Applying a Macro-Finance Yield Curve to UK Quantitative Easing

Transcription

1 Applying a Macro-Finance Yield Curve to UK Quantitative Easing Jagjit S. Chadha and Alex Waters This Draft: August 2012 Abstract We estimate a macro-finance yield curve model of both the nominal and real forward curve for the UK from 1993 to Our model is able to accommodate a larger number of macroeconomic variables than previously seen in other literature. We use the model to estimate the supply effects from debt issuance on both forward rates and so gauge the impact of Quantitative Easing on forward rates; we find that 10 year nominal interest rates on average are lower by 46 basis points which can largely be explained by the portfolio balance channel, the liquidity premium channel and the signalling channel but there is no statistical impact on the real rates. JEL Classifications: E43; E44; E47; E58; E65 Key Words: Term Structure of Interest Rates; Monetary Policy; Quantitative Easing 1 Introduction Over the past decade several different methodologies have been adopted by researchers to try and uncover which macroeconomic factors are driving the dynamics of the nominal term structure such as Ang and Piazzesi (2003), Rudebusch and Wu (2008) and Dewachter and Lyrio (2006) to name but a few. Quite often these researchers focus on the three macroeconomic variables that guide monetary policy which are inflation, real output and the policy interest rate. Although work is starting to extend beyond these three macro variables such as Afonso and Martins (2012) who study the effects of fiscal variables on the term structure or financial variables examined by Dewachter and Iania (2011), the number of macroeconomic variables that are used to explain the dynamics of the yield curve within affi ne term structure models are very limited. The primary reasons for this is that these models quite often have to be solved under both historical and risk neutral measures making it computationally burdensome to estimate all the parameters within these models (Borgy et al, 2011). In this paper we propose a new methodology that allows us to explore the macroeconomic underpinnings of the UK s nominal and real term structure of interest rates which can accommodate a much larger number of macroeconomic variables. In total we examine 31 different macroeconomic The authors would like to thank Evren Caglar, Eddie Gerba, Jack Meaning and James Warren for many helpful comments and discussions. Also we would like thank Mark Deacon and his staff on the Quantitative Analysis and Economic Research Desk at the UK Debt Management offi ce for comments on an earlier draft and providing us with some helpful data. Furthermore, we would like to thank participants at the Society of Computational Economics Conference, San Francisco in June 2011 as well as the participants at the Computational and Financial Economics conference, London, December Professor of Economics, Chair in Banking and Finance, School of Economics, University of Kent, Canterbury and Centre for International Macroeconomics and Finance, University of Cambridge. jsc@kent.ac.uk. School of Economics, University of Kent, Canterbury. address: aw365@kent.ac.uk. 1

2 variables spread across five key groups of data that may drive both the nominal and real term structures. These five groups are inflation, real activity, monetary and fiscal policy, financial and international. The estimation is performed in two stages; firstly adopting the state space methodology similar to that of Diebold et al (2006) and Afonso and Martins (2012). We estimate a four factor Svensson (1994, henceforth referred to as Svensson) forward curve of both the UK nominal and real term structure using a Kalman filter and maximum likelihood estimation. Then using seemingly unrelated regression (henceforth, SUR) we test down from 31 different macroeconomic and financial variables to determine whether or not macroeconomic factors can provide a good fit of both the nominal and real forward curve. Our main contribution to the literature is a detailed analysis of which macroeconomic and financial factors drive not only the nominal term structure but also the real term structure of interest rates. For the nominal curve we can identify a net supply effect for government debt at various maturities, that inflation expectations are more important than actual inflation, and that the exchange rate and macroeconomic announcements from Germany have a significant impact on the term structure but there is no apparent effect from the U.S. For the real curve we also identify a net supply effect that suggests that at different maturities the liquidity of the bond market matters and that there is some evidence of anchoring of inflation expectations with no change in long-term real forwards. Of the 31 variables, 9 are needed to explain movements in the nominal curve. Amongst these are the variables that concern monetary policy makers, debt-to-gdp and international variables such as the effective exchange rate and measures of German real activity. For the real dynamic factors the macroeconomic variables do less well and there are only four variables that drive the real curve, debt-to-gdp, inflation expectations, the Libor spread and notes and coins. To examine the performance of the model, we examine an out-of-sample forecast over the period when Quantitative Easing (QE) was first used as a tool for monetary policy by the Bank of England from March 2009 to January We define QE as large scale purchases of government bonds with Central Bank reserves when the policy rate is at its effective zero lower bound. It is probable that QE will impact on the curve through three channels, the portfolio balance channel, the liquidity premium channel and the signalling channel. The portfolio balance channel reflects a supply effect; where the imperfect substitutability of different assets can allow the relative supply of bonds determine the price of the asset. The liquidity premium channel exists because the Central Bank intervenes and becomes a large scale purchaser of bonds which can improve the functioning of the bond market and reduce the liquidity premium. Both of these channels should exert downward pressure on longer-term yields to ease financial conditions and stimulate growth. The signalling channel refers to the market s expectations of the future path of interest rates based on the signals the market receives from both the monetary authorities and wider the macroeconomy leaving the impact on the yield curve ambiguous. The signalling effect can be uncovered and analysed through risk-adjusted market interest rates used to gauge the expected path of interest rates. If the macroeconomic and financial variables are able to determine the path of the nominal and real term structure then we are able to provide a counterfactual path of forward rates at different maturities which we are then able to compare to the observed forward rates. The out-of-sample forecast, given the macroeconomic variables overestimates the observed data by a similar amount to the event study analysis of Caglar et al (2011) and Meaning and Zhu (2012) with the five year forward and the ten year forward on average overestimating the actual curve by 67 and 46 basis points respectively. We find the overestimate of the forward rates is correct both in terms of timing and maturities targeted; the overestimate occurs as of March 2009 and maturities greater than 24 months show an over prediction. The forecast of the real curve does not demonstrate any persistent deviation from the realised path of interest rates: a particularly appealing result as the Bank of England did not undertake any QE operations with real bonds. 2

3 We decompose the forecast error using the estimated and calculated effects of the portfolio balance channel, the liquidity premium channel and the signalling channel. Overall all three channels exert downward pressure on the yield curve and we uncover that it is the signalling channel that plays the most prominent role when QE was first implemented but this effect dissipates as QE purchases are extended. The portfolio balance channel has the largest effect at the end of the sample as the amount of purchases increases. This channel alone is found to reduce yields by as much as 136 basis points at 10 years. The liquidity premium channel does play a small role in reducing yields at the 5 year forward but not for the 10 year forward. On average the forecast error at 5 and 10 years is 67 and 46 basis points respectively and the average estimated impact that the three channels have on the yield curve are 88 and 86 basis points respectively. 1 This paper is further divided into five sections. Section Two outlines the methodology used to fit the term structure, the econometric specification and estimation techniques as well as the impulse responses. Section Three outlines the forward curve data, the estimated yield curve factors and the macroeconomic variables and data and the overall fit of the estimated term structure across the sample period. Section Four presents the empirical results from the impulse responses and post estimation analysis. Section Five outlines the forecast exercise performed over the QE period as well as the decomposition of the forecast error and section Six concludes. 2 Methodology Presented below is the two-stage methodology. The first stage begins with outlining the Svensson (1994) methodology that fits the term structure and employing an approach similar to that of Diebold et al (2006) and Afonso and Martins (2012). We estimate four latent forward curve factors which we call the level, slope and two curvatures by means of the Kalman filter. The second stage is to then take these estimated latent factors and using Seemingly Unrelated Regression with macroeconomic and financial variable we produce impulse responses that demonstrate upon announcement or revelation how each variable affects the shape of the forward curve. 2.1 Term Structure Model The functional form that has been used to fit the term structure of interest rates in this analysis is that of Svensson (1994). This parametric model is simple to implement and provides a parsimonious description of the term structure of interest rates. Svensson as a function of different factor weights and parameters β i produces a smooth fit of the yield curve. The functional form of Svensson is: ( 1 e τλ 1 y(τ) = β 1 + β 2 τλ 1 ) ( 1 e τλ 1 ) ( + β 3 e τλ 1 1 e τλ 2 ) + β τλ 4 e τλ 2. (1) 1 τλ 2 The Svensson methodology representation can then be interpreted as a dynamic latent factor model where β 1, β 2, β 3 and β 4 become time varying parameters that capture the level L t, slope S t, the first curvature C 1,t and the second curvature factor C 2,t of the yield curve at time t. The functional form of Svensson can then be expressed as: ( 1 e τλ 1 y(τ) = L t + S t τλ 1 ) ( 1 e τλ 1 ) ( + C 1,t e τλ 1 1 e τλ 2 ) + C 2,t e τλ 2. (2) τλ 1 τλ 2 1 That the impact of the three channels sum to more than the forecast error suggests a further factor such as credit risk, which we leave to further research. 3

4 The yield is denoted as y(τ) and τ is the maturity. Where L t, S t, C 1,t and C 2,t are time varying parameters and the parameters λ 1 and λ 2 remain constant throughout estimation. Lambda is used to determine the maximum loading of C 1,t and C 2,t. Together they determine the rate of exponential decay of the factor weights; the smaller the value of lambda, the slower is the decay and the greater the fit of longer maturities and larger values of lambda produce a better fit of short-term maturities. The parameters L t, S t, C 1,t and C 2,t are the parameters which correspond to their appropriate factor loadings. The loading on L t is 1 at all maturities and is regarded to be a long-term factor and is known as the level, any shift in the level has an equal effect across all yields. The factor loadings of S t has a functional form that starts at 1 but decays monotonically and quickly to 0 and is viewed as a short-term factor and is called the slope. The final two factors C 1,t and C 2,t have loadings that begin at 0, which increase and then decay back to zero. These factors are medium-term factors and are known as the curvatures. Any changes in C 1,t and C 2,t will change medium-term yields and therefore change the curvature of the yield curve. We assume that L t, S t, C 1,t and C 2,t follow a first order vector autoregressive process which allows the model to form a state-space system and by means of the Kalman filter we are able to obtain the maximum-likelihood estimates of the parameters and the implied estimates of L t, S t, C 1,t and C 2,t. The transition equation, which determines the dynamics of the state vector is: L t S t C 1,t C 2,t = µ L µ S µ C1 µ C2 + a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 L t 1 S t 1 C 1,t 1 C 2,t 1 + η L,t η S,t η C1,t η C2,t. (3) Where t = 1,..., T, in this case µ L, µ S, µ C1 and µ C2 are constants and η L,t, η S,t, η C1,t and η C2,t are the disturbances of the autoregressive process of the latent factors. The measurement equation relates a set of N observed zero-coupon yields of different maturities to the four latent factors of the yield curve and is given by: y t (τ 1 ) y t (τ 2 ). y t (τ N ) = ( ) ( ) ( ) 1 e τ 1 λ 1 1 e τ 1 λ 1 τ 1 λ 1 τ 1 λ 1 e τ 1λ 1 1 e τ 1 λ 2 τ 1 λ 2 e τ 1λ 2 ( ) ( ) ( ) 1 e τ 2 λ 1 1 e τ 2 λ 1 τ 2 λ 1 τ 2 λ 1 e τ 2λ 1 1 e τ 2 λ 2 τ 2 λ 2 e τ 2λ 2 (... ) ( ) ( ) 1 e τ N λ 1 1 e τ N λ 1 τ N λ 1 τ N λ 1 e τ N λ 1 1 e τ N λ 2 τ N λ 2 e τ N λ 2 L t ε τ 1,t S t C 1,t + ε τ 2,t. C 2,t ε τn,t. (4) Where t = 1,..., T and ε τ 1,t, ε τ 2,t,..., ε τn,t are the measurement errors of the observed forwards at every maturity at time t and the implied yields determined by the shape of the fitted Svensson curve. In matrix notation, the state-space form of the model can be written as: X t = µ + AX t 1 +η t η t i.i.d.(0, Ω). (5) Y t = BX t +ε t ε t i.i.d.(0, σ 2 I). (6) Where A and B are the transition and measurement matrices respectively. For the Kalman filter to be the optimal linear filter it is assumed that the initial conditions of the state vector are 4

5 uncorrelated with the innovations in both systems. The disturbances of both the measurement and the transition equations are i.i.d. and uncorrelated. The variance-covariance matrix of the disturbances for the transition equation Ω is non-diagonal and the variance-covariance matrix of the measurement equation σ 2 I is diagonal. Implying that at any maturity the residual between the fitted and the zero-coupon yield is not correlated with the residual at any other maturity. Given some set of initialisations for the four latent factors and the parameters (the coeffi cients of the A and B matrices, the variance-covariance matrices and the choice of lambdas) the one step-ahead prediction errors and the variance of these prediction errors are used to compute the log-likelihood function. The Kalman filter produces the maximum likelihood estimates and optimal filtered and smoothed estimates of the underlying latent factors L t, S t, C 1,t and C 2,t. 2.2 Macro-Finance Specification and Estimation The joint macro-finance model is specified as a system of regression equations with a lagged dependent variable. The dependent variables are the estimated forward curve factors for the nominal and real curve and only the lagged dependent variable of the respective forward curve factor will appear on the right hand side. The system of equations has the following form: Y t =α + γt + ρy t 1 +X t θ + d t δ + ε t. (7) Y is the vector of dependent variables, α is a vector of constants and γ is a time trend, all of which are 4 1 vectors. X is a x 1 vector of the independent variable; the macroeconomic variables and θ is the 4 x matrix of their coeffi cients where x can be as large as 31 (explained below). We use dummy variables to explain any large residuals, so d is an m 1 vector of dummy variables and δ is a 4 n matrix of coeffi cients for the dummy variables. The transition equation of the Kalman filter assumes that the error terms of the four factors are correlated and so it is appropriate to take account of this in the econometric analysis and so we use seemingly unrelated regression. We estimate the system of equations with feasible generalised least squares (henceforth, FGLS) and the standard errors are bootstrapped. FGLS is preferable to OLS for two reasons, the greater the correlation between the residuals in each equation, the greater the effi ciency gain obtained by FGLS. Secondly, the less correlation there is between the X matrices, the greater the gain to FGLS. Thus, estimating SUR with all of the information within the system of equations makes it more effi cient than estimating each individual equation. Within each yield curve factor equation we adopt a general-to-specific selection criterion for each macroeconomic variable. Only those macroeconomic variables that remain significant will remain in each equation. Post-estimation we perform exclusion restrictions on the coeffi cients within each equation as well as testing each macroeconomic variable across each equation to determine if they are jointly significant. After initial estimations we found that the residuals werei(1) for both the nominal and real equations (not shown). So to deal with these I(1) residuals and any trending independent variables we include a deterministic time trend, rendering the residuals stationary. 2.3 Impulse Responses We calculate impulse responses to show how each macroeconomic variable alters the dynamics of the yield curve, this being the relative change in the shape of the yield curve following the announcement of the variable and by how many basis points the curve can be expected to shift. The effect of the macroeconomic variables on the latent variables is determined by the coeffi cients of the SUR estimations. Using equation 2 we substitute the coeffi cients from the SUR estimations in for the parameter values that fit the curve to produce the impulse response as shown below: 5

6 IR j = ˆθ j,l + ˆθ j,s ( 1 e τλ 1 τλ 1 ) ( + ˆθ 1 e τλ 1 ) ( j,c1 e τλ 1 + τλ ˆθ 1 e τλ 2 ) j,c2 e τλ 2. (8) 1 τλ 2 Where IR are the impulse responses and ˆθ j are the coeffi cients of any one of the macroeconomic variables j, where j = 1, 2,.., 31. The lambda values used for the impulse response are the same as those used to estimate the forward curve factors. One set back of these impulse responses is the weighting placed on the level. As it has the largest factor weighting, any possible level effect may dominate the impulse response (albeit, this may be appropriate for some variables) but given this, the overall dynamics of the impulse responses do not change. The impulse responses are presented along with 95% confidence intervals. The confidence intervals are calculated as 2 : I.R ± 1.96 S where S = ( ( ) ˆX X ˆΩ 1 X) 1 ˆX. The confidence interval of the impulse response is calculated from the estimated coeffi cients and standard deviation of each independent variable, ˆΩ. The components of the vector X are the four coeffi cients from each one of the forward curve factors with respect to each separate macroeconomic variable. 3 Data The choice of UK government liability forwards we use to define the nominal curve are 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108 and 120 maturities expressed in months giving a total of 15 different nominal forwards. To fit the real curve we use 10 different index-linked forwards with maturities of 48, 54, 60, 66, 72, 78, 84, 96, 108 and 120 months. We start at 9 months for the nominal and 48 months for the real as the shorter maturities have too many missing observations. All forward rates are zero-coupon forwards 3 and all observations are end-of-month as the majority of all macroeconomic announcements are made towards the latter end of month. The estimation sample spans from March 1993 to December 2008 giving a total of 190 observations. This sample period starts shortly after the Bank of England s use of inflation targeting but also covers the financial crises and the recession but ends two months before the start of the Bank of England s use of Quantitative Easing as a monetary policy tool. The purpose of this is to then forecast across the QE period to determine whether or not macroeconomic variables can provide a counterfactual nominal and real term structure that may highlight the effects of QE. The macroeconomic variables are divided into 5 groups, inflation, real activity, policy, financial and international. We test down from 31 different macroeconomic time series but present only the significant results. The full list of variables is shown in Table 1. The initial set of macroeconomic and financial variables are guided by Clarke and Mortimer-Lee (2008) who analyses the relationship between UK interest rates and key announcements of macroeconomic variables. For inflation we use 2 The degrees of freedom are the number of data observations minus the number of variables on the right-hand-side of equation 8, which gives a t-value of 1.96 at 95% confidence. See Draper and Smith. (1998), pages The methodology on how these forwards curves were constructed can be seen in Anderson and Sleath (1999) where they have been adjusted for coupon payments (so all forward rates here are zero coupon) and in the case of the index-linked bonds, the indexation lag. The estimates for the forwards are derived from a cubic spline based technique. 6

7 inflation expectations which are the average one-year-ahead inflation forecast from HM Treasury. 4 Real activity has two variables, a real activity index and unemployment. For policy we include the Bank rate and for a measure of fiscal policy we use debt-to-gdp expressed as a percentage. To measure money supply we use a series of notes and coins. 5 For foreign variables we include the effective exchange rate, German retail sales, the IFO index of business climate, U.S. Non- Farm Payrolls expressed as year-on-year changes and the Fed Funds Rate. The final group of macroeconomic variables are the financial group and for this we use an index of the annual returns of three different equity series and we use a measure of the Libor spread, which is calculated as the difference between the three month Libor rate and the monetary policy rate of the Bank of England. Normally the Libor spread is the difference between the 3 month Libor and Overnight Interest Rate Swaps but as the OIS data does not go back to 1993 we use the policy interest rate as a proxy. 3.1 Principal Component Analysis For the measures of real activity and financial returns we created two indexes using the first principal component of correlated series. The index of real activity is extracted from three different measures of real activity; UK production, UK retail sales (the volume of sales measure is used which removes the effects of price changes) and the claimant count which were previously transformed to year-onyear changes. The first principal component explains 69% of the variance of the three variables. The NIESR construct a monthly estimate of real GDP from monthly indicators such as industrial production and retail sales. The NIESR measure was included amongst the initial set of variables but was found to be statistically insignificant in each equations but our constructed index was found to be significant. Both share a correlation of 93% and the relative standard deviation of the index to the NIESR measure is The correlation between this variable and the three measures of real activity are 88% with production, -95% with the claimant count and 48% with that of retail sales. The index of financial returns is created using the annualised returns of three different equity series namely, the Standard and Poor s index of the 500, the DAX 30 and the FTSE 100. Just like the real activity index we take the first principal component of the three series. This first principal component explains 90% of the variance of the constituent variables. The correlation between the principal component and the individual series are 97% with the FTSE 100, 95% for the S & P 500 and 93% with the DAX Fitting the Forward Curve For the nominal curve there are 15 maturities, the vectors Y t and ε t are B is a 15 4 matrix and σ 2 I is a matrix. For the transition equation there are 30 parameters that need to be estimated. The 16 elements of the 4 4 transition matrix A, the four elements of µ and the 16 elements of the 4 4 variance-covariance matrix Ω. The 15 diagonal elements of σ 2 I also need to be estimated for the nominal forwards and 10 for the real forwards. So in total 45 parameters must be estimated for the nominal curve and 40 for the real. The values of lambda are 4 We refer the reader to Forecasts for the UK Economy: a comparison of independent forecasts for more information. 5 This is a measure of real money and we use the Bank of England s notes and coins in circulation (series code LPMAVAB) deflated by RPI. Notes and coins are used to represent narrow money where M0 was the Bank s main narrow money measure. But when the Bank introduced its Money Market Reform in May 2006, the Bank ceased publication of M0 and instead began publishing a series for reserve balances to accompany Notes and Coin in circulation. Notes and coins are the longest available measure of narrow money. 7

8 fixed prior to estimation. 6 For the nominal curve we use the values for λ 1 and 0.03 for λ 2 and for the real curve λ 1 and for λ 2. Table 2 presents the estimates of the A and Ω matrices and the vector of constants µ. Knowing that the transition equation follows an AR(1) process the estimates of A for the nominal curve show that the level and slope are non-stationary but the two curvatures are not. As with the Nelson-Siegel literature the diagonal elements of A show declining persistence as we move from the level through to the second curvature but the shock volatility in Ω increases. The estimates of the real factors do not show the same relationship as the nominal in A. The level and first curvature are non-stationary unlike the slope and curvature but there is increasing volatility through the Ω matrix. The Svensson methodology fits the data very well for the nominal forwards; Table 3 shows the mean and standard deviation of the fitting error in basis points at each maturity. The average fitting error is less than a basis point except at 9 months and the fit improves as the maturity increases. 7 The estimated real factors fit the real curve very well at the short-end but do less well from 108 months onwards with the 120 month forward being over fitted on average by 3 basis points. The estimated nominal factors and real factors are presented in Figures 1 and 2 respectively. The level factors for both the nominal and the real are very persistent. With the nominal level remains positive throughout and the real level is positive expect for January, August and November Over the period of Quantitative Easing the two curvature factors of the nominal curve display considerable volatility. For the real curve, the slope and the second curvature behave in a similar fashion to those of their nominal counterparts. The first curvature looks as if there is a structural break in the data given that the series declines when the government introduced the Pensions Act in 1995 which became effective as of April 1997 and was responsible for reducing long-term yields dramatically. 8 A simple Chow test confirms that there is no structural break in the data at either one or 5 percent significance so we feel that we are able to continue our analysis with these estimated factors. 4 Results The results presented below are by means of impulse responses that were outlined in Section 2.3. Rather than directly using the coeffi cients of the macroeconomic variables from the SUR estimations we use the total effects, ˆθ i /1 ˆρ. All impulse responses can be read as being increasing or decreasing in hundreds of basis points and all impulse responses are shown with 95% confidence intervals. Figures 3a and 3b present the impulse responses of the nominal curve. 3a presents the three fundamental variables for monetary policy plus the fiscal policy measure 3b presents the financial and foreign variables. Figure 4 presents the impulse responses of the macroeconomic variables that have an effect on the real curve. 6 We solve for the optimal values of β 1, β 2, β 3, β 4, λ 1 and λ 2 using constrained optimisation by minimising the residual sum of squares between the average UK term structure across the entire sample period and the fitted curve from the Svensson methodology. The imposed constraints are that λ 1 and λ 2 have to be greater than zero. 7 The authors find that the Svensson methodology has an improved fit over the Nelson and Siegel methodology, particularly at the longer term maturities but the differences are not statistically different at 5% or 1% for the nominal curve. For the real curve the difference in the fitting errors between the Nelson-Siegel function form and Svensson are statistically significant so we continue the analysis for both curves with the functional form of Svensson. 8 This is the Minimum Funding Requirement of the Pension Act. This requirement was introduced by the government to ensure the solvency of UK pension funds which led to an increase in demand for long-dated nominal and index-linked bonds by institutional investors. 8

9 4.1 Nominal Curve Responses Inflation Expectations A 1% shock to inflation expectations increases the nominal curve by about 100 basis points at the short-end and 140 basis points at the long-end. The results of Gürkaynak et al (2005) suggest that post-independence of the Bank of England that announcements of various inflation measures have no impact on the curve which would suggest that the Bank of England is credible whereas our explicit measure of inflation expectations finds a very significant and increasing effect through the nominal term structure. The results of Diebold et al (2006) find the level increases as a result of an increase in inflation expectations but the results of the slope are different. They find that the short end of the term structure increases, which is consistent with the actions of monetary policy makers. They use spot rates so this then accounts for the difference in the results as noted in Gürkaynak et al (2005); the behaviour of forwards and spot yields can vary to the same macroeconomic shock. This could then further explain why there is a difference in the results between those found here and also those of Ang and Piazzesi (2003) and Ang et al (2009) who find a hump in the medium term yields as a result of a positive inflation shock which simultaneously leads to a negative term spread as short rates are pushed up higher than long-term rates Real Activity With a 1% shock to real activity we find that the long end of the curve increases over the short end, this is the result of an increase in the second curvature factors. A shock to real activity increases the long end of the curve by about 50 basis points but it is on the medium term yields that real activity has the largest impact, with the 60 month forward increasing up to 70 basis points. Diebold et al (2006) find that a shock to real activity that the yield curve behaves in a similar fashion to that of inflation with their spot rates. Both the level and the slope increases, particularly at the short end which is again consistent with a rise in the policy rate to try and reduce the growth rate of the economy. The results of Ang and Piazzesi (2003) and Ang et al (2009) find that the yield curve dynamics are similar for a positive output shock as they found for an inflation shock. Our results show that interest rates move more in response to inflation expectations rather changes in real activity as suggested by a Taylor rule. As with the other literature we examine the correlation between the estimated level and slope and the measures of inflation and real activity. We find that the inflation and the nominal level has a correlation of 0.49 suggesting that there is a link between inflation expectations and the level. The real activity index and the slope have a correlation of 0.65 which suggests that the slope factor is indeed a business cycle factor as suggested by the other literature and long-term forwards rise when real activity increases Bank of England Policy Rate A 100 basis point increase in the policy rate leads to a fall in the forward rates. The effect of a policy rate increase reduces the level and the first curvature and increases the slope but the curvature and level effect dominates leading to a fall in the impulse response. A similar result was found by Gürkaynak et al (2005), they find that their ten year forward falls with an increase in the policy rate. One suggestion put forward is that an increase in the policy rate of a credible policy maker will cause inflation expectations to fall. Responses to by the long end of the yield curve to macro variables are consistent with more persistent representations of the monetary transmission mechanism, see for example Chadha and 9

10 Holly (2010) Debt-to-GDP A 1% increase in debt relative to GDP is likely to increase the 10 year maturity by 9 basis points. 10 There is very little change in short-term rates but the supply effect becomes larger as we move along the curve. This may capture the differences in the relative liquidity of the sections of the yield curve, as the short-end of the curve is more liquid than the long-end Notes and Coins A 1 billion increase in notes and coins suggests that there is a fall in all yields with the greatest effect being on the short-term yields of about 5 basis points. This may be a slight overestimate in the elasticity of the effect in notes and coins but we would expect these dynamics to be correct as private agents substitute their excess balances of money for nominal bonds Effective Exchange Rate The effects of an appreciation of the effective exchange rate reduce the level of the term structure causing long-term rates to fall relative to short-term rates. This is consistent with a slowdown in real activity and an easing of inflationary pressures. It appears that a one point increase in the effective exchange rate reduces the nominal 120 month forward by about 20 basis points and with very little impact on the short-term, so any changes in future interest rates are quite slow to an appreciation of the exchange rate IFO and German Retail Sales For the IFO business climate index for Germany, we estimate an increase in the short-term maturities as the slope effect comes to dominate the impulse responses. The short-term interest rates rise relative to the long-term interest rates. A one percent increase in German Retail Sales has a similar result to that of the IFO index, with the slope effect being the only significant result. At the 120 month forward the results for both variables are similar but at the short-end the impact varies with the forward curve being influenced more by the impact of retail sales than business manager s sentiment on the prospects of the economy. These results suggest then that with an increase in foreign real activity and foreign interest rates rising one would expect a capital to flow out of the UK, leading to a sterling depreciation and an increase in domestic interest rates. This demonstrates that foreign factors can have an impact on the UK term structure through an exchange rate transmission mechanism. This result is somewhat corroborated by Ehrman et al (2011) find that an increase in the IFO index lead to an increase in yields of Euro area yields. 9 The authors examine the impact of the three important macroeconomic variables for monetary policy on the yield curve using a DSGE model with three different sticky-price models. The key difference between the models is the timing conventions that output and inflation follow. One model being purely forward looking, the second a hybrid model that has both leads and lags and the third a purely backward looking model. Their results show that the second moments of the backward model are closest to the second moments of the data and the greater the persistence of long-term interest rates after a macroeconomic shock. 10 Using the total net debt for the UK over the sample period this would suggest an average issuance of approximately 10bn of bonds at a time. 10

11 4.1.8 Libor Spread The effects of an increase in the Libor spread by 100 basis points only impacts on the slope. There is a dominant effect on the short-end of the curve but there is pass through to longer-term maturities with the long-end increasing 17 basis points but the short-end increases by 112 basis points. The Libor spread is often used as a measure of health of the banking sector because an increase in the Libor rate is destabilising for the economy as lending to other banks and business becomes more expensive. Generally a decrease in the policy rate will do little to stem banks lending to other banks and businesses. However these results are very similar to that of Dewachter and Iania (2010). They call a shock to their Libor spread a credit crunch shock. They find that the level increases as a result of this shock, the slope increase, leading to a flattening of the yield curve, and an increase in the short-end of the curve reduces the curvature. They find that the credit crunch factors appear to have a stagflationary effect on the economy. 4.2 Real Curve Responses Inflation Expectations For the real curve a shock to inflation increases the curve at medium-term yields by approximately 130 basis points but longer term yields show very little movement suggesting that there may be some anchoring of inflation expectations. The difference between the nominal and the real curve at 120 months is 124 basis points, so at this maturity inflation expectations and break-even rates increase by a similar amount with some possible inflation premium being attributed to the break-even rate Debt-to-GDP A one percent increase in the debt-to-gdp ratio causes an increase in all yields by approximately 9 basis points. This is an appealing result for the following reason; the nominal bond market is deeper and more liquid in every segment of the yield curve and so should carry less liquidity premia. A lot of the available yields below 5 years are of remaining maturity as opposed to direct issuance and so the larger supply effect for the real curve at maturities less than 120 months is readily explained but as the real market becomes more liquid, especially around maturities of 120 months the supply effect is almost identical Notes and Coins With an increase in real money of 1bn we find that the real term structure flattens and there is an increase in yields. This suggests that real money balances and real bonds are not substitutes as one would find with nominal bonds. Although the rate of return of real bonds are inflation protected it maybe that the opportunity cost is not large enough relative to nominal bonds to encourage people to switch out of money and into real bonds and this maybe also the cause to why the fall in nominal bonds is larger in absolute terms when compared to the negligible rise in real yields Libor We find that a 100 basis point Libor shock has very a very similar affect to that of the nominal curve. It is only the slope factor that is affected which leads to a flattening of the term structure. Also, the size of the responses between the nominal and real curves is almost identical at the same maturities. We find then, that if there was a credit shock, it does not suggest that there is any stagflationary pressure, in fact at 120 months the real term structure is higher than that of the nominal by a few basis points suggesting a fall in inflation over longer horizons. 11

12 4.3 Within Sample Fit and Residuals Post estimation we perform two types of exclusion restrictions on the independent variables with the results presented in Table 4. The first test is shown in the top section of the table and tests whether or not all macroeconomic variables in each equation are significant. As the result show, the macroeconomic variables in each equation are jointly significant. The second exclusion restriction is shown in the lower section of the table sand this tests whether or not each macroeconomic variable is significant across equations. The results show that each macroeconomic variable are jointly significant across the different equations in which they appear. Together these results imply that macroeconomic variables can explain changes in forward rates and alter the dynamics of the term structure of interest rates. The descriptive statistics of the estimations are shown in Table 5. The R 2 is high across all four of the nominal equations, particularly the level and slope equations. As found in the other literature, we struggle to try and find some better macroeconomic explanations for the curvature factors. These results are somewhat reinforced with the root-mean-squared-errors, with the results being smaller for the very well explained level and slope. For the real equations the R 2 are not as high as those as the nominal except for the first curvature which has an R 2 value of The second curvature is poorly explained by the macroeconomic variables. The RMSE of the real curve is larger than that of the nominal which highlights that the real curve is harder to predict. The tests for autocorrelation, heteroscedasticity and normality show that the residuals do not suffer with autocorrelation. Heteroscedasticity is a concern for the first curvature factor of the real curve. The only problem that is apparent is normality, particularly for the nominal level and the second curvature but also for the first curvature factor of the real curve. The estimated residuals have been shown to be stationary around a trend, rendering the forward curve factors stationary. 11 Dummy variables were used to take account of some of the larger residuals. For the nominal curve only one dummy variable was used and this was for the November of For the real curve there were three dummy variables used: October, November and December of Assessment of Quantitative Easing by Out-of-Sampling Forecasting On March 5 th 2009 the Bank of England s Monetary Policy Committee (MPC) announced, along with its interest rate cut that it would partake in large scale asset purchases whereby the Bank would create central bank reserves to purchase financial assets such as gilts and corporate bonds. Over the next ten months the MPC held its offi cial interest rate at 0.5% (the effective zero lower bound) and purchased almost 200bn pounds worth of nominal government liabilities. 12 Only nominal paper was purchased because the nominal market is much larger than the inflation index-linked market 13 as there were fears that large asset purchases might distort segments of the real yield curve because issuance at these maturities are often less than 4 billion. 14 The actions undertaken by the MPC were initially intended to lower the yields of gilts with a remaining maturity greater than 5 years but lower than 25 years but from the August that same year the maturities purchased were extended to any remaining maturity greater than 3 years. This provides a natural testing ground for our model. We hope to find that the nominal forward curve overestimates the actual 11 Table available upon request. 12 By the end of QE1 the Bank of England had also purchased billion of sterling corporate bonds. 13 In 2009 the amount of nominal gilts in issue was approximately 4 times larger than the amount of inflation-indexed gilts in issue. See the UK Debt Management Offi ce website for more details: 14 On the 5 th March 2009 the Asset Purchase Facility issued a notice to the market stating which gilts were eligible for purchase. This note can be found at: 12

13 curve and that there is no significant difference between the forecasted real curve and the actual real curve. Using a dynamic forecast we project the estimated model of the previous section over the period of QE to produce a counterfactual forward curve that can aid in the identification of the total impact of QE on the nominal and real curve. 15,16 Table 6 gives the point estimates of the forecast error (in basis points) for the nominal forward curves for the macro-finance yield curve when it is projected out-of-sample from January 2009 to January The first column in Table 6 reports the one-step ahead standard error. Figure 5 shows the 1, 5 and 10 year fitted forward rates and forecast errors. In general we find that the forward curve over predicts the actual curve from March onwards from 21 and 24 month forwards. The over prediction averages approximately 67 basis points at the 60 month forward and 46 basis points for the 120 month forward. This result is somewhat consistent with the implementation of QE with regards to its timing. That maturities greater than 2 years are affected by QE and the over prediction begins in the March with the start of purchases. Prior to QE the model under predicts the curve on average at 60 months by approximately 45 basis points and 80 basis points at 120 months. There may have been solvency issues regarding the UK government over this time that may increase yields and our fiscal policy variable is unable to capture the markets forward looking concerns. Given this, it is reasonable to ascribe the total change in yields from the pre-qe period to the QE period to the QE event, thus the overall change in model residuals around the QE event from March 2009 to January 2010 relative to the January and February of 2009 is a reduction in the 120 month nominal forwards of some bp. This results fall in line with the event study results of Caglar et al (2011) and Meaning and Zhu (2012). The forecast of the 60 and 120 month fitted and forecast errors of the real curve are shown in Figure 6 and the forecast errors and standard errors are shown in Table 7. Figure 6 shows that the actual forwards move in a very narrow range whereas the forecasted curve is somewhat more volatile. At 60 months, the average error forecast error is an over prediction of 6 basis points and given that the average forecast error from the March to January 2010 is only 3 basis points, the impact on the real curve is negligible at best. For the 120 month forward the average forecast error is smaller than that of the 60 months and the average error over the QE purchases is an overestimate of 19 basis points. For any maturity there is no persistent deviation away from the actual path of real bonds which is a result that is consistent with the Bank of England not purchasing any index-linked bonds as part of their QE program. 5.1 Forecast Error Decomposition The forecasts of the nominal and real forward curves offer intuitive results because the deviation between the actual and the forecasted nominal curve and the timing of the overestimate suggests that QE has reduced actual yields. Secondly, the forecast of the real curve shows no persistent deviation from actual rates which are consistent with there being no purchases of index-linked debt. However it would not be appropriate to attribute all of the difference between the actual and forecasted curve to QE. One reason is that both the real and nominal forecasts in Figures 5 and 6 are within one standard deviation of the actual path of interest rates and over the same period there were a multitude of events that would impact on the term structure that cannot be captured by our model but we are able to provide some intuition for. To explain how QE 15 With this methodology we are unable to test whether or not QE had an effect on macroeconomic variables. But if QE leads to higher output, that would raise both the actual and the fitted curve and so not have any impact on the residual. 16 Our model is able to forecast the path of interest rates at each maturity quite accurately over other sample periods (not shown). We feel this allows us to perform the forecasting exercise over QE. Any deviation from the estimated path of interest rates then can be attributed to QE. 13

14 may have impacted on the forward curve we provide a decomposition of the 60 and 120 month forecast errors over the QE period based on some rather strong assumptions about QE and the channels that it was expected to have worked its way through the transmission mechanism. For the UK s experience, Dale (2011) outlines three channels in which QE was intended to have worked through the transmission mechanism and impacted on the yield curve: portfolio balance, liquidity and signalling. All of these channels would have an immediate impact on the yield curve through the daily operations and announcement effects, thus allowing us to show the immediate financial market impact of QE on the nominal term structure. A related analysis is undertaken by Breedon et al (2012) who examine the impact of debt purchases on both monthly and daily bond yields over the same period as well as duration effects of the changing nominal bond portfolio brought about by the large scale asset purchases Debt-to-GDP Effect The variable debt-to-gdp captures a pure supply effect on the forward curve and not any other factors such as perceived credit risk and that the purchases of gilts undertaken by the APF for QE are the same as a reduction of the debt-to-gdp ratio as the purchases made reduces the amount of available supply to the private sector. This implies that the overall impact of QE reduces the debt-to-gdp ratio. We calculate the total number of bonds purchased each month at the reverse auctions and determine the equivalent size of debt-to-gdp, for example in the March the Bank of England purchased 1.1% of the total supply of bonds relative to GDP and by the January 2010 they had purchased approximately 14% of the total supply of bonds relative to GDP. Treating the purchases as a stock effect, we take the cumulative amount of purchases made from one month to the next. Given this we can estimate the impact of QE on bond yields at five and 10 years using the impulse response in Figure 3a Liquidity Channel The second channel in which QE may have worked through is the liquidity channel. The asset purchases made by the central bank should ease any liquidity issues that may have arisen from the financial crises and the recession. The Bank of England s behaviour as a large purchaser of gilts should improve the functioning of the gilt market and reduce the liquidity premium that is embedded within the term premia. The size of the liquidity premium will vary from one maturity to another depending on how deep and liquid the market is at that particular maturity, in this case bonds that have a remaining maturity of five and ten years. This premium is calculated 17 and the liquidity effect is subtracted from forecast error. The liquidity premium is there as a guide only to suggest which way the liquidity premium (if any exists) as the liquidity premium is determined by spot rates as opposed to the forward rates that we use but the affects would pass through to forwards rates Signalling Channel To measure the signalling channel we use a risk adjusted measure of a market interest rate. Statistically the chosen interest rate should be able to predict the policy rate but the presence of term 17 One way to measure the liquidity premium is to look at the spread between an off-the-run bond (a bond in the secondary market that is at 10 years to maturity) and the on-the-run bond (a bond that is issued into the primary market). Any difference between the two would represent a liquidity premium. In this case we subtract the average weighted accepted yield at each auction for 5 and 10 year nominal bond purchases and subtract this from the closing price of the same day. We sum the results across each month and as this is a flow variable and so we do not take the cumulative impact. We would like to thank the UK DMO for providing closing prices for five and ten year nominal bonds. 14

15 premia and differences in the maturity, liquidity and credit quality may blur the signal for the future path of the Bank rate. The policy rate can be forecasted using a simple regression of the form: i t,t+1 = α + β(f t,t+1 i t ) + ε t,t+1. (9) Where i t,t+1 is the realised Bank rate one year ahead, i t is Bank rate observed today, f t,t+1 is the forward rate observed today for one year ahead and ε t,t+1 is an error term. The parameters α and β should be equal to 0 and 1 respectively if the pure expectations hypothesis were to hold. A more relaxed version of the expectations hypothesis 18 would expected α 0 which implies there is some constant term premium and if β is statistically different to 1 suggests that there is some time-varying term premium. Gürkaynak et al (2007) and Piazzesi and Swanson (2008) use similar equations and analysis to examine market-based measures of the expected path of the Fed rate in the U.S. The data we use is the Bank of England s own commercial bank liability curve which is based on Libor interest rates and other market rates of various derivative instruments linked to the Libor such as forward rate agreements, short-term interest rate futures and Libor swaps. 19 This measure is based on an unsecured lending and so the interest rate at one year should include default risk and as Libor is a money market rate some liquidity risk will also be present. We look at the expected path of interest rates for one year only and we use end of month data over the same sample period as that of Sections 3 and 4. We correct the standard errors for heteroscedasticity and autocorrelation using Newey-West procedure with 5 lags. The R 2 over this sample period 0.37 suggesting at a year out that our chosen measure can explain some of the variation in Bank rate. The coeffi cient α has a value of which suggests that the forward rates in general overestimate the actual outcome by 85 basis points, this result is consistent with a positive forward premium. The coeffi cient on β of 0.96 is very close to one and with a general restriction test with a null hypothesis that β = 1, we do not reject the null hypothesis which implies that there is no time-varying term premium at this maturity for this particular instrument. What we call the signalling effect here is the difference between the policy rate and the markets expected path of interest rates. Taking the forward rate for one year ahead of the bank liability curve we subtract the estimated constant term premium (which is alpha in equation 9) as well as the policy rate at each month. This provides a measure of where the market expects the policy rate is supposed to be relative to today. The signalling effect is expected to occur within the month and should be different from one month to the next and we assume it is the same impact on both the 5 and 10 year forwards A Trial Decomposition The forecast error can then be explained by three channels of QE. The portfolio balance channel where the imperfect substitutability of different assets can allow the relative supplies determine the price of the asset, this is the debt-to-gdp effect, the liquidity channel is explained by the measure of liquidity premium and the signalling channel by forwards. This channel capture the markets expectations of the future path of the Bank rate or any other macroeconomic information that may determine the path of interest rates that is not already explained within the model and captured by the forecasts. The portfolio and liquidity channel are expected to put downward pressure on the yield curve but the signalling effect may be ambiguous because while the Bank of England undertake QE the market may believe that the Bank will hold interest rates at their current level or maybe reduce them further. But if there are inflationary pressures the market may expect the 18 Which still requires well behaved residual. We remove a unit root by taking the policy rate away from both sides. 19 For more information we direct the reader to the following note by the Bank of England: 15

16 Bank to pursue lower and stable inflation and change interest rates if the market is to believe what they observe to be the monetary authorities reaction function. Tables 8 and 9 show the decomposition of the residual at both five and ten years respectively from the March of 2009 from when QE was implemented. The debt-to-gdp effect becomes larger as the amount of gilt purchases increases and can explain as much as 98 to 126 basis points at 5 and 10 years respectively. For almost all of the gilt purchases at five years there is a fall in the liquidity premium. Suggesting that non-bank financial institutions were willing to substitute their 5 year bonds for cash and that the Bank of England were willing to pay a premium to ensure this was so. For ten year government paper the liquidity premium is positive (except for the March and April) although the reverse auctions were well covered. 20 Although market participants were willing to sell ten year bonds at a premium, the Bank of England s Asset Purchase Facility was not willing to pay a premium like they were for the five year bonds, which may simply be because the medium-term segment of the curve was the focus of the purchase strategy. 21 Whereas the debt-to-gdp effect becomes larger and exerts more downward pressure on the yield curve as time progresses and the amount of purchases increases, it is the signalling channel that has the largest impact at the start of QE and has less of an impact as the amount of bond purchases increases. In the March when QE was announced the MPC made the decision to cut the policy rate to 0.5% and our model already takes account of this change in the out-of-sample forecast. Initially, the signalling channel has a large impact on the curve with the market expecting interest rates to fall a further 48 basis points below the new policy rate of 0.5% suggesting that market participants were pricing the policy rate to be closer to zero. The largest impact of the signalling channel occurs in the March when QE was first announced but in the following months when the Bank announces further bond purchases the market expected interest rates to be lower than the current level of the base rate (except for August). The smaller size of these impacts suggests that market participants had already been expecting further rounds of purchases. Over this period inflation expectations had begun to rise from 1.9% in the March to 2.4% in December. The market may then feel that if inflation follows its trend it would be close to 3% in a years time which the market may then expect some response from the Bank of England to contain the rise in inflation. But with the announcement of further asset purchases the market can expect the Bank to hold interest rates steady for the foreseeable future. Clearly the signalling impact has had a large initial impact on the curve which explains some of the downward pressure on actual yields but in the later stages of QE it was the debt effect that became dominant Credit Effects The trial decomposition suggests that QE had a large impact on the forward curve but there remains a persistent residual to be explained. In effect the three channels more than account for the forecast residual, which suggests that another factor may have been driving up nominal yields. The remaining residual might well reflect some change in the perceived credit worthiness of UK bonds or some time variation in estimated impact of the other channels, in particular that of our debt-to-gdp variable that captures supply effects. Any default risk premium would be a source of upward pressure on the curve and so experimented with CDS spreads and the spread of Libor 20 Individual auction data is available for each bond at each one of its auctions from the QE data set from the Bank of England conference Learning the lessons from QE and other unconventional monetary policies. The data set is available at 21 The short-term is 3 to 7 year maturities and medium-term maturities are from 7 to 15 years. Data regarding the breakdown of the UK government liability portfolio is available from the UK Debt Management Offi ce and their website page 16

17 rate forwards over nominal government forwards at the same maturity. We find that the Liborforward spread has a correlation with the remaining residuals at 5 and 10 years of and respectively. The last two columns of Tables 8 and 9 shows the remaining forecast residual yet to be explained and the Libor-forward spread and we present the spread at both five and ten years in Figure 7. We observe that the Libor-forward spread at ten years is negative implying that the perceived likelihood of default by the government has increased over this period. Furthermore, it appears that the impact is greatest on long-term rates but due to data limitations we cannot estimate the impact the spread has on the yield curve so we are unable to make specific assertions about this channel but we want to develop a fully specified model of default risk to assess this in future work. 6 Conclusion We study a variety of macroeconomic and financial announcements and how they impact on the nominal and real term structure of UK government liabilities. The results are consistent with recent work undertaken by others in a joint macroeconomic and financial economic approach to understand the dynamics of the forward curve but our methodology is able to accommodate more a larger number of variables than the present literature. The impulse responses give results similar to the other macro-finance literature for the three key macroeconomic variables, policy rate, inflation and real activity but we identify nine key macroeconomic variables that impact on the nominal yield curve, all of which provide intuitive results such as inflation expectations leading to an increase in yields and a positive supply effect in the gilt market. Further to this, we find that key economic indicators from Germany impact on the UK curve but announcements from the U.S do not. For the real curve we find that there are only four variables that shift the real curve. Overall we find that the nominal curve is well explained by macroeconomic variables but the real curve does less well suggesting that something other than the state of the economy may determine the dynamics of the real yield curve such as government policy. We perform an out-of-sample forecasting exercise over the first round of QE in the UK (March 2009 to January 2010) to determine the future path of the forward rates conditional on our macroeconomic variables. We found that the nominal curve over-estimated the by 67 and 46 basis points for the five and ten year forwards. For the real curve there is no persistent over or under prediction of the curve which is consistent with both the timing and implementation of QE. These results are similar to those found by Caglar et al (2011) and Meaning and Zhu (2012) who examine the immediate financial market impact of QE in the UK. We find that the forecast error can be readily explained by the portfolio balance, liquidity premium and signalling channels. When the Bank of England purchased gilts they were willing to pay a premium to purchase 5 year bonds from the market but not for the 10 year bonds suggesting that the Bank was willing to pay a higher price for bonds in relatively less liquid segments of the curve. The liquidity channel alone can explain up to as much as 30 basis points in any one month but it was the portfolio balance channel that had the greatest impact on the nominal yield curve which can account for as much as 125 basis points in January From when QE was implemented it was the signalling channel that exerted downward pressure on the forward curve as the market expected the policy rate to be cut further or at least to be hold at the zero lower bound for at least in the foreseeable future with this channel explaining around 50 basis points of the decomposition. Our work has to be extended to consider credit risk effects on bonds and also the possibility of time variation but the identification of a supply effect on bond yields at this frequency allows us to have some confidence that the first round of QE did have some impact on lowering bond yields. 17

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20 Tables Table 1. Initial Macroeconomic and Financial Variables. Inflation Real Activity Policy Foreign Financial Inf. Expectations Real Activity* BoE UK Eff. ER Returns* ( RPI Unemployment M ) P IFO Index Libor CPI NIESR GDP PSNCR German R.S Nationwide House Prices RPIX Terms of Trade Debt-to-GDP U.S. NFP Oil RPIY Trade Balance FFR Gold PPI ECB Policy Vix Vol. Spot E.R: Note: Those variables marked with an asterisk are calculated as principal components. For further details see Subsection 3.1. $/ / / Table 2. Coeffi cients of the A, Ω and µ Matrices Nominal Parameter Estimates Real Parameter Estimates Estimated A Matrix Estimated A Matrix L t 1 S t 1 C 1,t 1 C 2,t 1 L t 1 S t 1 C 1,t 1 C 2,t 1 L t L t S t S t C 1,t C 1,t C 2,t C 2,t Estimated Ω Matrix Estimated Ω Matrix L t S t C 1,t C 2,t L t S t C 1,t C 2,t L t L t S t S t C 1,t C 1,t C 2,t C 2,t Estimated µ Matrix Estimated µ Matrix L t L t S t S t 1.00 C 1,t C 1,t C 2,t C 2,t Note: These estimates correspond to the transition equation of Equation 5, which explains the dynamics of the estimated latent factors. The estimated standard deviations of Equation 6 are not shown. 20