Economic models
at the
Bank of England
September 2000 update
Bank of England
Any enquiries about this publication should be addressed to:
Meghan Quinn
Monetary Analysis
Telephone 020-7601 5269
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email: [email protected]
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email: [email protected]
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Printed by Park Communications Ltd
© Bank of England 2000
ISBN 1 85730 182 X
Contents
Page
1
1.1
1.2
1.3
1.4
Introduction
Models, policy analysis and forecasting
General characteristics of the macroeconometric model
Changes to the MM
Other models added to the suite
5
5
6
7
8
2
Overview of the macroeconometric model
9
3
3.1
3.2
3.3
The structure of the macroeconometric model
The long-run real growth path
The long-run nominal growth path and inflation
Short-run dynamics and inflation
10
10
13
14
4
Recent changes to the model specification
15
5
5.1
5.2
5.3
Simulation properties
Rationale for simulations
Temporary interest rate shock
Exogenous change in a price level target
16
16
17
18
6
6.1
6.2
6.3
6.4
6.5
6.6
Detailed equation listing
Money, financial and wealth variables
Demand and output
Labour market
Prices
Fiscal policy
Income
20
21
27
36
42
54
56
7
Variables listing
59
Bibliography
68
The macroeconometric model
1
5
Introduction
In April 1999, the Bank of England published Economic Models at the Bank of England, setting
out the economic modelling tools that help the Monetary Policy Committee (MPC) in its work.
That volume included a complete listing of the Bank’s main macroeconometric model (MM), and
outlined the other members of the suite of models used for various aspects of monetary policy
analysis. It was made clear at the time that neither the MM nor the other models in use should be
thought of as fixed in form or content. Indeed, many aspects of the models are regularly reviewed,
and new approaches to modelling aspects of the economy are continually investigated.
The purpose of this publication is to provide an update of the changes incorporated in the MM
over the past 18 months. It provides a written listing of the MM, to accompany the simultaneous
release of the model code in electronic form. At the same time, we reference some other work
within the Bank that has added to the range of models in the suite and that is already publicly
available.
In this section we outline the Bank’s modelling philosophy (set out more fully in the earlier
volume), describe the key features of the MM, highlight the main ways in which the MM has
changed since April 1999, and outline some other relevant modelling work. Sections 2 and 3
explain the structure of the MM in more detail. Section 4 outlines the changes to the MM.
Section 5 discusses the MM simulation properties. Finally, Sections 6 and 7 provide a complete
model listing, including diagnostics on estimated equations and data sources.
1.1
Models, policy analysis and forecasting
The MM is the main tool for producing projections of GDP growth and inflation shown in the
Inflation Report. The MM is built around a number of estimated econometric relationships, but
some of the model properties—notably the long-run properties—are imposed in the form of
parameter restrictions for theoretical consistency. There is a continual need to evaluate and update
various components of the MM. Estimated MM econometric relationships may have broken down
or have changed in some way, so that research is required to investigate the causes and to test
alternatives that may eventually be incorporated in the MM itself.
The Bank continues to use a range of models. Some provide inputs into the quarterly projections,
while others are used to analyse specific policy questions that cannot be handled adequately within
the MM. Some research may prove difficult or impossible to incorporate in the MM—for
example, it may involve a different level of aggregation. It would then be run in parallel to
provide a comparison with MM outputs, or to provide insights into aspects of the economy that the
MM cannot address.
Occasionally, specific new policy issues arise that cannot be analysed using the existing
framework, and models are set up specifically to examine the key features of the issue at hand.
Examples have included the impact of the National Minimum Wage, the implementation of the
Working Time Directive, and the assessment of the impact on consumer spending of the windfalls
6
Economic models at the Bank of England
from building society demutualisations. In some cases, a purpose-built model may cease to be of
use once the issue it addresses no longer has monetary policy significance. But in other cases the
work is incorporated in tools that are used to assess issues of continuing relevance.
Each forecasting round requires assumptions to be made about a wide range of exogenous
variables. Auxiliary models are often used to inform these judgments. Some relate, for example,
to the world economy or some element of it, such as commodity prices or the level of world trade.
For the assessment of world economic activity and inflation, the MPC uses a model(1) of the world
economy provided by the National Institute of Economic and Social Research to help form its
judgments. Other models relate to aspects of the domestic economy that are not formally
modelled in the MM, but where parameters may be varied or restricted as a result of the auxiliary
analysis. In all cases, the assumptions incorporated in any specific forecast are a combination of
those suggested by the auxiliary model and the application of the MPC’s judgment. Profit margins
and house prices are examples of areas where forecast assumptions are influenced by both
supplementary modelling and MPC judgment.
Where the Bank does not have the tools to hand for analysing a specific issue, it will seek out the
best available analysis from the academic literature or from the research work of other central
banks and research institutes. For this reason, Bank staff are encouraged to keep abreast of the
relevant academic literature and to contribute to it by publication of working papers, contributions
to professional journals, and presenting their work at conferences. The Bank also runs a seminar
series, addressed both by outside experts and by internal staff. The general philosophy with which
the Bank approaches modelling and forecasting in particular, and monetary policy analysis in
general, is one of pluralism and openness. We are happy to receive comments on this and on any
other publication, particularly suggestions on how things could be improved.
1.2
General characteristics of the macroeconometric model
The core macroeconometric model (MM) consists of about 20 key equations determining
endogenous variables. There are a further 90 or so identities defining relationships between
variables, and there are about 30 exogenous variables whose paths have to be set, as discussed
above.
GDP is determined in the short term by the components of aggregate demand—private
consumption, investment (including inventory investment), government consumption, and net
exports.
In the longer term GDP is determined by supply-side factors, which determine potential output.
Domestic firms are modelled as producing a single composite good using an aggregate production
function of the Cobb-Douglas form. So output is determined in the long run by the evolution of
the capital stock, the labour supply and total factor productivity. These variables are assumed to
be unaffected by the price level or the inflation rate (so the model exhibits long-run monetary
neutrality and super-neutrality).
(1)
The National Institute Global Economic Model (NiGEM).
The macroeconometric model
7
Price level dynamics and the adjustment of actual output towards potential are broadly determined
by the interaction between aggregate demand and supply, augmented by explicit relationships for
aspects of wage and price-setting. These relationships are consistent with the view that firms set
domestic output prices as a cyclically varying mark-up over unit labour costs. RPIY is determined
by an equation linking retail prices to domestic output prices and import prices. Firms are also
assumed to determine the level of employment, and real wages are determined by bargaining in an
imperfectly competitive labour market. Inflation expectations have an explicit role in wage
determination. But price responses are sluggish, so there is slow adjustment towards both real and
nominal equilibria.
The appropriate assumptions under which to run the model depend on the exercise at hand. For
example, short-run forecasting typically requires different assumptions from those used for
long-run simulations, and for either purpose a wide range of alternative assumptions could be
made. For the main Inflation Report forecasts, nominal short-term interest rates are assumed to be
constant over the forecast period, but an alternative is also presented in which rates follow the path
implied by market expectations. When using the MM for simulation purposes, the short rate can
be set according to a policy rule linking short-term nominal interest rates to the monetary policy
target, but the nature of this rule can take many different forms. Different exchange rate
assumptions can be used in both the construction of projections and for simulations. A range of
possible treatments is also available for the evolution of net financial wealth, and for inflation
expectations.
A further example of where different assumptions may be used for different purposes relates to
government spending. The Inflation Report projections incorporate announced government
spending plans, but some alternative assumption is needed in longer-term simulations, as spending
plans are not announced for more than a few years at a time. In this case, a common assumption
is that government consumption growth is fixed in either nominal or real terms. The properties of
the model when used for simulation purposes are shown in Section 5 below.
1.3
Changes to the MM
The main areas of the MM in which changes have been introduced since April 1999 are:
●
The consumption function now incorporates a new measure of labour income, which
includes self-employment incomes (mixed incomes). Gross housing wealth and net financial
wealth now have a separate role in the dynamics. And the real (short) interest rate matters in
the long run, while nominal short rates affect the dynamics.
●
There is a new equation for house prices, which depend on average earnings and the long
real rate in the long run, while GDP enters the dynamics (in addition to earnings).
●
Both export and import equations have been modified as a result of estimation on new data,
the main effect being to lower slightly the relative price elasticities.
8
●
Economic models at the Bank of England
RPIY is determined by a modified relationship that weights domestic and import prices.
In addition, there are other small modifications resulting from data revisions and definitional
changes affecting the capital stock, investment, trade prices, earnings, employment, non-labour
income and the GDP deflator. There are minor changes to the treatment of value-added tax and
special duties (affecting the link from RPIY to RPIX), and a new equation for the government
expenditure deflator has been introduced. These changes are discussed more fully in Section 4.
The detailed specification of the current model is set out in Section 6.
The simulation properties of the MM, in terms of both timing and scale of responses, have not
been affected substantially by the recent changes. For example, an unanticipated change in the
short-term interest rate for four quarters still has its maximum impact on inflation after about nine
quarters, and the order of magnitude is similar to that shown in the earlier publication. The current
MM suggests that unanticipated changes in interest rates have a slightly faster impact on real GDP
than the earlier version, with the peak impact being felt after four rather than five quarters. The
size of the impact is comparable with the earlier version of the MM.
1.4
Other models added to the suite
There has been a large amount of work within the Bank of England over the past two years
designed to throw light on specific monetary policy related issues. Some of this research feeds
into the background analysis prepared as input to the quarterly forecasting round, while other work
feeds into monthly briefings to highlight specific issues on an ad hoc basis. Specific examples can
be found in the papers published in the Bank’s working paper series. A selection of such research
is highlighted here.
●
A series of papers has investigated the impact of model uncertainty on actual and optimal
monetary policy.(1)
●
Further work using structural vector autoregressions has been done, following on from
research discussed in Chapter 5 of the 1999 volume. One example was aimed at identifying
monetary policy shocks from the many other shocks that hit the economy, by imposing a
priori restrictions.(2) Another example used related methods to investigate the empirical
relationship between different measures of ‘gaps’ (output, employment and capacity
utilisation) by the imposition of restrictions implied by economic theory.(3)
●
Small-scale aggregated models (as discussed in Chapter 4 of the 1999 volume) have also
been used to investigate the relationship between optimal monetary policy and inflation
projections.(4)
(1)
(2)
(3)
(4)
Hall, Salmon, Yates and Batini (1999); Martin and Salmon (1999); and Martin (1999).
Dhar, Pain and Thomas (2000).
Astley and Yates (1999). (This paper was mentioned in the April 1999 publication, but was published subsequently.)
Batini and Nelson (2000).
The macroeconometric model
9
●
Optimising models (of the type outlined in Chapter 6 of the 1999 volume) have been used to
investigate several issues of relevance to monetary policy. For example, one model has been
used to investigate the determinants of the changing behaviour of mark-ups over time.(1)
Another paper has examined the potential impact of the labour market reforms of the 1980s
on the wage-setting and employment decisions of firms.(2)
●
Further work has been done on the Phillips curve type models discussed in Chapter 3 of the
1999 volume; this work may be published in due course.
●
There has also been considerable work on developing tools for extracting and interpreting
information from financial markets, for example about interest rate and inflation
expectations.(3)
For the remainder of this publication we focus entirely on the MM, how it has been developed and
its properties.
2
Overview of the macroeconometric model
The macroeconometric model contains around 20 estimated econometric equations. These are
supplemented by identities, transformations and linking equations. Including exogenous variables,
there are about 140 variables in the model database.(4)
The MM has three key features.
(i)
A long-run growth path for real output, consistent with the following properties:
●
price level neutrality: the long-run real growth path is independent of the price level. This is
ensured by restricting equations containing nominal variables to exhibit static homogeneity.
●
inflation neutrality: the long-run real level of output and the unemployment rate are
independent of the inflation rate, ie the long-run Phillips curve is vertical. To ensure that this
holds, equations containing nominal variables are restricted to satisfy dynamic homogeneity,
and inflation expectations are constrained to converge on actual inflation in the long run.
(ii)
A long-run growth path for nominal (ie money-denominated) variables, determined by an
anchor specified in terms of a target for a nominal variable (for example an inflation or a
price level target). This nominal anchor is usually secured by choosing a feedback rule for
monetary policy linking the instrument of monetary policy (short-term nominal interest
rates) to the selected target. Though the quantity of money does not have a causal role in
this set-up (over and above the inter-temporal price of money, ie interest rates) the money
(1)
(2)
(3)
(4)
Britton, Larsen and Small (2000).
Millard (2000).
Anderson and Sleath (1999); Clews, Panigirtzoglou and Proudman (2000); and Bliss and Panigirtzoglou (2000).
See the variables listing in Section 7.
10
Economic models at the Bank of England
supply will move in line with nominal output in the long run, in the absence of persistent
shifts in velocity.
(iii) Sluggish adjustment of nominal and real variables to economic shocks. Goods and labour
markets are characterised by both real inertia (quantities take time to adjust to economic
shocks) and nominal inertia (prices do not move immediately in response to changing
economic conditions), which results in slow adjustment of the economy towards its long-run
growth path. The speed of adjustment reflects the degree of inertia in the wage-price system,
and the costs of adjusting employment or the capital stock.
3
The structure of the macroeconometric model
This section discusses the structure of the MM in more detail. We follow the three stages
described above, by characterising the real and nominal long-run growth paths and the dynamic
response of the economy when it is away from these paths. All variables are expressed in logs,
unless otherwise stated.(1) Equation coefficients are written as Greek letters; for example intercept
coefficients are written as βz for the equation in which z is the dependent variable. A full
description of these and all the other model equations is given in Section 6.
3.1
The long-run real growth path
The core of the model’s supply side consists of four variables: output, labour (defined in terms of
hours worked), the capital stock (defined in terms of non-housing capital), and real wages.
Long-run output is determined by a simple Cobb-Douglas production function with constant
returns to scale and diminishing marginal returns to each factor of production, which can be
written as:
y = βy + αµ T + α l + (1 – α)k
(3.1.1)
where output (y) is produced by combinations of capital (k), labour (l) and labour-augmenting
technical progress (µ T), which is assumed to be exogenous.(2)
Firms are assumed to choose labour and capital so as to maximise their profits. This implies the
following marginal revenue/product conditions with respect to labour and capital:
y – l = βl + w – pd
(3.1.2)
y – k = βk + rc
(3.1.3)
where w is the nominal wage, pd is the domestic product price (ie the GDP deflator), rc is the real
cost of capital, and the long-run constants capture, among other things, the production technology
and the degree of competition in product and labour markets.
(1)
(2)
The convention adopted throughout is for variables to be assigned upper-case and their logs to be assigned lower-case letters.
The parameter α is set at 0.7, consistent with the observed average share of income going to labour.
The macroeconometric model
11
Equation (3.1.1) can be expressed as a labour demand equation as in (6.3.11) below by writing
employment on the left-hand side. Equation (3.1.2) can be expressed as either a labour demand
curve, by writing employment on the left-hand side, or as a price mark-up equation, with prices on
the left-hand side. In the equation listing in Section 6, it is expressed as a price mark-up over unit
labour costs (6.4.1). In the long run this mark-up is assumed constant. Together with the
assumption of Cobb-Douglas technology this implies that long-run factor shares are also constant.
Equation (3.1.3), expressed as the business investment equation (6.2.12), and the capital
accumulation equation (6.2.11) together describe the investment/demand-for-capital relationship.
Wage-setting behaviour is consistent with an imperfectly competitive model of the labour market,
where firms bargain with workers over real wages, but set goods prices and employment levels
(see, for example, Layard, Nickell and Jackman (1991)). The real wage equation (3.1.4) which is
estimated in equation (6.3.3) implies that, in the long run, real unit labour costs depend positively
on a set of structural variables (Zs) (such as union power and the replacement ratio) and negatively
on the unemployment rate (U):(1)
w – pd = βw + y – l + θ1Zs – θ2U
(3.1.4)
where the rate of unemployment is defined as the active working population minus employment as
a proportion of the active working population by headcount (6.3.8). The link between employment
in terms of total hours worked and in terms of the number of people in work is given by a
relationship for average hours (6.3.13).
Equation (3.1.2) and (3.1.4) both define alternative expressions for the labour share that must be
consistent in the long run. By substituting equation (3.1.2) into (3.1.4) and rearranging, the
long-run unemployment rate (U*) can be expressed as:
U* = (βl + βw + θ1Zs) / θ2
(3.1.5)
In a closed economy, aggregate demand and supply cannot, by definition, diverge from each other.
But in an open economy such as the United Kingdom, aggregate demand can be met from
overseas as well as from domestic supply, and domestic supply can be sold overseas to meet
foreign demand. So a stylised IS-curve model of aggregate demand can be written as:
yd = βy + γ1y + γ2yw + γ3r + γ4x
d
(3.1.6)
where aggregate demand, yd, depends on domestic income (y), overseas income (yw), the real
interest rate (r) and the real exchange rate (x), which will affect the share of domestic and overseas
demand being spent on UK output.
(1)
In all the supply-side equations, labour costs include employers’ taxes.
12
Economic models at the Bank of England
Summing the expenditure components of GDP determines aggregate demand:
●
consumption (6.2.8) is modelled in the long run as a function of labour income, wealth and
real interest rates; nominal interest rates are included in the short-run dynamics to proxy for
confidence and cash-flow effects; and the change in unemployment is used to capture
influences on precautionary saving;
●
investment (6.2.11 and 6.2.12) is consistent with equation (3.1.3) above, and determines the
stock of capital;
●
stockbuilding (6.2.16 and 6.2.17) over the medium term is consistent with a downward trend
in the long-run stock-output ratio;
●
real government spending (6.2.15) is treated as exogenous in the simulations outlined in
Section 5. But announced nominal government spending plans provide the basis for
forecasting this component (with real government spending influenced by the projection for
inflation);
●
exports (6.2.18) depend on world trade volumes (weighted by UK market shares) and a
measure of the real exchange rate, which reflects the competitiveness of UK exports; and
●
imports (6.2.19) are modelled as a function of domestic demand, a competitiveness term, and
a proxy measure for the increase in gross trade flows relative to world demand arising from
increased globalisation and international specialisation.
As the United Kingdom is an open economy, aggregate demand is brought into line with potential
supply in the long run by movements in the real exchange rate, via a combination of changes in
the nominal exchange rate and domestic price level for a given path for foreign prices.
For simulation purposes, the nominal exchange rate (st) (6.1.6) is typically determined by the
uncovered interest parity condition.(1) This relates expected changes in the exchange rate between
two currencies to differences in their interest rates and a risk premium. This can be written as:
st = set+1 + it – iwt – ρt
(3.1.7)
where st is defined as the number of units of foreign currency per unit of domestic currency, se is
the expected value of the nominal exchange rate, i and iw are the domestic and foreign one-period
nominal interest rates respectively, and ρ is the risk premium. This relationship can be written
equivalently as a relationship expressing the real exchange rate in terms of real interest rates:
xt = xet+1 + rt – rwt – ρt
(1)
(3.1.8)
Since November 1999 the central projection in the Inflation Report has assumed that the nominal exchange rate evolves along a path halfway
between a constant rate and the path implied by the uncovered interest parity condition, conditioned on constant UK interest rates, with a
zero risk premium.
The macroeconometric model
13
where rw is the world real interest rate. This implies that, in the long run, UK real interest rates
would differ from world real interest rates by the risk premium, if the long-run real exchange rate
were constant.
3.2
The long-run nominal growth path and inflation
The long-run levels of output and employment described above are independent of both the price
level and the rate of inflation. This reflects the theoretical presumption and empirical evidence
that there is no long-run trade-off between inflation and the level of output or employment.
The MM has several measures of prices:
●
the domestic output price index (the GDP deflator at factor cost, pd) (6.4.1) is modelled as a
mark-up over unit labour costs, as described above;
●
the RPIY price index (RPI excluding indirect and council taxes, and mortgage interest
payments) (6.4.5) is modelled as a weighted average of domestically consumed output prices
and import prices;
●
RPIX (6.4.6) is then modelled by adding indirect and council taxes to RPIY;
●
the retail price index (RPI) (6.4.11) is obtained by adding mortgage interest payments to
RPIX;
●
the consumers’ expenditure deflator (6.4.13) is assumed to grow in line with RPIX excluding
council taxes;
●
the government consumption deflator (6.4.14) is assumed to grow in line with the average of
retail prices, excluding mortgage interest payments and council taxes, and unit labour costs;
●
import prices (6.4.2) are assumed in the long run to be set by foreign suppliers in world
markets, with their sterling value determined by the nominal exchange rate; and
●
export prices (6.4.3) reflect a weighted average of domestic costs, exchange rate adjusted
world prices, and oil prices.
The level and rate of change of the overall price level are determined by monetary policy in the
medium term. This can be modelled in the form of a policy rule, linking the instrument of
monetary policy (short-term nominal interest rates) to the monetary policy target. In principle, any
nominal variable can act as a target for monetary policy, for example an inflation target or a price
level target.
A variety of policy rules can be used. In the United Kingdom, the government’s inflation target is
defined in terms of RPIX inflation. The MM allows this type of policy regime to be captured by a
14
Economic models at the Bank of England
rule in a variety of ways. One common formulation is that developed by Taylor (1993). A version
of this is expressed in equation (3.2.1), where nominal interest rates (i) are set with reference to
some ‘equilibrium’ real interest rate (rt*), the current annual inflation rate πt, the deviation of the
current annual inflation rate from the inflation target (πt – πt*), and the excess of actual output
over potential (y – y*):(1)
it = πt + rt* + λ1(πt – πt*) + λ2(yt – yt*)
(3.2.1)
In this simple rule, the responsiveness of nominal interest rates to deviations of inflation from
target and output from potential is determined by the weights λ1 and λ2. Such rules usually ensure
that inflation converges on its target rate in the long run.
An alternative simple price level target rule is also used in some of the simulations described later
in Section 5. This specifies that nominal interest rates (it) are set with reference to the equilibrium
real interest rate (rt*), the current annual inflation rate πt and the deviation of the actual price level
from some target price level (pt – pt*). One form of such a rule is:
it = πt + rt* + λp (pt – pt*)
(3.2.2)
where λp determines the responsiveness of nominal interest rates to the deviation of the price level
from target. As long as λp is above zero, this rule should ensure that the price level converges on
its target trajectory in the long run.
The monetary policy rules adopted in the simulations described below are purely illustrative, and
can be varied according to the question being considered. They have no status as a guide to
monetary policy in practice.
3.3
Short-run dynamics and inflation
When on its long-run growth path, output grows in line with supply potential (determined by
technology and factor supplies), and inflation is determined by the stance of monetary policy. The
relationships that characterise this long-run growth path are embedded in equilibrium-correction
terms in dynamic equations that ensure that these relationships reassert themselves gradually in the
face of economic shocks. Often, theoretical reasoning has more to say about long-run
relationships than about how quickly the economy should return to its long-run growth path
following shocks. So, unlike the long-run properties, which are almost always imposed to reflect
economic theory, the short-run dynamics are more often empirically estimated to match average
historical behaviour.
Output and inflation diverge from their long-run growth paths because of different kinds of inertia,
which are reflected in the equations of the MM. There are two main types:
(1)
The ‘output gap’ measure used in the Taylor rule in the MM defines potential output as the output that would be implied by a
Cobb-Douglas production function at the current level of capital input, potential labour inputs, and exogenous labour-augmenting technical
progress.
The macroeconometric model
15
●
real inertia, which restrains real variables from moving immediately to their long-run values,
for example because of the costs of adjusting employment or the capital stock; and
●
nominal inertia, which restrains prices from adjusting immediately in the face of shocks.
One example is the assumption that wage contracts are fixed for one year, and so do not
respond quickly to unanticipated developments (see Section 6.3.1). Another is that there
may be costs associated with changing prices.
When inertia is present, the economy can deviate for some time from its long-run real and nominal
growth paths in the face of shocks. Deviations of output from its long-run growth path will tend to
be associated with changes in inflationary pressure. If the economy is operating above its
potential output level, inflationary pressure will tend to rise, other things being equal, and
conversely if activity is below potential. In practice, it is difficult to judge with any precision
where activity stands in relation to capacity. One useful conceptual framework for thinking about
inflationary pressure is a short-run Phillips curve.(1) Although it is not possible to derive a Phillips
curve analytically from the complex dynamic wage-price system in the MM, it is possible to
explore influences on inflation in the MM by running a variety of simulations.
4
Recent changes to the model specification
As with any macroeconometric model, the MM does not have a fixed specification and is being
continually developed. These developments may reflect new data or, more generally, new analysis
of a particular sector of the economy.
Since the publication of the MM in April 1999 there have been a series of small revisions to
equation specifications in the light of new data. These new data reflect not only the regular stream
of quarterly data, but also periodic revisions to past series, for example the re-grossing of the
labour market statistics and revisions to the National Accounts data contained in the Blue Book.
Such changes to the data require the relevant equations to be re-estimated to check that historical
relationships remain valid.
Often these revisions do not require alterations to the basic structure of the equation, but can lead
to different estimated coefficients in the equation. Equations where coefficients have been
modified slightly due to new data include: real wages (6.3.3), domestic output prices (6.4.1),
import prices (6.4.2), export prices (6.4.3), households’ non-labour income (6.6.4), business
investment (6.2.12), employment in hours (6.3.11), average hours (6.3.12), and trend output
(6.2.5). In addition, there have been minor changes to the treatment of value-added tax and special
duties tax, and a simple equation for the government expenditure deflator (6.4.14) has been
introduced. These slight modifications have had no material impact on the simulation properties
of the MM.
Equations where there have been more significant changes to the coefficients include the volume
equations for exports (6.2.18) and imports (6.2.19). A combination of new data and the more
(1)
For a more complete discussion of Phillips curve models see Chapter 3 of Bank of England (1999).
16
Economic models at the Bank of England
recent behaviour of trade volumes suggests that the long-run real exchange rate elasticities for UK
exports and imports are now lower than previously estimated. These modifications to the trade
volume elasticities somewhat reduce the impact of changes in exchange rates on net trade.
In addition to data revisions since the previous publication of the MM, there has been further
analysis of the role of real wealth and interest rates in determining consumers’ real spending and
the behaviour of house prices. The current version of the MM includes both a separate role for
gross housing and net financial wealth in the short-run dynamics of the consumption function and
a role for real interest rates in the long run (6.2.8). Complementing these changes to the
consumption equation, the current version of the MM also includes a more detailed account of
house price determination (6.4.15), although it is still relatively simple compared with
single-equation specifications available in the academic literature. These modifications have
marginally strengthened the channel from real interest rates to domestic demand.
A further modification was aimed at simplifying the use of the MM in forecasting. In the
April 1999 version, both RPIY and GDP deflator equations were jointly dependent on unit
labour costs. This joint determination sometimes made the dynamics of the MM difficult to
understand. For clarity, the MM specification was altered to simplify the transmission of certain
shocks by separately identifying the price of output produced in the United Kingdom, including
exports (6.4.1), and the price of output both consumed and produced in the United Kingdom,
excluding exports (6.4.4). For some simulations the RPIY inflation is now modelled as a weighted
average of domestic and import prices directly rather than relying on unit labour costs as a proxy
for domestic prices. This modification marginally increases the speed of the MM’s price
dynamics.
5
Simulation properties
5.1
Rationale for simulations
Some of the properties of the MM can be assessed by examining sub-systems within it, or by
analysing single equations. However, the properties of the model as a whole can best be
understood by simulation analysis.
The illustrative simulations presented here concentrate mainly on the behaviour of the nominal
side of the model, and focus on the effects of nominal interest rate changes on the price level,
inflation and output.
As explained earlier, it is possible to run the model under a range of different assumptions. In
particular, those adopted for forecasting may not be the same as those that are appropriate for
longer-term simulations. For example, in MPC forecasts, it will typically be assumed that the
government’s spending plans are set in cash terms.(1) But in longer-run simulations, it is more
appropriate to assume that government spending is either exogenous in real terms or fixed as a
share of output.
(1)
For example, see the discussion on page 53 of the May 2000 Inflation Report.
The macroeconometric model
17
Other important assumptions relate to how expectations are formed. In the simulations described
below, expectations of the exchange rate one period ahead are assumed to be formed in a
forward-looking manner, consistent with the predictions of the model itself. This implies that the
exchange rate will jump in response to unexpected changes in interest rate differentials or in the
long-run exchange rate level. Other asset prices are not treated in a forward-looking manner, but
are assumed to move in ways that are broadly consistent with the long-run growth path of the
economy (for example, equity prices are assumed to grow in line with nominal GDP). Inflation
expectations are assumed to exhibit a degree of inertia: wage-setters, for example, take time to
respond to new information (see equation (6.4.16b)).
Various assumptions can be made about the behaviour of monetary policy. For example, different
interest rate rules can be used. Neither the stance of monetary policy nor the assumptions about
how expectations are formed affects the real properties of the simulations in the long run, but they
are both likely to have important effects on the dynamic responses of inflation and real activity in
the shorter run.
5.2
Temporary interest rate shock
In attempting to understand the transmission mechanism of monetary policy, it is helpful to
simulate the effects of a change in nominal interest rates. In practice, interest rates are not
changed in isolation but are altered in response to economic developments that require a monetary
policy response. And the effect of any resulting change in interest rates will also depend on the
policy regime—since this, in turn, will affect the behaviour of agents in the economy, for example
via its effect on long-run inflation expectations and other asset prices.
Given these considerations, we analyse below the effect of an unanticipated one-year positive
1 percentage point deviation in interest rates from a monetary policy rule. In these simulations,
nominal interest rates continue to be affected beyond the first year, as interest rate changes are
determined by the monetary policy rule. The monetary policy rule used for the purpose of this
benchmark simulation is one in which nominal interest rates are set according to a Taylor rule of
the type shown in equation (3.2.1).
To illustrate the point that the simulation responses of inflation and output will depend on the
specific assumptions made, we show three different simulations:
●
First, the coefficients in the Taylor rule on the deviations of inflation from target and output
from base are set at 0.5.(1)
●
Second, the coefficient in the Taylor rule on the deviation of inflation from target is
increased to 1.0, suggesting that the monetary authority responds more strongly to inflation
deviations from target.
(1)
This coefficient choice was originally adopted in Taylor (1993).
18
Economic models at the Bank of England
Third, the coefficient on the deviation of inflation from target in the Taylor rule is increased
further to 1.5, suggesting that the monetary authority will respond even more strongly to
inflation deviations from target.
●
It would also be possible to vary other aspects of the simulation, for example by altering the
assumptions about how expectations are formed. However, it should be emphasised that, even for
a given set of assumptions, the effects of a change in interest rates are highly uncertain, because of
uncertainty about the value of the parameters underlying the model and about the specification of
the model itself.
Charts 1(a) and (b) show the results for inflation and GDP respectively. The maximum effect of
the temporary interest rate increase on real activity occurs after about one year, and the maximum
effect on inflation occurs after about two years. For the benchmark simulation, where the Taylor
rule with a weight of 0.5 on the deviation of inflation from target is adopted, the level of GDP falls
by about 0.3% at the end of the first year, recovering to base after three years. Inflation remains
broadly unchanged during the first year, reflecting the degree of nominal inertia in the economy,
but by the beginning of the third year has fallen by just over 0.3 percentage points. Thereafter, it
returns slowly to base.
Simulation (i): Response to 1 percentage point rise in nominal interest rates
for one year
Chart 1(a)
Chart 1(b)
Annual RPIX inflation rate
GDP level
Difference from base, percentage points
Difference from base, per cent
0.2
Taylor rule (1.0)
0.1
Taylor rule (1.5)
0.1
+
+
0.0
0.0
–
–
Taylor rule (1.5)
0.2
Taylor rule (0.5)
0.1
0.1
0.2
0.2
0.3
0.3
Taylor rule (1.0)
Taylor rule (0.5)
1
2
3
4
5
6
7
8
9 10
Quarters
11
12
13
14
15
16
0.4
1
2
3
4
5
6
7
8 9 10 11 12 13 14 15 16
Quarters
0.4
When a Taylor rule with a greater weight on deviations of inflation from target is adopted, the
peak effect on inflation is similar to the benchmark case. But thereafter inflation returns to base
more quickly as interest rates are adjusted more strongly in response to the temporary deviation of
inflation from target. However, this more vigorous response of monetary policy results in a more
volatile path for output, with GDP rising further above its long-run level before stabilising.
5.3
Exogenous change in a price level target
The next simulation shows what could happen if there were an unexpected 1% permanent decrease
in a nominal target for monetary policy defined in terms of the (RPIX) price level. Although this
The macroeconometric model
19
simulation does not correspond to the current UK monetary policy framework, which is defined in
terms of an inflation target, it is useful for testing and illustrating the nominal neutrality of the
model (ie that the change in the nominal anchor has no long-run effect on real variables such as
output and employment). The impact of such a decrease in a price level target will partly depend
on the rule that monetary policy is assumed to follow. We assume that nominal interest rates are
set according to a simple rule of the type shown in equation (3.2.2).
To illustrate the impact of different policy rules, Table A shows simulation results when the
coefficient on the deviations of the price level from target is set at 0.25, 0.50 and 0.75 respectively.
The initial price level gap is reflected directly in nominal interest rates, which rise in all cases by
between 1/4 and 3/4 percentage points immediately, before falling below base by the third year. Real
interest rates remain above base levels for as long as RPIX is above its new target level. Output
falls by a maximum of between 0.1% and 0.3% at the end of two years, before returning close to
base level after about four years. The MM exhibits a smooth and gradual adjustment to its
long-run path when the Taylor rule is used in the simulations above, but this is not the case when
the simple price level target rule is used to close the MM. While RPIX falls to its new target level
over four years it does not do so smoothly. Instead, the MM response to this shock is
characterised by a dampened cyclical pattern. Despite this cyclical pattern, the simulation results
show the long-run nominal neutrality of the model: all real variables return to their long-run
values and all nominal variables fall by 1%.
Table A
Simulation (ii): Response to 1% fall in price level target
Percentage or percentage point (pp) difference from base
1 quarter
Feedback coefficient on p-p*
Variable
RPIX level (%)
RPIX annual inflation (pp)
Nominal interest rate (pp)
Real interest rate (pp) (b)
GDP (%)
Real exchange rate (%)
Consumer spending (%)
Investment (%)
Exports (%)
Imports (%)
2 years
4 years
Steady state (a)
0.25
0.50
0.75
0.25
0.50
0.75
0.25
0.50
0.75
-0.03
-0.03
0.24
0.27
-0.03
0.75
-0.04
0.00
-0.05
-0.04
-0.04
-0.04
0.48
0.52
-0.05
1.01
-0.08
0.00
-0.07
-0.09
-0.05
-0.05
0.72
0.77
-0.06
1.21
-0.12
0.00
-0.09
-0.13
-0.16
-0.12
0.13
0.24
-0.14
0.15
-0.15
-0.31
-0.06
-0.14
-0.22
-0.18
0.27
0.45
-0.22
-0.10
-0.28
-0.56
-0.05
-0.30
-0.27
-0.23
0.40
0.63
-0.30
-0.41
-0.40
-0.80
-0.02
-0.45
-0.66
-0.28
-0.19
0.09
-0.07
-0.35
-0.06
-0.40
0.03
-0.08
-0.98
-0.42
-0.40
0.02
-0.05
-0.83
-0.03
-0.61
0.11
-0.11
-1.24
-0.53
-0.72
-0.19
0.01
-1.21
0.05
-0.73
0.20
-0.07
-1.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
(a) The MM has been constructed to ensure that these steady-state responses are obtained.
(b) Here the real interest rate is defined as rt = it – π et, where π et = πt.
All the simulations shown above illustrate the monetary transmission mechanism within the model
by which changes in nominal interest rates bring about changes in the price level. There are three
main channels:
20
Economic models at the Bank of England
●
First, the real exchange rate appreciates on impact by around 1%, before depreciating back
towards its base value.(1) This initial jump reflects the anticipated cumulative
policy-induced rise in the real interest rate over the entire simulation. The short-term
appreciation has a direct effect on domestic prices, partly through the influence of import
prices on retail prices and partly through the temporary effect of the real exchange rate on
real wages. There is also an indirect effect on prices via net trade. Exports initially fall by
around 0.1% before recovering as the real exchange rate falls back. Imports are stimulated
by the fall in competitiveness, but this is outweighed by the effects of the fall in domestic
demand. The net impact is for imports to fall by 0.3% after two years.
●
Second, higher nominal interest rates result in the stock of net financial wealth being
devalued. That, together with the short-term confidence effects, results in lower
consumption in the short run. Consumption falls by about 0.3% after two years. Further
out, this is offset by a terms-of-trade effect on income and by a wealth effect associated with
the exchange rate appreciation.
●
Third, there is the effect of real interest rates on investment and consumption, with
investment falling by about 0.6% at the end of two years. In the longer run, investment
recovers to bring the capital stock back towards its long-run level, but this occurs slowly
because of the protracted response of investment in the model. Similarly, as real interest
rates and wealth return to their long-run level, consumption also returns to its long-run level.
6
Detailed equation listing
This section describes, in detail, the equations that form the Bank’s macroeconometric model
(MM). It is important to note that the equations outlined in this section have been developed as
part of the overall structure of the MM. The structure of the equations, the short-run and long-run
restrictions placed on the coefficients, the available set of variables and the choice of data have all
been dictated by the structure of the MM. The criteria for selection of a particular equation
include the equation’s statistical properties and its impact on the overall MM simulation properties.
For these reasons the individual equations outlined in this section should not be viewed as the
‘best’ single equations estimated: they are part of a system of equations that form the MM. From
time to time alternative equations are developed and examined as part of the MM structure. They
may be used to examine the impact of alternative economic theories, or the impact of relaxing
some of the restrictions placed on the equations.
Several conventions are used in the presentation of the equations in this section. Lower-case
letters indicate natural logs. The subscript t denotes time: data are quarterly. ∆ indicates a first
difference. Q1, Q2, Q3 and Q4 denote seasonal dummies. Each relationship is written so as to
distinguish the long-run solution of the equation from the short-run dynamics. Long-run solutions
(1)
This description of the transmission mechanism applies to all the simulations shown, but the quantitative results apply to the price level
target shock with a monetary policy rule of the form (3.2.2), with the feedback parameter set to 0.5. Altering assumptions under which the
simulations are performed may alter the transmission mechanism as described here. For example, the transmission mechanism might differ
if inflation expectations were assumed to evolve differently or if real interest rates were assumed to be affected by temporary monetary
policy shocks.
The macroeconometric model
21
appear in square brackets, and follow from the usual practice of estimating error-correction
models. T-values are shown in parentheses, where applicable. Single-equation dynamic responses
are given for some key variables. A standard set of diagnostic tests is shown, with the probability
values for the diagnostic tests given in square brackets. Definitions and data sources for all model
variables are given at the end of the equation listing. Bold numbers in brackets are
cross-references to equations.
6.1
6.1.1
Money, financial and wealth variables
Three-month interest rate (RS)
The usual settings are either an exogenous constant nominal rate or market interest rate path (the
alternative conditioning assumptions in Inflation Report projections), or a simple form of the
Taylor rule linking the short-term nominal interest rate to the monetary policy target variable.
RSt = INFt + Θ 0 + Θ1( gdpt − gdptt ) + Θ 2 ( INFt − ZPSTAt )
(6.1.1)
where
RS = base rate of interest (6.1.1).
INF = four-quarter inflation rate of RPIX.
GDPT = trend output (6.2.5).
GDP = GDP(A) at factor cost (6.2.1).
ZPSTA = Government inflation target (exogenous).
Θ0, Θ1 and Θ2 are parameters to be set by the user.
A variant of the rule links short nominal rates to a price level target.
(
RSt = INFt + Θ 0 + Θ1( gdpt − gdptt ) + Θ 2 rpixt − rpixt*
)
(6.1.1a)
where
RS = base rate of interest (6.1.1).
INF = four-quarter inflation rate of RPIX.
GDPT = trend output (6.2.5).
GDP = GDP(A) at factor cost (6.2.1).
RPIX = RPI excluding mortgage interest payments (6.4.6).
RPIX* = target level for RPIX (exogenous).
6.1.2
Twenty-year bond rates (RL)
For simulation purposes, long rates are set by reference to current short rates.
RLt = RSt
where
(6.1.2)
22
Economic models at the Bank of England
RL = 20-year bond yield (6.1.2).
RS = base rate of interest (6.1.1).
6.1.3
Mortgage interest rates (RMM4)
Mortgage interest rates affect the mortgage interest payment sub-index of the RPI. They are
proxied by a constant mark-up over a weighted average of current and lagged short-term interest
rates. Lagged rates are used to reflect some of the delay between official rate changes and their
pass-through to mortgage borrowers. Both the weights and the mark-up were chosen to
approximate past experience.
()
()
RMM4t = 23 RSt + 13 RSt −1 + 1.0
(6.1.3)
where
RMM4 = mortgage interest rate (6.1.3).
RS = base rate of interest (6.1.1).
6.1.4
Deposit interest rates (RD)
Combined with the short-term interest rate, the deposit rate defines the opportunity cost of holding
money. In the long run, it is equal to a proportion of the short-term rate (6.1.1).
∆RDt = 0.0572 − 0.28 ∆ RDt −1 + 0.75 ∆RSt + 0.24 ∆RSt −1 − 0.13[ RDt −1 − 0.7 RSt −1 ]
(1.4)
(−2.7)
(17.4)
(3.0)
(6.1.4)
(−2.8)
where
RD = deposit rate (6.1.4).
RS = base rate of interest (6.1.1).
Adjusted R2 = 0.80
Equation standard error = 0.38
LM test for serial correlation: F-stat. = 1.79 [0.17]
Normality test: χ2(2) = 57.12 [0.00]
Heteroscedasticity: F-stat. = 1.12 [0.36]
Sample period: 1977 Q3–1998 Q2
6.1.5
Real cost of capital (RCC)
Investment is affected by the real cost of capital (RCC). This reflects the rate of depreciation (on a
quarterly basis), and the rates of both corporate taxation and investment allowances. As elsewhere
in the model, inflation expectations relate to RPIX.
(
(
 1 − PVICt 
4
RCCt = 0.0025 
 RLt − INFEt + 100 1 − BETAt
 1 − RCt 
where
))
(6.1.5)
The macroeconometric model
23
RCC = real cost of capital (6.1.5).
PVIC = present value of investment allowances (exogenous).
RC = effective corporate income taxation rate (exogenous).
RL = 20-year bond yield (6.1.2).
INFE = expectations of annual RPIX inflation (6.4.16).
1–BETA = business sector net capital stock depreciation rate (exogenous).
6.1.6
Nominal effective exchange rate (EER)
For simulation purposes, the exchange rate is modelled using uncovered interest parity (UIP), with
model-consistent expectations.
eert = xeert + RSDF
(6.1.6)
where
EER = sterling effective exchange rate index (6.1.6).
XEER = expected sterling effective exchange rate (one period ahead).
RSDF = interest rate differential (6.1.7).
The interest rate differential is defined by:
RS
WRSt 
RSDF = log1 + t  − log1 +



400
400 
(6.1.7)
where
RS = base rate of interest (6.1.1).
WRS = world nominal interest rate (exogenous).
But under certain forecasting conditions the exchange rate is assumed to evolve along a path
halfway between a constant rate and the path implied by the UIP condition, with backward-looking
expectations and conditional on constant interest rates.
For simulation purposes, the sterling-dollar exchange rate (EDS) is simply related to the effective
rate. The constant reflects different units of measurement.
edst = – 4.02 + eert
(6.1.8)
EDS = sterling-US dollar exchange rate (6.1.8).
EER = sterling effective exchange rate index (6.1.6).
6.1.7
Real exchange rate (RXRX, RXRM and RRX)
The exporters’ real exchange rate (RXRX) is the effective exchange rate deflated using relative
export prices. The importers’ relative price (RXRM) is the price of imports relative to the GDP
deflator. A further measure (RRX) relates the GDP deflator to world export prices.
24
Economic models at the Bank of England
RXRXt =
EERt .PXt
WPXt
(6.1.9)
where
RXRX = exporters’ real exchange rate (6.1.9).
EER = sterling effective exchange rate index (6.1.6).
WPX = world export prices (exogenous).
PX = export price deflator (6.4.3).
RXRMt =
PMt
PGDPt
(6.1.10)
where
RXRM = importers’ relative price (6.1.10).
PM = import price deflator (6.4.2).
PGDP = GDP deflator at factor cost (6.4.1).
RRXt =
EERt .PGDPt
WPXt
(6.1.11a)
where
RRX = real exchange rate (GDP deflator measure) (6.1.11).
EER = sterling effective exchange rate index (6.1.6).
WPX = world export prices (exogenous).
PGDP = GDP deflator at factor cost (6.4.1).
To solve the model in forward-looking mode, it is convenient to express the UIP condition in real
terms.
rrxt = xrrxt + RSDFt + xwpxt + pgdpt − wpxt − xpgdpt
(6.1.11b)
where
RRX = real exchange rate (GDP deflator measure) (6.1.11).
XRRX = expected real exchange rate (one period ahead).
RSDF = interest rate differential (6.1.7).
XWPX = expected M6 export prices (one period ahead) (exogenous).
PGDP = GDP deflator at factor cost (6.4.1).
WPX = world export prices (exogenous).
XPGDP = expected GDP deflator at factor cost (one period ahead).
When the real exchange rate is determined in this way, the nominal exchange rate is obtained by
inverting the identity for RRX in (6.1.11a).
The macroeconometric model
6.1.8
25
Household sector wealth (WEL)
Household sector wealth (WEL) is modelled in two parts: gross housing wealth and net financial
wealth (GHW and NFW).
WELt = GHWt + NFWt
(6.1.12)
Nominal housing investment is the flow into nominal housing wealth. The stock of nominal
housing wealth is revalued in line with changes in house prices.
 PHSEt 
GHWt = ( PGDPt . IHt ) + GHWt −1 

 PHSEt −1 
(6.1.13)
where
GHW = gross housing wealth (6.1.13).
PGDP = GDP deflator at factor cost (6.4.1).
IH = private sector dwellings investment (6.2.13).
PHSE = UK house prices (6.4.15).
Saving is the flow into nominal net financial wealth. The stock of net financial wealth is revalued
in line with changes in asset prices (REV).
NFWt = PCt ( RHPIt − Ct − IHt ) + NFWt −1. REVt
(6.1.14)
where
NFW = net financial wealth (6.1.14).
PC = total final consumers’ expenditure deflator (6.4.13).
RHPI = real household post-tax income (6.6.5).
C = consumers’ expenditure (6.2.8).
IH = private sector dwellings investment (6.2.13).
REV = wealth revaluation term (6.1.15).
The weights in the revaluation term were derived from the household sector balance sheet (in
Financial Statistics) and reflect the relative importance of foreign assets, gilts, interest-bearing
deposits and equities in net financial wealth.
 EERt −1   WEQPt 
 EERt −1 
 RLt −1 
REVt = 0.045 + 0.12 

 + 0.01
 + 0.15 

 EERt   WEQPt −1 
 EERt 
 RLt 
 EERt −1   USRLt −1 
 EQPt 
+ 0.04 

 + 0.64 

 EERt   USRLt 
 EQPt −1 
where
REV = wealth revaluation term (6.1.15).
EER = sterling effective exchange rate index (6.1.6).
(6.1.15)
26
Economic models at the Bank of England
WEQP = world equity prices (exogenous).
RL = 20-year bond yield (6.1.2).
USRL = US long bond rate (exogenous).
EQP = equity prices (6.1.16).
An alternative, used in forecasting, is to allow NFW to evolve in line with nominal GDP.
Equity prices either evolve in line with a simple dividend growth model.


(1 + ∆gdplt )
EQPt = 0.0007 
 (GDPLt )
 ( RLt / 100 − ∆gdplt ) 
(6.1.16a)
where
EQP = equity prices (6.1.16).
GDPL = GDP at factor cost in current prices (6.2.4).
RL = 20-year bond yield (6.1.2).
Or in line with nominal GDP, supplemented by an exogenous constant (risk premium).
∆eqpt = ∆gdplt + 0.02
(6.1.16b)
where
EQP = equity prices (6.1.16).
GDPL = GDP at factor cost in current prices (6.2.4).
RL = 20-year bond yield (6.1.2).
The simpler nominal GDP growth equation (6.1.16b) was adopted for the simulations.
6.1.9
Broad money demand (M4)
Real broad money holdings respond to activity, net financial wealth and interest rates. Money can
be specified as the nominal anchor in a monetary policy rule for interest rates.
∆( m 4 − pgdp)t = 0.0077 + 0.33 ∆( m 4 − pgdp)t −1 + 0.45 ∆gdpmt − 0.28 ∆gdpmt −1
(4.2)
(3.3)
(3.1)
(−1.8)
− 0.037[ m 4t −1 − pgdpt −1 − gdpmt −1
(−1.9)
− 0.60 (nfwt −1 − pgdpt −1 − gdpmt −1 )
(−77.6)
− 0.022 ( RDt −1 − RSt −1 )]
(−4.8)
(6.1.17)
The macroeconometric model
27
where
M4 = broad money (6.1.17).
GDPM = GDP(A) at constant market prices (6.2.3).
PGDP = GDP deflator at factor cost (6.4.1).
NFW = net financial wealth (6.1.14).
RD = deposit rate (6.1.4).
RS = base rate of interest (6.1.1).
Adjusted R2: 0.28
Equation standard error: 0.011
LM test for serial correlation: F-stat. = 2.12 [0.13]
Normality test: χ2(2) = 1.73 [0.42]
Heteroscedasticity: F-stat. = 1.60 [0.14]
Sample period: 1977 Q3–2000 Q1
6.2
6.2.1
Demand and output
Gross domestic product (GDP, GDPM, GDPL)
Expenditure-based GDP is an accounting identity, including domestic demand, imports, exports
and the factor cost adjustment.
GDPt = DDt + Xt – Mt – FCAt
(6.2.1)
where
GDP = GDP(A) at factor cost (6.2.1).
DD = total domestic expenditure (6.2.7).
X = exports of goods and services (6.2.18).
M = imports of goods and services (6.2.19).
FCA = factor cost adjustment (6.2.2).
The factor cost adjustment is an (estimated) function of GDP.
fcat = –1.87 + gdpt
(6.2.2)
where
FCA = factor cost adjustment (6.2.2).
GDP = GDP(A) at factor cost (6.2.1).
GDPM is expenditure-based GDP at constant 1995 market prices.
GDPMt = GDPt + FCAt
(6.2.3)
28
Economic models at the Bank of England
where
GDPM = GDP(A) at constant market prices (6.2.3).
GDP = GDP(A) at factor cost (6.2.1).
FCA = factor cost adjustment (6.2.2).
Nominal GDP (GDPL) at factor cost is the product of GDP (6.2.1) and the GDP deflator (PGDP)
(6.4.1).
GDPLt = PGDPt .GDPt
6.2.2
(6.2.4)
Trend GDP and capacity utilisation (GDPT, CAPU)
Trend output is determined by an estimated Cobb-Douglas production function. The direct inputs
are the non-residential capital stock, the level of population of working age and exogenous
labour-augmenting technical progress captured by the time trend.
gdptt = 0.063 + 0.3 knht + 0.7 powat + (0.70)(0.004) TIME
(3.9)
(6.2.5)
(18.7)
where
GDPT = trend output (6.2.5).
KNH = capital stock excluding residential housing (6.2.10).
POWA = population of working age (exogenous).
TIME = time trend.
Adjusted R2: 0.94
Equation standard error: 0.033
LM test for serial correlation: F-stat. = 455.7 [0.00]
Normality test: χ2(2) = 4.84 [0.09]
Heteroscedasticity: F-stat. = 6.34 [0.00]
Sample period: 1978 Q1–1997 Q4
Capacity utilisation is proxied by the residuals from the estimation of a production function where
the direct inputs are employment measured in hours, the non-residential capital stock and
labour-augmenting technical progress captured by the time trend.
CAPUt = gdpt − 2.66 − 0.70 empht − 0.30 knht − (0.70)(0.004) TIME
(288.0)
where
CAPU = capacity utilisation (6.2.6).
GDP = GDP(A) at factor cost (6.2.1).
EMPH = total employment in hours (6.3.11).
( −34.1)
(6.2.6)
The macroeconometric model
29
KNH = capital stock, excluding residential housing (6.2.10).
TIME = time trend.
Adjusted R2: 0.98
Equation standard error: 0.018
LM test for serial correlation: F-stat. = 171.6 [0.00]
Normality test: χ2(2) = 2.87 [0.24]
Heteroscedasticity: F-stat. = 4.94 [0.00]
Sample period: 1979 Q1–1998 Q4
6.2.3
Domestic demand (DD)
Domestic demand is determined by an accounting identity, as the sum of consumers’ expenditure,
investment, government consumption and stockbuilding.
DDt = Ct + It + Gt + IIt
(6.2.7)
where
DD = total domestic expenditure (6.2.7).
C = consumers’ expenditure (6.2.8).
I = fixed investment (6.2.9).
G = general government final consumption expenditure (exogenous), (see Section 6.2.10).
II = net stockbuilding (6.2.16).
6.2.4
Consumers’ expenditure (C)
In the long run, consumers’ expenditure is a function of wealth, labour income and real interest
rates. Nominal rates are included in the short run of the equation to capture confidence and
cash-flow effects of changes in the stance of monetary policy. Changes in the unemployment rate
are also included, to capture precautionary saving influences.
∆ct = − 0.036 + 0.19 ∆lyt + 0.052 ∆( ydijt −1 − pct −1 ) − 0.068 ∆urt −1 + 0.14 ∆( ghwt − pct )
( −2.7)
(3.6)
( −3.5)
(4.0)
(4.8)
+ 0.014 ∆(nfwt − pct ) − 0.0016 ∆RSt − 0.0017 ∆RSt −1
(1.3)
( −2.4)
( −2.8)
− 0.17[ct −1 − 0.89 lyt −1 − 0.11( welt −1 − pct −1 ) + 0.0028 ( RSt − 2 − INFEt − 2 )]
( −4.3)
(3.6)
(2.3)
where
C = consumers’ expenditure (6.2.8).
LY = real post-tax labour income (6.6.6).
YDIJ = non-labour income (6.6.4).
PC = total final consumers’ expenditure deflator (6.4.13).
(6.2.8)
30
Economic models at the Bank of England
UR = rate of unemployment (6.3.8).
GHW = gross housing wealth (6.1.13).
NFW = net financial wealth (6.1.14).
WEL = total household sector wealth (6.1.12).
RS = base rate of interest (6.1.1).
INFE = expectations of annual RPIX inflation (6.4.16).
Adjusted R2: 0.73
Equation standard error: 0.006
LM test for serial correlation: F-stat. = 0.01 [0.99]
Normality test: χ2(2) = 0.20 [0.91]
Heteroscedasticity: F-stat. = 1.82 [0.03]
Sample period: 1975 Q1–1998 Q1
Single-equation dynamic responses:
Response of consumers’ expenditure level to a 1% shock to RHS variables
Per cent
Quarters ahead
Real labour
Real net
Real gross
income
financial wealth
housing wealth
1
0.3
0.02
0.12
4
0.6
0.04
0.09
8
0.7
0.06
0.07
Long run (LR)
0.9
0.07
0.05
50% of LR by
3 quarters
3 quarters
o/s
90% of LR by
11 quarters
12 quarters
o/s
Quarters ahead
1
4
8
Long run (LR)
Per cent
Nominal
interest rate
-0.003
-0.002
-0.0
0.0
Unemployment
rate
-0.07
-0.04
-0.02
0.0
Real interest rate
0.000
-0.001
-0.002
-0.003
6 quarters
15 quarters
Real non-labour
income
0.05
0.03
0.01
0.0
o/s = overshoots eventual long-run response in short term.
6.2.5
Fixed investment (I)
Fixed investment is the sum of business investment, housing investment and government
investment.
It = IBUSt + IHt + IGt
where
(6.2.9)
The macroeconometric model
31
I = fixed investment (6.2.9).
IBUS = business investment (6.2.12).
IH = private sector dwellings investment (6.2.13).
IG = general government investment (exogenous), (see Section 6.2.9).
6.2.6
The capital stock (KNH, KBUSNH)
The capital stock is the previous period’s stock (allowing for depreciation at the exogenous rate
(1–BETA)) plus current investment. 21% of government investment is assumed to be housing,
consistent with the average historical relationship.
KNHt = BETANH.KNHt–1 + It – IHt – 0.21IGt
(6.2.10)
KBUSNHt = BETA.KBUSNHt–1 + IBUSt
(6.2.11)
where
KNH = capital stock, excluding residential housing (6.2.10).
KBUSNH = business non-residential capital stock (6.2.11).
1–BETANH = whole-economy net capital stock net of housing depreciation rate (exogenous).
1–BETA = business sector net capital stock depreciation rate (exogenous).
IG = general government investment (exogenous), (see Section 6.2.9).
IBUS = business investment (6.2.12).
6.2.7
Business investment (IBUS)
The assumed Cobb-Douglas production technology implies that the capital-output ratio is a
function of the user cost of capital with a unit coefficient in the long run. But adjustment is
sluggish.
∆ibust = − 0.054 + 0.11 ∆ibust −1 + 0.19 ∆ibust − 2 + 0.18 ∆ibust − 3
(−1.9)
(1.0)
(1.7)
(1.8)
+ 0.17 ∆ibust − 4 + 1.0 ∆gdpt −1
(1.7)
(1.5)
− 0.047 [3.09 + ibust −1 − kbusnht − 2 + 0.27 ( kbusnht − 2 − gdpt − 2 + rcct −1 )]
(−1.9)
+ dummy
where
IBUS = business investment (6.2.12).
GDP = GDP(A) at factor cost (6.2.1).
KBUSNH = business non-residential capital stock (6.2.11).
RCC = real cost of capital (6.1.5).
DUMMY = 1985 Q2.
(6.2.12)
32
Economic models at the Bank of England
Adjusted R2: 0.36
Equation standard error: 0.028
LM test for serial correlation: F-stat. = 1.35 [0.27]
Normality test: χ2(2) = 2.28 [0.32]
Heteroscedasticity: F-stat. = 1.60 [0.10]
Sample period: 1983 Q2–1999 Q3
Single-equation dynamic responses:
Response of business investment level to a 1% shock to RHS variables
Per cent
Quarters ahead
Real cost
GDP output
of capital
1
-0.01
1.0
4
-0.06
1.4
8
-0.01
1.6
Long run (LR)
-0.27
0.27
The structure of the entire supply side of the MM (including capital accumulation (6.2.11) and the
production function (6.3.11)) ensures that in the long run a 1% increase in the real cost of capital
will result in a 1% decrease in the level of business investment and the capital-output ratio.
6.2.8
Private sector dwellings investment (IH)
Private sector dwellings investment (IH) follows the growth of business investment.
∆iht = ∆ibust
(6.2.13)
where
IH = private sector dwellings investment (6.2.13).
IBUS = business investment (6.2.12).
6.2.9
Government investment (IG)
In the simulations described in Section 5, real government investment (IG) is assumed to be
exogenous. Nominal government investment (IGL) is derived by identity using the overall GDP
deflator (6.4.1). But announced nominal government spending plans provide the basis for
forecasting this component, with real spending defined by the inverse of equation (6.2.14) below.
IGLt = IGt . PGDPt
6.2.10
(6.2.14)
Government consumption (GL)
In the simulation described in Section 5, real government consumption (G) is assumed to be
exogenous. Nominal government consumption (GL) is derived by identity using the government
The macroeconometric model
33
consumption deflator (PG) (6.4.14). But announced nominal government spending plans provide
the basis for forecasting this component, with real spending then defined by the inverse of
equation (6.2.15) below.
GLt = Gt . PGt
6.2.11
(6.2.15)
Stockbuilding (II)
Net stockbuilding is defined as the change in total stocks.
IIt = KIIt – KIIt–1
(6.2.16)
where
II = net stockbuilding (6.2.16).
KII = stock level (6.2.17).
In the medium term, the stock-output ratio follows a downward (time) trend.
∆kiit = 0.03 + 0.14 ∆gdpt −1 + 0.26 ∆gdpt − 2 − 0.13[kiit −1 − gdpt −1 + 0.004 TIMEt −1 ]
(1.9)
(1.2)
(1.8)
(6.2.17)
(−2.5)
where
KII = stock level (6.2.17).
GDP = GDP(A) at factor cost (6.2.1).
TIME = time trend.
Adjusted R2: 0.30
Equation standard error: 0.006
LM test for serial correlation: F-stat. = 2.15 [0.09]
Normality test: χ2(2) = 1.07 [0.58]
Heteroscedasticity: F-stat. = 1.05 [0.41]
Sample period: 1982 Q1–1998 Q2
6.2.12
Export volumes of goods and services (X)
Both of the trade volume equations are modelled as a function of relative prices and total demand
for exports or imports (proxied by world trade (TRAD) (exogenous) and domestic demand (DD)
(6.2.7) respectively).
∆xt = 0.72 − 0.33 ∆ xt −1 − 0.10 ∆ rxrxt + 0.30 ∆ tradt
(2.9)
(−2.8)
(−1.3)
(1.5)
− 0.11 [ xt −1 − tradt −1 + 0.69rxrxt −1 ] + dummy
(−2.9)
(1.7)
(6.2.18)
34
Economic models at the Bank of England
where
X = exports of goods and services (6.2.18).
RXRX = exporters’ real exchange rate (6.1.9).
TRAD = world trade (exogenous).
DUMMY = 1991 Q1.
Adjusted R2: 0.33
Equation standard error: 0.016
LM test for serial correlation: F-stat. = 1.35 [0.37]
Normality test: χ2(2) = 1.95 [0.38]
Heteroscedasticity: F-stat. = 0.51 [0.90]
Sample period: 1985 Q3–1997 Q4
Single-equation dynamic responses:
Response of export volume level to a 1% shock to RHS variables
Per cent
Quarters ahead
Exporters’ real
World trade
exchange rate
1
-0.1
0.3
4
-0.3
0.5
8
-0.5
0.6
Long run (LR)
-0.7
1.0
50% of LR by
7 quarters
5 quarters
90% of LR by
26 quarters
24 quarters
6.2.13
Import volumes of goods and services (M)
∆mt = − 0.25 + 1.73 ∆ddt − 0.21 [ mt −1 − ddt −1 + 0.22 rxrmt −1 − 0.90 SPECt −1 ]
(−3.3)
(8.3)
(−3.3)
(1.1)
+ dummy
where
M = imports of goods and services (6.2.19).
DD = total domestic expenditure (6.2.7).
RXRM = importers’ relative price (6.1.10).
SPEC = trade specialisation term (exogenous).
DUMMY = 1981 Q1.
Adjusted R2: 0.64
Equation standard error: 0.018
LM test for serial correlation: F-stat. = 0.12 [0.95]
Normality test: χ2(2) = 106.1 [0.0]
(−5.6)
(6.2.19)
The macroeconometric model
35
Heteroscedasticity: F-stat. = 1.12 [0.36]
Sample period: 1980 Q2–1997 Q4
Single-equation dynamic responses:
Response of import volume level to a 1% shock to RHS variables
Per cent
Quarters ahead
Importers’ relative Domestic demand
Specialisation
price
1
0.0
1.7
0.0
4
-0.1
1.3
0.5
8
-0.1
1.1
0.8
Long run (LR)
-0.2
1.0
0.9
50% of LR by
3 quarters
o/s
3 quarters
90% of LR by
10 quarters
o/s
10 quarters
o/s = overshoots eventual long-run response in short term.
6.2.14
Balance of payments (BAL, BALT, BIPD and BTRF)
The current account balance (BAL) is determined by an accounting identity. It is the sum of the
nominal trade balance (BALT), the balance of interest, dividends and profits (BIPD) and the
balance of transfers (BTRF) (exogenous).
BALt = BALTt + BIPDt + BTRFt
(6.2.20)
The nominal balance of trade is also defined by an accounting identity.
BALt = Xt . PXt – Mt . PMt
(6.2.21)
where
BALT = nominal balance of trade (6.2.21).
X = exports of goods and services (6.2.18).
M = imports of goods and services (6.2.19).
PX = export price deflator (6.4.3).
PM = import price deflator (6.4.2).
The balance of interest payments, dividends and profits (BIPD) is proxied by the product of the
exogenous world nominal interest rate (WRS) and the average net external assets (NEA) over the
current and past quarter.
WRSt   NEAt + NEAt −1 
BIPDt = 
 400  

2
where
(6.2.22)
36
Economic models at the Bank of England
BPID = balance of interest, dividends and profits (6.2.22).
WRS = world nominal interest rates (exogenous).
NEA = net external assets (6.2.23).
6.2.15
Net external assets (NEA)
Net external assets are defined as the difference between gross UK holdings of foreign assets
(UKA) (6.2.24) and foreign holdings of UK assets (FOH) (6.2.25).
NEAt = UKAt – FOHt
(6.2.23)
Both gross UK and foreign asset holdings are modelled using a simple stock/flow framework. The
current account balance (assumed equal to the capital account balance) gives the flow apportioned
to the two stocks according to their average ratio across the 1990s.
 WPXt WPXt −1 
UKAt = 0.51BALt + UKAt −1 

 EERt EERt −1 
(6.2.24)
where
UKA = UK holdings of foreign assets (6.2.24).
BAL = current account balance (6.2.20).
WPX = world export prices (exogenous).
EER = sterling effective exchange rate index (6.1.6).
 PGDPt 
FOHt = − 0.49 BALt + FOHt −1 

 PGDPt −1 
(6.2.25)
where
FOH = foreign holdings of UK assets (6.2.25).
BAL = current account balance (6.2.20).
PGDP = GDP deflator at factor cost (6.4.1).
6.3
6.3.1
Labour market
Average earnings (EARN)
The labour market earnings equation is based on a forward-looking model of contract dynamics.
The desired or target wage (W*) is defined as:
wt* = pgdpt + gdpt – empt – remt – 0.013 URt + ZPROXY
where
PGDP = GDP deflator at factor cost (6.4.1).
UR = rate of unemployment (6.3.8).
(6.3.1)
The macroeconometric model
37
ZPROXY = measure of structural effects of labour market developments (exogenous).
GDP = GDP(A) at factor cost (6.2.1).
EMP = level of employment in heads (6.3.12).
REM = effective employers’ social contribution tax rate (exogenous).
ZPROXY is designed to capture movements in unobservable structural variables that affect the
labour market (such as union power and the replacement ratio), and is derived by fitting a
Hodrick-Prescott filter through data on the labour share of income and the unemployment rate.
Following Moghadam and Wren-Lewis (1994), the pure theoretical model of contract dynamics
sets earnings each period equal to a backward and a forward convolution of expected wages,
wstarc, where:
wstarct = (1 / 16)( w∗t ,t + w∗t ,t +1 + w∗t ,t + 2 + w∗t ,t + 3
+ w∗t −1,t −1 + w∗t −1,t + w∗t −1,t +1 + w∗t −1,t + 2
+ w∗t − 2,t − 2 + w∗t − 2,t −1 + w∗t − 2,t + w∗t − 2,t +1
(6.3.2)
+ w∗t − 3,t − 3 + w∗t − 3,t − 2 + w∗t − 3,t −1 + w∗t − 3,t ),
and w*i,j is the expectation of w*j formed at time i.
In estimation, these forward-looking terms are treated as rational. The estimated equation also
embodies the assumption that wage-setters partly adopt a rule of thumb in setting their contracts
(using a combination of lagged wage growth and current growth of retail prices and productivity to
determine their wage settlements) in addition to explicitly forward-looking considerations.
earnt = 0.79 {0.63(0.25)[(earnt −1 + earnt − 2 + earnt − 3+ earnt − 4 )
(26.7) (6.0)
+ (2.5) 0.65 ( ∆earnt −1 + ∆earnt − 2 + ∆earnt − 3 + ∆earnt − 4 )
(10.1)
+ (2.5) (1 − 0.65)( ∆rpt + ∆rpt −1 + ∆rpt − 2 + ∆rpt − 3 )]
+ (1 − 0.63)[earnt −1 + 0.65∆earnt −1 + (1 − 0.65)∆rpt ]} + (1 − 0.79)wstarc t
where
EARN = average earnings index (6.3.3).
RP = RPIX plus productivity (6.3.4).
Adjusted R2: 0.99
Equation standard error: 0.005
LM test for serial correlation: F-stat. = 0.71 [0.586]
Normality test: χ2(2) = 0.58 [0.747]
Heteroscedasticity: F-stat. = 6.17 [0.016]
Sample period: 1980 Q4–1997 Q4
(6.3.3)
38
Economic models at the Bank of England
rpt = rpixt + gdpt – empt
(6.3.4)
where
RP = RPIX inflation adjusted by productivity (6.3.4).
RPIX = RPI excluding mortgage interest payments (6.4.6).
GDP = GDP(A) at factor cost (6.2.1).
EMP = level of employment in heads (6.3.12).
In simulations, expected future values of w* (where w*ij is the expectation of w*j formed at time i)
can be treated in different ways. Either as a model-consistent expectation.
w*t,t+1 = w*t+1 ; w*t,t+2 = w*t+2 ; w*t,t+3 = w*t+3
(6.3.5)
Or as the result of a backward-looking rule, as adopted in the simulations.
w∗t ,t + m = pgdpt + gdpt − empt − remt − 0.013URt + ZPROXYt
m . INFEt  

+ log1 + 

  400  

 (GDPt / EMPt ) − (GDPt − 4 / EMPt − 4 )  
+ log1 + m 4 

(GDPt − 4 / EMPt − 4 )



(6.3.6)
for m = 1, 2 and 3.
Single-equation dynamic responses:
Response of the level of earnings to a 1% shock to RHS variables
Per cent
Quarters ahead
GDP deflator
Productivity
Unemployment
1
0.2
0.5
-0.2
4
0.6
1.1
-0.7
8
0.9
1.2
-1.2
Long run (LR)
1.0
1.0
-1.3
50% of LR by
4 quarters
o/s
4 quarters
90% of LR by
8 quarters
o/s
8 quarters
o/s = overshoots eventual long-run response in short term.
6.3.2
Unemployment level, unemployment rate, participation rate (UN, UR, PA)
The level of unemployment (UN) is modelled as an identity, relating unemployment to changes in
employment (EMP), the population aged 16 and over (POP) (exogenous) and the participation
rate (PA).
UNt = PAt . POPt – EMPt
(6.3.7)
The macroeconometric model
39
where
UN = level of unemployment (6.3.7).
PA = participation rate (6.3.9).
POP = population aged 16 and over (exogenous).
EMP = level of employment in heads (6.3.12).
The rate of unemployment (UR) is an identity.


UNt
URt = 100

 EMPt + UNt 
(6.3.8)
where
UR = rate of unemployment (6.3.8).
UN = level of unemployment (6.3.7).
EMP = level of employment in heads (6.3.12).
and the participation rate is given by:
POWA 
POWA 
PA = 1 − INWA 
− INOLD 1 −
 POP 

POP 
(6.3.9)
where
PA = participation rate (6.3.9).
INWA = proportion of population of working age that is inactive (6.3.10).
POWA = population of working age (exogenous).
POP = population aged 16 and over (exogenous).
INOLD = proportion of population over retirement age that is inactive (exogenous).
INWA can be assumed invariant to the economy cycle or modelled by a simple cyclical influence
derived from lagged employment (as a proportion of the working-age population). The latter
assumption was used in the simulations described in Section 5.
 EMPt −1 
 EMPt − 2 
INWAt = 0.06 − 0.25
 + 0.20
 + 0.00006TIME + 0.87 INWAt −1
 POWAt −1 
 POWAt − 2 
(6.3.10)
(2.3)
(−5.6)
(4.2)
(2.9)
(16.6)
where
INWA = proportion of population of working age that is inactive (6.3.10).
EMP = level of employment in heads (6.3.12).
POWA = population of working age (exogenous).
TIME = time trend.
40
Economic models at the Bank of England
Adjusted R2: 0.98
Equation standard error: 0.001
Sample period: 1984 Q1–1998 Q4
6.3.3
Employment in hours (EMPH)
Long-run employment in hours is derived from an inverted production function. Changes in GDP
appear in the dynamic hours equation.
∆empht = − 0.0008 + 0.43∆empht −1 + 0.28∆gdpt + 0.17[CAPUt −1 ]
(−1.3)
(4.9)
(4.3)
(4.6)
(6.3.11)
where
EMPH = total employment in hours (6.3.11).
GDP = GDP(A) at factor cost (6.2.1).
CAPU = capacity utilisation (6.2.6).
Adjusted R2: 0.76
Equation standard error: 0.004
LM test for serial correlation: F-stat. = 0.35 [0.846]
Normality test: χ2(2) = 1.61 [0.445]
Heteroscedasticity: F-stat. = 0.10 [0.753]
Sample period: 1980 Q1–1998 Q1
Single-equation dynamic responses for equation:
Response of employment in hours level to a 1% shock to RHS variables
Per cent
Quarters ahead
GDP output
Capital stock
1
0.3
0.0
4
0.9
-0.2
8
1.3
-0.4
Long run (LR)
1.4
-0.4
50% of LR by
3 quarters
5 quarters
90% of LR by
7 quarters
10 quarters
6.3.4
Employment in heads (EMP)
Employment in heads (EMP) can be written as:
EMPt =
EMPHt
AVH t
where
EMP = level of employment in heads (6.3.12).
EMPH = total employment in hours (6.3.11).
(6.3.12)
The macroeconometric model
41
AVH = average hours per worker (6.3.13).
6.3.5
Average hours per worker (AVH)
Average hours per worker is a decreasing function of the extent of part-time working, and is
cyclical about a long-run trend.
∆avht = − 0.21 − 0.06 avht −1 + 0.32 ∆avht −1
(−2.4)
(−2.4)
(3.0)
− 0.61( ∆PTSHt − 0.32 ∆PTSHt −1 + 0.06 PTSHt −1 ) + 0.09CAPUt −1
(6.3.13)
(3.2)
where
AVH = average hours per worker (6.3.13).
PTSH = part-time share of employment (exogenous).
CAPU = capacity utilisation (6.2.6).
Adjusted R2: 0.30
Equation standard error: 0.004
LM test for serial correlation: F-stat. = 0.86 [0.49]
Normality test: χ2(2) = 0.27 [0.87]
Heteroscedasticity: F-stat. = 4.75 [0.033]
Sample period: 1980 Q1–1998 Q1
Single-equation dynamic responses for equation:
Response of average hours level to a 1% shock to RHS variables
Per cent
Quarters ahead
Part time share
Capacity utilisation
1
-0.6
0.0
4
-0.6
0.3
8
-0.6
0.7
Long run (LR)
-0.6
1.4
50% of LR by
1 quarter
9 quarters
90% of LR by
1 quarter
25 quarters
6.3.6
Unit labour costs (ULC)
Nominal unit labour costs (ULC) are defined by the following identity:
ULCt =
EARNt . REMt . EMPt
GDPt
where
ULC = nominal unit labour costs (6.3.14).
(6.3.14)
42
Economic models at the Bank of England
EARN = average earnings index (6.3.3).
REM = effective employers’ social contribution tax rate (exogenous).
EMP = level of employment in heads (6.3.12).
GDP = GDP(A) at factor cost (6.2.1).
6.4
6.4.1
Prices
Domestic output prices—the GDP deflator (PGDP)
The PGDP equation is modelled as a long-run mark-up over unit labour costs. Short-run
movements in the mark-up are captured by the capacity utilisation term.
∆pgdpt = − 0.19 + 0.24 ∆pgdpt −1 + 0.48∆ulct −1 + 0.27∆ulct − 2 + 0.25CAPUt −1
(−0.7)
(1.9)
(4.0)
− 0.07[ pgdpt −1 − ulct −1 ]
(2.7)
(6.4.1a)
(−1.1)
where
PGDP = GDP deflator at factor cost (6.4.1).
ULC = nominal unit labour costs (6.3.14).
CAPU = capacity utilisation (6.2.6).
Adjusted R2: 0.24
Equation standard error: 0.012
LM test for serial correlation: F-stat. = 1.02 [0.36]
Normality test: χ2(2) = 0.11 [0.94]
Heteroscedasticity: F-stat. = 3.37 [0.001]
Sample period: 1975 Q2–1997 Q1
Single-equation dynamic responses:
Response of GDP deflator level to a 1% shock to RHS variables
Per cent
Quarters ahead
Unit labour costs
Capacity utilisation
1
0.0
0.0
4
1.1
0.8
8
1.1
1.8
Long run (LR)
1.0
0.0
50% of LR by
o/s
o/s
90% of LR by
o/s
o/s
o/s = overshoots eventual long-run response in short term.
An alternative, used to account for possible terms-of-trade effects in recent years, includes import
prices.
The macroeconometric model
43
∆pgdpt = − 0.27 + 0.56 ∆ulct −1 + 0.19 ∆ulct − 2 + 0.26 ∆pmt −1 + 0.18CAPUt −1
(5.4)
(5.7)
(2.3)
− 0.09 [ pgdpt −1 − ulct −1 ]
(6.4.1b)
(−1.5)
where
PGDP = GDP deflator at factor cost (6.4.1).
CAPU = capacity utilisation (6.2.6).
ULC = nominal unit labour costs (6.3.14).
PM = import price deflator (6.4.2).
Adjusted R2: 0.40
Equation standard error: 0.0088
LM test for serial correlation: F-stat. = 0.25 [0.77]
Normality test: χ2(2) = 0.95 [0.62]
Heteroscedasticity: F-stat. = 1.20 [0.31]
Sample period: 1980 Q1–1997 Q1
6.4.2
Goods and services import deflator (PM)
In the short run import prices are affected by unit labour costs, non-oil commodity prices, oil
prices and M6 export prices. In the long run, import prices are determined by non-oil commodity
prices, oil prices and M6 export prices.
∆pmt = − 0.07 + 0.08∆(comnust −1 − edst −1 ) + 0.48∆( wpxt − eert )
(−3.1) (3.3)
(10.5)
+ 0.03∆( petspott − edst ) + 0.41∆ulct −1
(2.9)
− 0.18[ pmt −1 − 0.93 ( wpxt −1 − eert −1 ) − 0.04 ( petspott −1 − edst −1 )
(−3.1)
− 0.03 (comnust −1 − edst −1 )]
where
PM = import price deflator (6.4.2).
COMNUS = world non-oil commodity prices (index, US$) (exogenous).
EDS = US dollar-sterling exchange rate (6.1.8).
EER = sterling effective exchange rate index (6.1.6).
PETSPOT = oil spot prices (US$) (exogenous).
ULC = nominal unit labour costs (6.3.14).
WPX = world export prices (exogenous).
Adjusted R2: 0.69
Equation standard error: 0.012
(6.4.2a)
44
Economic models at the Bank of England
LM test for serial correlation: F-stat. = 0.29 [0.75]
Normality test: χ2(2) = 6.22 [0.07]
Heteroscedasticity: F-stat. = 0.98 [0.53]
Sample period: 1984 Q1–1997 Q4
Single-equation dynamic responses:
Response of import price level to a 1% shock to RHS variables
Per cent
Quarters ahead Sterling M6
Sterling non-oil
Sterling oil
export prices
commodity prices
prices
1
0.5
0.08
0.03
4
0.7
0.06
0.03
8
0.8
0.04
0.04
Long run
0.9
0.03
0.04
50% of LR by
1 quarter
o/s
1 quarter
90% of LR by
9 quarters
o/s
6 quarters
Unit labour
costs
0.4
0.2
0.1
0.0
o/s = overshoots eventual long-run response in short term.
An alternative specification sometimes used in forecasting includes an additional dynamic term for
the dollar-sterling exchange rate and relaxes the dynamic homogeneity constraint.
∆pmt = − 0.07 + 0.02 ∆(comnust −1 − edst −1 ) + 0.95∆wpxt − 0.28∆eert
(−4.3)
(1.6)
(3.9)
(5.7)
+ 0.02 ∆( petspott − edst ) − 0.14 ∆edst
(1.8)
( −4.0)
− 0.19 [ pmt −1 − 0.93 ( wpxt −1 − eert −1 ) − 0.04 ( petspott −1 − edst −1 )
( −4.5)
− 0.03 (comnust −1 − edst −1 )]
where
PM = import price deflator (6.4.2).
COMNUS = world non-oil commodity prices (index, US$) (exogenous).
EDS = Sterling-US dollar exchange rate (6.1.8).
EER = sterling effective exchange rate index (6.1.6).
PETSPOT = oil spot prices (US$) (exogenous).
ULC = nominal unit labour costs (6.3.14).
WPX = world export prices (exogenous).
Adjusted R2: 0.83
Equation standard error: 0.009
LM test for serial correlation: F-stat. = 0.84 [0.50]
Normality test: χ2(2) = 4.9 [0.09]
(6.4.2b)
The macroeconometric model
45
Heteroscedasticity: F-stat. = 1.22 [0.28]
Sample period: 1984 Q1–1997 Q4
6.4.3
Goods and services export deflator (in sterling) (PX)
Export prices are modelled as a function of M6 export prices in sterling terms, UK unit labour
costs and oil prices.
∆pxt = − 0.46 + 0.30 ∆( wpxt − eert ) + 0.67∆ulct + 0.03∆( petspott − edst )
( −5.8)
(13.9)
(3.0)
[
]
− 0.33 pxt −1 − 0.55 ( wpxt −1 − eert −1 ) − 0.40 ulct −1 − 0.05 ( petspott −1 − edst −1 ) (6.4.3)
( −5.9)
( −11.2)
+ dummy
where
PX = export price deflator (6.4.3).
WPX = world export prices (exogenous).
EER = sterling effective exchange rate index (6.1.6).
ULC = nominal unit labour costs (6.3.14).
PETSPOT = oil spot prices (US$) (exogenous).
EDS = US dollar-sterling exchange rate (6.1.8).
DUMMY = 1989 Q2.
Adjusted R2: 0.58
Equation standard error: 0.012
LM test for serial correlation: F-stat. = 0.33 [0.72]
Normality test: χ2(2) = 0.03 [0.99]
Heteroscedasticity: F-stat. = 2.03 [0.04]
Sample period: 1985 Q3–1997 Q4
Single-equation dynamic responses:
Response of export price level to a 1% shock to RHS variables
Per cent
Quarters ahead
Unit labour costs
Oil prices
Sterling M6
export prices
1
0.7
0.03
0.3
4
0.5
0.04
0.5
8
0.4
0.05
0.5
Long run (LR)
0.4
0.05
0.6
50% of LR by
o/s
1 quarter
1 quarter
90% of LR by
o/s
5 quarters
5 quarters
o/s = overshoots eventual long-run response in short term.
46
6.4.4
Economic models at the Bank of England
Domestic price index (DPP)
An index for the price of domestic output produced and consumed in the United Kingdom is
constructed by removing export prices from the GDP deflator.
dppt = pgdpt – 0.20 pxt
(6.4.4)
where
DPP = domestic price index (6.4.4).
PGDP = GDP deflator at factor cost (6.4.1).
PX = export price deflator (6.4.3).
6.4.5
RPIY
RPIY is a weighted combination of the domestic price index and import prices.
∆rpiyt = 0.68 + 0.41 ∆( dppt ) + 0.33 ∆( dppt −1 ) + 0.21 ∆( dppt − 2 )
(0.7)
(4.8)
(4.1)
(2.6)
+ 0.14 ∆pmt + 0.07 ∆pmt −1 + 0.04 ∆pmt − 2
(4.1)
(1.9)
− 0.14 [rpiyt −1 − 1.0 dppt −1 − 0.20 pmt −1 ]
( −2.3)
+ seasonal dummies
where
RPIY = RPI excluding MIPS and indirect taxes (6.4.5).
PM = import price deflator (6.4.2).
DPP = domestic price index (6.4.4).
Adjusted R2: 0.85
Equation standard error: 0.003
LM test for serial correlation: F-stat. = 3.00 [0.07]
Normality test: χ2(2) = 0.26 [0.87]
Heteroscedasticity: F-stat. = 0.59 [0.86]
Sample period: 1987 Q1–1997 Q1
(6.4.5a)
The macroeconometric model
47
Single-equation dynamic responses:
Response of RPIY level to a 1% shock to RHS variables
Per cent
Quarters ahead
Import prices
Domestic prices
1
0.1
0.4
4
0.2
1.1
8
0.2
1.0
Long run (LR)
0.2
1.0
50% of LR by
1 quarter
2 quarter
90% of LR by
2 quarter
o/s
o/s = overshoots eventual long-run response in short term.
An alternative specification used for some simulations models RPIY as a function of import prices
and unit labour costs:
∆rpiyt = 0.11 + 0.62 ∆(rpiyt −1 ) + 0.30 ∆(remt + earnt ) + 0.07∆( pmt )
(2.1)
(8.8)
(4.4)
+ 0.04 CAPUt −1 − 0.04 [rpiyt −1 − 0.20 pmt −1 − 0.8 ulct −1 ]
(1.4)
(6.4.5b)
( −2.0)
+ seasonal dummies
where
RPIY = RPI excluding MIPS and indirect taxes (6.4.5).
REM = effective employers’ social contribution tax rate (exogenous).
EARN = average earnings index (6.3.3).
PM = import price deflator (6.4.2).
CAPU = capacity utilisation (6.2.6).
ULC = nominal unit labour costs (6.3.14).
Adjusted R2: 0.82
Equation standard error: 0.003
LM test for serial correlation: F-stat. = 6.50 [0.01]
Normality test: χ2(2) = 0.71 [0.70]
Heteroscedasticity: F-stat. = 1.17 [0.32]
Sample period: 1982 Q2–1999 Q1
6.4.6
RPIX
RPIX is constructed in two stages. First, indirect taxes are added to RPIY to calculate RPIX
excluding council taxes (RPXC). Then council taxes are added to give RPIX itself. None of the
equations is behavioural, so any wedge between RPIY and RPIX inflation rates is driven solely by
assumptions about indirect taxes and council tax. Throughout, price changes are calculated
relative to the first quarter of the year, a feature designed to approximate chain-linking.
48
Economic models at the Bank of England
RPIX is calculated as a weighted average of RPXC (6.4.7) and RPCC (6.4.10).
RPIX = (Q1. RPIXt − 4 + Q2. RPIXt −1 + Q3. RPIXt − 2 + Q4. RPIXt − 3 ).

  WCC  

RPCC
  1 − WMIP  . Q1. RPCC + Q2. RPCC + Q3. RPCC + Q4. RPCC  +  (6.4.6)
t −4
t −1
t −2
t −3


 

WCC   
RPXC
 1 − 
.


   1 − WMIP    Q1. RPXCt − 4 + Q2. RPXCt −1 + Q3. RPXCt − 2 + Q4. RPXCt − 3  
where
RPIX = RPI excluding mortgage interest payments (6.4.6).
RPCC = council tax and rates sub-index of RPI (6.4.10).
RPXC = RPI excluding MIPS and council taxation (6.4.7).
WCC = weight of council tax and rates in RPI per 1,000 (exogenous).
WMIP = weight of MIPS in RPI per 1,000 (exogenous).
6.4.6.1
Indirect taxes
The retail price index excluding the mortgage interest payment sub-index, local authorities and
indirect taxes (RPIY) omits two types of indirect taxation, value-added tax and excise duties, both
of which are added to calculate RPXC. RPXC is modelled as a weighted sum of changes in an
index of excise duties (DUTY) and RPIY, scaled to allow for the effects of changes in VAT.
RPXC = VAR . (Q1. RPXCt − 4 + Q2. RPXCt −1 + Q3. RPXCt − 2 + Q4. RPXCt − 3 ).
 

WDUT
DUTY

  WDUT + WRPY   Q1. DUTY + Q2. DUTY + Q3. DUTY + Q4. DUTY  
t −4
t −1
t −2
t − 3  (6.4.7)



WDUT
RPIY

 + 1 −


  WDUT + WRPY   Q1. RPIYt − 4 + Q2. RPIYt −1 + Q3. RPIYt − 2 + Q4. RPIYt − 3  
where
RPXC = RPI excluding MIPS and council taxation (6.4.7).
DUTY = index of duty element of RPI (6.4.9).
RPIY = RPI excluding MIPS and indirect taxes (6.4.5).
WDUT = weight of duty element in RPI (exogenous).
WRPY = weight of RPIY in RPI (exogenous).
The VAT multiplier (VAR) is a weighted sum of changes to the value-added taxation rates, where
the weights reflect the proportion of the RPIX basket covered by VAT at each rate.
The macroeconometric model
49


1 + RVAT / 100
VAR = 0.40 + 0.54 

Q
1
1
+
RVAT
/
100
Q
2
1
RVAT
/
100
Q
3
1
RVAT
/
100
Q
4
1
RVAT
/
100
+
+
+
+
+
+
(
)
(
)
(
)
(
)
 
t −4
t −1
t −2
t −3



1 + FVAT / 100
+ 0.04 

Q
1
1
FVAT
/
100
Q
2
1
FVAT
/
100
Q
+
+
+
+
3
1
+
FVAT
/
100
+
Q
4
1
+
FVAT
/
100
(
)
(
)
(
)
(
)

 
t −2
t −3
t −4
t −1

(6.4.8)


1 + IVAT / 100
+ 0.02 

Q
1
1
IVAT
/
100
Q
2
1
IVAT
/
100
Q
3
1
IVAT
/
100
Q
4
1
IVAT
/
100
+
+
+
+
+
+
+
(
)
(
)
(
)
(
)

 
t −4
t −1
t −2
t −3

where
VAR = the VAT multiplier (6.4.8).
RVAT = standard rate of VAT (exogenous).
FVAT = rate of VAT charged on domestic fuel and light (exogenous).
IVAT = rate of VAT charged on insurance (exogenous).
Excise duties (DUTY) are indexed to the RPI, changing in the first quarter of each year, in line
with the prevailing four-quarter RPI inflation rate.
DUTY = (1 – Q1) DUTYt–1 + Q1 . DUTYt–1 . (RPIt / RPIt–4)
(6.4.9)
where
DUTY = index of duty element of RPI (6.4.9).
RPIY = RPI excluding MIPS and indirect taxes (6.4.5).
6.4.6.2
Council tax
The council tax component of the RPI (RPCC) is held flat except for in Q2, when council taxes
are set. Council taxes are indexed to RPXC, and so rise by the four-quarter growth rate of RPXC
prevailing in Q2 of each year.
RPCCt = (1 – Q2) RPCCt–1 + Q2 . RPCCt–1 . (RPXCt / RPXCt–4)
(6.4.10)
where
RPCC = council tax and rates sub-index of RPI (6.4.10).
RPXC = RPI excluding MIPS and council taxation (6.4.7).
6.4.7
RPI
The RPI is calculated as a weighted average of RPIX (6.4.6) and the index of mortgage interest
payments (MIPS) (6.4.12). As with RPIX, price changes are calculated relative to the first quarter
of the year, a feature designed to approximate chain-linking.
50
Economic models at the Bank of England
RPI = (Q1. RPIt − 4 + Q2. RPIt −1 + Q3. RPIt − 2 + Q4. RPIt − 3 ).
MIPS
 WMIP
+ 


Q1. MIPSt − 4 + Q2. MIPSt −1 + Q3. MIPSt − 2 + Q4. MIPSt − 3


RPIX
 (1 − WMIP )



Q1. RPIXt − 4 + Q2. RPIXt −1 + Q3. RPIXt − 2 + Q4. RPIXt − 3 

(6.4.11)
where
RPI = retail price index (6.4.11).
MIPS = mortgage interest payments sub-index of RPI (6.4.12).
RPIX = RPI excluding mortgage interest payments (6.4.6).
WMIP = weight of MIPS in RPI (exogenous).
The ONS calculates MIPS from the average debt outstanding on mortgages. The debt was
previously subdivided between that eligible for tax relief and the rest. Around 75% was eligible
for tax relief, hence the multiplier attached to RMI. Since the removal of mortgage interest relief
in April 2000, RMI has been set to zero. We model MIPS as a function of gross housing wealth,
the mortgage interest rate and the average rate at which mortgage interest relief was claimed (RMI)
(exogenous). GHW is used as a proxy for the stock of nominal housing debt.
RMM4t  
RMIt  
GHWt 
1 − 0.75

 100  
 100  
MIPSt =
64005.36
(6.4.12)
where
MIPS = mortgage interest payments sub-index of RPI (6.4.12).
GHW = gross housing wealth (6.1.13).
RMM4 = mortgage interest rate (6.1.3).
RMI = average rate at which MIRAS was claimed (exogenous).
6.4.8
The consumers’ expenditure deflator (PC)
The consumers’ expenditure deflator (PC) is derived from RPXC (retail price index excluding
MIPS and council tax), because council tax payments are not counted as part of consumption. The
expenditure deflator is seasonally adjusted by applying quarterly dummies to RPXC. This is
necessary because national accounts data are seasonally adjusted, but retail prices data are not.
The constant is imposed to correct for differences in indexation between the two series.
pct = rpxct – 5.01 + seasonal dummies
where
PC = total final consumers’ expenditure deflator (6.4.13).
RPXC = RPI excluding MIPS and council taxation (6.4.7).
(6.4.13)
The macroeconometric model
6.4.9
51
Government consumption deflator (PG)
The government expenditure deflator is proxied by movements in unit labour costs and the retail
price index excluding council taxation payments. The weights reflect the fact that the breakdown
of government consumption is roughly half on goods and half on services.
pgt = – 4.0 + 0.5rpxct + 0.5ulct
(6.4.14)
where
PG = government expenditure deflator (6.4.14).
RPXC = RPI excluding MIPS and council taxation (6.4.7).
ULC = nominal unit labour costs (6.3.14).
6.4.10
House prices (PHSE)
House prices are determined by average earnings and the long real rate in the long run, while GDP
enters the dynamics (in addition to earnings).
∆phset = − 0.01 + 0.35 ∆( phset −1 ) + 0.92 ∆(earnt ) + 0.92 ∆( gdpmt − 2 )
( −0.8) (3.5)
(2.2)
(2.6)
− 0.034 [ phset −1 − earnt −1 + 0.17 RLRt −1 ] + dummy
( −2.0)
(1.2)
where
PHSE = UK house prices (6.4.15).
EARN = average earnings index (6.3.3).
GDPM = GDP(A) at market prices (6.2.3).
RLR = 10-year index-linked bond yields (exogenous).
DUMMY = 1998 Q3.
Adjusted R2: 0.53
Equation standard error: 0.018
LM test for serial correlation: F-stat. = 4.43 [0.02]
Normality test: χ2(2) = 1.27 [0.53]
Heteroscedasticity: F-stat. = 1.03 [0.43]
Sample period: 1982 Q2–1998 Q4
(6.4.15a)
52
Economic models at the Bank of England
Single-equation dynamic responses:
Response of house price level to a 1% shock to RHS variables
Per cent
Quarters ahead
Earnings
GDP
Real interest
rates
1
1.2
0.0
-0.01
4
1.4
1.3
-0.03
8
1.3
1.1
-0.06
Long run (LR)
1.0
0.0
-0.17
50% of LR by
o/s
o/s
13 quarters
90% of LR by
o/s
o/s
42 quarters
o/s = overshoots eventual long-run response in short term.
An alternative specification that imposes dynamic homogeneity is used in the simulations
described in Section 5.
∆phset = − 0.01 + 0.28 ∆( phset −1 ) + 0.72 ∆(earnt ) + 0.92 ∆( gdpmt − 2 )
− 0.034 [ phset −1 − earnt −1 + 0.17 RLRt −1 ] + dummy
(6.4.15b)
where
PHSE = UK house prices (6.4.15).
EARN = average earnings index (6.3.3).
GDPM = GDP(A) at market prices (6.2.3).
RLR = 10-year index-linked bond yields (exogenous).
DUMMY = 1998 Q3.
A further alternative specification for the house price equation includes a measure of real user cost
of housing. The user cost of housing is calculated using the short-term mortgage interest rate,
adjusted for tax changes, offset by expected capital gains and allowing for other costs such as
depreciation. Over the estimation period expected capital gains were proxied by past house price
inflation.
∆phset = − 0.17 + 1.22 ∆(earnt ) + 0.56 ∆( gdpmt − 2 ) + 0.004 ∆ (nfwt −1 − pct −1 )
( −2.5) (4.1)
(2.0)
(4.9)
− 0.002 ∆(USERt −1 ) − 0.038 [ phset −1 − earnt −1 + 0.03USERt −1 ] + dummy
( −4.0)
( −2.6)
where
PHSE = UK house prices (6.4.15).
EARN = average earnings index (6.3.3).
GDPM = GDP(A) at market prices (6.2.3).
( −5.6)
(6.4.15c)
The macroeconometric model
53
NFW = net financial wealth (6.1.14).
PC = total final consumers’ expenditure deflator (6.4.13).
USER = real user cost of housing (exogenous).
DUMMY = 1998 Q3.
Adjusted R2: 0.65
Equation standard error: 0.015
LM test for serial correlation: F-stat. = 1.37 [0.26]
Normality test: χ2(2) = 3.25 [0.20]
Heteroscedasticity: F-stat. = 0.57 [0.86]
Sample period: 1982 Q2–1998 Q4
6.4.11
Four-quarter RPIX inflation expectations (INFE)
There are several possible options available within the model for inflation expectations
formulation.
Inflation expectations can be modelled as a simple autoregressive model, estimated using a
ten-year rolling regression. The rolling-regression estimation is designed to capture an element of
learning by agents. The model is generally re-estimated before each forecast, in order to reflect
recent inflation experience.
INFEt = 1.1 + 1.2 INFt −1 − 0.6 INFt − 2
(1.0) (9.7)
(−3.2)
(6.4.16a)
where
INFE = expectations of annual RPIX inflation (6.4.16).
INF = four-quarter inflation rate RPIX.
A second version of INFE is designed to be flexible, with both a backward and a forward-looking
element. The Government inflation target (ZPSTA) may also affect expectations directly.
INFEt = Z1INFt −1 + Z2 INFt + Z3 INFt + 4 + (1 − Z1 − Z2 − Z3 ) ZPSTA
(6.4.16b)
where
INFE = expectations of annual RPIX inflation (6.4.16).
INF = four-quarter inflation rate of RPIX.
ZPSTA = Government inflation target (exogenous).
The weights (Z1, Z2 and Z3) can be varied. Purely adaptive expectations are generated with
Z1=1, Z2=Z3 = 0. Reducing Z1, Z2 and Z3 raises the weight attached to the inflation target. In the
simulations described in Section 5, a simple (perfect foresight) variant is used: Z1=0, Z2=1, Z3=0.
54
6.5
Economic models at the Bank of England
Fiscal policy
The fiscal sector of the model has few estimated equations—it mainly comprises accounting
identities and adding-up constraints. Taxation receipts are usually calculated from exogenous
taxation rates and the relevant tax bases.
6.5.1
Taxation receipts
Total taxation receipts (TAX) (6.5.1) are an accounting identity. They are equal to the sum of tax
deductions from household income (TJL) (6.5.2); corporate income taxation receipts (TYC)
(6.5.6); indirect taxes less subsidies within the United Kingdom (FCAL) (6.5.7); subsidies from
the European Union (SBEU) (exogenous) less indirect taxes to the European Union (TXEU)
(exogenous).
TAXt = TJLt + TYCt + FCALt + SBEUt – TXEUt
6.5.2.
(6.5.1)
Tax deductions from household income
Total tax deductions from household income (TJL) (6.5.2) are defined by an accounting identity.
They are equal to the sum of receipts from taxes on household income (TYJ) (6.5.3); net social
contributions from household sector (YJC) (6.5.4); and other current taxes paid by households
(TCC) (6.5.5).
TJLt = TYJt + YJCt + TCCt
(6.5.2)
Taxes on household income (TYJ) (6.5.3) are the product of the exogenous household income
taxation rate (RJY) and wages, salaries and self-employed income (YE) (6.6.2).
TYJt = RJYt . YEt
(6.5.3)
Net social contributions from the household sector (YJC) (6.5.4) are the product of the exogenous
household sector social contribution rate (RJC) and wages, salaries and self-employed income
(YE) (6.6.2).
YJCt = RJCt . YEt
(6.5.4)
Other current taxes paid by households (TCC) (6.5.5) are estimated as a function of previous
receipts, the council tax component of the RPI (RPCC) (6.4.10) and a time trend (TIME).
tcct = 0.48 + 0.74tcct −1 + 1.23∆rpcct + 0.26rpcct −1 + 0.002TIMEt
(1.7)
6.5.3
(19.6)
(1.5)
(6.5.5)
(1.5)
Taxes on corporate income
Corporate income taxation receipts (TYC) (6.5.6) are calculated by the product of the exogenous
corporate income taxation rate (RC) and nominal GDP (GDPL) (6.2.4).
The macroeconometric model
TYCt = RCt . GDPLt
6.5.4
55
(6.5.6)
Indirect taxes less subsidies
The nominal factor cost adjustment (FCAL) (6.5.7) is given by total taxes on production and
imports (TE) (6.5.8) less government subsidies on products (SUBS) (6.5.11).
FCALt = TEt – SUBSt
6.5.5
(6.5.7)
Taxes on expenditure
Total taxes on production and imports (TE) (6.5.8) are the sum of value-added taxation receipts
(TVAT) (6.5.9) and other expenditure taxation receipts (TSD) (6.5.10).
TEt = TVATt + TSDt
(6.5.8)
Value-added taxation receipts (TVAT) are calculated by the product of the exogenous value-added
taxation rate (EVAT) and nominal consumption.
TVATt = EVATt . Ct . PCt
(6.5.9)
where
TVAT = value-added taxation receipts (6.5.9).
EVAT = value-added taxation rate (exogenous).
C = consumers’ expenditure (6.2.8).
PC = total final consumers’ expenditure deflator (1995 = 1) (6.4.13).
Other expenditure taxation receipts (TSD) are calculated by the product of the exogenous other
expenditure taxation rate (RSD) and nominal consumption.
TSDt = RSDt . Ct . PCt
(6.5.10)
where
TSD = other expenditure taxation receipts (6.5.10).
RSD = other expenditure taxation rate (exogenous).
C = consumers’ expenditure (6.2.8).
PC = total final consumers’ expenditure deflator (6.4.13).
6.5.6
Government subsidies
Government subsidies on products (SUBS) (6.5.11) are calculated as the product of nominal GDP
(GDPL) (6.2.4) and the exogenous effective government subsidies rate (RTS).
SUBSt = RTSt . GDPLt
(6.5.11)
56
6.5.7
Economic models at the Bank of England
Other government expenditure
Real government consumption (G) and investment (IG) are assumed exogenous in the simulations
outlined in Section 5 (see Sections (6.1.9) and (6.1.10)). In addition to these national accounting
concepts, nominal government expenditure includes current grants to the household sector
(YJG)—predominantly social security payments—and interest payments on general government
debt (DI).
Nominal current grants are assumed to be uprated in line with the total final consumers’
expenditure deflator (PC) (6.4.13).
∆yjgt = ∆pct
(6.5.12)
Interest payments on government debt are modelled as a default, assuming 95% bond finance.
RL
∆DIt = 0.95 t  PSNBt
 400 
(6.5.13)
where
DI = general government debt interest payments (6.5.13).
RL = 20-year bond yield (6.1.2).
PSNB = public sector net borrowing (6.5.14).
6.5.8
Public sector net borrowing (PSNB)
The PSNB is defined as the sum of the nominal expenditure components less taxation receipts.
PSNBt = GLt + IGLt + YJGt + DIt – TAXt
(6.5.14)
where
PSNB = public sector net borrowing (6.5.14).
GL = nominal general government final consumption expenditure (6.2.15).
IGL = nominal general government investment (6.2.14).
YJG = current grants to the household sector (6.5.12).
DI = general government debt interest payments (6.5.13).
TAX = total tax receipts (6.5.1).
6.6
6.6.1
Income
Income accounting
Total household sector pre-tax income (YJ) (6.6.1) is the sum of income from wages, salaries and
self-employment (YE) (6.6.2); current grants to the household sector (YJG) (6.5.12); non-salary
income, such as employers’ social contributions (YEC) (6.6.3); and other households’ non-labour
income (YDIJ) (6.6.4), such as income from dividends and interest.
The macroeconometric model
57
YJt = YEt + YJGt + YECt + YDIJt
(6.6.1)
Total wages, salaries and self-employed income (YE) is modelled as an identity. The data source
for employment is on an LFS rather than a national accounts basis, while YE is a national accounts
construct. So a scaling factor is used to reconcile these sources.
YEt = 0.0363EARNt . EMPt
(6.6.2)
where
YE = total wages, salaries and self-employed income (6.6.2).
EARN = average earnings index (6.3.3).
EMP = level of employment in heads (6.3.12).
Total employers’ social contributions (YEC) (6.6.3) is proxied by the product of the exogenous rate
of employers’ social contributions (REC) and wages, salaries and self-employed income (YE)
(6.6.2).
YECt = RECt . YEt
(6.6.3)
Growth in the ratio of households’ non-labour income (YDIJ) to nominal GDP (GDPL) is a
function of the nominal interest rate (RS) and a time trend.
∆( ydijt − gdplt ) = − 0.92(( ydijt −1 − gdplt −1 ) + 2.81
( −5.3)
(20.1)
RS


− 3.34 log1 +  t −1   − 0.0072TIMEt −1 )
  400  
( −2.5)
( −7.6)
where
YDIJ = households’ non-labour income (6.6.4).
GDPL = GDP at factor cost in current prices (6.2.4).
RS = base rate of interest (6.1.1).
TIME = time trend.
Adjusted R2: 0.4
Equation standard error: 0.04
LM test for serial correlation: F-stat. = 0.00 [0.96]
Normality test: χ2(2) = 0.79 [0.67]
Heteroscedasticity: F-stat. = 0.83 [0.58]
Sample period: 1988 Q3–1998 Q2
(6.6.4)
58
Economic models at the Bank of England
Single-equation dynamic responses:
Response of households’ non-labour income to a 1% shock to RHS variables
Per cent
Quarters ahead
Nominal interest
rate
1
3.1
4
3.3
8
3.3
Long run (LR)
3.3
50% of LR by
1 quarter
90% of LR by
1 quarter
6.6.2
Real household post-tax income and real labour income
Real household post-tax income (RHPI) (6.6.5) is defined as total household sector pre-tax income
(YJ) (6.6.1) less tax deductions from household income (TJL) (6.5.2), all deflated by the
consumers’ expenditure deflator (PC) (6.4.13).
RHPIt =
(YJt − TJLt )
PCt
(6.6.5)
Real post-tax labour income (LY) (6.6.6) is defined as total household sector pre-tax income (YJ)
(6.6.1) excluding tax deductions from household income (TJL) (6.5.2) and households’ non-labour
income (YDIJ) (6.6.4), all deflated by the consumers’ expenditure deflator (PC) (6.4.13).
LYt =
(YJt − YDIJt − TJLt )
PCt
(6.6.6)
The macroeconometric model
59
7
Variables listing
Name
Description
Source(1)
Code / Details
AVH
Average hours worked per person
per week / 1,000
ONS
YBUS.Q / MGRZ.Q
BAL
Current account balance
(£m current prices)
ONS
HBOP.Q
BALT
Nominal goods and services
trade balance
(£m current prices)
ONS
BOKI.Q + IKBD.Q
BETA
1 – business sector net
capital stock depreciation rate
BoE
BoE construction based on
ONS net business sector
capital stock and business
investment (NPEL.Q)
BETANH
1 – whole-economy net capital
stock net of housing depreciation
rate
BoE
BoE construction based on
ONS net whole-economy
capital stock net of housing
and non-dwelling investment
(NPQT.Q – DFEG.Q)
BIPD
Balance of interest, dividends
and profits
(£m current prices)
ONS
HBOJ.Q
BTRF
Balance of transfers
(£m current prices)
ONS
IKBP.Q
C
Consumers’ expenditure
(£m 1995 constant prices)
ONS
ABJR.Q + HAYO.Q
CAPU
Capacity utilisation
BoE
Log(YBHH.Q) – 2.7
– 0.7 log(YBUS.Q)
– 0.3 log(KNH)
– (0.7)(0.004)TIME
COMNUS
World non-oil commodity prices
(index, US$)
Datastream
ECALLI$
DD
Total domestic expenditure
(£m 1995 constant market prices)
ONS
YBIM.Q
DI
General government debt
interest payments
(£m current prices)
ONS
ROXY.Q
(1)
ONS: Office for National Statistics; BoE: Bank of England; HMCE: HM Customs and Excise Departmental Report, Section 3:
Government Policy Changes; DETR: Department of the Environment, Transport and the Regions; IMF: International Monetary Fund; and
BIS: Bank for International Settlements.
60
Economic models at the Bank of England
Name
Description
Source
Code / Details
DPP
Domestically consumed output
price index
ONS/BoE
Log(CGCB.Q / YBHH.Q)
– 0.2 log(IKBH.Q / IKBK.Q)
DUTY
Index of duty element of RPI
(January 1987 = 100)
BoE/ONS
Weighted chain-linked
average of basket
(DOBN.Q + DOBO.Q
+ DOBH.Q + DOBK.Q
+ DOCU.Q + DOCV.Q)
EARN
Average Earnings Index
(1995 = 100)
ONS
LNMQ.Q
EDS
US dollar-sterling exchange rate
(£:US$)
ONS
AJFA.Q
EER
Sterling effective exchange rate
index (1990 = 100)
ONS
AJHX.Q
EMP
Total employment in heads
ONS
(thousands; including self-employed)
MGRZ.Q
EMPH
Total employment in hours per week ONS
(millions)
YBUS.Q
EQP
Equity prices (FTSE All-Share)
ONS
AJMA.Q
EVAT
Value-added taxation rate
ONS
RUDR.Q /
(ABJQ.Q + HAYE.Q)
FCA
Factor cost adjustment
(£m 1995 constant prices)
ONS
ABMI.Q – YBHH.Q
FCAL
Factor cost adjustment
(£m current prices)
ONS
CMVL.Q
FOH
Foreign holdings of UK assets
(£m current prices)
ONS
HBQB.Q
FVAT
Rate of VAT charged on domestic
fuel and light
HMCE
G
General government final
consumption expenditure
(£m 1995 constant prices)
ONS
NMRY.Q
GDP
GDP(A) at factor cost
(£m 1995 constant prices)
ONS
YBHH.Q
GDPL
GDP at factor cost
(£m current prices)
ONS
CGCB.Q
The macroeconometric model
61
Name
Description
Source
Code / Details
GDPM
GDP(A) at market prices
(£m 1995 constant prices)
ONS
ABMI.Q
GDPT
Trend output
BoE
0.063 + 0.7 log(YBTF.Q)
+ 0.3 log(KNH)
+ (0.7)(0.004)TIME
GHW
Gross housing wealth
(£m current prices)
ONS
CGRI.A
GL
General government final
consumption expenditure
(£m current prices)
ONS
NMRP.Q
I
Fixed investment
(£m 1995 constant prices)
ONS
NPQT.Q + NPJR.Q
IBUS
Business investment
(£m 1995 constant prices)
ONS
NPEL.Q
IG
General government investment
(£m 1995 constant prices)
ONS
DLWF.Q
IGL
General government investment
(£m current prices)
ONS
RNCZ.Q + RNSM.Q
IH
Private sector dwellings investment
(£m 1995 constant prices)
ONS
DFEA.Q
II
Change in inventories
(£m 1995 constant prices)
ONS
CAFU.Q
INF
Four-quarter inflation rate of RPIX
ONS
Four-quarter percentage
growth of CHMK.Q
INFE
Expectations of annual RPIX
inflation
BoE
1.1 + 1.2 INF(–1)
– 0.6 INF(–2)
INOLD
Proportion of population over
retirement age that is inactive
ONS
(MGSL.Q – MGRZ.Q
– MGSC.Q – YBSN.Q) /
(MGSL.Q – YBTF.Q)
INWA
Proportion of population of
working age that is inactive
ONS
YBSN.Q / YBTF.Q
IVAT
Rate of VAT charged on insurance
HMCE
KBUSNH
Business non-residential capital
stock (£m 1995 constant prices)
BoE
BETA.KBUSNH(–1)
+ NPEL.Q
KII
Stock level
(£m 1995 constant prices)
ONS
ONS starting level
+ CAFU.Q
62
Economic models at the Bank of England
Name
Description
Source
Code / Details
KNH
Non-residential housing capital
stock (£m 1995 constant prices)
BoE
BETANH.KNH(–1)
+ (NPQT.Q – DFEA.Q)
LY
Real post-tax labour income
(£m 1995 constant prices)
ONS
(RPQK.Q – ROYL.Q
+ ROYT.Q – NRJN.Q
+ ROYH.Q) / ((ABJQ.Q
+ HAYE.Q) / (ABJR.Q
+ HAYO.Q))
M
Imports of goods and services
(£m 1995 constant prices)
ONS
IKBL.Q
M4
Break-adjusted stock of broad
money (£m current prices)
BoE
Statistical Abstract 1998,
Part 2, Table 10,
code: VUBR
MIPS
Mortgage interest payments
sub-index of RPI
(January 1987 = 1)
ONS
DOBQ.Q
NEA
Net external assets
(£m current prices)
ONS
HBQC.Q
NFW
Net financial wealth
(£m current prices)
ONS
NZEA.Q
PA
Participation rate of population
aged 16 and over
ONS
(MGRZ.Q + MGSC.Q)
/ MGSL.Q
PC
Total final consumers’
expenditure deflator (1995 = 1)
ONS
(ABJQ.Q + HAYE.Q )
/ (ABJR.Q + HAYO.Q)
PETSPOT
Oil spot prices (average of Brent
crude, West Texas, Dubai light;
US$)
Bloomberg
Brent crude: EUCRBRDT
Dubai light: PGCRDUBA
West Texas: USCRWTIC
PG
Government expenditure deflator
(1995 = 1)
ONS
NMRP.Q / NMRY.Q
PGDP
GDP deflator at factor cost
(1995 = 1)
ONS
CGCB.Q / YBHH.Q
PHSE
UK house prices
(1990 = 1, mixed adjusted)
DETR
PM
Import price deflator (1995 = 1)
ONS
IKBI.Q / IKBL.Q
POP
Population aged 16 and over
ONS
MGSL.Q
POWA
Population of working age
(men 16–64 years, women 16–59)
ONS
YBTF.Q
The macroeconometric model
63
Name
Description
Source
Code / Details
PSNB
Public Sector Net Borrowing
(£m current prices)
ONS
EQLD.Q
PTSH
Part-time share of employment
ONS
YCBH.Q / MGRZ.Q
PVIC
Present value of investment
allowances
BoE
Constructed using
approximate allowance
schemes for different
types of asset weighted by
the asset composition of
business investment.
PX
Export price deflator (1995 = 1)
ONS
IKBH.Q / IKBK.Q
RC
Effective corporate income
taxation rate
ONS
ACCD.Q / CGCB.Q
RCC
Real cost of capital
ONS/BoE
0.0025(((1 – PVIC) /
(1 – (ACCD.Q / CGCB.Q))
(AJLX.Q + (1 – BETA4)
100)) – INFE)
RD
Deposit rate
BoE
Σ di .wi
i
where wi = weights of
components in Divisia money
and di = rates of return.
Monetary and Financial
Statistics, Table 7.
REC
Total effective employers’ social
contribution rate
ONS
ROYK.Q / (ROYJ.Q
+ ROYH.Q)
REM
Employers’ effective tax rate
ONS
1 + (ROYK.Q / ROYJ.Q)
REV
Wealth revaluation term
Datastream/
ONS
0.12(AJHX.Q(–1) /
AJHX.Q) (WEQP /WEQP(–1))
+0.04(AJHX.Q(–1) / AJHX.Q)
(US10B.Q(–1) / US10B.Q)
+0.01(AJHX.Q(–1) / AJHX.Q)
+0.15(AJLX.Q(–1) / AJLX.Q)
+ 0.64(AJMA.Q /
AJMA.Q(–1)) + 0.045
RHPI
Real household post-tax income
(£m 1995 constant prices)
ONS
RPQK.Q((ABJR.Q
+ HAYO.Q) / (ABJQ.Q
+ HAYE.Q))
RJC
Personal sector social contribution
rate
ONS
(RPHU.Q – RVFH.Q) /
(ROYJ.Q + ROYH.Q)
RJY
Effective household income
taxation rate
ONS
RPHS.Q /
(ROYJ.Q + ROYH.Q)
64
Economic models at the Bank of England
Name
Description
Source
Code / Details
RL
Redemption yield on long-dated
British Government Securities
(20 years; per cent per annum)
ONS
AJLX.Q
RLR
10-year index-linked bond spot
yield (end quarter)
BoE
RMI
Average rate at which MIRAS is
claimed
HMCE
RMM4
Mortgage interest rate
BoE
Monetary and Financial
Statistics, Table 28.3, code:
quarterly average of
WBMG
RP
RPIX inflation adjusted by
productivity
ONS
(CHMK.Q.MGRZ.Q) /
YBHH.Q
RPCC
Council tax and rates sub-index
of RPI (January 1987 = 100)
ONS
DOBR.Q
RPI
Retail price index
(January 1987 = 1)
ONS
CHAW.Q
RPIX
RPI excluding mortgage interest
payments (January 1987 = 1)
ONS
CHMK.Q
RPIY
RPI excluding mortgage interest
payments and indirect taxes
(January 1987 = 100)
ONS
CBZW.Q
RPXC
RPI excluding mortgage interest
payments and council taxation
(January 1987 = 100)
ONS
DQAD.Q
RRX
Real exchange rate, GDP
deflator measure
Datastream/
ONS
(CGCB.Q.AJHX.Q) /
(YBHH.Q.WPX)
RS
London clearing banks’ base rate
ONS
AMIH.Q
RSD
Other expenditure taxation rate
ONS
(NTAB.Q – RUDR.Q) /
(ABJQ.Q + HAYE.Q)
RSDF
Interest rate differential
BoE/ONS
Log(1 + AMIH.Q / 400)
– log(1 + WRS / 400)
RTS
Effective government subsidy rate
ONS
AAXW.Q / CGCB.Q
RVAT
Standard rate of VAT
HMCE
RXRM
Importers’ relative price
ONS
(IKBI.Q.YBHH.Q) /
(IKBL.Q.CGCB.Q)
The macroeconometric model
65
Name
Description
Source
Code / Details
RXRX
Exporters’ real exchange rate
Datastream/
ONS
(IKBH.Q.AJHX.Q) /
(IKBK.Q.WPX)
SBEU
Subsidies from the European Union
ONS
AAXW.Q – ROXF.Q
SPEC
Trade specialisation term
BoE
Hodrick-Prescott filter of X
with λ = 10,000
X = (TRAD / WGDP)
SUBS
Government subsidies on products
(£m current prices)
ONS
AAXW.Q
TAX
Total tax receipts (including
private pension contributions)
(£m current prices)
ONS
RPHR.Q + ACCD.Q
+ RPHU.Q – RVFH.Q
+ CMVL.Q
TCC
Other current taxes paid by
households (£m current prices)
ONS
RPHT.Q
TE
Total taxes on production and
imports (£m current prices)
ONS
NTAB.Q
TIME
Time trend
BoE
1963 Q1 = 1
TJL
Tax deductions from household
income (£m current prices)
ONS
RPHU.Q – RVFH.Q
+ RPHR.Q
TRAD
World trade (1996 = 100)
ONS/IMF
Average of world import
volumes. Weighted by
countries’ shares in total UK
exports of goods and services
in 1996.
TSD
Other expenditure taxation
ONS
NTAB.Q – RUDR.Q
TVAT
Value-added taxation receipts
ONS
RUDR.Q
TXEU
Taxes to the European Union
ONS
FHLE.Q + QYZO.Q
+ QYZM.Q + CIOG.Q
TYC
Corporate income taxation
receipts (£m current prices)
ONS
ACCD.Q
TYJ
Household income tax payments
(£m current prices)
ONS
RPHS.Q
UKA
UK holdings of foreign assets
(£m current prices)
ONS
HBQA.Q
ULC
Nominal unit labour cost
ONS
(LNMQ.Q(1 + (ROYK.Q /
ROYJ.Q))MGRZ.Q) /
YBHH.Q
66
Economic models at the Bank of England
Name
Description
Source
Code / Details
UN
LFS/ILO unemployment level
ONS
MGSC.Q
UR
LFS/ILO unemployment rate
ONS
MGSC.Q / MGSF.Q
USER
Real user cost of housing
BoE
Based on average post-tax
mortgage rate (AJNL.Q)
USRL
US long bond rates
BoE
Monetary and Financial
Statistics, Table 28.2, code:
quarterly average of YD10US
WCC
Weight of council tax and
rates in RPI per 1,000
ONS
CZXF.A
WDUT
Weight of duty element in RPI
BoE
WEL
Total household sector wealth
(£m current prices)
BoE
NZEA.Q + GHW
WEQP
World equity prices (S&P
global equity price index)
Datastream
WIWRLDL(PI)
WGDP
World GDP (1996 = 100)
Datastream
Average world GDP
weighted by countries’ shares
in total UK exports of goods
and services in 1996.
WMIP
Weight of mortgage interest
payments in RPI per 1,000
ONS
CZXE.A
WPX
World export prices (1995 = 100)
Datastream
G7 (excluding UK) weighted
average of exports of goods
and services deflators;
effective exchange rate
weights applied.
WRPY
Weight of RPIY in RPI / 1,000
ONS/BoE
1,000 – sum of weights of
VAT, excise duties, local
taxation, insurance tax,
mortgage interest payments.
WRS
World nominal interest rate
BoE/BIS
G7 (excluding UK) weighted
average of three-month
interbank interest rates;
effective exchange rate
weights applied.
di
wi
rp
i
i
where wi = weight of good i in
RPI basket (parts per 1,000),
di = duty on good i and
rpi = retail price of good i.
∑
The macroeconometric model
67
Name
Description
Source
Code / Details
X
Exports of goods and services
(£m 1995 constant prices)
ONS
IKBK.Q
YDIJ
Households’ non-labour income
(£m current prices)
ONS
ROYL.Q – ROYT.Q
+ NRJN.Q – ROYH.Q
YE
Total wages, salaries and
self-employment income
(£m current prices)
ONS
ROYJ.Q + ROYH.Q
YEC
Total employers’ social
contributions (£m current prices)
ONS
ROYK.Q
YJ
Total household sector pre-tax
income (including benefits)
(£m current prices)
ONS
ROYJ.Q + ROYK.Q
+ RPHL.Q – RPIA.Q
+ RPHM.Q – RPIB.Q
+ RPQJ.Q + ROYL.Q
– ROYT.Q + NRJN.Q
YJC
Net social contributions from
household sector
(£m current prices)
ONS
RPHU.Q – RVFH.Q
YJG
Current grants to the household
sector (£m current prices)
ONS
RPHL.Q – RPIA.Q
+ RPHM.Q – RPIB.Q
+ RPQJ.Q
ZPROXY
Measure of structural effects
of labour market developments
BoE
Hodrick-Prescott filter of X
with λ = 10,000
X = log[(LNMQ.Q(1
+ (ROYK.Q / ROJY.Q)).
MGRZ.Q) / YBHH.Q]
+ 0.013(MGSC.Q / MGSF.Q)
ZPSTA
Government inflation target
Set equal to 2.5%
68
Economic models at the Bank of England
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