.8
0.6
0.4
0.2
0
2
4
6
Relativ e v olatility of traded productiv ity
to non-traded sector
0
2
4
6
Relativ e v olatility of traded productiv ity
to non-traded sector
0.2
0
-0.2
-0.4
-0.6
0
2
4
6
Relativ e v olatility of traded productiv ity
to non-traded sector
-0.8
Figure 11: Sensitivity - Traded Sector Volatility (
A = AN )
3
1
SD(RER)/SD(CC*)
Correlation (CC*,RER)
1.2
0.8
0.6
0.4
0.2
0
2.5
2
1.5
1
0.02
0.04
0.06
Volatility of pref erence shocks (S.D.)
0
0.02
0.04
0.06
Volatility of pref erence shocks (S.D.)
0
0.02
0.04
0.06
Volatility of pref erence shocks (S.D.)
0.6
0.4
Correlation (CA,Y)
Corr(CA, TB)
1.2
1
0.8
0.2
0
-0.2
0.6
0
0.02
0.04
0.06
Volatility of pref erence shocks (S.D.)
-0.4
Figure 12: Sensitivity - Preference Shock Volatility (
x)
-4
10
6
x 10
-4
MAE
RMSE
8
4
6
4
15
0
4
x 10
2
4
Cost of inv estment
2
6
5
0
2
4
Cost of inv estment
6
2
4
Cost of inv estment
6
2000
0
6
6
W-TEST
NAGAO
x 10
2
4
Cost of inv estment
4
2
0
6
1000
8
6
2
4
Cost of inv estment
3000
10
0
0
4000
KLIC
BARTLETT
x 10
0
2
4
Cost of inv estment
6
0
x 10
8
4
2
0
0
Figure 13: Distance Measure - Investment Cost (b)
-4
x 10
3.5
-4
3
5
4.5
0
0.01
0.02
0.03
0.04
x 10 Cost of f inancial intermediation
0.05
2.5
7
2
4
1.5
1
0.5
0
0
0.01
0.02
0.03
0.04
x 10 Cost of f inancial intermediation
0
0.05
8
0
0.01
0.02
0.03
0.04
12
x 10 Cost of f inancial intermediation
0.05
0
0.05
6
W-TEST
NAGAO
0.05
1
6
4
2
0
0.01
0.02
0.03
0.04
5
x 10 Cost of f inancial intermediation
2
12
8
0
3
KLIC
BARTLETT
x 10
5.5
MAE
RMSE
6
4
2
0
0.01
0.02
0.03
0.04
Cost of f inancial intermediation
0.05
0
0.01
0.02
0.03
0.04
Cost of f inancial intermediation
Figure 14: Distance Measure - Financial Intermediation Costs ( )
-4
x 10
3.5
x 10
3
5.5
5
-0.4
2.5
-0.4
-0.2
0
0.2
0.4
state NFA to Consumption ratio
x Steady
10
4
-0.2
0
0.2
0.4
Steady state NFA to Consumption ratio
500
1.5
400
KLIC
BARTLETT
2
1
300
0.5
-0.4
200
-0.4
10
10
-0.2
0
0.2
0.4
6
state NFA to Consumption ratio
x Steady
10
6
4
2
-0.4
-0.2
0
0.2
0.4
6
Steady
state NFA to Consumption ratio
x 10
8
W-TEST
8
NAGAO
-4
6
MAE
RMSE
6.5
6
4
-0.2
0
0.2
0.4
Steady state NFA to Consumption ratio
2
-0.4
-0.2
0
0.2
0.4
Steady state NFA to Consumption ratio
Figure 15: Distance Measure - NFA ratio (a)
0.04
0.015
0.01
MAE
RMSE
0.03
0.02
0.005
0.01
0
x 10
4
0.2
0.4
0.6
0.8
Persistence of UIP Shock
0
1
0
0.2
0.4
0.6
0.8
7
Persistence of UIP Shock
x 10
1
400
4
3
W-TEST
NAGAO
0
0.2
0.4
0.6
0.8
7
Persistence of UIP Shock
x 10
1
0
1
600
2.5
2
1
1
800
3
4
0.2
0.4
0.6
0.8
Persistence of UIP Shock
1000
3.5
2
0
KLIC
BARTLETT
4
0
0
0.2
0.4
0.6
0.8
Persistence of UIP Shock
1
3
2
1
0.2
0.4
0.6
0.8
Persistence of UIP Shock
Figure 16: Distance Measure - UIP Shock (
U IP )
x 10
-3
3
4
1
0
2
4
6
4
Relativ
e v olatility of traded productiv ity
x 10
10
to non-traded sector
0
8
2
4
6
Relativ e v olatility of traded productiv ity
to non-traded sector
2000
6
1500
4
1000
2
500
1.5
1.5
0
2
4
6
8
Relativ
e v olatility of traded productiv ity
x 10
2
to non-traded sector
W-TEST
0
2
4
6
8
Relativ
e v olatility of traded productiv ity
x 10
2
to non-traded sector
1
0.5
0
0
2500
KLIC
BARTLETT
-3
2
2
0
NAGAO
x 10
MAE
RMSE
6
1
0.5
0
2
4
6
Relativ e v olatility of traded productiv ity
to non-traded sector
0
0
2
4
6
Relativ e v olatility of traded productiv ity
to non-traded sector
Figure 17: Distance Measure - Traded Sector Volatility (
A = AN )
-3
2
x 10
1.5
1
1
0.5
0.5
0
0
0
0.02
0.04
0.06
x 10Volatility of pref erence shocks (S.D.)
0
5
1.5
4000
1
2000
0.5
0
8
0
0
0.02
0.04
0.06
8
x 10Volatility of pref erence shocks (S.D.)
8
0
0.02
0.04
0.06
8
x 10Volatility of pref erence shocks (S.D.)
6
W-TEST
NAGAO
6
4
2
0
0.02
0.04
0.06
Volatility of pref erence shocks (S.D.)
6000
KLIC
BARTLETT
2
-3
MAE
RMSE
1.5
x 10
4
2
0
0.02
0.04
0.06
Volatility of pref erence shocks (S.D.)
0
0
0.02
0.04
0.06
Volatility of pref erence shocks (S.D.)
Figure 18: Distance Measure - Preference Shock (
x)
Table 1 - Open Economy Stylized Facts
AUS
Countries
Standard
Deviation
Output
Relative
CPI
Volatility
CPI/PPI
to GDP
REER
RER(CPI)
Consumption
Investment
Employment
CA/GDP
TB/GDP
Correlation
CPI
with GDP
CPI/PPI
REER
RER(CPI)
Consumption
Investment
Employment
CA/GDP
TB/GDP
AR(1)
CPI
Coefficient
CPI/PPI
REER
RER(CPI)
Consumption
Investment
Employment
CA/GDP
TB/GDP
Correlation REER - CA/GDP
RER - CA/GDP
2.03
0.74
0.67
3.33
3.39
0.67
2.73
0.77
0.25
0.07
-0.43
-0.43
0.25
-0.08
0.04
0.78
0.45
-0.43
-0.46
0.89
0.66
0.81
0.83
0.85
0.87
0.92
0.71
0.57
0.02
-0.05
CAN
2.28
0.51
0.99
1.95
1.50
0.43
2.40
0.58
0.28
0.08
-0.78
-0.75
-0.27
0.17
0.42
0.70
0.87
-0.27
-0.33
0.80
0.94
0.88
0.90
0.73
0.90
0.90
0.69
0.72
0.20
-0.16
DEU
2.69
0.39
0.41
1.02
3.11
1.16
1.64
0.32
1.65
0.31
-0.35
-0.36
-0.27
-0.07
0.86
0.89
0.56
-0.52
-0.62
0.82
0.84
0.84
0.81
0.79
0.70
0.85
0.49
0.73
0.29
-0.19
ESP
1.51
0.73
1.06
2.54
5.53
0.82
3.01
1.25
1.58
0.29
-0.61
0.24
0.55
-0.48
0.44
0.77
0.84
-0.36
-0.53
0.82
0.86
0.85
0.83
0.78
0.92
0.97
0.06
0.53
-0.27
0.24
FIN
3.10
0.37
0.48
1.64
2.92
0.62
2.49
0.68
1.58
0.43
-0.18
-0.30
0.61
-0.41
0.68
0.84
0.74
0.02
-0.29
0.83
0.78
0.89
0.84
0.86
0.86
0.96
0.27
0.32
-0.13
0.00
FRA
1.07
1.08
1.70
2.09
7.61
1.59
2.97
0.69
0.72
0.16
-0.38
-0.36
0.01
-0.28
-0.04
0.81
0.66
-0.22
-0.40
0.81
0.90
0.76
0.81
0.67
0.90
0.93
0.17
0.62
-0.17
0.20
GBR
1.61
0.92
0.75
3.15
4.54
0.68
3.19
0.83
0.98
0.25
-0.72
-0.32
-0.03
-0.12
0.47
0.62
0.63
-0.35
-0.43
0.76
0.79
0.75
0.72
0.83
0.88
0.92
0.47
0.59
0.10
-0.09
ITA
JPN
1.24
1.21
0.96
3.35
6.60
1.14
2.68
0.86
1.42
0.28
-0.44
-0.12
0.46
-0.46
0.40
0.58
0.59
-0.12
-0.26
0.87
0.83
0.84
0.79
0.73
0.88
0.87
0.17
0.48
-0.22
0.11
1.31
0.65
0.73
6.11
7.21
0.77
2.60
0.36
0.17
0.04
-0.26
-0.46
-0.42
0.26
0.47
0.88
0.31
-0.42
-0.40
0.71
0.73
0.82
0.80
0.62
0.86
0.84
0.55
0.81
0.19
0.00
Note: 1. Data is HP filtered OECD countries quarterly data from 1981Q1 to 2004Q4 from IMF database.
Hodrick-Prescott lambda coefficient is chosen to be 1600.
2. REER means real effective exchange rate. RER(CPI) is the real exchange rate calculated
from bilateral exchange rate with US. CA/GDP is the percentage of current account over GDP.
TB/GDP is the percentage of trade balance over GDP. Other data results are available on request.
USA
1.39
0.56
0.93
3.49
0.00
0.52
2.14
0.62
0.12
0.03
-0.62
-0.18
-0.09
----0.67
0.83
0.83
-0.50
-0.46
0.76
0.86
0.79
----0.75
0.87
0.87
0.72
0.66
0.01
-----
Table 2 - Annual Calibration for Two-Country Model
Values
Description
β
0.96 Discount factor
δ
0.1 Depreciation factor
α
0.67 Labor share
ρ
2 CRRA
ηX
-4 Parameter of labor supply
θ
1.5 Elasticity: Home/Foreign traded goods
κ
0.44 Elasticity: Traded/Non-traded goods
υ,υ*
(0.73,0.02) Share of Yh,Yf, in Yt, Yt*
ω,ω*
(0.26,0.21) Share of Yt,Yt*, in Y, Y*
ε
0.004 Interest spread
ā
0 Steady state Net Foreign Asset
b
2 Cost of capital adjustment
σA,σA*
(2.12%,2.43%) Volatility of traded productivity shocks
ρA,ρA*
0.65 Persistence of traded productivity shocks
σAN,σAN*
(1.33%,3.43%) Volatility of nontraded productivity shocks
ρAN,ρAN*
0.57 Persistence of nontraded productivity shocks
σx,σx*
(1.59%,1.44%) Volatility of preference shocks
ρx,ρx*
0.77 Persistence of preference shocks
Note: We have an utility function similar to Holland and Scott (1996). The elasticity of
intertemporal substitution in leisure is 0.2 and the elasticity of labor supply in this
model is about 4. Y and Y* denote UK and US real GDP; Yt and Yt* denote traded
goods consumed in each country; Yh and Yf denote UK produced traded goods
consumed in home and overseas.
Table 3 - Variance decomposition
Persistent UIP shocks c(iii) No spillover
c(ii) Symmetric spillover
Shocks
CC*
RER
CA
CC*
RER
CA
Traded, H
0.6%
0.0%
0.0%
5.2%
0.4%
0.3%
Nontraded, H
1.0%
0.4%
0.5%
36.0%
1.3%
0.1%
Traded, F
1.3%
0.1%
0.0%
0.1%
0.0%
0.0%
Nontraded, F
60.9%
4.4%
0.6%
5.8%
0.3%
0.0%
Preference, H
2.5%
0.1%
0.0%
8.7%
0.3%
0.1%
Preference, F
1.7%
0.1%
0.0%
2.3%
0.1%
0.0%
UIP
32.1%
94.9%
98.7%
42.0%
97.5%
99.4%
d(ii) Symmetric spillover
Transitory UIP shocksd(iii) No spillover
Shocks
CC*
RER
CA
CC*
RER
CA
Traded, H
0.9%
0.4%
0.7%
8.7%
5.7%
4.8%
Nontraded, H
1.5%
4.3%
8.3%
60.8%
20.6%
1.7%
Traded, F
1.9%
0.6%
0.1%
0.2%
0.1%
0.2%
Nontraded, F
88.5%
49.1%
9.9%
9.8%
5.0%
0.8%
Preference, H
3.6%
1.3%
0.7%
14.7%
5.2%
1.8%
Preference, F
2.4%
0.9%
0.3%
3.8%
1.4%
0.4%
UIP
1.3%
43.3%
80.1%
2.0%
62.0%
90.3%
Persistent UIP shocks c(ii) No spillover
c(iii) Symmetric spillover
Shocks
CC*
RER
CA
CC*
RER
CA
Preference, H
6.8%
0.1%
0.0%
16.4%
0.3%
0.1%
Preference, F
4.6%
0.1%
0.0%
4.3%
0.1%
0.0%
UIP
88.6%
99.8%
99.9%
79.3%
99.6%
99.9%
Transitory UIP shocksd(ii) No spillover
d(iii) Symmetric spillover
Shocks
CC*
RER
CA
CC*
RER
CA
Preference, H
49.5%
3.0%
0.9%
71.8%
7.6%
1.9%
Preference, F
32.9%
2.0%
0.4%
18.7%
2.0%
0.4%
UIP
17.6%
95.0%
98.8%
9.5%
90.4%
97.6%
Note: The measure shown here is unconditional variance explained by corresponding shocks
denoted by percentage. We take two-country base case calibration as in Table-2.
Table 4 - Results of no spillover model
No preference shocks
Excluding productivity shocks
Productivity & preference shocks
UK Data a(i)
b(i)
c(i)
d(i)
a(ii)
b(ii)
c(ii)
d(ii)
a(iii)
b(iii)
c(iii)
d(iii)
Relative volatility to output (excluding interest rate and CA/Y)
0.78
Consumption
0.58
0.58
0.87
0.60
0.61
0.61
0.99
0.64
0.59
0.59
0.78
0.60
2.30
2.24
2.23
3.08
2.39
2.03
2.04
3.30
2.28
2.16
2.16
2.75
2.26
Investment
1.01
Interest rate
0.19
0.19
0.57
0.34
0.07
0.07
0.58
0.30
0.20
0.20
0.53
0.35
0.95
0.36
0.36
0.47
0.36
1.41
1.42
1.35
1.40
0.93
0.93
0.93
0.92
Labor
0.86
Wage
0.66
0.66
0.86
0.67
0.46
0.46
0.84
0.50
0.59
0.59
0.74
0.60
4.89
RER
2.16
2.03
8.46
3.29
1.01
0.98
9.99
3.26
1.80
1.70
6.79
2.65
1.66
ToT
2.09
2.14
8.64
3.33
0.70
0.67
10.20
3.31
1.69
1.72
6.91
2.64
1.06
CA/Y
0.20
0.38
1.99
0.44
0.06
0.15
2.24
0.42
0.21
0.42
1.80
0.45
1.27
CC*
1.56
1.58
1.63
1.56
0.82
0.82
1.13
0.84
1.32
1.33
1.39
1.32
Correlation with output
0.79
Consumption
0.98
0.98
0.81
0.96
0.99
0.99
0.81
0.97
0.98
0.98
0.85
0.97
0.79
Investment
0.88
0.88
0.78
0.85
0.99
0.99
0.80
0.93
0.92
0.92
0.82
0.90
0.19
-0.91
-0.92
-0.68
-0.62
-0.97
-0.97
-0.53
-0.38
-0.88
-0.88
-0.64
-0.60
Interest rate
0.78
Labor
0.97
0.97
0.51
0.93
0.99
0.99
0.78
0.97
0.82
0.82
0.71
0.81
0.13
Wage
0.99
0.99
0.88
0.98
-0.86
-0.86
-0.07
-0.71
0.41
0.41
0.46
0.42
-0.10
RER
0.33
0.35
-0.17
0.13
0.70
0.73
-0.25
0.08
0.40
0.42
-0.10
0.20
-0.13
ToT
0.00
-0.01
-0.25
-0.09
0.69
0.72
-0.26
0.01
0.11
0.10
-0.17
0.00
-0.30
CA/Y
0.12
0.26
-0.25
-0.05
0.81
0.57
-0.30
-0.03
0.22
0.32
-0.18
0.02
0.19
CC*
0.38
0.38
0.46
0.39
0.72
0.71
0.71
0.72
0.45
0.44
0.49
0.45
Correlation with RER
0.10
ToT
-0.35
-0.41
0.93
0.45
0.98
0.98
1.00
0.99
-0.23
-0.28
0.93
0.47
-0.61
CC*
0.98
0.98
-0.19
0.55
1.00
1.00
-0.66
0.04
0.98
0.98
-0.15
0.58
Note: Please refer to the note of Table 6.
Table 5 - Results of spillover model
No preference shocks
Excluding productivity shocks
Productivity & preference shocks
UK Data a(i)
b(i)
c(i)
d(i)
a(ii)
b(ii)
c(ii)
d(ii)
a(iii)
b(iii)
c(iii)
d(iii)
Relative volatility to output (excluding interest rate and CA/Y)
0.78
Consumption
0.58
0.57
0.73
0.59
0.59
0.60
1.00
0.63
0.58
0.58
0.71
0.59
2.30
2.23
2.23
2.68
2.30
2.04
2.04
3.35
2.30
2.19
2.19
2.56
2.24
Investment
1.01
Interest rate
0.37
0.36
0.63
0.46
0.07
0.07
0.59
0.30
0.37
0.37
0.61
0.47
0.95
0.35
0.35
0.41
0.35
1.43
1.43
1.36
1.42
0.74
0.74
0.75
0.73
Labor
0.86
Wage
0.66
0.66
0.77
0.67
0.47
0.47
0.86
0.51
0.62
0.62
0.72
0.63
4.89
RER
1.06
1.00
5.85
2.00
1.25
1.21
10.27
3.43
1.10
1.05
5.24
1.86
1.66
ToT
0.97
0.98
5.96
2.02
0.86
0.82
10.47
3.44
0.95
0.95
5.32
1.83
1.06
CA/Y
0.11
0.39
1.67
0.40
0.06
0.18
2.27
0.42
0.13
0.43
1.62
0.41
1.27
CC*
0.90
0.91
1.00
0.91
1.00
1.01
1.26
1.02
0.92
0.93
1.00
0.93
Correlation with output
0.99
Consumption
0.79
0.99
0.87
0.98
0.99
0.99
0.80
0.97
0.99
0.99
0.89
0.98
0.98
Investment
0.79
0.98
0.89
0.96
0.99
0.99
0.80
0.93
0.98
0.98
0.90
0.97
-0.50
Interest rate
0.19
-0.50
-0.50
-0.45
-0.98
-0.98
-0.54
-0.38
-0.51
-0.52
-0.50
-0.46
0.98
0.78
0.98
0.71
0.96
0.99
0.99
0.78
0.97
0.78
0.78
0.70
0.78
Labor
0.99
Wage
0.13
0.99
0.93
0.99
-0.88
-0.87
-0.06
-0.72
0.68
0.68
0.67
0.68
0.06
RER
-0.10
0.10
-0.16
-0.03
0.85
0.86
-0.23
0.17
0.25
0.29
-0.10
0.09
0.44
ToT
-0.13
0.47
-0.11
0.14
0.83
0.84
-0.26
0.06
0.52
0.54
-0.07
0.21
0.64
CA/Y
-0.30
0.17
-0.14
0.11
0.87
0.67
-0.30
-0.01
0.69
0.27
-0.10
0.16
0.18
CC*
0.19
0.18
0.26
0.19
0.86
0.86
0.82
0.86
0.34
0.34
0.38
0.35
Correlation with RER
0.10
ToT
0.26
0.23
0.98
0.81
0.98
0.98
1.00
0.99
0.42
0.39
0.98
0.82
-0.61
CC*
0.97
0.98
-0.32
0.39
1.00
1.00
-0.58
0.14
0.98
0.98
-0.23
0.48
Note: Please refer to the note of Table 6.
Table 6 - Results of Distance Measures
No preference shocks
a(i)
b(i)
c(i)
No Spillover
3.46E-04 3.56E-04 4.88E-03
RMSE
1.89E-04 1.94E-04 1.78E-03
MAE
5.68E+04
4.29E+04 3.33E+04
BARTLETT
1.30E+03 9.86E+02 7.67E+02
KLIC(B)
5.10E+07 3.85E+07 2.90E+07
NAGAO
4.73E+07 3.64E+07 2.78E+07
W TEST
Spillover
4.89E-04 4.95E-04 4.90E-03
RMSE
2.63E-04 2.68E-04 1.83E-03
MAE
2.08E+04 1.99E+04 4.11E+03
BARTLETT
4.82E+02 4.64E+02 1.04E+02
KLIC(B)
5.66E+06 7.67E+06 3.80E+05
NAGAO
5.15E+06 7.20E+06 3.59E+05
W TEST
d(i)
Excluding productivity shocks
a(ii)
b(ii)
c(ii)
d(ii)
Productivity & preference shocks
a(iii)
b(iii)
c(iii)
d(iii)
4.61E-04 3.49E-04 3.50E-04 4.83E-03 4.17E-04 4.22E-04 4.29E-04 4.90E-03 5.33E-04
2.33E-04 1.46E-04 1.47E-04 1.74E-03 2.03E-04 2.62E-04 2.70E-04 1.81E-03 2.95E-04
4.35E+04 6.21E+09 5.75E+09 1.20E+11 6.96E+08 4.54E+04 3.91E+04 3.05E+04 3.59E+04
9.99E+02 1.39E+08 1.31E+08 2.72E+09 1.58E+07 1.05E+03 9.09E+02 7.11E+02 8.34E+02
4.59E+07 1.00E+18 8.86E+17 3.86E+20 1.28E+16 3.67E+07 3.42E+07 2.58E+07 3.33E+07
4.38E+07 9.60E+17 8.48E+17 3.70E+20 1.23E+16 3.43E+07 3.24E+07 2.47E+07 3.18E+07
5.82E-04 3.50E-04 3.51E-04 4.84E-03 4.23E-04 5.91E-04 5.96E-04 4.93E-03 6.82E-04
3.15E-04 1.61E-04 1.62E-04 1.74E-03 2.08E-04 3.56E-04 3.64E-04 1.87E-03 3.94E-04
7.50E+03 5.73E+09 5.05E+09 8.66E+10 7.33E+08 1.22E+04 1.17E+04 2.25E+03 4.25E+03
1.81E+02 1.28E+08 1.15E+08 1.97E+09 1.67E+07 2.87E+02 2.76E+02 5.71E+01 1.03E+02
8.96E+05 8.48E+17 6.77E+17 2.03E+20 1.43E+16 1.82E+06 2.96E+06 1.03E+05 2.47E+05
8.28E+05 8.13E+17 6.48E+17 1.94E+20 1.37E+16 1.64E+06 2.79E+06 9.70E+04 2.24E+05
Notes for Table 4 - 6:
1. UK data is calculated from the HP filtered series with the HP coefficient 6.25. Data range is 1980-2004, annual.
2. Data is UK GDP, consumption, investment, wage, interest rate, hours worked, real exchange, terms of trade, CA/GDP, CC* (RER is real exchange rate, ToT is the terms of trade,
CA is current account balance and CC* is the relative consumption of UK to US.)
3. Model details
Parameter
a(i), a(ii), a(iii)
ā
UIP Shock?
Persistence of UIP Shocks
0
no
b(i), b(ii), b(iii)
0.2
no
-----
c(i), c(ii), c(iii)
0
yes
0.8
d(i), d(ii), d(iii)
-----
yes
0
0.32
ā is the steady state net foreign asset position as percentage of household consumption.