To estimate the model, we need to specify how the variables in the model are related to UK data. The fact that COMPASS equations are written in terms of de-trended variables means that we need to account for this de-trending process. This sub-section describes the data we use and explains how those data are related to COMPASS variables.
We use data for fifteen macroeconomic variables (described below).53 The sample period is 1993Q1–2007Q4.54 This is a relatively short sample, but is motivated by a desire to ensure that the estimation period excludes large changes in monetary policy regime. An alternative approach would be to use a longer sample period and account for the changes in monetary regime, though this creates other issues.55 We exclude the recent financial crisis from our sample period to avoid this episode having a disproportionate effect on the properties of the data and to avoid using the most recent data that is subject to larger revisions.
The data series that feed into the model are called ‘raw observables’, because they rep- resent the raw data prior to any transformation or de-trending.56 These raw observables are transformed into appropriate units using a set of ‘data transformation equations’, which de-trend the raw observables in the appropriate way to convert them into units that correspond to the units in which COMPASS variables are measured (typically 100 times a logarithmic deviation from steady state). The raw observables, their data sources and data transformation equations are shown in Table 1.
The information in Table 1 shows how raw data is mapped to model obervable units.
As an example consider the data transformation equation for the import price deflator:
dlnpmdeft= 100∆ ln pmdeft−Π∗,ttt −Πm,ttt (24) This equation defines a variable dlnpmdeftthat can be mapped into COMPASS variables measured as (100 times) the logarithm of the stationary variable. To arrive at dlnpmdeft, we take the first difference of the logarithm of the raw data for the import deflator and multiply it by 100.57 Then we subtract two ‘time-varying trends’, which are introduced to correct for deterministic movements in the trend of the import deflator over our sample that are not captured by the trends within the model itself.
In this example, the time-varying trends correct for the low-frequency decline in CPI inflation associated with the transition to inflation targeting over the early (and in the training) part of our sample, Π∗,ttt , and the fact that import prices have fallen less rapidly
53As is common in models like COMPASS, the number of data series used (treated as ‘observable’) is much smaller than the total number of endogenous variables in the model. In part, this reflects the fact that some of the equations in DSGE models are identities (for example, a market clearing condition specifying that total expenditure equals total output). But it also reflects that many of the endogenous variables do not have a measured counterpart in the data (eg TFP).
54Data from 1987Q3–1992Q4 is used as a ‘training sample’ to initialise the Kalman filter.
55Harrison and Oomen (2010) use UK data from the 1960s and account for regime shifts by using deterministic breaks in the trends for nominal variables. That exercise demonstrates the difficulties involved in producing a set of consistently measured UK macroeconomic data, even for a small number of series. Recent work by C´urdia and Finocchiaro (2013) proposes a method for incorporating regime shifts explicitly into the estimation of structural models.
56The term ‘observables’ comes from the state space modelling literature. We use state space methods to estimate the model parameters using the Kalman filter. For more details, see Section 6.2.2.
57Scaling by 100 is simply a normalisation. It means that the units of variables in COMPASS corre- spond (approximately) to percentage deviations from steady state, which is useful as a starting point in analysis of the model’s outputs when preparing the forecast.
24 Working Paper No. 471 May 2013
than predicted by the supply side structure of the model, Πm,ttt . The CPI inflation target trend is calculated relative to the current inflation target of 2% per annum for CPI inflation, so that Π∗,ttt = 0 over most of our sample.58 The trend Πm,ttt is computed as the average difference between the observed growth rate of import prices in the data and the rate predicted by the supply side structure of the model adjusted for the CPI inflation trend.59 The same approach is used for the other time-varying trends in Table 1. That is, they are used to ensure that the mean of the transformed observables over the estimation sample is consistent with the assumptions embodied in the model. These trends are: Πx,ttt , dlnxkpttt, dlnmkpttt , Πxf,ttt & dlnyfttt . The underlying economic rationale for these time-varying trends it that it is difficult to match the secular increase in global trade as a share of activity using standard supply side assumptions. This is a feature of the data in many countries (a similar de-trending approach is taken by Adolfson et al.
(2007) using euro area data).
Having defined observable data in appropriate units, a set of ‘measurement equations’
is used to relate the transformed observables to COMPASS variables. These are also shown in Table 1. For example, the measurement equation associated with import price inflation, dlnpmdeft is:
dlnpmdeft=πtM + 100 ln Π∗
ΓX +σme,pmmepmt (25)
which tells us that the detrended (log difference of the) import deflator is mapped into the model variable πM (the deviation of quarterly import price inflation from steady state) plus the steady-state trend growth in import prices implied by the model, plus an iid measurement error. The steady-state growth of import prices in the model is determined by the inflation target, Π∗, and the relative deterministic growth rate of technological progress in the import (and export) sector, ΓX (see the discussion in Section A.2 of Appendix A for more details). Measurement error, mepmt , is included to cope with the fact that import prices may be particularly prone to measurement issues.60 We also include measurement error for investment growth, wage growth, hours worked, export prices and import and export volumes.
58Over the earlier part of our sample, Π∗,ttt takes positive values, based on the implicit inflation target implied by official policy announcements (following the approach used in Batini and Nelson (2001)).
59The supply-side structure of the model implies that relative import prices should decline at the rate of relative productivity in the export retail sector, ΓX. Since it is not possible to calibrate simultaneously the trend growth rates of GDP, final output and all the expenditure components, the approach taken is to calibrate the relative trend growth of exports and imports, ΓX, to deliver target trend growth rates for final output, ΓZ, and GDP, ΓV. That means that ln ΓX= 1−ααVV ln ΓZ−ln ΓV
.
60As is the case for the structural shocks, measurement errors are treated asiidwith standard normal distributions. The parameterσme,pm is used to scale the measurement error and determines its standard deviation.
Table 1: Observables, data transformation and measurement equations
Variable Description ONS code Data transformation equation Measurement equation
gdpkp Real GDP ABMI dlngdpkpt≡100∆ ln gdpkpt dlngdpkpt= ∆vt+γZt + 100 ln
ΓZΓH ΓX−1−αVαV ckp Real consumption ABJR+HAYO dlnckpt≡100∆ ln ckpt dlnckpt= ∆ct+γtZ+ 100 ln ΓZΓH
ikkp Real business investment NPELa dlnikkpt≡100∆ ln ikkpt dlnikkpt= ∆it+γtZ+ 100 ln ΓZΓHΓI
+σImemeIt gonskp Real government spending NMRY+DLWFb dlngonskpt≡100∆ ln gonskpt dlngonskpt= ∆gt+γZt + 100 ln ΓZΓHΓG xkp Real exports IKBKc dlnxkpt≡100∆ ln xkpt−dlnxkpttt dlnxkpt= ∆xt+γtZ+ 100 ln ΓZΓHΓX
+σXmemeXt mkp Real imports IKBLc dlnmkpt≡100∆ ln mkpt−dlnmkpttt dlnmkpt= ∆mt+γtZ+ 100 ln ΓZΓHΓX
+σMmemeMt pxdef Export deflator IKBH/IKBLc dlnpxdeft≡100∆ ln pxdeft−Π∗,ttt −Πx,ttt dlnpxdeft= ∆pEXt −∆qt+πZt + 100 lnΓΠX∗ +σP XmemeP Xt pmdef Import deflator IKBI/IKBLc dlnpmdeft≡100∆ ln pmdeft−Π∗,ttt −Πm,ttt dlnpmdeft=πtM+ 100 lnΓΠX∗ +σP Mme meP Mt
awe Nominal wage per capita KAB9 dlnawet≡∆ ln awet−Π∗,ttt dlnawet= ∆wt+γtZ+πtZ+ 100 ln ΓZΠ∗
+σmeW meWt cpisa Seasonally adjusted CPI D7BTd dlncpisat≡100∆ ln cpisat−Π∗,ttt dlncpisat=πtC+ 100 ln Π∗
rga Bank Rate In-housee robst≡100 ln 1 +rga100t14
−Π∗,ttt robst=rt+ 100 lnR eer Sterling ERI In-housef dlneert≡100∆ ln eert dlneert= ∆qt−πZt
hrs Total hours worked YBUS dlnhrst≡100∆ ln hrst dlnhrst= ∆lt+ 100 ln ΓH+σLmemeLt yf World output In-houseg dlnyft≡100∆ ln yft−dlnyfttt dlnyft= ∆zFt +γtZ+ 100 ln ΓZΓH pxfdef World export deflator In-househ dlnpxfdeft≡100∆ ln pxfdeft−Πxf,ttt dlnpxfdeft= ∆pXt F + 100 lnΓΠX∗
aFollowing the 2011 ONS Blue Book publication, this series was unavailable from 1997 onwards. Bank staff constructed the series back to 1987 by projecting the pre-2011 Blue Book growth rates of business investment from 1987-1996 backwards from the published level of business investment in 1997Q1.
bData for government investment (DLWF) prior to 1997 was constructed by Bank staff in the same way as for business investment – see footnotea.
cIn-house adjustments are made to take account of the effects of MTIC fraud. See the box on pp 22-23 of the August 2006 Inflation Report for more details.
dThis CPI series is seasonally adjusted by Bank staff using the X-12 method after accounting for the effect of VAT changes. The resulting series is converted to a quarterly frequency by averaging the index numbers in each month of each quarter.
eAvailable from the Bank’s Statistical Interactive Database with the code IUQABEDR.
fComputed as a weighted average of individual bilateral exchange rates where the weights are determined by shares in UK trade. See, for example, Lynch and Whitaker (2004). Available from the Bank’s Statistical Interactive Database with the code XUQBK67.
gConstructed as a measure of world trade, taking the average of imports across countries weighted by those countries respective shares in UK trade.
hConstructed as the average of export prices across countries weighted by those countries respective shares in UK trade.
26Working Paper No. 471 May 2013