Working Paper No. 432
An estimated DSGE model of energy,
costs and inflation in the United Kingdom
Stephen Millard
July 2011
Working Paper No. 432
An estimated DSGE model of energy,
costs and inflation in the United Kingdom
Stephen Millard(1)
Abstract
In this paper, I estimate a dynamic stochastic general equilibrium (DSGE) model of the
United Kingdom. The basic building blocks of the model are standard in the literature. The main
complication is that there are three consumption goods: non-energy output, petrol and utilities; given
relative prices and their overall wealth, consumers choose how much of each of these goods to consume
in order to maximise their utility. Each of the consumption goods is produced according to a
sector-specific production function and sticky prices in each sector imply sector-specific New Keynesian
Phillips Curves. I show how this model, once estimated, could form a useful additional input within a
policymaker’s ‘suite of models’ by considering its implications for the responses of various
macroeconomic variables to different economic shocks and by decomposing recent movements of
energy and non-energy output and inflation into the proportions caused by each of the shocks.
Key words: Dynamic stochastic general equilibrium model, energy prices and inflation.
JEL classification: E13, E31.
(1) Bank of England. Email: stephen.millard@bankofengland.co.uk
The views expressed in this paper are those of the author, and not necessarily those of the Bank of England. I am grateful to
Maria Barriel and Ryland Thomas for putting together the data set used to estimate the model and for helpful discussions about
the model itself. I am also grateful to Neal Hatch, Simon Price and seminar participants at the Bank of England for useful
comments. This paper was finalised on 25 March 2011.
The Bank of England’s working paper series is externally refereed.
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Working Paper No. 432 July 2011 2
Contents
Summary 3
1 Introduction 5
2 The model 6
2.1 Households 7
2.2 Non energy producing firms 9
2.3 Value-added sector 10
2.4 Petrol producers 10
2.5 Utilities producers 11
2.6 Monetary and fiscal policy 12
2.7 Foreign sector 12
2.8 Market clearing 13
2.9 Shock processes 13
3 Estimation 14
3.1 Data 14
3.2 Priors 15
3.3 Estimation results 17
3.4 Effects of including/excluding energy 21
4 Impulse response functions 22
4.1 How does energy affect the responses of variables to shocks? 23
4.2 The effects of a rise in energy prices 31
5 Using the model to decompose movements in output and inflation 33
6 Conclusions 38
References 39
Working Paper No. 432 July 2011 3
Summary
The job of monetary policy makers is to set monetary policy so as to achieve their goal of low
and stable inflation. In order to carry this out, it is important to understand what drives inflation
and how changes in monetary policy feed through the economy into inflation. But no single
model can capture all aspects of reality. This is why many central banks have used, and
continue to use, a variety of macroeconomic models to help in their understanding of inflation.
The main purpose of this paper is to estimate a model of the United Kingdom that, unusually,
includes an energy sector. It could in principle be used as another input within a policymaker’s
‘suite of models’.
The standard model of inflation suggests that it is driven by lagged and future expected inflation
and movements in costs. One important cost for most producers is the cost of energy. So,
inflation will be affected by movements in energy prices. In addition, to the extent that
consumers use energy themselves, movements in energy prices will have a direct, and
immediate, effect on consumer price inflation, which is not necessarily captured by standard
models. The novelty of this paper, relative to previous work, is that the model takes seriously
the effects of movements in energy and other costs on inflation. The goal is to produce a
macroeconomic model that can be used to analyse quantitatively the effects on inflation of many
temporary shocks, including but not limited to energy prices as well as how monetary policy can
respond to such shocks. Furthermore, estimating the model enables us to evaluate how these
shocks have evolved over time and the implications of this for explaining movements in output
and inflation.
The basic building blocks of the model are standard. The main complication is that there are
three consumption goods: non-energy output, petrol and utilities (which can be thought of as a
combination of gas and electricity). Each of these consumption goods is produced using
different combinations of five inputs: labour, capital, imported (non-energy) intermediates, oil
and gas. The prices set by the producers of these goods are sticky. Demand for oil and gas over
and above what we produce has to be met from abroad. The central bank affects aggregate
demand via movements in interest rates. How this level of aggregate demand translates into
demand for each of the goods is determined by consumers’ preferences and relative prices.
Finally, the model adds a government that ‘eats up’ some of the non-energy good and levies
taxes as well as a specific duty on petrol.
The estimates suggest, not surprisingly, that petrol prices are highly flexible, utility prices are
quite flexible, while non-energy prices, on the other hand, are very sticky. The relative
stickiness of prices in the three sectors are in line with survey and other evidence for the United
Kingdom. In terms of the shocks, the estimates suggest that the productivity shock is fairly
persistent but the others much less so; the model is able to explain persistence in the data
without having to resort to extremely persistent shocks. The estimated standard deviation of
monetary policy shocks is very low, not altogether surprising given that the model was
estimated over the inflation-targeting period. But, the domestic demand and investment-specific
technology shocks are highly volatile over this period. Finally, the estimates suggest that the
Working Paper No. 432 July 2011 4
model including energy prices is better able to explain UK macroeconomic data than an
otherwise identical model that does not include energy prices.
Given these estimates, it is possible for the model’s user to apply the model quantitatively to UK
policy issues. The paper has shown how this could be done by examining the effects of many
different shocks on inflation and by decomposing recent movements in output and inflation into
those parts caused by each of the model’s structural shocks. It found that the fall in gross nonenergy
output from 2008 Q2 to 2009 Q3 was driven by three shocks: to productivity, to world
demand and to the domestic risk premium, proxying the effects of the recent financial crisis.
The risk premium shock also put downwards pressure on inflation during this period while the
productivity shock was putting upwards pressure on inflation. The world demand shock, by
contrast, was much less important in explaining the behaviour of inflation over this period.
Working Paper No. 432 July 2011 5
1 Introduction
The job of monetary policy makers is to set monetary policy – by which I typically mean
interest rates, though currently many central banks are operating directly on bank reserves
through quantitative easing – so as to achieve their goal of low and stable inflation. But in order
to carry out this job, it is important to understand what drives inflation and how changes in
monetary policy feed through the economy into inflation. This is why many central banks have
used, and continue to use, a variety of macroeconomic models to help in their understanding of
inflation. The main purpose of this paper is to estimate a dynamic stochastic general
equilibrium (DSGE) model of the United Kingdom that could be used as another input within a
policymakers ‘suite of models’.
Previous authors have estimated DSGE models for the United Kingdom, eg, Di Cecio and
Nelson (2007), Harrison and Oomen (2010), Kamber and Millard (2010) and Faccini et al
(2011). The standard model of inflation – as embodied in the models estimated by all of these
authors – is based around the ‘New Keynesian Phillips Curve’ (NKPC), which suggests that
inflation is driven by lagged and future expected inflation and real marginal cost. Typically in
these models, real marginal cost will be equivalent to real unit labour costs (the ‘labour share’),
although, as shown by Faccini et al and Kamber and Millard, this is not the case in models
where hiring and firing costs are important and real marginal cost has to be amended
accordingly.
But importantly for this paper, when labour and energy are complementary inputs to production,
real marginal cost will also be affected by movements in energy prices. Hence, given NKPC
theory, movements in energy prices will be important for inflation. In addition, to the extent
that consumers use energy themselves, movements in energy prices will have a direct effect on
consumer price inflation, which is not necessarily captured by the NKPC. This effect was
clearly seen recently in the United Kingdom as the rise in oil prices from $75 a barrel in 2007
Q3 to $121 a barrel in 2008 Q2 was associated with a rise in CPI inflation from 1.8% in 2007
Q3 to 4.8% in 2008 Q3. So, the novelty of this paper, relative to those of Di Cecio and Nelson
(2007), Harrison and Oomen (2010), Kamber and Millard (2010) and Faccini et al (2011) is that
the goal is to estimate a model that takes seriously the effects of movements in all the elements
within firms’ costs – labour, capital, imported intermediates and energy – on inflation, and that
can be used to analyse how a central bank should respond to movements in energy prices in
order to achieve its inflation target.
There is a large literature that seeks to understand the effects of movements in energy prices on
output and inflation.1
1
See Blanchard and Gali (2007) for a review of the relevant empirical literature.
Most of this literature uses a structural VAR approach in which shocks to
oil prices have typically been identified as in Rotemberg and Woodford (1996). The idea is that
the nominal price of oil is determined by the worldwide demand and supply of oil and, so, can
be thought of as exogenous to output and inflation (and other variables) within any given
economy. This implies that, to examine the effects of an exogenous oil price shock, all the
researcher needs to do is to run a VAR and calculate the impulse response functions based on
Working Paper No. 432 July 2011 6
the oil price being ordered first in the VAR. An alternative approach to identifying oil price
shocks has been to consider specific dates on which the oil price moved in a dramatic (that is,
‘exogenous and unforeseeable’) way. Hamilton (1985) came up with a list of dates on which
such ‘oil shocks’ had happened and this list was extended by Hoover and Perez (1994) who used
monthly data. Most recently, Cavallo and Wu (2006) develop measures of exogenous oil-price
shocks for the period 1984 to 2006 based on market commentary (specifically that found in Oil
Daily and Oil and Gas Journal) on daily oil price fluctuations. They then regress output and
inflation on these measures to find out how they respond to oil price shocks.
All of these empirical approaches find that oil shocks have large effects on output and inflation.
But, constructing a model in which oil has large effects has proven to be difficult. Hamilton
(2008), for example, shows that given the share of energy in production in the United States and
the elasticity of output with respect to a change in energy use, movements in oil prices can only
explain a small fraction of the falls in GDP typically seen after oil price rises. Kim and
Loungani (1992) and Dhawan and Jeske (2008) show the same thing in DSGE models. Against
this, Rotemberg and Woodford (1996) argue that under imperfect competition with
countercyclical desired mark-ups it is possible to generate falls in output in line with the
empirical results.
In this paper, I take a DSGE model and estimate it using UK data. The model itself is not
original: it is that developed in Harrison et al (2011) to analyse the effects of the large rise in oil
prices between 2003 and 2008 on UK inflation. But, the emphasis in the two papers is different.
Harrison et al are interested in analysing the effects of energy theoretically, with a particular
emphasis on the implications of permanent energy price shocks for economies with declining
stocks of natural resources, such as the United Kingdom. In contrast, the goal of this work is to
produce a macroeconomic model that can be used to analyse quantitatively the effects of many
temporary shocks, including but not limited to energy prices, on inflation as well as how
monetary policy can respond to such shocks. Furthermore, estimating the model enables us to
evaluate how these shocks have evolved over time and the implications of this for explaining
movements in output and inflation.
The rest of the paper is structured as follows. Section 2 lays out the model of Harrison et al
(2011). Section 3 discusses the data and the estimation procedure and presents the estimation
results. Section 4 discusses the implications of the estimates for the responses of
macroeconomic variables to the shocks in the model and Section 5 shows the evolution of these
shocks over time and decomposes recent movements in output and inflation among them.
Section 6 concludes.
2 The model
The basic building blocks of the Harrison et al (2011) model are standard in the literature. The
main complication is that there are three consumption goods: non-energy output, petrol and
utilities (which can be thought of as a combination of gas and electricity). This approach is
similar to that of Kim and Loungani (1992) and Dhawan and Jeske (2008), who allow for
consumption of energy and non-energy; the current model goes further by splitting energy into
Working Paper No. 432 July 2011 7
petrol and utilities. The central bank operates a Taylor rule that affects aggregate demand via an
IS curve. How this level of aggregate demand translates into demand for each of the goods is
determined by preferences and relative prices. Wage inflation depends on total hours worked in
a ‘Phillips Curve’ relationship. Each of the consumption goods is produced according to a
sector-specific production function and sticky prices in each sector imply sector-specific New
Keynesian Phillips Curves (NKPCs). The production functions themselves involve different
combinations of five inputs: labour, capital, imported (non-energy) intermediates, oil and gas.2
At the margin, demand for oil and gas has to be met by reducing our net exports of these goods
(increasing our net imports). Finally, the model adds a government that ‘eats up’ some of the
non-energy good and levies taxes as well as a specific duty on petrol. This model incorporates
nominal rigidities in the goods and labour markets and real rigidities such as habit formation in
consumption, investment adjustment cost and variable capital utilisation. In what follows, I just
present the log-linear equilibrium conditions; a detailed derivation can be found in the technical
appendix to Harrison et al
2.1 Households
Households consume the three final goods and supply differentiated labour to the firms. They
are also assumed to own the capital stock and make decisions about capital accumulation and
utilisation. This assumption, now standard in the business cycle literature, is done in order to
simplify the firms’ decision problem. The following set of equation determines the household’s
choice of consumption, capital accumulation and utilisation:
( )
( ) ( ) ( ) 







+





−−−
−+
−
−+
+
−+
−
= 1++− tbtctt
chab
c
tt
chab
t
chab
chab
t EicEcc ,,11 1
1
11
ˆ
11
1
ˆ
11
1
ˆ ε
β
π
σψ
σ
σψσψ
σψ
(1)
( )
( )
tinvtkt
z
z
tkk
tt
z
k
tk
z
k
tkk
z
k
tbtctt
wEk
kEkkEi
,1,2
11,,
ˆ
1
ˆ
ˆ
1
ˆ1
1
1ˆ1
1
1
1
ε
χδ
χ
εχ
χδ
χ
χ
χδ
ε
χε
χδ
ε
ε
β
π
+
+−
+−
+−
+





+
+−
+
−





++
+−
=







+





−−−
+−
+−1+
(2)
tztk zw ˆˆ , φ= (3)
where c is consumption, i is the nominal interest rate, πc is the inflation rate of consumer prices,
εb is best thought of as a risk premium shock, wk is the rental rate for capital, εinv is an
investment-specific technology shock, z is the capital utilisation rate and k is the capital stock.3
2
This represents a difference to the approach of Kim and Loungani (1992) and Dhawan and Jeske (2008) who assume that energy –
petrol and utilities in the current model – is not produced but rather is directly imported.
Variables without time subscripts refer to their steady-state values and ‘hatted’ variables
represent log deviation from trend. In terms of the parameters, ψhab represents the degree of
habit formation in consumption, σc is the intertemporal elasticity of substitution, β is the
discount rate, χk scales the costs of adjusting the capital stock, χz scales the effect of capital
3
The investment-specific technology shock reduces the costs of adjusting the capital stock and so means that a given level of
investment will add more to the capital stock.
Working Paper No. 432 July 2011 8
utilisation on the depreciation rate, δ is the steady-state depreciation rate and φz is the inverse
elasticity of the capital utilisation cost function.
Equation (1) is the consumption Euler equation. Consumption depends on past consumption
due to external habit formation. As a result, the elasticity of consumption to the interest rate
depends not only on the elasticity of substitution but also on the degree of habit formation
parameter. Equation (2) is the capital accumulation equation in which lagged capital appears
due to the assumption of capital adjustment costs.4
Aggregate consumption is composed of consumption of petrol, utilities and ‘non-energy’.
Consumption of ‘energy’ will be given by:
Equation (3) determines capacity utilisation
as a function of the rental rate of capital.
( ) tPptUptE ccc ,,, ˆˆ1ˆ ψψ +−= (4)
and, hence, aggregate consumption will be given by:
( ) teetnet ccc ,, ˆˆ1ˆ ψψ +−= (5)
Relative prices will be given by:
tU
p
tE
ep
tn
e
tU cccp ,,,,
ˆ
1
ˆ
11
ˆ
1
ˆ
σσσσ
−








−+= (6)
and
tP
p
tU
p
tPtU ccpp ,,,, ˆ
1
ˆ
1
ˆˆ
σσ
+−=− (7)
Households also have the option of holding either foreign or domestic bonds but trade in foreign
bonds incurs quadratic costs. This results in the UIP condition:
trftfbftttt bissE ,,1 1
1
ˆˆ εχ
β
+−













−−−=−+ (8)
where s is the nominal exchange rate, χbf is a parameter determining the cost of holding foreign
bonds and εrf is a shock to world real interest rates. As a normalisation, I denote foreign bond
holdings as a proportion of non-energy output and I assume, without loss of generality, that the
supply of domestic government bonds is zero in all periods; that is, the government balances its
budget via lump-sum taxes on consumers.
4
Note that, following Harrison and Oomen (2010), I assume capital adjustment costs rather than the investment adjustment costs, more
often used in the literature. Although this formulation is much more intuitive than the more standard formulation, it means that the
model is unable to capture the hump-shaped dynamics of investment.
Working Paper No. 432 July 2011 9
Each household is a monopoly supplier of differentiated labour. Thus, they set their wages as a
mark-up over their marginal rate of substitution between leisure and consumption (percentage
deviation denoted by mrs), subject to nominal wage stickiness and partial indexation of wages to
inflation. Hence, wage inflation will be given by:
( )( )
( )( )
( ) twtt
ww
h
w
ww
tt
w
t
w
w
t mrswWEWW ,11
ˆ
111
11
11
ε
βξψ
σ
σ
ψβψ
βξ
β
βξ
ξ
+−
+−





+
−−
−
+
+
+
= +−
 (9)
where W is nominal wage inflation and εw is a wage mark-up shock. Here ψw is the share of
household members able to reoptimise their wages and ξw governs the extent to which non
optimised wages are indexed to past inflation. The steady-state wage mark-up is given by
1−w
w
σ
σ
and σh denotes the Frisch elasticity of labour supply. The equations defining the
marginal rate of substitution and the real consumption wage are:
( )( )1ˆ1ˆ
1ˆ1
−−++= tchabt
c
t
h
t cchmrs σψ
σσ
(10)
tcttt wWw ,1ˆˆ π−+= −
 (11)
2.2 Non energy producing firms
The representative non-energy producing firm is assumed to have the following production
function:
( ) tatqtqt eBq ,ˆˆ1ˆ εαα ++−= (12)
where q denotes output of non-energy, and εa represents a shock to this. B denotes a bundle of
value-added, Vn, and intermediate imported goods, Mn:
( ) tnBtnBt MVB ,,
ˆˆ1ˆ αα +−= (13)
and e denotes energy input in this sector, which will be given by:
tutpt IIe ,,
ˆˆˆ == (14)
where Ip is input of petrol, and Iu is input of utilities, both to the non-energy sector.
Cost minimisation implies the following demand curves for value-added, imports and energy:
ta
q
q
t
q
q
t
q
tvcttn BqpV ,,,
1
ˆ
1
ˆ
1
ˆˆˆ ε
σ
σ
σ
σ
σ
µ
−
+
−
++−= (15)
Working Paper No. 432 July 2011 10
ta
q
q
t
q
t
q
tmttn BqpM ,,,
1
ˆ1
1
ˆ
1
ˆˆˆ ε
σ
σ
σσ
µ
−
+








−−+−= (16)
( )( ) ( ) taqtUntpnqttqt ppqe ,,, 1ˆ1ˆˆˆˆ εσψψσµσ −+−+−+= (17)
where µ is real marginal cost and pvc is the ‘competitive’ price of value-added (the marginal cost
of producing it). Firms in the non-energy sector are also subject to nominal rigidities in their
price-setting. In particular, each period they are only allowed to set their price optimally with a
probability of 1-χp. If they cannot change their price optimally, they partially index their price
to lagged inflation. The resulting NKPC is:
( )( )
( ) tt
p
pp
tttt E ,11 ˆ
1
11
11
µεµ
χβε
βχχ
π
βε
ε
π
βε
β
π +
+
−−
+
+
+
+
= −+ (18)
where ε is the degree of indexation and εµ is a price mark-up shock.
2.3 Value-added sector
‘Value-added’ producers use labour and capital to produce value-added, V:
( ) ( )ttvtvt zkhV ++−= −1
ˆˆ1ˆ αα (19)
The term in z shows that the capital effectively used in production depends on the intensity of
capital utilisation. Unlike Harrison et al (2011), I assume value-added producers need to borrow
the money to finance a proportion, ψwc, of their wage bill. This can be motivated by the fact that
firms typically need to borrow to finance their working capital needs: that is, the need for funds
to cover the gap between production and when the firm is able to sell its output. This
assumption has been used by many others, eg, Fuerst (1992) and Christiano and Eichenbaum
(1992, 1995), and implies a ‘cost channel’ of monetary transmission.
Cost minimisation by value-added producers implies the following demand curves for capital
and labour:
















+





−−−−+= tbtwcttvcVtt iwpVh ,, 1
1
ˆˆˆˆ ε
β
ψσ (20)
( )tktvcVttt wpVzk ,,1 ˆˆˆˆˆ −+=+− σ (21)
2.4 Petrol producers
Petrol, qp, is produced using inputs of crude oil, Io, and value-added, Vp. I assume a simple
Leontieff production function:
tptotp VIq ,,,
ˆˆˆ == (22)
Working Paper No. 432 July 2011 11
The motivation for this choice of production function is that it is not clear how adding more and
more workers to a given amount of oil could physically increase the amount of petrol that can be
produced from it. Firms in this sector are also subject to nominal rigidities in their price-setting.
In this case, they are able to optimally change their price in any given quarter with probability 1χpp
and εpp represents the degree of indexation. The resulting NKPC is:
( )( )
( ) tp
pppp
pppp
tpb
pp
pp
tpbt
pp
tpb E ,1,1,, ˆ
1
11
11
µ
χβε
βχχ
π
βε
ε
π
βε
β
π
+
−−
+
+
+
+
= −+ (23)
where real marginal cost in this sector will be given by:
( ) tpbtoqptvcqptp ppp ,,,, ˆˆ1ˆˆ −−+= ψψµ (24)
where po is the price of oil and ppb is the basic (pre-duty) price of petrol. Finally, I can note that
by definition:
1,,, ˆˆ −−+= tpbtpbttpb ppππ (25)
2.5 Utilities producers
Output of utilities, qu, is produced using inputs of gas, Ig, and value-added, Vu. I assume a
simple Leontieff production function:
tutgtu VIq ,,,
ˆˆˆ == (26)
Again, the motivation for this choice of production function is that it is not clear how adding
more and more workers to a given amount of natural gas could physically increase the amount
of gas and electricity that can be produced from it. Firms in this sector are also subject to
nominal rigidities in their price-setting. In this case, they are able to optimally change their
price in any given quarter with probability χu and εu represents the degree of indexation. The
resulting NKPC is:
( )( )
( ) tu
uu
uu
tu
u
u
tut
u
tu E ,1,1,, ˆ
1
11
11
µ
χβε
βχχ
π
βε
ε
π
βε
β
π
+
−−
+
+
+
+
= −+ (27)
where real marginal cost in this sector will be given by:
( ) tutgutvcutu ppp ,,,, ˆˆ1ˆˆ −−+= ψψµ (28)
where pg is the price of gas and pu is the price of utilities. Finally, I can note that by definition:
1,,, ˆˆ −−+= tututtu ppππ (29)
Working Paper No. 432 July 2011 12
2.6 Monetary and fiscal policy
Monetary policy is assumed to follow a Taylor rule with the central bank responding to
deviations of inflation from target and value-added from flexible-price value-added:
( )( ) titytcpdotrgtrgt yii ,,1
ˆ11
1
1
1
εθπθθ
β
θ
β
++−+













−−=





−− − (30)
where εi is a monetary policy shock. Flexible-price value-added is defined as what value-added
would be in a flexible-price version of the model given the estimated values of the shocks.
The fiscal authority levies a duty on petrol. In my estimation, I assume that this is not changed
over time. Given that, I obtain:
( ) tpbdtp pp ,, ˆ1ˆ ψ−= (31)
That is, since it is held constant, the petrol duty has no role other than to reduce the impact of a
change in petrol producers’ other costs on the final price of petrol paid by consumers.
Since, I assume, as said earlier, that the government balances its budget using lump-sum taxes
on consumers (denoted by T), we can write the government’s budget constraint as
ttptpdt TqPG += ,,ψ . Unanticipated changes in government spending will form part of the
‘domestic demand’ shock that I discuss below.
2.7 Foreign sector
I assume that the United Kingdom is a small open economy. Hence, world prices are
exogenous. Oil and gas prices adjust immediately to their world prices:5
tpto sp o
ˆˆ , −= ε (32)
tptg sp g
ˆˆ , −= ε (33)
where opε is a shock to world oil prices and gpε is a shock to world gas prices.
UK import prices, on the other hand, take time to adjust to purchasing power parity. This results
in the NKPC for import prices:
( )( )
( ) ( )tmtp
pmpm
pmpm
tmt
pm
tm
pm
pm
tm psE mf ,1,1,,
ˆˆ
1
11
11
−−
+
−−
+
+
+
+
= +− ε
ξβι
βξξ
π
βι
β
π
βι
ι
π (34)
where mpε is a shock to the world price of our imports.
Finally, I assume the following demand function for our exports of non-energy goods:
5
For simplicity, I ignore issues about different varieties of crude oil as well as refining costs.
Working Paper No. 432 July 2011 13
( )( )txyztnztn sxx f
ˆ1ˆˆ 1,, ηεψψ −−+= − (35)
where fyε is a world demand shock.
2.8 Market clearing
I close the model with the following market-clearing conditions:
( ) ( )tPtP
c
uu
c
n
tUtU
c
uu
tn
c
n
ttc cp
cp
cp
cp
c
cp
cp
cp
c
cp
c
cp ,,,,,, ˆˆ1ˆˆˆˆˆ +







−−+++=+ (36)
tp
un
tu
u
tn
n
t V
V
V
V
V
V
V
V
V
V
V
V ,,,
ˆ1ˆˆ






−−++= (37)
tP
P
P
tP
P
P
tP I
q
c
c
q
c
q ,,,
ˆ1ˆˆ 





−+= (38)
tU
U
U
tU
U
U
tU I
q
c
c
q
c
q ,,,
ˆ1ˆˆ








−+= (39)
tO
o
o
tO X
I
X
I ,,
ˆˆ −= (40)
tG
g
g
tG X
I
X
I ,,
ˆˆ −= (41)
( )
tg
g
tn
n
t
z
tttn
n
t
q
c
x
q
x
z
q
k
k
q
k
k
q
k
c
q
c
q ,,1, ˆˆˆ1ˆˆˆ ε
χδ
+++
−
−+= − (42)
( ) ( ) ( )tntm
n
toto
o
tgtg
g
tn
n
tftf Mp
q
M
Xp
q
X
Xp
q
X
x
q
x
bb ,,,,,,,1,,
ˆˆˆˆˆˆˆ
1
+−+++++= −
β
(43)
where εg is a shock to the exogenous components of domestic demand shock. This can be
thought of as combining shocks to government spending, stockbuilding and the part of
investment that cannot be explained via the cost of capital.
2.9 Shock processes
As is common in the literature, I suppose that the shocks follow AR(1) processes:
tataata ,1,, ηερε += − (44)
tbtbbtb ,1,, ηερε += − (45)
tgtggtg ,1,, ηερε += − (46)
tItiiti ,1,, ηερε += − (47)
ttt ,1,, µµµµ ηερε += − (48)
tinvtinvinvtinv ,1,, ηερε += − (49)
twtwwtw ,1,, ηερε += − (50)
tytyyty ffff ,1,, ηερε += − (51)
tptpptp mfmfmfmf ,1,, ηερε += − (52)
tptpptp oooo ,1,, ηερε += − (53)
Working Paper No. 432 July 2011 14
tptpptp gggg ,1,, ηερε += − (54)
trftrfrftrf ,1,, ηερε += − (55)
where the η’s are all assumed to be iid normal processes, whose standard deviations are to be
estimated.
3 Estimation
3.1 Data
The model was estimated using Bayesian techniques on data for the period 1996 Q2 (the earliest
quarter for which data on wholesale gas prices were available) to 2009 Q3. As there are ten
shocks in the BBWE model, ten data series were used in the estimation: five domestic and five
‘world’. In terms of domestic data, I used data on final output of the non-energy producing
sector, consumption, the consumption deflator, investment, total hours worked in the private
sector, real wages and the Bank Rate. Consumption was defined as the sum of the ONS chained
volume measures of final consumption expenditure by households (ABJR) and non-profit
institutions (HAYO). The consumption deflator was calculated by dividing the sum of the ONS
measures of final consumption expenditure at current market prices by households (ABJQ) and
non-profit institutions (HAYE) by the volume measure. Investment was defined as ‘business
investment’ (NPEL). How the series for final output of the non-energy producing sector was
constructed is discussed at length in Harrison et al (2011). Data on total hours worked in the
private sector were taken from the Bank of England Quarterly Model (BEQM) and is described
in Harrison et al (2005). The real wage was calculated by dividing private-sector wages and
salaries including self-employment income (again as described in Harrison et al (2005)) by total
hours worked in the private sector and then again by the consumption deflator. Finally, the
ONS publish a series for the ‘London clearing banks: Base rate’ as an annual rate (Code:
AMIH) and this was converted to a quarterly rate. Prior to the estimation, all data were
detrended using the Hodrick-Prescott filter with the smoothing parameter, λ, set to 1,600.
For world data I used series for UK-weighted world trade, world export prices and world
interest rates taken from BEQM described in Harrison et al (2005). I used the dollar oil price,
available on a daily basis from Datastream (Code: OILBRNP_P) converted to its quarterly average.
Finally, I calculated a world wholesale gas price by multiplying the wholesale gas price in
sterling (available on a daily basis from Bloomberg: Code: NBPGDAHDBBSW) by the
sterling exchange rate index, published daily by the Bank of England. Following Harrison and
Oomen (2010), these data were used to estimate the foreign shock processes separately and the
results were hard-coded into the model that was estimated. The estimation results were:6
0142.0,9061.0 ,1,, =+= − ffff ytytyty σηεε (56)
0075.0,8991.0 ,1,, =+= − mfmfmfmf ptptptp σηεε (57)
6
These equations – with the exception of the equation for world gas prices – were estimated over the period 1977 Q1 to 2009 Q3. The
equation for world gas prices was estimated over the period 1996 Q2 – the earliest date for which we have data on wholesale gas prices
– to 2009 Q3.
Working Paper No. 432 July 2011 15
14100,7283.0 ,1,, .σ oooo ptptptp =+= − ηεε (58)
2544.0,5940.0 ,1,, =+= − gggg ptptptp σηεε (59)
0012.0,8738.0 ,1,, =+= − ffff rtrtrtr σηεε (60)
One thing we can note here is that the shocks to world oil and gas prices have high volatility and
low persistence relative to the other foreign shocks.
3.2 Priors
I followed Harrison and Oomen (2010) and split my parameters into two groups: those that
were most important in determining the steady state of the model and, hence, average ratios, and
those that determine the dynamics of the model. Parameters in the first group were set so as to
match the steady-state values used in Harrison et al (2011). When I came to estimate the model,
I held these parameters fixed. The values I used for this first group of parameters, and the
relevant steady-state ratios I fixed, are shown in Table A.
Table A: First group parameter values
Parameter Value Description Motivation
β 0.9925 Discount factor Assumption
χbf 0.001 Cost of adjusting portfolio of
foreign bonds
Normalisation
δ 0.013 Depreciation rate Assumption
χz =1/β-(1-
δ)
Scales the effect of capital
utilisation on the depreciation rate
Ensures capital utilisation equals 1 in
steady state
σw 3.8906 Elasticity of demand for
differentiated labour
Implies a wage mark-up of 1.35 (that
is, 35%) in steady state
σe 0.4 Elasticity of substitution between
non-energy and energy in
consumption
Assumption
σp 0.1 Elasticity of substitution between
petrol and utilities in energy
consumption
Assumption
σv 0.5 Elasticity of substitution between
labour and capital in value-added
production
Assumption
σq 0.15 Elasticity of substitution between
energy and everything else in nonenergy
production
Assumption
ηx 1.5 Elasticity of demand for exports Harrison et al (2011)
ψe 0.0526 Share of energy in consumption
Implies 0215.0=
cp
cp
c
uu
ψp 0.5913 Share of petrol in energy
consumption Implies 03.0=
cp
cp
c
pp
αq 0.0528 Cost share of energy in non-energy
output
Implies
( ) 016.0=
+++
+
pppuu
ggoo
Xcpcpq
ipip
αB 0.3154 Cost share of imports in ‘bundle’
Implies
( ) 25.0=
+++ pppuu
nm
Xcpcpq
mp
Working Paper No. 432 July 2011 16
Table A (continued): First group parameter values
Parameter Value Description Motivation
αv 0.1701 Cost share of capital in value-added
Implies 75.0=
Vp
wh
v
ψn 0.3096 Cost share of petrol in energy
output Implies 82.1=
uu
pp
ip
ip
ψqp 0.1844 Cost share of value-added in petrol
output Implies 82.1=
uu
pp
ip
ip
ψu 0.4834 Cost share of value-added in
utilities output Implies 82.1=
uu
pp
ip
ip
ψd 0.617 Share of duty in petrol prices
Implies
( ) 617.0
1
=
+
p
pp
p
d τ
cp
c
c
n 0.9474 Share of non-energy consumption
in total consumption
To match data in Harrison et al (2011)
cp
cp
c
uu 0.0215 Share of utility consumption in total
consumption
To match data in Harrison et al (2011)
V
Vn 0.9815 Share of value-added used as input
in non-energy goods
To match data in Harrison et al (2011)
V
Vu 0.0145 Share of value-added used as input
in utilities
To match data in Harrison et al (2011)
p
p
q
c 0.4204 Share of petrol output going to
consumption
To match data in Harrison et al (2011)
u
u
q
c 0.4054 Share of utilities output going to
consumption
To match data in Harrison et al (2011)
o
o
I
X 0.4551 Ratio of oil exports to oil inputs To match data in Harrison et al (2011)
g
g
I
X -0.0792 Ratio of gas exports to gas inputs To match data in Harrison et al (2011)
q
cn 0.5802 Share of private consumption in
non-energy output
To match data in Harrison et al (2011)
q
k 4.7202 Ratio of capital to non-energy
output
To match data in Harrison et al (2011)
q
cg 0.1032 Share of government consumption
in non-energy output
To match data in Harrison et al (2011)
q
xn 0.2552 Share of exports in non-energy
output
To match data in Harrison et al (2011)
q
M n 0.2581 Ratio of imports of non-energy
goods to output of non-energy
goods
To match data in Harrison et al (2011)
q
X o 0.0035 Ratio of oil exports to output of
non-energy goods
To match data in Harrison et al (2011)
q
X g -0.0007 Ratio of gas exports to output of
non-energy goods
To match data in Harrison et al (2011)
For the second group of parameters, I generally took my priors from Harrison and Oomen
(2010). In particular, I set the priors for the inverse of risk aversion in consumption, σc, the
scale parameter for the costs of adjusting the capital stock, χk, the elasticity of capital utilisation
Working Paper No. 432 July 2011 17
costs, φz, and the elasticity of labour supply σh, exactly in line with Harrison and Oomen. My
prior for the coefficient on inflation in the Taylor rule is normal with a mean of 1.5 (as in the
original Taylor paper) and a standard deviation of 0.25. For the remaining parameters I used
beta distributions – since, by definition, they have to lie between 0 and 1 – with relatively loose
priors. In all cases, I set my prior means to 0.5 and my prior standard deviations to 0.2. My
priors are shown in Table B. In terms of the parameters governing the shock processes, I use
beta distributions for the autocorrelation coefficients with means of 0.5 and standard deviations
of 0.2, and I use inverse gamma distributions for the standard deviations with means of 1% and
two degrees of freedom.
Table B: Priors for second group parameters
Parameter Description Prior
distribution
Prior
mean
Prior
standard
deviation
σc Intertemporal elasticity of substitution Normal 0.66 0.198
ψhab Degree of habit persistence in consumption Beta 0.5 0.2
εk Degree of persistence in investment adjustment costs Beta 0.5 0.2
χk Scale of capital adjustment costs Normal 201 60.3
φz Inverse elasticity of capital utilisation costs Normal 0.56 0.168
ψwc Share of wage bill paid financed by borrowing Beta 0.5 0.2
σh Frisch elasticity of labour supply Normal 0.43 0.107
ψw Probability of being able to change wages Beta 0.5 0.2
ξw Degree of wage indexation Beta 0.5 0.2
χp Probability of not being able to change price: nonenergy
sector
Beta 0.5 0.2
χu Probability of not being able to change price: utilities Beta 0.5 0.2
χpp Probability of not being able to change price: petrol Beta 0.5 0.2
εp Degree of indexation: non-energy sector Beta 0.5 0.2
εu Degree of indexation: utilities sector Beta 0.5 0.2
εpp Degree of indexation: petrol sector Beta 0.5 0.2
ψx Degree of persistence in export demand Beta 0.5 0.2
ψpm Probability of not being able to change price:
importers
Beta 0.5 0.2
εpm Degree of indexation: importers Beta 0.5 0.2
θpdot Taylor rule coefficient on inflation Normal 1.5 0.25
θy Taylor rule coefficient on output Normal 0.125 0.05
θrg Degree of interest rate smoothing in Taylor rule Beta 0.5 0.2
3.3 Estimation results
As is now standard in the literature, I first estimated the mode of the posterior distribution by
maximising the log posterior function, which combines the priors with the likelihood given by
the data, and then used the Metropolis-Hastings algorithm (as implemented in Dynare) to obtain
the full posterior distribution. I used a sample of 250,000 draws (dropping the first 50,000
draws), obtaining an acceptance rate of 0.31. To test the stability of the sample, I used the
Brooks and Gelman (1998) diagnostic (as implemented by Dynare), which compares within and
between moments of multiple chains. Table C shows the posterior mode and means for the
model parameters together with a 90% confidence interval.
Working Paper No. 432 July 2011 18
Table C: Estimation results
Parameter Description Posterior
mode
Posterior
mean
Confidence interval
σc Intertemporal elasticity of substitution 0.7103 0.6256 0.4777 0.7775
ψhab Degree of habit persistence in
consumption
0.6019 0.5876 0.4204 0.7564
εk Degree of persistence in investment
adjustment costs
0.1887 0.1871 0.0793 0.2920
χk Scale of capital adjustment costs 106.05 116.52 64.47 172.96
φz Inverse elasticity of capital utilisation
costs
0.4554 0.4591 0.3207 0.5980
ψwc Share of wage bill paid financed by
borrowing
0.5548 0.4974 0.2551 0.7067
σh Frisch elasticity of labour supply 0.3547 0.3423 0.2724 0.4172
ψw Probability of being able to change wages 0.4630 0.4719 0.3487 0.5972
ξw Degree of wage indexation 0.1708 0.1882 0.0389 0.3212
χp Probability of not being able to change
price: non-energy sector
0.8904 0.8968 0.8297 0.9668
χu Probability of not being able to change
price: utilities
0.5618 0.5760 0.4478 0.7047
χpp Probability of not being able to change
price: petrol
0.2192 0.2371 0.0948 0.3853
εp Degree of indexation: non-energy sector 0.2832 0.1491 0.0307 0.2581
εu Degree of indexation: utilities sector 0.2769 0.2073 0.0829 0.3359
εpp Degree of indexation: petrol sector 0.5021 0.4622 0.2479 0.7901
ψx Degree of persistence in export demand 0.4175 0.4152 0.3150 0.5234
ψpm Probability of not being able to change
price: importers
0.4283 0.4169 0.2817 0.5654
εpm Degree of indexation: importers 0.8937 0.8296 0.7141 0.9481
θpdot Taylor rule coefficient on inflation 1.2190 1.1951 0.9904 1.4646
θy Taylor rule coefficient on output 0.1528 0.1494 0.1196 0.1813
θrg Degree of interest rate smoothing in
Taylor rule
0.7800 0.7640 0.6952 0.8318
In terms of the parameter values themselves, the posterior mean estimate for the inverse
coefficient of relative risk aversion, σc, is slightly lower than its prior mean, though it is not well
identified by the data. The posterior mean estimate for εk is much lower than its prior mean,
suggesting little persistence in investment. The inverse Frisch elasticity of labour supply is
lower than its prior mean and is well identified. The degree of habits in consumption is not well
identified and, at 0.59, is a little lower than previous estimates on UK data.7
I find that about
50% of the wage bill has to be financed using working capital, though again this parameter is
not well identified. There appears to be little persistence in export demand.
In terms of nominal rigidities, the posterior mean estimates suggest, not surprisingly, that petrol
prices are flexible, being changed roughly every four months on average. Utility prices are also
quite flexible being changed roughly every seven months on average. Non-energy prices, on the
other hand, are very sticky, being changed roughly every 29 months on average, respectively.
The relative stickiness of prices in the three sectors are in line with survey and other evidence
for the United Kingdom.8
7
See, eg, Banerjee and Batini (2003).
But the absolute degree of price stickiness in the non-energy sector
8
See Greenslade and Parker (2010) and Bunn and Ellis (2009).
Working Paper No. 432 July 2011 19
seems much too high, although this result has often been found in estimated DSGE models.9
Indexation in all sectors is quite low suggesting little inflation persistence. The posterior
estimates suggest that import prices are relatively flexible, changing on average every five
months, and the degree of indexation of import prices is high. Surprisingly, wages are estimated
to be fairly flexible, changing roughly every six months on average. Wage changes are hardly
indexed to lagged wage inflation, as might be expected given the lack of formal indexation of
wage bargains in the United Kingdom at present.
Turning to the shocks, Table D shows the estimated posterior mode and means for the
autocorrelation coefficients and standard errors of the domestic shocks, together with a 90%
confidence interval. The posterior estimates suggest that the productivity shock is fairly
persistent but the other shocks much less so; the model is able to explain persistence in the data
without having to resort to extremely persistent shocks. The mean posterior estimate for the
standard deviation of monetary policy shocks is only 15 basis points. This is not altogether
surprising given that the model was estimated over the inflation-targeting period. But, the
domestic demand and investment-specific technology shocks are highly volatile over this period
with posterior mean estimates for their standard deviations of 9.2% and 6.1%, respectively.
Table D: Estimation results for the domestic shock processes
Autocorrelation coefficients
Posterior mode Posterior mean Confidence interval
Productivity, εa 0.8906 0.8747 0.8176 0.9311
Risk premium, εb 0.7217 0.6656 0.5760 0.7544
Domestic demand, εg 0.7306 0.6621 0.5235 0.8263
Monetary policy, εi 0.3381 0.3174 0.1888 0.4458
Investment-specific
technology, εinv
0.4269 0.4323 0.3025 0.6055
Wage mark-up, εw 0.2569 0.2247 0.0924 0.3407
Price mark-up, εµ 0.2615 0.2398 0.0842 0.3935
Standard deviations
Productivity, εa 0.0120 0.0123 0.0103 0.0142
Risk premium, εb 0.0032 0.0041 0.0030 0.0052
Domestic demand, εg 0.0901 0.0923 0.0777 0.1068
Monetary policy, εi 0.0015 0.0015 0.0013 0.0018
Investment-specific
technology, εinv
0.0551 0.0612 0.0320 0.0952
Wage mark-up, εw 0.0093 0.0098 0.0079 0.0118
Mark-up, εµ 0.0025 0.0028 0.0023 0.0034
Tables E and F show the importance of each of the shocks in terms of how much each explains
the variance in the endogenous variables. The productivity shock is clearly the most important
in explaining consumption, accounting for almost two thirds of the variation in consumption.
The investment-specific technology shock contributes little to the variation in all variable except
for investment, where it explains 83% of the variance. The bulk of the variation in GDP, total
hours and gross output is explained by a combination of the productivity, domestic demand,
monetary policy and risk premium shocks, which together account for 79%, 77% and 81% of
9
See Smets and Wouters (2003) and Gali et al (2001) for the euro area and Di Cecio and Nelson (2007), Harrison and Oomen (2010)
and Kamber and Millard (2010) for the United Kingdom.
Working Paper No. 432 July 2011 20
their variance, respectively. Turning to nominal variables, 64% of the variation in price
inflation is explained by the price mark-up shock, with the productivity shock accounting for
about 18%. Similarly, 80% of the variation in wage inflation is explained by the wage mark-up
shock. And, so, a combination of the mark-up and productivity shocks explain 83% of variation
in the real wage. 80% of the variation in the nominal interest rate is explained by the
productivity, monetary policy and price mark-up shocks. Perhaps surprisingly, the foreign
shocks account for little of the variation in UK data with the exception of the real exchange rate,
33% of whose variation is explained by them. Another 30% of the variation in the real
exchange rate is explained by the productivity shock.
Table E: Variance decompositions – domestic shocks
Productivity Monetary
policy
Domestic
demand
Investment-
specific
technology
Wage
mark-up
Price
mark-up
Risk
premium
Consumption 62.7% 6.8% 3.3% 0.2% 1.2% 3.0% 10.6%
Investment 2.1% 3.8% 1.7% 83.3% 0.5% 2.7% 5.1%
GDP 20.5% 13.8% 22.5% 4.7% 4.4% 5.5% 21.8%
Gross output 38.3% 9.3% 20.2% 4.0% 0.3% 4.8% 13.5%
Total hours 21.4% 12.7% 21.8% 4.5% 8.0% 4.2% 21.0%
Real wage 11.0% 5.2% 1.3% 0.2% 57.5% 14.2% 4.3%
Real
exchange
rate
29.7% 11.2% 1.4% 0.2% 1.4% 6.1% 17.5%
Nominal
interest rate
35.4% 26.6% 2.5% 0.4% 0.3% 17.8% 8.6%
Annual
inflation rate
17.8% 2.0% 0.3% 0.0% 5.7% 63.7% 0.9%
Annual wage
inflation rate
1.4% 5.8% 1.4% 0.2% 79.7% 3.6% 5.7%
Table F: Variance decompositions – foreign shocks
Oil price Foreign
demand
Foreign
export price
Foreign
interest rate
Gas price
Consumption 0.3% 4.8% 2.8% 3.6% 0.7%
Investment 0.0% 0.4% 0.1% 0.1% 0.0%
GDP 0.1% 2.5% 0.2% 3.8% 0.2%
Gross output 0.1% 6.7% 1.8% 0.8% 0.2%
Total hours 0.1% 2.3% 0.2% 3.5% 0.2%
Real wage 0.4% 4.0% 0.6% 0.6% 0.7%
Real
exchange
rate
0.1% 10.8% 1.3% 20.3% 0.2%
Nominal
interest rate
0.9% 0.7% 1.3% 4.0% 1.6%
Annual
inflation rate
2.8% 0.5% 1.0% 2.2% 3.2%
Annual wage
inflation rate
0.0% 0.9% 0.1% 1.0% 0.1%
Working Paper No. 432 July 2011 21
3.4 Effects of including/excluding energy
This section discusses how the inclusion of energy price effects – the main contribution of this
model over, say, the Smets and Wouters (2007) model – affects the estimated coefficients of the
model and whether, overall, it forms a better description of UK macroeconomic data than the
simpler model. In order to do this, I consider an otherwise identical model in which
consumption consists of only one good, which is produced using labour, capital and imports
only. I drop the world oil and gas prices from my set of observed variables in the estimation and
shocks to world oil and gas prices from my set of shocks in the model. I leave my fixed
parameters and priors unchanged. The results are shown in the 4th
column of Table G and the
3rd
column of Table H.
Table G: Parameter estimates for model with and without energy
Parameter Description Baseline Ex-energy
σc Intertemporal elasticity of substitution 0.6256 0.6481
ψhab Degree of habit persistence in
consumption
0.5876 0.5310
εk Degree of persistence in investment
adjustment costs
0.1871 0.3698
χk Scale of capital adjustment costs 116.52 219.03
φz Inverse elasticity of capital utilisation
costs
0.4591 0.5560
ψwc Share of wage bill paid financed by
borrowing
0.4974 0.5629
σh Frisch elasticity of labour supply 0.3423 0.3943
ψw Probability of being able to change wages 0.4719 0.4672
εw Degree of wage indexation 0.1882 0.4864
χp Probability of not being able to change
price: non-energy sector
0.8968 0.5344
χu Probability of not being able to change
price: utilities
0.5760 χpp
Probability of not being able to change
price: petrol
0.2371 εp
Degree of indexation: non-energy sector 0.1491 0.3149
εu Degree of indexation: utilities sector 0.2073 εpp
Degree of indexation: petrol sector 0.4622 ψx
Degree of persistence in export demand 0.4152 0.4274
ψpm Probability of not being able to change
price: importers
0.4169 0.4489
εpm Degree of indexation: importers 0.8296 0.4788
θpdot Taylor rule coefficient on inflation 1.1951 1.3542
θy Taylor rule coefficient on output 0.1494 0.1413
θrg Degree of interest rate smoothing in
Taylor rule
0.7640 0.3018
Working Paper No. 432 July 2011 22
Table H: Shock process parameters for the model with and without energy
Autocorrelation coefficients
Baseline Ex-energy
Productivity, εa 0.8747 0.5252
Risk premium, εb 0.6656 0.4652
Domestic demand, εg 0.6621 0.4810
Monetary policy, εi 0.3174 0.5148
Investment-specific
technology, εinv
0.4323 0.3414
Wage mark-up, εw 0.2247 0.5641
Price mark-up, εµ 0.2398 0.5600
Standard deviations
Productivity, εa 0.0123 0.0157
Risk premium, εb 0.0041 0.0074
Domestic demand, εg 0.0923 0.0920
Monetary policy, εi 0.0015 0.0035
Investment-specific
technology, εinv
0.0612 0.1382
Wage mark-up, εw 0.0098 0.0076
Price mark-up, εµ 0.0028 0.0071
As can be seen, some of the parameter estimates look quite different. In particular, the model
with no explicit energy effects estimates capital adjustment costs to be much larger, wage and
price indexation to be much higher and suggests that prices are quite flexible. This result
probably reflects ‘averaging’ of the degrees of stickiness estimated for the individual sectors in
the model with explicit energy effects. Given the lack of energy price shocks, the model
without energy requires more volatility in the other shocks to explain the data. Finally, we can
note that the model with energy price effects included explains the data much better than the
model that does not include them. In particular, the estimated log data density for the
benchmark model is 1818 whereas for the model excluding energy effects it is only 1623.
4 Impulse response functions
This section presents some impulse response functions for the estimated model. In particular,
the results in this section are brought to bear on two questions: to what extent does the inclusion
of energy effects alter the estimated responses of variables to shocks in the model, and, more
specifically, how do variables respond to world oil and gas price shocks. The variables
considered are value-added output, aggregate consumption, inflation, the base rate, the real
wage and the exchange rate.10
10
Throughout this section ‘output’ refers to ‘value-added’ output (that is, GDP) and not to the gross output of any sector or the economy
as a whole.
Working Paper No. 432 July 2011 23
4.1 How does energy affect the responses of variables to shocks?
Chart 1 shows the responses of output, aggregate consumption, inflation, the base rate, the
exchange rate and the real wage to a one standard deviation monetary policy shock. In each
case, the chart shows the responses implied by the estimated model together with the responses
implied by the estimated model that excluded energy. In the benchmark estimated model, the
shock represents an exogenous increase in the base rate of 51 basis points; in the version of the
model estimated without energy effects, it represents a shock to interest rates of 84 basis points.
Taking the results of the benchmark model first, we can note that the shock has quite a large
effect on output, which falls by about 0.45%, while inflation falls by only about 0.03 percentage
points on impact. Consumption also falls. The maximum response of real variables to the
shock occurs immediately. Output and consumption then return to base with the shock having a
minimal effect on either of them after about one year. Inflation continues to fall for three
quarters after the initial shock, reaching a minimum -0.10 percentage points below base, before
returning back to base. The effect is basically zero after about two years. Interest rates take
about a year to return to base. The exchange rate follows the path of the interest rate – as a
result of UIP – with the initial impact of the shock being an appreciation of 0.6%. The shock
has a significant effect on real wages in the benchmark model as nominal wages fall quickly
relative to the price level – given the estimated degrees of wage and price stickiness. These
responses are in line with the empirical results of, eg, Di Cecio and Nelson (2007), Kamber and
Millard (2010) and Christiano et al (2005), except that the responses of output and consumption
are not ‘hump-shaped’. This results from the assumption of ‘capital adjustment costs’ in the
current paper rather than ‘investment adjustment costs’. Christiano et al (2005) makes clear that
it is investment adjustment costs that are key to generating hump-shaped responses in real
variables to a monetary policy shock. So, replacing these with capital adjustment costs, which
Smets and Wouters (2007) argue would not generate a hump-shaped response of investment to
shocks, is likely to result in a lack of hump-shaped responses in all real variables. The response
of real wages is large and hump-shaped as a result of the high degree of wage flexibility
estimated in the current model, particularly relative to the price of non-energy goods.
We can note that, with the exception of inflation, real wages and interest rates, the responses of
variables are similar to their responses in the estimated version of the model without energy.
The inflation response is much stronger in the model without energy effects given that prices
were estimated to be much more flexible in this model. For the same reason, the real wage
response is much smaller in the model that excluded energy effects.
Working Paper No. 432 July 2011 24
Chart 1: Effects of a monetary policy shock
Working Paper No. 432 July 2011 25
Chart 2: Effects of a productivity shock
Chart 2 shows the responses of the same variables to a positive productivity shock for the
benchmark model and for the version of the model without energy. Perhaps surprisingly, valueadded
output falls. This is because the productivity shock affects gross output (of the nonenergy
good) given value-added input. With sticky prices, demand for gross output will not
respond much to the increase in productivity, so producers will cut down on inputs – including
value-added. Consumption rises as the shock makes households wealthier and habit persistence
is large enough to ensure a hump-shaped response. Inflation and interest rates fall and the
Working Paper No. 432 July 2011 26
exchange rate depreciates as UK goods are now cheaper to produce vis-à-vis foreign goods. We
can note that the inclusion of energy within the model has a large impact on the responses; in
particular, value-added oscillates for the first couple of years after the shock before returning to
base and the effect of the shock on inflation (and, as a result, interest rates) is much stronger.
Again these results come about because prices are estimated to be quite flexible in the version of
the model that ignores energy prices.
Chart 3: Effects of a domestic risk premium (financial) shock
Chart 3 shows the responses of value-added output, aggregate consumption, investment,
inflation, the base rate, the exchange rate, the real wage, employment and wage inflation to a
positive domestic risk premium shock (ie, a shock that raises the interest rate faced by
consumers relative to the policy rate). This is an important shock to consider since we would
Working Paper No. 432 July 2011 27
expect it to pick up the recent financial crisis. As expected, an increase in risk premia caused
by, say, a credit tightening, would lead to falls in consumption, output and price inflation.
However, the exchange rate appreciates as demand falls in the United Kingdom relative to
abroad; the effect on the exchange rate of changes in the relative risk of UK versus foreign
assets would, in this model, come through movements in the foreign exchange risk premium
shock. In this case, the responses of output, consumption and the exchange rate are
quantitatively similar to their responses in the estimated version of the model without energy.
Real wages respond less in the model with no explicit energy effects since they are estimated to
be stickier. Inflation, on the contrary responds by more – since prices are estimated to be more
flexible – and the interest rate responds more given the central bank’s Taylor rule.
Chart 4 shows the effects of a one standard deviation (about 9%) domestic demand shock. Such
a shock leads to an increase in output of about 0.85% but a fall in consumption of about 0.15%,
as the increase in output is much smaller than the increase in the exogenous components of
domestic demand. The increase in demand leads to a rise in inflation and interest rates. The
real wage rises on account of the increased demand for labour, though the magnitude of this rise
again depends on which version of the model is used. Finally, the increase in domestic demand
relative to foreign leads to an appreciation of the exchange rate.
Finally in this subsection, Charts 5 and 6 show the effects of a world demand and a world export
price shock, respectively. A world demand shock leads to an increase in output, real wages and,
eventually, consumption. The increase in demand also pushes up on inflation and interest rates
rise in response but the effect on both these variables is small. The rise in relative demand for
the home economy’s exports leads to an appreciation of the exchange rate. Now, a shock to
world demand might be expected to lead to rises in world commodity prices, such as oil and gas,
with a dampening effect on domestic output. However, this channel is not present in this model
since world oil and gas prices are assumed to be exogenous and are unaffected by world
demand.
A shock to world export prices leads to a rise in domestic import prices, which, in turn, feeds
into domestic inflation. Domestic consumption falls as final output becomes more expensive.
The response of value-added is interesting. A rise in import prices leads to a reduction in gross
output and, other things equal, lowers the demand for value-added. But in the model with
explicit energy effects, the fall in the relative price of energy will lead to an increase in demand
for energy – both for production and consumption – and, in turn, an increase in the demand for
value-added in energy production, outweighing the effect coming from the fall in demand for
value-added in the production of non-energy goods. We can note, however, that in both cases
the effect on value-added is small. Finally, the exchange rate appreciates in response to the
increased demand for the home economy’s exports.
Working Paper No. 432 July 2011 28
Chart 4: Effects of a domestic demand shock
Working Paper No. 432 July 2011 29
Chart 5: Effects of a world demand shock
Working Paper No. 432 July 2011 30
Chart 6: Effects of a world export prices shock
Working Paper No. 432 July 2011 31
4.2 The effects of a rise in energy prices
The key difference between this model and more standard macroeconomic models is the
presence of a supply chain linking movements in world oil and gas prices, to movements in
petrol and utilities prices, to movements in the overall level of consumer prices. Given that, it is
instructive to consider the effects of shocks to the world prices of oil and gas.
Chart 7: Effects of a world oil price shock
Working Paper No. 432 July 2011 32
Chart 7 shows the responses of output, aggregate consumption, investment, inflation, the base
rate, the exchange rate, the real wage, employment and wage inflation to a temporary exogenous
increase in the world price of oil of 14% (a one standard deviation shock). The effects of such a
shock on real variables are small.11
Output has fallen by just under 0.04% after one quarter,
consumption by roughly 0.08% and investment by roughly 0.06%. Inflation is 0.11 percentage
points higher after one quarter but then falls back quickly, being close to its steady-state rate
after two years and beyond. Workers take a hit in their real consumption wage in the year and a
half following the shock; that is, there is little evidence of real wage resistance.
Why are the effects of an oil shock estimated to be so low? The answer is the result of the
relatively low autocorrelation of this shock (estimated to be 0.73). Given the persistence of
movements in oil prices, this seems like a surprising result. The answer to the puzzle can be
seen in Chart 8. The chart shows that the large rises in the oil price in recent years have been
interpreted as trend increases by the model; it is only the additional movements in the oil price
that are interpreted as shocks.12
Chart 8: World oil price, actual and trend
This model can also be used to analyse the effects of shocks to world gas prices. Chart 9 shows
the responses of output, aggregate consumption, investment, inflation, the base rate, the
exchange rate, the real wage, employment and wage inflation to a temporary exogenous increase
in the world price of gas of 25% (a one standard deviation shock). The effects of this shock are
similar to those of an oil price shock. The effects on real variables are, again small and, again,
this is because the shock has low persistence (estimated to be 0.59). Output has fallen by 0.06%
after one quarter, consumption by 0.11% and investment by 0.15%. Inflation is 0.11 percentage
points higher after two quarters but then falls back quickly. Again, workers take a hit in their
real consumption wage in the two years following the shock.
11
Although this might seem surprising, it is, as I said earlier, a fairly common result within the DSGE literature on oil (eg, Dhawan and
Jeske (2008) and Rotemberg and Woodford (1996)) and it also matches the recent experience of large oil price movements with little
obvious effect on output.
12
It would be instructive to consider the effects of permanent movements in energy prices within this model. But, in order to do this
properly, it is necessary to take a stand on the degree of self-sufficiency of oil and gas production in the long run, as is done in Harrison
et al (2008), but which is really beyond the scope of this paper. The interested reader is referred to their paper for further elaboration on
these issues.
Working Paper No. 432 July 2011 33
Chart 9: Effect of a world gas price shock
5 Using the model to decompose movements in output and inflation
A major motivation for estimating this model is that it is important for monetary policy makers
to understand what drives output and inflation in different periods. Given that the estimation
produces time series for the shocks, it is possible to decompose movements in output and
inflation into those fractions caused by each of the shocks. Doing this enables us to ascertain
what shocks have been the main drivers of these variables.
Working Paper No. 432 July 2011 34
Chart 10: Domestic shocks
Working Paper No. 432 July 2011 35
Chart 10 shows the estimated time series for the domestic shocks within the model and Chart 11
shows the time series for the world shocks. As implied by the estimation results, we can see that
the shocks to domestic demand, investment-specific technology and world oil and gas prices
have been highly volatile over this period whereas monetary policy shocks have been small.
Chart 11: Foreign shocks
Working Paper No. 432 July 2011 36
Concentrating on the recent past, we can see that the economy has been affected by large
negative shocks to productivity and domestic and world demand and a large positive shock to
the domestic risk premium. This positive shock to the domestic risk premium is what the model
will have picked up as the financial crisis of 2007-09. The increase in risk premium occasioned
by the financial crisis would act to reduce demand in the model creating a recession, as was seen
in the United Kingdom from 2008 Q2 to 2009 Q3. The world demand shock reflects what
happened to world trade around the end of 2008 and the beginning of 2009. The negative
productivity shock is harder to explain; it is likely to reflect, at least partly, the negative impact
of the financial crisis on the ability of firms to raise working capital.
To investigate further which shocks have been driving the UK economy over this sample period,
Chart 12 shows a decomposition of movements in gross non-energy output between 2005 Q1
and 2009 Q3 into the portions caused by each of the shocks. We can see that, as expected, the
recent slowdown in gross non-energy output has been driven by the negative productivity shock,
the domestic risk premium shock and world demand.13
What may come as a little surprising is
that energy prices are not major drivers of movements in non-energy output, despite being an
input into production of that good and despite having moved substantially over this period.
Again, this can be explained by the fact that the bulk of the movement in oil and gas prices was
treated as a ‘trend increase; the remaining movements are quite volatile and, as a result, nonenergy
producers seem to smooth through movements in this component of costs. Monetary
policy shocks were mildly supporting output in 2009 since interest rates were cut by more than
would have been suggested by the Taylor rule in the model. Of course, the ‘systematic’
response of monetary policy would have been supporting output in 2009 as would have the
additional monetary stimulus coming from quantitative easing.
Chart 12: Shock decomposition for gross non-energy output
13
Recall, a negative productivity shock would result in a rise in value-added output but a fall in gross output.
Working Paper No. 432 July 2011 37
Of course, we might expect movements in energy prices to be key determinants of movements
in energy (ie, petrol and utilities) output. This is illustrated in Chart 13. As can be seen, the
high energy prices of 2008 substantially pulled down on energy output. By the end of the
sample, low energy prices were pushing up on energy output. In addition, low productivity of
non-energy inputs in the non-energy sector were pushing up on the demand for energy as nonenergy
producers substituted towards energy. Against this, low world demand and the risk
premium shock (proxying the effects of the financial crisis) were pulling energy output down.
Chart 13: Shock decomposition for gross energy output
Chart 14: Shock decomposition for the CPI inflation rate
Turning to inflation, Chart 14 suggests that the price mark-up shock was pushing down
substantially on inflation in 2009 with the financial shock and energy prices also contributing.
Against this, the negative productivity shock was acting to push inflation up. Energy prices
seem to have a much larger effect on inflation than they do output. The high oil prices in early
Working Paper No. 432 July 2011 38
2008 and gas prices throughout 2008 were pushing up on inflation in 2008. As oil and gas
prices fell in 2009, they again acted to push down on inflation.
6 Conclusions
This paper has estimated a DSGE model of the United Kingdom developed originally by
Harrison et al (2011). The basic building blocks of the model are standard in the literature. The
main complication is that there are three consumption goods: non-energy output, petrol and
utilities; given relative prices and their overall wealth, consumers choose how much of each of
these goods to consume in order to maximise their utility. Each of the consumption goods is
produced according to a sector-specific production function and sticky prices in each sector
imply sector-specific New Keynesian Phillips Curves. This model, once estimated, forms a
useful additional input within a policymaker’s ‘suite of models’.
Estimating the parameters of this model using Bayesian techniques enables the user to apply the
model quantitatively to UK policy issues. The paper has shown how this could be done by
examining the effects of many different shocks on inflation and by decomposing recent
movements in output and inflation into those parts caused by each of the structural shocks. It
found that the fall in gross non-energy output from 2008 Q2 to 2009 Q3 was driven by three
shocks: to productivity, to world demand and to the domestic risk premium. The risk premium
shock also put downwards pressure on inflation during this period while the productivity shock
was putting upwards pressure on inflation. The world demand shock was much less important
in explaining the behaviour of inflation over this period.
Working Paper No. 432 July 2011 39
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