Output gaps, inflation and financial cycles in the UK

Abstract

This paper aims at constructing potential output and output gap measures for the UK which are pinned down by macroeconomic relationships as well as financial indicators. The exercise is based on a parsimonious unobserved components model which is estimated via Bayesian methods where the time-paths of unobserved variables are extracted with the Kalman filter. The resulting measures track current narratives on macroeconomic cycles and trends in the UK reasonably well. The inclusion of summary indicators of financial conditions leads to a more optimistic view on the path of UK potential output after the crisis and adds value to the model via improving its real-time performance. The models augmented with financial conditions have some real-time wage inflation forecasting ability over the monetary policy-relevant 2- to 3-year horizon during the last 15 years. Finally, we also introduce a new approach to construct financial conditions indices, with emphasis on their real-time performance and ability to track the evolution of macro-financial imbalances. Our results can be relevant from both monetary and macro-prudential policy perspectives.

Appendices

Appendix 1: Technical details of the unobserved components model

1.1 State-space form of the UCM

The state-space form gathers the structure of the model into a form consisting of a measurement equation and a state equation. The state-space form can be easily handled by the Kalman filter:

$$ \begin{array}{*{20}l} {Y_{t} = C'X_{t} + V_{t} } \hfill & {V_{t} \,\sim\,N(0,R)} \hfill \\ {X_{t} = AX_{t - 1} + FW_{t} } \hfill & {W_{t} \,\sim\,N(0,Q)} \hfill \\ \end{array} $$

Based on the structure introduced above, the measurement equation consists of the following matrices (for B2, the last row and column are excluded):

$$ \left[ {\begin{array}{*{20}c} {y_{t} } \\ {\pi_{t} } \\ {u_{t} } \\ {{\text{dltu}}_{t} } \\ {{\text{fci}}_{t} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & { - 1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\hat{y}_{t} } \\ {\bar{y}_{t} } \\ {g_{t} } \\ {\hat{\pi }_{t} } \\ {\bar{\pi }_{t} } \\ {\hat{u}_{t} } \\ {\bar{u}_{t} } \\ {\bar{u}_{t - 1} } \\ {{\text{fcy}}_{t} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ {\eta_{t}^{\text{ltu}} } \\ {\eta_{t}^{\text{fci}} } \\ \end{array} } \right] $$

The state equation is the following (for B2, the last row and column are excluded):

$$ \left[ {\begin{array}{*{20}c} {\hat{y}_{t} } \\ {\bar{y}_{t} } \\ {g_{t} } \\ {\hat{\pi }_{t} } \\ {\bar{\pi }_{t} } \\ {\hat{u}_{t} } \\ {\bar{u}_{t} } \\ {\bar{u}_{t - 1} } \\ {\widehat{\text{fci}}_{t} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\alpha_{1} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\alpha_{2} } \\ 0 & 1 & 1 & 0 & 0 & 0 & \iota & \iota & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ {\beta_{2} } & 0 & 0 & {\beta_{1} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ {\gamma_{2} } & 0 & 0 & 0 & 0 & {\gamma_{1} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \rho \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\hat{y}_{t - 1} } \\ {\bar{y}_{t - 1} } \\ {g_{t - 1} } \\ {\hat{\pi }_{t - 1} } \\ {\bar{\pi }_{t - 1} } \\ {\hat{u}_{t - 1} } \\ {\bar{u}_{t - 1} } \\ {\bar{u}_{t - 2} } \\ {{\text{fcy}}_{t - 1} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varepsilon_{t}^{{\hat{y}}} } \\ 0 \\ {\varepsilon_{t}^{g} } \\ {\varepsilon_{t}^{{\hat{\pi }}} } \\ {\varepsilon_{t}^{{\bar{\pi }}} } \\ {\varepsilon_{t}^{{\hat{u}}} } \\ {\varepsilon_{t}^{{\bar{u}}} } \\ 0 \\ {\varepsilon_{t}^{\text{fci}} } \\ \end{array} } \right] $$

1.2 Conditional forecast information

The actual real-time versions of the FCI models are also augmented with Bank of England CPI and GDP forecasts to examine whether this improves the real-time performance. The forecasts are added to the models in a similar fashion as in Blagrave et al. (2015):

$$ \begin{aligned} Y_{t + j}^{F} & = Y_{t + j} + \varepsilon_{t + j}^{Yf} \\ \pi_{t + j}^{F} & = \pi_{t + j} + \varepsilon_{t + j}^{\pi f} \\ \end{aligned} $$

for $j = 1, \ldots ,12,$ for the level of GDP and rate of (quarterly) inflation, respectively. Hence, the forecasts are seen as imprecise estimates of future GDP and inflation, with error terms $\varepsilon_{t + j}^{Yf}$ and $\varepsilon_{t + j}^{\pi f}$ accounting for the noise and forecast errors. The forecast variables are added to the models in three formats: pure GDP, pure and CPI and both forecasts. As detailed in the main text, the pure GDP format adds the most information.

Appendix 2: Details of the financial conditions indices

It is not obvious which variables should be included in a financial conditions index (FCI) nor is there agreement on which estimation method should be used to derive the index. There is a wide existing literature on the estimation of FCIs.²⁸ In these studies, FCI’s are typically derived by using simple averages, principal components analysis or vector autoregressions. The main focus of many of these studies is to examine the ability of FCI’s to predict near-term GDP dynamics. However, there is surprisingly little (if any) analysis in the literature on the real-time performance of FCI models, which is a crucial metric for our purposes. Consequently, we refrain from using any of the existing FCI’s for the UK due to our different emphases on the construction of the FCI, as well as to avoid having to re-estimate and update FCI’s introduced in earlier studies. We are also primarily interested in building a financial conditions index with relevant information on the business cycle rather than just financial markets, which is a slightly different aim compared to more traditional FCI studies.

For the current study, 22 financial indicators for the UK were collected. Different combinations of these indicators were tested, and given the uncertainty related to the estimation of FCIs, we use two different FCIs (denoted FCI1 and FCI2), derived by different methods from different variable sets.

2.1 FCI1

For FCI1, four key UK financial indicators are combined using a simple factor model structure, where the FCI is the only state variable (i.e. a common factor driving the fluctuation in the observable indicators), and the four indicators are the observable variables with four idiosyncratic single shock terms. The four indicators were chosen with the relevant theoretical literature in mind (see the Introduction section). The factor model is estimated using a sample from 1988 to 2014, and the resulting FCI1 index relevant for the sample of the current study (i.e. from 1995 to 2014) is presented in Fig. 12. FCI1 performs well in real time and the contributions of the different indicators to it are intuitive (see Fig. 12c, e). The choice of the relevant indicators depends on the real-time performance of the model. The financial indicators (all demeaned and divided by their respective standard deviations) in the model are:

Fig. 12

FCI indices and variables. Demeaned series. Series are normalised so that a tightening of conditions is a negative movement

• Mortgage spread (2-year fixed mortgage rate (75% LTV) minus Bank Rate) (y/y growth rate). (Source: Bank of England) (mgage_spread)

• Composite UK house price index (average of the Halifax and Nationwide House Price Indices) (y/y growth rate) (Bank of England/Halifax/Nationwide) (houseprice)

• Credit to households in real terms (monetary financial institutions’ sterling net lending to the household sector deflated by CPI) (y/y growth rate). (Bank of England/ONS) (rcred_hhold)

• Credit to firms in real terms (monetary financial institutions’ sterling net lending to the non-financial corporate sector deflated by CPI) (y/y growth rate). (Bank of England/ONS) (rcred_firms)

2.2 FCI2

For the second factor model (FCI2), a more mechanical approach was taken. First, the performance of different combinations of the most relevant financial indicators was examined to extract the best-performing common factor series in real time. In practice, the following procedure was followed:

• 1. The general suitability and lengths of available samples was examined for all the series. Based on this analysis, nine of the 22 series were dropped either because the time series were too short or bore no relation to macroeconomic cycles.

• 2. Real-time performance of all possible combinations of the remaining 13 variables was examined by running a real-time experiment from 2001 to 2014, recording the sum of standard deviations of the different vintages of the resulting FCI series at every quarter, and then, the combinations were ranked based on this sum. This experiment was carried out for an FCI consisting of four, five and six variables. Based on this analysis, it was decided that concentrating on 5-variable FCI’s would strike an optimal balance between the best real-time performance and maintaining as parsimonious a model as possible.

• 3. The 13 variables entering into the 5-variable FCI’s were further ranked by their average position in terms of the standard deviations of the models that a particular variable entered in. The prevailing selection (above) was then made based on the best-performing variables, but also at the same time ensuring that the index would include interest rate spread variables, volatility variables and lending variables.

For the calculation of the index, the variables were first detrended and demeaned and then smoothed with a 4-quarter moving average measure of them.²⁹ The resulting index is again shown in Fig. 12a. Similarly to the FCI1 model, the FCI2 model also performs well in real time (Fig. 12d). A dynamic-factor model for FCI2 was derived using the following variables:

• Net credit flow to households (monetary financial institutions’ sterling net lending to the household sector) (y/y growth rate). (Bank of England)

• Net credit flow to non-financial corporations (monetary financial institutions’ sterling net lending to private non-financial corporations) (y/y growth rate). (Bank of England)

• Money market spread (3-month GBP LIBOR minus Bank Rate) (Bloomberg/Bank of England)

• Short gilt spread (2-year UK Gilt yield minus 3-month GBP LIBOR) (Bloomberg/Bank of England)

• Bond market volatility (3-month rolling volatility of 10-year UK Gilt yield) (Bloomberg/Bank of England)

2.3 Summary

While the selection of any FCI is always contentious and is not the main focus of the current study, we believe that our FCI indices strike a balance between a good real-time performance, parsimoniousness and the inclusion of relevant variables. The FCIs are also not only relatively close to each other, but also relatively close over the relevant time horizon to the UK FCI introduced by OECD (see Guichard et al. 2009).

Appendix 3: Data

The following data is used in the models:

• GDP: ESA2010 chain linked volumes, seasonally and working day adjusted (Source: ONS), in log level

• Unemployment rate: LFS unemployment rate of 16–64-year-olds, in proportion of the labour force. (ONS)

• Inflation: CPI excluding food and energy. Seasonally adjusted annualised q-o-q log changes (Bank of England/ONS)

• Wage inflation: average weekly earnings, regular pay, seasonally adjusted annualised q-o-q changes (ONS)

• Long-term unemployment rate: proportion of the labour force that has been unemployed for more than 12 months, based on claimant count statistics for the UK³⁰ (Bank of England/ONS)

• Financial conditions indices (see details in Appendix 2).

Appendix 4: Additional Figures

See Figs. 13 and 14.

Fig. 13

FCI model—trend unemployment rate and trend price/wage variables. a Unemployment rate as  %, bq/q change in  %, cq/q change in  %

Fig. 14

Prior and posterior distributions of the FCI2w model