Abstract. particularly at intermediate and long-term horizons. JEL classications: C13; C32; E43; E44; E58.

Transcription

1 Forecasting U.K. yield curve dynamics using macroeconomic and asset pricing factors: A just-identied no-arbitrage approach Alena Audzeyeva a,, Robin C. Bladen-Hovell a, Suranjan S. Jayathilaka b a Keele Management School, Keele University, UK b Department of Management and Economics, University of Sheeld International College, UK Abstract We propose a no-arbitrage ane Gaussian model that links term structure dynamics to the evolution of asset pricing measures along with traditional macroeconomic and latent factors. In contrast to FAVAR-type models, our model exploits factor-specic rich data. Applying the methodology in Hamilton and Wu (2012) avoids endemic identication and estimation issues. We nd that medium and long-term yields respond more to shocks in asset pricing than real activity. During periods of heightened economic uncertainty, models employing concept-specic factors produce forecasts superior to those using generic FAVAR-type factors; asset pricing measures contribute to forecast accuracy particularly at intermediate and long-term horizons. JEL classications: C13; C32; E43; E44; E58. Keywords: Yield curve; Ane Gaussian term structure model; Macroeconomic factors; No-arbitrage restrictions; Zero lower bound; Forecasting Corresponding author. address: a.audzeyeva@keele.ac.uk 1

2 1 Introduction A number of recent studies put forward no-arbitrage ane models describing the joint dynamics of the term structure of interest rates and the macroeconomy; examples are Ang and Piazzesi (2003), Hördahl et al. (2006), Dewachter and Lyrio (2006), and Moench (2008). They all nd that over and above latent factors, observed macroeconomic variables account for a large share of the variation in the term structure and help to better forecast the underlying set of interest rates. 1 Another important advantage of joint modeling of the term structure and the macroeconomy over term structure specications using latent factors only, proposed by Nelson and Siegel (1987), Litterman and Scheinkman (1991), Pearson and Sun (1994), Dai and Singleton (2000) inter alia, is that they directly model how interest rates respond to macroeconomic variables. Furthermore, the class of noarbitrage models are particularly attractive as they impose theoretically important cross-sectional restrictions ruling out arbitrage opportunities thereby making these models especially useful to economic policy makers and investors. However, the information used to capture the state of the economy in many widely-used dynamic term structure models is typically limited to macroeconomic indicators measuring past real and nominal economic activity. Furthermore, some macroeconomic aggregates, for example, GDP growth, only become available with long delays. Consequently, forward-looking economic information does not enter into the model which may result in inferior securities pricing and interest rate forecasts. The second shortcoming of many extant models is that they exploit only a small number of economic variables for term structure modeling; see, for example, Ang and Piazzesi (2003), Diebold et al. (2006), Hördahl et al. (2006), and Kaminska (2008). The implicit assumption here is that theoretically important economic drivers can be directly observed via just a few macroeconomic aggregates. But this contradicts an observation in Bernanke and Boivin (2003) and Bernanke et al. (2005) that nancial institutions operate in a "data rich environment", so that a single, as in many models, or even a few macroeconomic aggregates may not give an accurate representation of key macroeconomic concepts. Consequently, preferred models may omit important information, resulting in biased estimates. An exception is Moench (2008) who proposes a data-rich approach to term 1 In a more recent study, Koopman and van der Wel (2013) nd that, even without imposing no-arbitrage restrictions, augmenting the ane term structure model employing yield factors with macroeconomic factors improves the accuracy of yield forecasts for most maturities and forecasting horizons. 2

3 structure modeling using a factor-augmented autoregression (FAVAR) specication put forward by Bernanke et al. (2005). However, an issue with the Moench (2008) approach is that economic factors are extracted from a broad pool of variables, not permitting interest rate evolution to relate directly to specic economic drivers and, thus, limiting model inference and practical application. This study aims to address these issues by proposing a formulation of a no-arbitrage vector autoregression (VAR) model of interest rates that relates term structure dynamics to a market asset pricing factor in addition to typically studied macroeconomic measures such as real activity and ination and latent variables. The idea behind the introduction of the asset pricing factor is that asset prices contain investors' expectations of future economic performance. Importantly, unlike measures of realized economic activity, they reect evaluations of future losses and other costs associated with recessions. This property of asset prices is captured in the Tallarini (2000) standard dynamic stochastic general equilibrium model and the Diercks (2015) New Keynesian model. Tallarini (2000) shows that the cost of business cycles reected in the equity prices is high. A higher equity premium is associated with increased risk aversion that drives up market prices of risk. Diercks (2015) further points out that asset pricing matters for monetary policy analysis. This is because in the presence of long-run uncertainty in economic growth, rms characterized by recursive risk preferences strongly dislike recessions. Consequently, the monetary authorities may be willing to accommodate substantially higher prices chosen by forward-looking rms in the face of negative long-run news shocks to productivity despite this driving up ination because higher ination reduces the average mark-up and stimulates output and employment. Asset pricing measures in this respect may add a new forward-looking dimension associated with the costs of recessions, not reected in typical macroeconomic indicators. In addressing the second issue, our VAR formulation explicitly relates the term structure dynamics to specic economic concepts in a data-rich modeling environment. In our modeling approach economic factors summarize information from a large set of economic variables separately under each heading, so that the impact of each broad factor can be directly linked to shocks to the real side of the economy, the nominal side of the economy, or market asset valuation. This contrasts with Moench (2008) who follows the FAVAR literature by employing a broad pool of variables for factor construction which requires a researcher to make a post-hoc factor interpretation. Mumtaz (2010) is the only exception that, similar to our study, employs rich economic concept-specic fac- 3

4 tors in a FAVAR setting. However, the purpose of the analysis in Mumtaz (2010) is dierent from ours as it focuses on the mutual dynamics of the U.K. economy and a single policy rate instrument and disregards the interest rates forecasting aspect altogether. In addition, we relax a common assumption, for example, in Ang and Piazzesi (2003) and Kaminska (2008), that economic factors are independent, thus allowing for more realistic variable dependencies. We refer to our term structure model using data-rich macroeconomic and asset pricing factors as MAP-FVAR. Our methodology for model identication and estimation follows Hamilton and Wu (2012) in addressing an important critique that these authors raised with regards to a number of extant ane Gaussian term structure models using reduced-form representations; examples of such models are Dai and Singleton (2000), Ang and Piazzesi (2003) and Pericoli and Taboga (2008). Hamilton and Wu (2012) show that these models, that have been a work-horse of the current term structure literature, are in fact unidentied. The underidencation issue arises when multiple values of the structural parameter vector are associated with the same reduced-form parameter vector in the term structure model that is assumed to price exactly a sub-set of linear combinations of observed yields through a restricted vector autoregression in the observed yields and macroeconomic variables. This issue cannot be resolved using data. We specify our model as just identied to address this issue. Furthermore, parameter estimation in our model, which is based on no-arbirtage restrictions, uses the minimum-chi-square method advocated for ane Gaussian term structure models by Hamilton and Wu (2012). They show that the minimum-chi-square method improves upon the typically used maximum-likelihood (ML) based and other non-linear numerical search methods that suer from a tendency of arriving to sub-optimal parameter estimates. This is because models of joint term structure and macroeconomic dynamics represent a highly non-linear system that depends on a large number of parameters which are non-linearly related to yields and in which high persistence in yields leads to at likelihood surfaces. Consequently, the routinely used ML based numerical search methods struggle to converge and, if they do, are likely to converge to a local rather than global maximum; other studies documenting this issue for ML methods are Christensen et al. (2011) and Kim and Orphanides (2012) inter alia. Important advantages of the minimum-chi-square method over traditionally used ML estimation methods are (1) considerably simpler numerical computations associated with nding structural model parameters and (2) the modeler's ability to distinguish with certainty between global and local maximum of the likelihood surface. Consequently, from a 4

5 methodological standpoint, our model identication and estimation represent an improvement in the robustness of parameter estimates relative to the previous literature on no-arbitrage ane Gaussian term structure models. The empirical part of the study analyzes the impact of various state factors on the yield curve. Further, in our forecasting exercise we provide new evidence on the comparative forecasting accuracy out-of-sample for the proposed model vis-á-vis a number of alternative widely used approaches for incorporating the economic factor dynamics into no-arbitrage ane term structure models. A focus on the out-of-sample (or real time) predictive ability is of practical importance as a good in-sample model t may not be indicative of useful out-of-sample predictive ability. This is because typical model estimation approaches using the full sample data, by construction, avoid large in-sample prediction errors and, therefore, are prone to over-tting. For consistency, we employ the minimum-chi-square method to estimate the just-identied representations of the competitor modeling approaches. Our analysis provides new empirical evidence for the joint term structure of interest rates and macroeconomic modeling in two important aspects. First, we compare the forecast accuracy of alternative ane Gaussian term structure specications in the U.K. context. Most existing empirical evidence for the term structure forecasting performance is U.S.-specic. Among the scant literature analyzing out-of-sample forecasts of the entire yield curve, Ang and Piazzesi (2003) and Moench (2008) focus on the U.S. economy and Hördahl et al. (2006) is German economy-based. Since evidence from these analyses is related to periods prior to the economic crisis, our second empirical contribution subsequently extends the empirical evidence to two recent episodes of change in economic environment, not previously analysed in this context. To this end, our forecasting analysis covers three separate periods, namely, the period of "great stability" in the U.K., the recent economic crisis and a post-crisis period, characterized by near-zero interest rates that dierentiate it from earlier periods of stability. 2 The respective periods are January 2002 to December 2006, January 2007 to December 2010, and January 2011 to March The analysis reveals that yields of all maturities respond positively to shocks in asset pricing. 2 Steeley (2014) is the only known study that examines changes in the UK term structure dynamics during the economic crisis and the subsequent period of near zero interest rates. He establishes that the change in the UK interest rate regime to a near-zero level in 2008 has considerable implications for the modeling and forecasting of the UK yield curve. However, the latent factor term structure models used in Steeley (2014) do not permit directly relating the change in the term structure dynamics to underlying economic factors. 5

6 Furthermore, both macroeconomic and asset pricing factors are important determinants of time variation in yields but their relative impact is term-specic. The real activity factor is the strongest single economic determinant of the variation in short-term yields but its inuence virtually disappears for medium and long-term yields. Ination and asset pricing jointly determine the variation in medium and long-term yields; the ination factor exhibits a stronger inuence on medium-term yields whereas the asset pricing factor is the strongest single economic determinant at the long end of the yield curve. The analysis of the model predictive ability provides evidence that asset pricing measures contain useful information about future yields. Particularly at intermediate and long-term horizons, the proposed term structure model generates more accurate forecasts than several traditional no-arbitrage ane Gaussian modeling approaches during the recent crisis and post-crisis period characterized by a high variation in yields and the shape of the yield curve. The contribution of the asset pricing factor to forecast accuracy is most pronounced for medium and long yields. Furthermore, beyond the nding in Moench (2008), indicating superior forecasting ability of term structure models using rich economic information sets, we nd that employing economic concept-specic, as opposed to broad (as in Moench, 2008) information-rich factors further enhances forecast accuracy during crisis and post-crisis periods. The remainder of the paper is structured as follows. Section 2 sets out the proposed model and Section 3 details the estimation procedure. The data and state variables are described in Section 4. Sections 5 and 6 discuss the empirical model estimation and the out-of-sample forecasting results, respectively. Finally, Section 7 concludes the paper. 2 The model We employ a term structure formulation that builds upon the seminal work of Ang and Piazzesi (2003) and related analysis of the model parametrization in Pericoli and Taboga (2008). Our choice of the modeling framework is motivated by the primary focus of these authors on the joint evolution of the macroeconomy and the entire term structure of interest rates as opposed to a single policy rate in the FAVAR formulation proposed in Bernanke et al. (2005). However, we augment the model specication in Ang and Piazzesi (2003) and Pericoli and Taboga (2008) by introducing information- 6

7 rich economic state variables. In the economic state variable construction, we build upon the FAVAR literature; see, for example, Bernanke et al. (2005) and Moench (2008). The motivation stems from an observation in Bernanke and Boivin (2003) and Bernanke et al. (2005) that while many variables, available to both the econometrician and the policy maker, contain similar information, the information content of a single variable is noisy due to either measurement errors or idiosyncratic movements. Therefore, theoretically relevant economic concepts such as real activity and ination cannot be directly observed and ought to be treated as unobservable. The factor-augmented approach in Bernanke et al. (2005) permits summarizing the comovement in a large number of economic time series by boosting the economically-relevant signal and diminishing the impact of noise from individual time series. The macroeconomic literature highlights the multidimensional nature of the macroeconomy and emphasizes the importance of identifying and separately studying the eects of shocks from various specic sources; see, for example, Wu and Zhang (2008). Consequently, in our approach we step beyond the FAVAR literature by constructing data-rich factors representing specic economic concepts, as opposed to using a broad pool of variables requiring a researcher to make a post-hoc factor interpretation. Additionally, our model introduces a third dimension, asset pricing, not typically studied in the literature. Specically, our asset pricing factor summarizes information contained in an array of stock market indexes and dividend yield data series that enter along with macroeconomic variables into a FAVAR model data set of, for example, Moench (2008) and Bernanke et al. (2005), but whose impact is not separately studied. The real activity factor is based on measures of gross domestic product and industrial production, both for various sectors of the economy, private consumption, public and private investment and stock building, and employment whereas the ination factor is constructed using a retail price index, a consumer price index and its sector-specic components, a producer price ination index as well as export and import price data. 3 We further augment our set of economic and market factors by two latent yield variables, motivated by evidence in Pericoli and Taboga (2008) indicating a superior ability of more richly parameterized models, with both economic and latent factors, to capture the joint dynamics of long term interest rates and the macroeconomy. Latent factors are thought to capture information about an economic environment that may be omitted in the set of observed factors either because of omitted 3 Appendix A lists factor-specic data variables and Section 4.2 further describes the data. 7

8 variables in the data set or because of information loss as a result of the necessary reduction in the data dimensionality for modeling purposes. The inclusion of latent variables further dierentiates our model from Moench (2008) and Mumtaz (2010) who exploit economic factors solely in their FAVAR formulations. 2.1 State dynamics Suppose movements in the yield curve are governed by N state variables that enter into a state vector X t. X t follows a rst order Gaussian vector autoregression: X t = c + ρx t 1 + Σɛ t, (1) where c is a N 1 parameter vector, ρ and Σ are N N parameter matrices, with ɛ t i.i.d N(0, I N ). The specication implies that X t X t 1,..., X 1 N(µ t, ΣΣ ), with µ t = c + ρx t. In what follows we omit the upper subscript P in this and other specications which are given under the historical (physical) probability measure and use the upper subscript Q to distinguish specications given under the risk-neutral (pricing) probability measure. Further, the vector of state variables contains m macroeconomic and asset pricing factors ft m and l latent factors ft: l Xt T = (ft m, ft), l m + l = N. We depart from the Ang and Piazzesi (2003) specication of X t by including contemporaneous (but not lagged) state factors, motivated by the evidence in Pericoli and Taboga (2008) favoring a model with no lags when there are no overidentifying restrictions. Similar to Pericoli and Taboga (2008), we specify ρ and Σ so as to allow for interdependency among the state factors. Consequently, we obtain: f m t = c m + ρ mm f m t 1 + ρ ml f l t 1 + Σ mm ɛ m t f l t = c l + ρ lm f m t 1 + ρ ll f l t 1 + Σ lm ɛ m t + Σ ll ɛ l t, (2) where c m and c l are (m 1) and (l 1) vectors, respectively. An advantage of our factor dynamics specication in (2) is that it relaxes a theoretically and empirically over-restrictive assumption in Ang and Piazzesi (2003) and Kaminska (2008) inter alia of state factor independence under both 8

9 the historical and risk-neutral probability measures. 2.2 The term structure of interest rates The instantaneous short rate r t is assumed to be an ane function of the state variables: r t = δ 0 + δ 1X t, (3) where δ 0 is a scalar and δ 1 is an N 1 vector. All assets in the economy are priced using a pricing kernel for which we adopt a standard conditionally log-normal form, consistent with the absence of arbitrage opportunities; see, for example, Due and Kan (1996): M t+1 = exp ( r t 12 λ tλ t λ tɛ ) t+1 (4) where λ t is an N 1 vector of market prices of risk associated with various shocks in an N 1 vector ɛ t+1. We parametrize the market prices of risk to be time-varying and an ane function of the state variables: λ t = λ 0 + λ 1 X t (5) Here λ 0 is an N 1 vector and λ 1 is an N N matrix. When all elements in λ 1 are zero, market prices of risk become constant, resulting in time invariant risk premia. Specifying prices of risk as timevarying (as opposed to xed) allows the model to reproduce many established empirical properties of the yield curve dynamics, including various stylized deviations from the expectation hypotheses; see, for example, Dai and Singleton (2002). Further, in the absence of arbitrage opportunities in the bond market, the model parameters under the historical probability measure P and the risk-neutral measure Q are uniquely related: c = c Q + Σλ 0, ρ = ρ Q + Σλ 1 (6) 9

10 The price of an (n + 1)-period zero-coupon bond can be computed recursively by: p n+1 t = E t (M t+1 p n t+1) (7) The specications of the state dynamics (1), the dynamics of the short rate (3) and the pricing kernel (4) are consistent with bond prices that are an exponentially ane function of the state vector: p n t = exp(ā n + b nx t ), (8) where ā n and b n are specic to bond maturity n. Furthermore, the yield of a n-period zero-coupon bond is ane in the state variables: y n t = log pn t n = a n + b nx t, (9) where a n = ān n, b n = b n n. Using (8), one can obtain by induction analytical expressions for a n and b n : 4 ( ) a n = δ 0 + b 1 + 2b (n 1)b c Q n 1 n 1 ) (b 2n 1ΣΣ b b 2ΣΣ b (n 1) 2 b n 1ΣΣ b n 1 b n = 1 n [(ρ Q ) n IN ] [ ρ Q I N ] 1 δ1 (10) Equations (9) and (10) allow obtaining the yield of a bond of any maturity n and, thus, modeling the entire term structure of bond yields. The yield term structure dynamics is governed by the variables in the state vector X t via parameters c Q, ρ Q, δ 0, δ 1 and Σ. 3 Model estimation Term structure models employing a large array of economic variables are typically estimated using a two-step approach; see, for instance, Bernanke et al. (2005), Moench (2008), and Boivin et al. (2009). In the rst step, economic common factors that enter the state vector are constructed using 4 The derivation of (10) is outlined in Ang and Piazzesi (2003), and further details are given, for example, in Hamilton and Wu (2012). 10

11 principal components techniques. To this end, we construct the rst principal component separately for each factor-specic variable group from monthly data observations using the Stock and Watson (2002) methodology. In the second step, we employ the Chen and Scott (1993) methodology that allows the values of the latent state variables to be inferred from observed bond prices to formulate the measurement equations for the term structure model and to estimate the model. Accordingly, we assume l observed bond yields to be measured without errors and denote a vector of two yields as Y1,t n ; the number of bonds here is set equal to the number of latent factors l. The remaining k bond yields enter vector Y2,t n where they are assumed to be measured with error. The measurement equations for the term structure model are: Y n 1,t = A 1 + B 1m f m t + B 1l f l t Y n 2,t = A 2 + B 2m f m t + B 2l f l t + Σ k ɛ k t, (11) where A 1 and A 2 are (l 1) and (k 1) vectors of constants and B 1m, B 1l, B 2m, B 2l are (l m), (l l), (k m), and (k l) matrices of factor loadings, respectively. Σ k is the variance of the measurement error, with ɛ k t i.i.d N(0, I k ). A 1, A 2, B 1i and B 2i, i = {m, l}, are calculated recursively using (10). In the the term structure model ((2) and (11)) estimation, we depart from the standard approach, which relies on non-linear numerical search methods to estimate structural model parameters, by using an approach based on minimum-chi-square estimation proposed by Hamilton and Wu (2012). In this approach, the term structure model is rst re-parameterized into a reduced form, with orthogonal error terms with the reduced form parameters estimated using OLS. 5 The structural parameters implied by the reduced-form are obtained by minimizing the chi-square statistic under the reducedform parameter restrictions. Specically, if π denotes a vector of reduced-form parameters, L(π; Y ) is the log-likelihood based on the entire sample and R is the information matrix, then an estimate ˆπ coincides with a hypothetical function of structural parameters g(θ). Consequently, the optimal estimate of θ is found by minimizing the chi-square-statistic T : min θ = T (ˆπ g(θ)) R (ˆπ g(θ)) (12) 5 The reduced form parametrization is detailed in Appendix B and the model estimation procedure is described in Appendix C. 11

12 We further adopt the normalization conditions suggested by Pericoli and Taboga (2008) and applied in Hamilton and Wu (2012): Σ mm is lower triangular, Σ lm = 0, Σ ll = I l, ω ɛ is diagonal, δ 1l 0 and c Q l = 0. Further, Hamilton and Wu (2012) propose that c b in ρ Q ll = a b which avoids c a multiple solutions and helps ensure that the model is just-identied. 6 The Hamilton and Wu (2012) approach oers several advantages. First, the re-parameterization allows using simple OLS to maximize the likelihood function with respect to the reduced-form parameters, thus avoiding the singularity problem caused by a typically extremely at surface of the log-likelihood function when maximizing it directly with respect to structural parameters in the traditional approach. Second, unlike in the standard approach, the modeler can identify with certainty whether a structural parameter estimate corresponds to the global or a local maximum. Third, the approach ensures that the model is identied, thus, addressing identication issues with respect to a number of widely-used ane models with latent factors and models with macroeconomic and latent factors. 4 Data and state variables 4.1 Yield data We employ end-of-month yield data for zero-coupon bonds of short, medium and long maturities: 1, 2, 3, 5, 7 and 10 years, provided by the Bank of England. 7 The data is from January 1993 to March 2015 (267 monthly observations). Employing data on long (7 and 10 year) maturity bonds, which are frequently excluded in previous studies, enables the analysis of the interaction between the state factors and a more complete characterization of the entire term structure of interest rates. [Insert Figure 1 around here] Figure 1 gives the evolution of bond yields of short, medium and long maturity in Panel A and the slope and the curvature of the yield curve in Panel B. The gure indicates that the term structure 6 The number of structural parameters in ρ Q and δ 1 sum up to 29, which is equal to the number of the associated reduced-form parameters. 7 Anderson and Sleath (2001) document that in the construction of yields on zero-coupon bonds from prices of coupon paying bonds, the Bank of England employs a spline model with approximately 14 parameters which exceeds a number of bonds of various maturities used in our term structure model estimation. 12

13 dynamics is broadly characterized by a number of stylized facts. The average term structure is upward sloping. The term structure attened in the second half of the 1990s and also in the second half of the 2000s, preceding economic recessions in both cases. The two periods of the yield curve attening ended with the inversion of the curve indicated with a negative sign of the yield curve slope in Panel B. An interesting observation is that on both occasions the yield curve remained predominantly inverted over lengthy periods of more than 40 months: September February 2001 and February August 2008, respectively. The term structure regained its upward sloping shape towards the end of 2008 which coincided with the introduction of the quantitative easing policy by the Bank of England. The evident change in dynamic properties of both specic yields and the shape of the entire yield curve across periods is likely to have implications for yield curve modeling and forecasting. Of primary interest in this analysis is the forecast period that spans the great stability, crisis and postcrisis periods. Panel A shows that the yield variability fells to low levels for all yields during the great stability period, with the standard deviation at 0.50%, 0.37% and 0.31% for 1, 5 and 10-year yields, respectively. During the crisis period that followed, the yield variability surged more than four-fold and three-fold reaching its highest levels at 2.21% and 1.18% for 1 and 5-year yields, respectively, and two-fold reaching 0.62% for the 10-year yield all signaling heightened economic uncertainty. Post-crisis the variability of the 5-year yield subsided only moderately, remaining notably higher than during the great stability period, and the variability in the 10-year yield remained at the crisis-level. Only the variability in the 1-year yield plummeted to 0.18%, inuenced by the quantitative easing policy activities that induced a prolonged period of near-zero short interest rates. Further, Panel B reveals that reecting the variability in short and long-term yields, the variability in the slope of the yield curve reached its highest level during the crisis period, with the slope standard deviation (equal to 1.67%) four-fold higher than in the preceding period of great stability. The variability in the slope declined post-crisis but it remained notably higher than during the great stability period. The time variation in the curvature followed a similar pattern. 4.2 Economic state variables To give some intuition about economic state variables, Table 1 lists a selection of variables from each factor-specic group that have the highest R-squared with the relevant factor. Appendix A 13

14 lists the variables under factor-specic headings together with the data sources. 8 The real activity factor is most closely correlated with the GDP growth rate; its second highest correlation is with the services sector GDP growth rate. The ination factor is highly, roughly equally, correlated with the consumer and producer price ination indexes whereas the asset pricing factor is most closely associated with the stock market price indices FTSE 100 and FTSE All-Share and only marginally less correlated with the Dividend Yield of FTSE All-Share index. [Insert Table 1 around here] [Insert Figure 2 around here] Figure 2 gives time series of the real activity, ination and asset pricing state variables. The gure indicates that there is a substantial correlation between the state variables. In particular, the highest in magnitude unconditional correlation at is observed between real activity and ination. The unconditional correlations between ination and asset pricing and between real activity and asset pricing are also negative, at and -0.05, respectively. However, it is apparent that conditional correlations between the state variables vary in both magnitude and sign. Furthermore, real activity is closely, generally positively, associated with various yields along the yield curve; the unconditional correlation is high at 0.44 with the 1-year yield, declining to 0.26 for the 10-year yield. Ination is unconditionally negatively correlated with short and medium yields but notably less so with long yields whereas the unconditional correlation of asset pricing with various yields is relatively weak. In contrast, conditional correlations between the state variables and yields vary considerably in magnitude and switch signs between various sub-periods for all three state variables which provides initial evidence that both the strength and the nature of the relationship between the state variables and the yield curve may vary across periods with dierent economic environments. 5 Empirical estimation results 5.1 In-sample t Table 2 reports the mean and standard deviation of the observed and the MAP-FVAR modelimplied yields and Figure 3 provides an illustration of the average model-implied yields vis-á-vis 8 Applying a log-dierence transformation produces stationary variables which are then normalized to have zero mean and unit variance. Consequently, the resulting principal components are unit free. 14

15 the observed yields. The results indicate that the model is able to generate short, medium and long yields that are extremely close to the observed values throughout the sample period, including periods of heightened volatility during two periods of economic and nancial crises in our sample. Our absolute mean tting errors of 0.12, 0.15, 0.14 and 0.09 percent per 100 of the bond face value for 2, 3, 5 and 7-year bonds compare favorably to 0.41, 0.46, 0.51 and 0.56 percent, respectively, in Moench (2008). The standard deviation of the absolute tting errors are tiny: 0.09, 0.11, 0.10, and 0.07 percent relative to 0.50, 0.57, 0.65 and 0.72 percent, respectively, reported in Moench (2008). The results indicate the model's ability to generate exceptionally accurate yields even at long maturity. [Insert Table 2 around here] [Insert Figure 3 around here] 5.2 Model parameter estimates This section reports the estimation results for our no-arbitrage term structure model ((2) and (11)). In (11) we assume that 1 and 10-year bond yields are measured without errors, Y1,t n = (y1 t, yt 10 ), and that the remaining k = 4 bond yields are measured with errors, Y n 2,t = (y2 t, y 3 t, y 5 t, y 7 t ). Table 3 reports term structure parameter estimates under the risk-neutral probability measure in Panel A and under the historical probability measure in Panel B; standard errors obtained using bootstrap are in parentheses. An interesting observation is that in addition to the commonly documented large diagonal values for all factors in ˆρ Q and ˆρ, except for the asset pricing factor in ˆρ, we obtain a number of sizable o-diagonal elements suggesting a non-trivial conditional correlation between the related state factors. The standard errors of many of these o-diagonal elements are relatively large. Nevertheless, the conditional correlation ˆρ between some o-diagonal elements, for example, the asset pricing factor and the ination factor is both economically and statistically signicant which implies that restricting a conditional correlation matrix to be diagonal may be over-restrictive. The constant term ˆδ 0 in the instantaneous interest rate function (3), which measures the long-run mean of the instantaneous rate, and the elements of ˆδ 1, which can be interpreted as factor loadings on the instantaneous interest rate, are all positive. Estimates of the market prices of risk parameters ˆλ 0 and ˆλ 1, which we infer from (6), are 15

16 reported in Panel C of Table 3. The elements of the vector λ 0 governing the unconditional mean of the market prices of risk that inuences the long-run mean of yields through the constant term in (9) are sizable and similar in the absolute value across factors. Only one element, corresponding to the second latent factor, is somewhat smaller in magnitude. None of the individual λ 0 elements are statistically signicant. A similar picture emerges for the λ 1 elements that govern time variation in risk premia. This result is common in the ane term structure literature; see, for example, Moench (2008) and Wu and Zhang (2008). Moench (2008) suggests that lack of statistical signicance does not necessarily indicate a poor model t but rather cautions about making inference on the basis of statistical signicance of individual risk premium parameters. [Insert Table 3 around here] [Insert Figure 4 around here ] By combining the factor loadings δ 1 on the short (instantaneous) rate and the market prices of risk λ 1, we can obtain the risk premium on the short rate, jointly implied by all factors in the state vector; Figure 4 plots the risk premium. Here the negative average risk premium in the 1990s is in line with the nding in Wu and Zhang (2008) for a similar period in the U.S. economy. For the UK, the negative risk premium is driven by a combined eect of relatively high real output growth and asset pricing and low ination, accompanied by negative average market prices of risk for the asset pricing and ination factors in this sub-period. The average risk premium becomes less negative during the great stability and subsequently turns positive, increasing in magnitude, during the crisis and post-crisis periods. A positive risk premium is jointly driven by a sizable drop in both real output growth and asset pricing along with escalated ination, accompanied by a switch to negative in the average market price of real activity risk, a positive average market price of ination risk and a surge in the magnitude of all three, real activity, ination and asset pricing market prices of risk. 5.3 Factor loadings From (9) one observes that the factor loadings b n determine the impact of a state factor on the n-year yield y n. Factor loadings also indicate a contemporaneous response of y n to a one-unit shock from the state factors. Figure 5 plots the loadings of the ve state factors across bond maturities; 16

17 coecients b n are annualized and reect the movements of one standard deviation in the state factors. [Insert Figure 5 around here] Figure 5 indicates that the short-term (1-year) yield responds positively to contemporaneous shocks in all three economic factors. The response to the real activity growth factor is the strongest among the factors but it weakens rapidly, becoming negligible at medium and long maturities. In line with these observations, Ang and Piazzesi (2003) and Wu and Zhang (2008) also document in the context of the U.S. economy that short-term yields respond strongly and positively to shocks in real activity, with this response diminishing with maturity albeit somewhat slower than in the U.K. context. In contrast, a positive response to the asset pricing factor persists with maturity. This response is at its strongest at short and long maturities. Interestingly, the response to the ination factor turns from positive to negative at medium maturities but the negative response dissipates as maturity increases from medium to long. Such response can be associated with the attening and inversion of the yield curve that precede and accompany economic recessions, a feature observed in our yield data; see Figure 2. All-in-all, the analysis suggests that all three economic factors are important but the real activity factor is the strongest single determinant of the variation in yields at short maturity. The variation in the medium and long-term yields is jointly determined by the ination and asset pricing factors. But their relative importance varies with maturity: ination exhibits a stronger inuence at medium maturity whereas asset pricing is more inuential at long maturity. Our observation about a higher relative sensitivity of medium maturity yields to ination than real activity shocks is interesting, particularly given the U.S. based ndings in Ang et al. (2011) that medium maturity yields respond more strongly to ination policy shifts than to output gap policy shifts by the Federal Reserve. They further document that medium maturity yields are more responsive than short maturity yields to ination policy shifts, an observation which also aligns well with our nding of a greater sensitivity of medium than short maturity yields to ination shocks. Consequently, our evidence points to potentially similar policy implications in the context of the U.K. economy, highlighting a worthwhile avenue for further analysis. 9 9 The analysis is Ang et al. (2011) focuses on short and medium-term yields and does not extend beyond 5-year maturity. 17

18 Following Litterman and Scheinkman (1991), the latent factors are typically interpreted as "level", slope", or "curvature", given the eects of these factors on the yield curve. Indeed, Latent Factor 1 is highly correlated with the empirical level of the yield curve measured as (y 1 +y 5 +y 10 )/3; the correlation coecient is Latent Factor 2 covaries moderately with the empirical slope measured as (y 10 y 1 ) and is more strongly associated with the empirical curvature calculated as (2y 5 y 1 y 10 ), the respective correlations are and Consequently, we term this latent factor as "shape". The impact of the level and shape factors is comparable in magnitude to that of the economic factors, suggesting that the latent factors may be useful for picking up relevant information omitted in the economic factors. The response of the yield curve to a positive unit shock in the level factor is positive across yields which is consistent with the factor interpretation as "level". The response to a positive unit shock in the shape factor is negative for all yields but it is more than three-times stronger for medium and long maturities than short maturities. Consequently, a shock to the shape factor can contribute to atenning and inversion of the yield curve. 6 Out-of-sample forecasting A good in-sample t by a term structure model may not necessarily translate into accurate predictions of future interest rates. Consequently, this section focuses on examining the predictive ability of the no-arbitrage MAP-FVAR model in a recursive out-of-sample predictive exercise. To this end, we compare the model-implied forecasts vis-á-vis forecasts generated by a number of competitor no-arbitrage models. We analyze forecasts at h = 1, 3, 6 and 12 month-ahead forecasting horizons for six yields of 1, 3, 5, 7 and 10-year maturity. In the forecasting exercise, we split the sample period into an initial estimation period and a holdout period. The rst window of length T 0 enables a rst forecast based on data up to month t = T 0 ; the next window of length T enables a second forecast, and so forth. The initial estimation period is from January 1993 to December 2001 containing 108 observations; the initial holdout period is from January 2002 to March 2015 containing 159 observations. At each month-t, we estimate the models using data up to and including month t and then forecast yields in month t + h. The forecasting exercise contains two elements. The rst element focuses on the predictive 18

19 accuracy at each horizon over the entire holdout-period. The second element subsequently analyzes the predictive accuracy separately in three sub-samples that are characterized by dierent economic environments, which we term great stability, crisis and post-crisis. The MAP-FVAR model implied forecasts for month t + h are constructed as follows: ŷ t+h t = â n + ˆb n ˆF t+h t (13) Here coecients â n and ˆb n are recursively calculated according to (10), using as input the estimates ĉ Q, ˆρ Q, ˆδ 0, δ 1 and ˆΣ obtained by estimating the reduced-form representation using the minimumchi-square-based methodology of Hamilton and Wu (2012). 10 The state factor for the month t + h forecasts is obtained as: h 1 ˆF t+h t = ˆρ h F t + j=0 ˆρ j ĉ, (14) where economic state variables in F t constructed via principal components are re-estimated each month t using data up to and including month-t. 6.1 Competitor models This section outlines three competitor models used in the analysis. The models are selected based on their formulation representing joint modeling of the term structure and the macroeconomy precluding arbitrage opportunities. In each case we address the Hamilton and Wu (2012) critique regarding identication issues by analyzing just-identied forms of the competitor models. Furthermore, to ensure a consistent comparison across the model, we employ the minimum-chi-square methodology outlined in Section 3 in model estimation. 11 Model with real output growth and ination factors The Ang and Piazzesi (2003) no-arbitrage model employing two standard factors, real output growth and ination, and latent variables to capture the macroeconomy provides a natural competitor for our model that allows examining the contribution of the asset pricing factor to forecast accuracy. Hence, 10 The model estimation methodology is detailed in Section 3 and Appendices B and C. 11 In the forecasting exercise, statistically insignicant elements of ρ are set to zero to keep the models reasonably parsimonious. 19

20 our rst competitor model, denoted as M-F V AR, employs a state factor F M F V AR containing two macroeconomic factors measuring real activity and ination, m = 2, and three latent factors, l = 3, employed by Ang and Piazzesi (2003). Each macroeconomic factor is constructed as the rst principal component from a factor-specic variable group, and thereby represents a specic economic concept. To ensure that the model is just-identied in terms of the parameter count, a further restriction x is applied: c b in ρ Q ll = x 2 a b, in addition to the standard normalization conditions: Σ mm is x 3 c d lower triangular, Σ lm = 0, Σ mm = I l, δ 1 0, and c Q l = 0. The forecasts are constructed according to: ŷ t+h t = â n + ˆb n ˆF M F V AR t+h t, (15) where â n and ˆb n are recursively calculated using (10) and parameters therein are estimated as before. In (11), 1, 5 and 10-year yields are assumed to be measured without errors, Y1,t n = (y1 t, yt 5, yt 10 ), and the remaining k = 3 yields measured with errors enter Y n 2,t = (y2 t, y 3 t, y 7 t ). Model with real output growth, ination and asset pricing variables The second competitor model follows the traditional term structure literature in employing individual variables to represent relevant economic concepts in the state vector; see, for example, Kaminska (2008) and Pericoli and Taboga (2008). Consequently, this formulation allows examining whether employing rich data sets in the construction of economic concept-specic factors translates into more accurate future yield forecasts. In the model specication, abbreviated as M AP -V AR, the state vector F MAP V AR contains three individual economic variables, m = 3, and two latent variables, l = 2. We employ real GDP growth, CPI index and FTSE All Share stock market index as measures of real activity, ination and market asset pricing, respectively, the concepts that are also employed in our M AP -F V AR formulation using information-rich economic factors. The three selected variables are highly representative of their respective factor groups as indicated by their high correlation with the relevant factors; see Table 1. The size of the state factor components (m = 3, l = 2) is as in MAP -F V AR, therefore the normalization conditions in MAP -F V AR also warrant the identication for the model at hand. 20

21 The model-implied forecasts are generated as: ŷ t+h t = â n + ˆb n ˆF MAP V AR t+h t, (16) where â n and ˆb n and parameters therein are obtained as before. Model with generic economic factors The third competitor model builds on the Moench (2008) FAVAR term structure specication that employs generic economic factors summarizing information from a large pool of economic variables to model the economy. In FAVAR models, economic factors entering the state vector are typically represented by several common factors obtained via principle components summarizing information from a vast pool of data, that do not have a direct economic interpretation. 12 In the spirit of this approach, we pool our macroeconomic and asset pricing variables together to construct three common factors as the rst three principle components with the largest eigenvalues. We augment the state vector F G F V AR containing the three generic economic factors by two latent factors to keep the size of the state vector equivalent across the models. This model, abbreviated as G-F V AR, helps establishing whether direct economic interpretation of the factors in the MAP -F V AR, relating them to specic theoretically-relevant concepts, enhances the forecasting accuracy relative to a model with generic economic factors such as G-F V AR. The size of the state factor components (m = 3, l = 2) is as in MAP -F V AR, the normalization conditions are as for M AP -F V AR. The model-implied forecasts are generated according to: ŷ t+h t = â n + ˆb n ˆF G F V AR t+h t, (17) where â n and ˆb n and parameters therein are obtained as before. 6.2 Forecasting results Table 4 summarizes the root mean squared errors (RMSE) obtained from out-of-sample yield forecasts implied by MAP -F V AR and the three competitor models. The smallest RMSE for a given yield and forecast horizon is highlighted in bold. The results unequivocally indicate that the asset 12 As the economic factors in the state vector constructed via principal components are up to a non-singular rotation, these factors cannot be given a direct structural economic representation. 21

22 pricing factor contributes to a substantial reduction in the forecast errors at the intermediate and long forecast horizons. Specically, the M AP -F V AR forecast RMSEs improve upon the competitor models by 2% and 4% for M-F V AR and G-F V AR, respectively, on average, across yields at the 3-month-ahead horizon. The RMSE reduction aorded by M AP -F V AR relative to the competitor models further improves with the forecast horizon; it becomes, respectively, 19% and 8% at 6-month-ahead horizon and 27% and 12% at 12-month-ahead horizon. Interestingly, at the intermediate and long forecast horizons M AP -V AR, also beneting from the asset pricing information, is the second best performing model for 5, 7 and 10-year yield forecasts which suggests that asset pricing information may be particularly useful for forecasting medium and long-term yields. Only at the 1-month-ahead forecast horizon does the M-F V AR model employing the conventional real activity and ination factors, building on Ang and Piazzesi (2003), generate the smallest forecast errors. [Insert Table 4 around here] Furthermore, a consistently superior forecasting ability across yields and forecast horizons of MAP -F V AR relative to MAP -V AR corroborates the nding in Moench (2008) that employing data-rich economic factors in term structure models leads to more accurate out-of-sample yield forecasts relative to models exlpoiting only a small information set. To formally assess if the observed dierences in the model forecast accuracy are signicant, we apply the White (2000) "Reality Check" test. The test, which is based on a squared forecast error loss function, allows yield forecasts generated by MAP -F V AR, which we select as a benchmark, to be compared bilaterally with the competitor models; the results are reported in Table 5. A negative test statistic indicates that the average squared forecast loss of the MAP -F V AR is lower than that of a competitor model at hand; signicant test statistics are given in bold. 13 Consistent with our previous ndings summarized in Table 4, the formal comparison results indicate a signicantly superior forecast accuracy of MAP -F V AR relative to all competitor models, particularly at 6 and 12-monthahead horizons. Furthermore, consistently smaller average forecast losses at all forecast horizons of the MAP -F V AR vis-á-vis MAP -V AR, with many signicant test statistic values, conrm our 13 Signicance is determined by comparing the average forecast loss dierential with the 10% percentile of the empirical distribution of the loss dierential series obtained by applying bootstrap with 1000 resamples. 22