Real-time forecasting with macro-finance models in the presence of a zero lower bound. Leo Krippner and Michelle Lewis. March 2018

Transcription

1 DP2018/04 Real-time forecasting with macro-finance models in the presence of a zero lower bound Leo Krippner and Michelle Lewis March 2018 JEL classification: C43, E43 Discussion Paper Series ISSN

2 DP2018/04 Real-time forecasting with macro-finance models in the presence of a zero lower bound Abstract Leo Krippner and Michelle Lewis We investigate the forecasting performance of a joint model of macroeconomic and yield curve components for the United States, using data as would have been available in real time. Relative to a standard macroeconomic model benchmark, our results show a clear benefit from including yield curve information when forecasting inflation and the Federal Funds Rate, for horizons up to the four years that we tested. We find some real-time forecast improvement for capacity utilization, our variable representing real economic activity, but only for longer horizons, and similarly when using macroeconomic data to help forecast yield curve components. Using a shadow/lower-bound term structure model allows the ready extension of our forecasting framework to include the unconventional period of monetary policy, and we obtain very similar results to those already mentioned for the conventional period. The Reserve Bank of New Zealand s discussion paper series is externally refereed. The views expressed in this paper are those of the author(s) and do not necessarily reflect the views of the Reserve Bank of New Zealand. We would like to thank participants at the 2015 New Zealand Econometrics Study Group, the 2015 New Zealand Association of Economists conference, Glen Rudebusch, Ken West, and colleagues at the Reserve Bank of New Zealand for helpful discussion and comments. Leo Krippner: Economics Department, Reserve Bank of New Zealand, 2 The Terrace, PO Box 2498, Wellington, New Zealand. address: leo.krippner@rbnz.govt.nz. Michelle Lewis: Economics Department, Reserve Bank of New Zealand, 2 The Terrace, PO Box 2498, Wellington, New Zealand. address: michelle.lewis@rbnz.govt.nz ISSN c Reserve Bank of New Zealand

3 Non-technical summary We investigate the real-time forecasting performance of macro-finance vector autoregression models, which incorporate macroeconomic data and yield curve component estimates as would have been available at the time of each forecast, for the United States. Our results show a clear benefit from using yield curve information when forecasting macroeconomic variables, both prior to the Global Financial Crisis and continuing into the period where the lower-bound constrained shorter-maturity interest rates. The forecasting gains, relative to traditional macroeconomic models, for inflation and the Federal Funds Rate are generally statistically significant and economically material for the horizons up to the four years that we tested. However, macro-finance models do not improve the real-time forecasts over shorter horizons for capacity utilization, our variable representing real economic activity. This is in contrast to the related recent macro-finance literature, which establishes such results (as do we) with pseudo real-time, i.e. truncated final-vintage, data. Nevertheless, for longer horizons that are more relevant for central bankers, yield curve information does improve activity forecasts. Overall, our results suggest that the yield curve contains fundamental information about the likely evolution of the macroeconomy. We find less convincing evidence for the reverse direction, which is likely because expectations of macroeconomic variables are already reflected in the yield curve. However, for longer horizons, we find there are still some gains from using macroeconomic variables to forecast the yield curve.

4 1 Introduction In this paper, we investigate the forecasting performance of a joint model of macroeconomic and yield curve data for the United States (US), using real-time data and including the lower-bound period. We are motivated by previous related literature, to be outlined shortly below, that has found relationships between macroeconomic and yield curve data that appear to be empirically useful to forecasters. However, these studies have generally used in-sample or pseudo-real-time data (i.e. truncated final-vintage data), rather than the genuine real-time data that would actually have been available at the time. Furthermore, for reasons we outline further below, the methods employed in previous studies are not strictly applicable to the lower-bound environments experienced by many developed economies since the Global Financial Crisis. We show how the lower-bound constraint may readily be allowed for in such analysis. The initial work linking macroeconomic and yield curve data began in the late 1980s with the observation that flatter yield curve slopes (i.e. the spread between long-term and short-term rates, such as the 10-year government bond and three-month Treasury bill) provided a leading indicator for slower future output growth or recessions. For example, Estrella and Hardouvelis (1991) provided the first comprehensive statistical study into the relationship for the US, finding that the bond spread is useful for forecasting economic activity, particularly 4-6 quarters ahead. Subsequent studies have generally confirmed such predictive power, albeit with variation across countries and/or among different sample periods. The related literature is far too vast to cite here, and we refer readers to Wheelock and Wohar (2009) for a comprehensive survey. 1 Regarding inflation, a parallel literature tests the yield curve as a predictor of inflation; see Stock and Watson (2003) for a survey. The more recent literature has employed term structure models to investigate the joint dynamics of macroeconomic and yield curve data. Term structure models offer the advantage of summarizing all yield curve data with just several factors, rather than selecting a given spread. Furthermore, they do so in a theoretically consistent manner if the arbitrage-free condition is imposed. The seminal article of Ang and Piazzesi (2003) investigates the relationship from macroeconomic variables to arbitrage-free latent factors of the yield curve within a structural VAR. It finds that a large proportion of the variation 1 Also see for an extensive bibliography. Rudebusch and Williams (2009) is the latest confirmation, for the US, on the predictive power of the yield curve slope that we are aware of. 1

5 in yields can be explained by output and inflation data. Extending that to a forecasting perspective using pseudo real-time data, the authors find significant improvements when forecasting yields. Similarly, Moench (2008) uses an arbitrage-free term structure model with a large number of macroeconomic variables in a factor augmented VAR (FAVAR) model to forecast US bond yields. Up to 12 quarters ahead, the pseudo real-time forecasts from that macro-finance FAVAR outperform those from yield curve models, such as the three-factor Duffee (2002) model and the Diebold and Li (2006) dynamic Nelson and Siegel (1987) model. In the reverse direction, Ang, Piazzesi and Wei (2006) finds that using an arbitrage-free term structure model, rather than just the bond spread, in conjunction with pseudo real-time GDP growth data improves US GDP forecasts out to 12 quarters. Ang, Bakaert, and Wei (2006) obtain similar results. Related literature has used term structure models to investigate the joint dynamics of macroeconomic and yield curve data, but not in a forecasting context. 2 However, Ghysels, Horan, and Moench (2012) warns that using in-sample or pseudo-real-time data may overstate Treasury yield forecast improvements. Specifically, much of the predictive power of macroeconomic data for bond yields disappears when using real-time data. Our exercise therefore tests the potential for forecasting improvements from a joint model of macroeconomic and yield curve data (our macro-finance model) using real-time data. Given the literature discussed earlier, we also test for potential forecast improvements from yields to the macroeconomy. The framework we use is analogous to that in Diebold, Rudebusch, and Aruoba (2006), which considers the bidirectional relationship between US macroeconomic data (capacity utilization, inflation, and the Federal Funds Rate) and the yield curve, where the latter is summarized using estimated Nelson Siegel (1987) Level, Slope, and Bow factors. 3 Our application has three main differences. First, our focus is on the forecasting performance from the model, rather than establishing the in-sample relationships between the macroeconomic variables and yield curve components as in Diebold et al. (2006). Second, in the period prior to the lower-bound constraint on 2 Examples include Kozicki and Tinsley (2001), Piazzesi (2005), Ang and Piazzesi (2003), Dewachter and Lyrio (2006), Balfoussia and Wickens (2007), Ludvigson and Ng (2009), Joslin, Priebsch, and Singleton (2014), Bikbov and Chernov (2010), and Wright (2011). 3 Those authors find yield curve factors explain a significant proportion of variation in macroeconomic variables, and the reverse relationship is also important but to a lesser degree. The authors also link the Level component to inflation, the Slope to economic activity, while the Bow appears unrelated to the key macroeconomic variables. 2

6 nominal interest rates, we use an arbitrage-free version of the Nelson-Siegel model (ANSM), as detailed in Krippner (2006) and Christensen, Diebold, and Rudebusch (2011). Third, to accommodate the period where shorter-maturity interest rates were constrained by the near-zero policy rate setting in the US following the Global Financial Crisis but additional monetary accommodation was delivered via unconventional means, we incorporate the ANSM within a shadow/lower-bound framework, as detailed in Krippner (2011, 2015) and Christensen and Rudebusch (2015). The reason for using the shadow/lower-bound model in our full-sample exercise is to avoid the distortion that would otherwise be present in the yield curve components between the non-lower-bound and lower-bound periods. That is, whether using a 10-year less 3-month spread or estimated latent slope factor, the constraint on shorter-maturity interest rates would lead usual measures of the yield curve slope to understate the degree of monetary accommodation in the lower-bound period. That in turn would distort associated macroeconomic outcomes relative to the unconstrained period. Using yield curve components estimated from a shadow/lower-bound term structure model overcomes that potential distortion, because the components of the shadow yield curve move exactly like the arbitrage-free version of the Nelson-Siegel model in the prelower-bound period, and continue to move freely in the lower-bound period. At each point in time, we estimate the model to obtain Level, Slope, and Bow state variables. We then use those in a vector autoregression (VAR) model with the capacity utilization, inflation, and Federal Funds Rate data that was also available at the time (allowing for publication lags) to produce joint forecasts of macroeconomic and yield curve data. Our results first confirm the pseudo-realtime forecasting results from the literature. That is, the forecasts from our joint macro-finance model generally outperform the forecasts of macroeconomic variables from the models estimated using only macroeconomic data and, to a lesser extent, the forecasts of yield curve variables from the models estimated using only yield curve data. However, the results are weaker when the stricter real-time considerations are imposed. Specifically, the forecast improvement from including the term structure almost disappears for capacity utilization, with gains remaining only for longer forecast horizons. Nevertheless, the information gain at the longer-horizons should still be of use to policy makers given this horizon is consistent with medium-term objectives. The forecast improvement is more robust for inflation, even when incorporating real-time economic activity in the forecasting model. The remainder of the paper proceeds as follows. Section 2 details the macro- 3

7 finance VAR models that we use for our forecasting comparison exercises. In section 3, we outline the data used in the models, including the models we use to estimate the state variables that summarize the yield curve data. Section 4 presents and discusses the results, and section 5 concludes. 2 Forecasting models In this section we detail the models to be used in our forecasting exercises, including the yields-only and macro-only subset models to be used as benchmarks. All of the models are VARs estimated using ordinary least squares, and the appropriate lag length for the VARs are selected using the Bayesian Information Criterion (BIC). 4 In all cases, for the repeated estimations for the pseudo-real-time and real-time forecasting, we find the optimal lag length of 1, which conveniently allows us to present the models in their full form below. 2.1 Macro-only model The macro-only subset model using the Diebold et al. (2006) data takes the form of a small traditional VAR containing economic activity, inflation, and the policy rate, i.e.: y t a 10 a 11 a 12 a 13 π t = a 20 + a 21 a 22 a 23 r t a 30 a 31 a 32 a 33 y t 1 π t 1 r t 1 + e a1t e a2t e a3t (1) where y t, π t, and r t are respectively capacity utilization, core CPI inflation, and the Federal Funds Rate rate, which are discussed in section 3.1. The macro-only model serves as the benchmark for forecasts of macroeconomic variables from the macro-finance models below. 5 4 The BIC results are not reported here but are available upon request. 5 Obviously, many alternative benchmark models could be used. We use VAR forecasts to be consistent with the yields-only VAR model and the macro-finance models. 4

8 2.2 Yields-only model The yields-only subset model is an unconstrained VAR as follows: L t b 10 b 11 b 12 b 13 L t 1 e b1t S t = b 20 + b 21 b 22 b 23 + (2) B t b 30 b 31 b 32 b 33 S t 1 B t 1 e b2t e b3t where L t, S t, and B t are respectively the Level, Slope, and Bow state variables estimated from the arbitrage-free Nelson-Siegel model or the lower-bound augmented version of that model. We provide the details of those models in sections 3.1 and The yields-only model serves as a benchmark for forecasting the yield curve with only yield curve information, which will be compared against the macrofinance models below. 2.3 Unrestricted macro-finance model The simplest macro-finance model is obtained by using the macroeconomic variables and yield curve state variable within a single unconstrained VAR, i.e.: 7 y t c 10 c 11 c 12 c 13 c 14 c 15 c 16 π t c 20 c 21 c 22 c 23 c 24 c 25 c 26 r t L t = c 30 c 40 + c 31 c 32 c 33 c 34 c 35 c 36 c 41 c 42 c 43 c 44 c 45 c 46 S t c 50 c 51 c 52 c 53 c 54 c 55 c 56 B t c 60 c 61 c 62 c 63 c 64 c 65 c 66 y t 1 π t 1 r t 1 L t 1 S t 1 B t 1 + e c1t e c2t e c3t e c4t e c5t e c6t (3) The bolded parameters in the top-left quadrant of equation 3 relate to the macro-only VAR, while the bolded parameters in the bottom-right quadrant relate to yields-only VAR. The parameters in the top-right quadrant allow the yield curve components to help forecast the macroeconomic variables, and vice-versa for the parameters in the bottom-left quadrant. 6 Yield curve forecasts could be obtained directly from the estimated term structure model, but we use VAR forecasts to be consistent with the macro-only VAR model and the macro-finance models. 7 Our macro-finance VAR models are a therefore the result of a two-step estimation, first to obtain the yield curve state variables and then to estimate the VAR model itself. A one-step estimation would be possible within a full state space formulation, but the repeated estimations required for our real-time forecasting application would be too computationally burdensome. Ang, Piazzesi, and Wei (2006) also use a two-step estimation for the same reason. 5

9 2.4 Restricted macro-finance model Restrictions may be applied to the macro-finance model to obtain parsimony that may improve forecast performance. The first set of restrictions we impose is so the policy rate does not affect other variables (but it can affect itself). The motivation is that the Level, Slope, and Bow variables already summarize the information from the full yield curve, including an implied policy rate estimate, so estimating an additional influence from the policy rate is redundant. The second set of restrictions we impose is so the Bow state variable does not affect any of the macroeconomic variables. This restriction reflects the lack of theoretical and empirical evidence, as discussed in Diebold et al. (2006), about a relationship between the Bow and macroeconomic variables. The final restriction we impose is a zero steady-state value for L t, given that that its persistence is consistent with a near-unit-root process. The final restricted model we use is therefore as follows: y t d 10 d 11 d 12 0 d 14 d 15 0 y t 1 π t d 20 d 21 d 22 0 d 25 d 26 0 π t 1 r t L t = d d 31 d 32 d 33 d 34 d 35 0 r t 1 d 41 d 42 0 d 44 d 45 d 46 L t 1 S t d 50 d 51 d 52 0 d 54 d 55 d 56 S t 1 B t d 60 d 61 d 62 0 d 64 d 65 d 66 B t 1 + e d1t e d2t e d3t e d4t e d5t e d6t (4) 3 Data and forecast production In this section we detail the data used in the macrofinance models. Section 3.1 details the source of the real-time macroeconomic data. Section 3.2 discusses the real-time estimation of the Level, Slope, and Bow state variables for our pre-lb exercise, and section 3.3 discusses the estimation for the full sample including the LB period. In section 3.4 we detail how the data are used in our forecasting exercises, and the evaluation of the forecasts is discussed in section Macroeconomic data As noted in section 2.1, the macroeconomic variables we use in our models are the following monthly series: 6

10 capacity utilization, which we demean to create a similar concept to the output gap often used by central banks; 8 core CPI inflation, which is calculated as annualized monthly log differences of the price level; 9 and the effective Fed Funds rate. For the pseudo real-time forecasting exercise, we use the final vintages of the series above. These were obtained from the Federal Reserve Bank of St. Louis FRED website ( at the end of 2015, when we started our project. For the real-time macroeconomic data, i.e. the data that actually would have been available to the forecaster at each point in time, we obtain the vintages from the Federal Reserve Bank of St. Louis ALFRED website ( Capacity utilization and core CPI inflation are subject to minor changes between vintages, which mostly reflects seasonal adjustment, but it may also be due to a fuller set of information, changes in methodology, or technical reasons. Our real-time demeaning of capacity utilization, to reflect the amount of capacity pressure as would have been gauged in real time, also results in changes to the series over time. Real-time monthly data vintages are used from December 1996 and the final vintage is December Figure 1 plots the capacity utilization and inflation data, and also serves to illustrate the extent of revisions as time evolves. Note that the Federal Fund Rate is observed at the end of the month and that series has no revisions relative to the historical vintages. 8 In principle, it would be more ideal to have used the actual output gap. However, to our knowledge, no real-time output gap vintages series are available, and trying to create such a series would be open to subjectivity on how to do so (e.g. on model specification and parameter choices). However, we have undertaken in-sample and pseudo-realtime forecast exercises using the official Congressional Budget Office (CBO) 2014 measure of potential output and 2014Q1 GDP data. These obtain similar results to those we report for capacity utilization. 9 For robustness, we also tested core PCE, headline PCE, and headline CPI inflation. All results were similar, but strongest overall for headline CPI inflation. We use core CPI inflation because it is used in much of the literature, and it is more conceptually consistent to expect financial market prices to reflect the trend component of inflation rather than the idiosyncratic components that are sometimes present in headline inflation measures. 7

11 Figure 1: Real-time macroeconomic data Demeaned US capacity utilization US annualized core inflation Note: Vintages from St. Louis ALFRED, with means subtracted for capacity utilization. 3.2 Arbitrage-free Nelson-Siegel model Prior to the Global Financial Crisis (GFC), nominal yields were not constrained by the lower bound. Hence it is valid to apply the standard arbitragefree Nelson-Siegel model (hereafter ANSM) developed in Krippner (2006) and Christensen, Diebold, and Rudebusch (2011). The heart of the ANSM is the following expression for forward rates: f(t, u) = L t + S t exp ( φu) + B t φτ exp ( φu) + V E f (u) (5) where f(t, u) is the instantaneous forward rate at time t as a function of forward horizon u, L t, S t, and B t are the state variables, 1 and the functions of φ are the forward rate factor loadings associated with each state variable, and V E f (u) represents the volatility effect for forward rates required to make the model arbitrage free (see Krippner (2015) for the full expression). The interest rates, R (t, τ), at observation data t and as a function of time to maturity τ, are then given by: R(t, τ) = 1 τ τ 0 f(t, u) du = L t + S t ( 1 exp ( φτ) φτ +B t ( 1 exp ( φτ) φτ ) ) exp ( φτ) + V E R (τ) (6) where the state variables remain as for the forward rate expression, 1 and the functions of φ are now the interest rate factor loadings associated with 8

12 1 Figure 2: Nelson-Siegel interest rate factor loadings Level Slope Bow Time to maturity (τ) Note: Example of arbitrage-free Nelson-Siegel factor loadings with φ = each state variable, and V E R (τ) represents the volatility effect for interest rates required to make the model arbitrage free (see Christensen, Diebold, and Rudebusch (2011) or Krippner (2015) for the full expression). 10 The intuition underlying the names Level, Slope, and Bow comes from the shape of the factor loadings, which are plotted in figure 2. The first loading is constant by maturity and in practice reflects the long-horizon Level of the yield curve. The second loading represents long-maturity yields relative to short-maturity yields, so it reflects the Slope of the yield curve. The third loading represents mid-maturity yields relative to short- and long-maturity yields, so it reflects the Bow (or Curvature) of the yield curve. Note that φ is an estimated parameter, and it determines the decays of the Slope and Bow factor loadings and the position of the local maximum for the Bow factor loading. We estimate the ANSM using the Kalman filter, as detailed in Christensen, Diebold, and Rudebusch (2011) and Krippner (2015), with end-of-month zero coupon bond data, from the end of 1985 until the end of 2007, which is prior to the events associated with the GFC and the subsequent lower-bound period. The start of the sample period is chosen to capture the period of the great moderation and inflation stability in the US, and it also avoids the 10 Ignoring the volatility effect gives the original Nelson-Siegel model, in forward rate or interest rate terms, which is not arbitrage-free. 9

13 structural break in the early 1980s. 11 The maturities we use are 0.25, 0.5, 1, 2, 3, 5, 7, and 10 years, which follows Diebold et al. (2006) and is consistent with much of the literature. To be consistent with the real-time vintages of macroeconomic data, we recursively estimate the ANSM with the sample period expanding monthby-month. Hence, the first vintage is the estimation using the sample up to December 1996, and the final vintage uses the sample up to December Figure 3 contains the real-time estimates of the ANSM Level, Slope, and Bow. It turns out that the real-time updates show little variation relative to historical estimates, so plotting all of the vintages together looks like a single series. For the Level state component, the largest difference relative to the last vintage is 24 basis points and the absolute mean difference is just three basis points. The Slope and Bow components also show little real-time variation, with absolute mean differences of four and 12 basis points respectively Shadow/lower-bound arbitrage-free Nelson-Siegel model After the GFC, the US Federal Reserve cut the policy rate to near-zero levels, and so the lower bound for nominal yields became a material constraint. As detailed in Krippner (2015), it is no longer valid to apply the ANSM in such an environment, essentially because the ANSM would be mis-specified relative to the properties of the observable data. 13 However, it is valid to use the ANSM as the shadow term structure representation within the shadow/lower-bound framework developed by Krippner (2011, 2015), and we will call this model 11 Estrella, Rodrigues, and Schich (2003) and Joslin, Priebsch, and Singleton (2014) find a structural change around that period. 12 The realtime estimation exercise was also done for the yield curve out to 30 years. These estimates showed more realtime variation, with the largest difference peaking at 111 basis points. 13 The ANSM implies that short-maturity rates are free to move below the lower bound and they maintain a constant volatility over the sample period, whereas the shortmaturity interest rate data is constrained by the lower bound and has markedly lower volatility relative to the pre-lb period. Also see Christensen and Rudebusch (2015), Bauer and Rudebusch (2015), and Wu and Xia (2016) for further discussion on the inconsistencies of non-lower-bound models when applied near the lower bound. 10

14 11 Figure 3: ANSM Level ANSM real-time estimates ANSM Slope 10 Realtime estimates 0 9 Percent 8 7 Percent Years ANSM Bow Years ANSM Phi Percent 2 Percent Years Years Note: Real-time estimates of ANSM state variables. The sample starts December 1985 and the realtime vintages start in December The sample ends in December 2007, which is prior to the onset of the GFC and the lower-bound environment. 11

15 the KANSM. 14 The intuition underlying the KANSM is that the observed short rate r (t) = max {r (t), r LB }, where r (t) is a Gaussian diffusion for the shadow short rate, and r LB is the lower-bound parameter. With a lower bound value of zero imposed, 15 the lower bound forward rate curve is then as follows: f (t, u) = f (t, u) Φ [ ] f (t, u) 1 + ω (u) exp ω (u) 2π ( 1 2 [ ] ) 2 f (t, u) ω (u) where f (t, u) is the lower bounded forward rate, ω (u) is the shadow short rate volatility function, and Φ [ ] is the cumulative normal density function, and f (t, u) is the ANSM forward rate expression from equation 5. The LB interest rate curve is obtained by the straightforward univariate numerical integration of the following expression: R(t, τ) = 1 τ τ 0 (7) f(t, u) du (8) The first plot in figure 4 provides an example of applying the KANSM in a lower-bound period, i.e. where the lower-bound constrains shorter-maturity interest rates. The second plot is an example for a non-lb period, where all interest rates are sufficiently high for the lower bound to be an immaterial constraint. Across both non-lb and LB periods, the KANSM still obtains Level, Slope, and Bow state variables, and these are plotted in figure 4. The associated forward rate and interest rates factor loadings, and their interpretation, remain as for the ANSM, except they now represent the shadow yield curve. Importantly, the shadow term structure estimated from the KANSM is essentially coincident with the ANSM estimates in the pre-lb period, as in the second plot of figure 2. Correspondingly, the Level, Slope, and Bow estimates in figures 3 and 5 are almost identical over the period The Krippner (2011, 2015) framework is developed in continuous time, which accommodates the continuous-time ANSM. Wu and Xia (2015) develop a discrete-time equivalent to the Krippner (2011, 2015) framework. Both frameworks are very tractable approximations to the shadow/lower-bound framework suggested in Black (1995). 15 It is possible to estimate a lower bound parameter, but we obtained implausibly high estimates when the recursive samples span only a short time in the lower bound period (because the models effectively uses the free parameter as an extra degree of freedom). Imposing a LB value of zero for all recursive samples avoids this issue. 16 When the lower bound is not a mateial constraint on r (t) or its expectations, then r (t) = max {r (t), r LB } may be treated as r (t) = r (t), and so the KANSM becomes the standard ANSM. 12

16 Percentage points Figure 4: Examples of applying the KANSM 5 Lower-bound period 5 Non-LB period Time to maturity (years) yield curve data model yield curve shadow yield curve option effect Time to maturity (years) But after the onset of the LB period, the shadow term structure can adopt negative values for shorter maturities, and so the Slope estimate in particular can continue to vary freely. Hence, the KANSM allows us to extend the sample period from December 2007 to December 2014, without the yield curve components being subject to the distortion that the ANSM would incur between the non-lb and LB periods. As an interesting aside, figure 6 plots the final vintage of annualized core CPI inflation with the KANSM Level estimate, and the final vintage of capacity utilization (and the CBO output gap) with the KANSM Slope estimate. The correlations in both plots are highly significant, at 0.62 and 0.59 respectively, which is consistent with the in-sample results from Diebold et al. (2006). The respective correlations for the ANSM model to 2007 are very similar. It is these sort of inter-relationships that the forecasting exercise tries to exploit, but in real time. 3.4 Producing the forecasts We undertake four sets of model forecasts, i.e. pseudo real-time forecasts and genuine real-time forecasts for the pre-gfc sample, and then we repeat those exercises for the full sample. The pseudo real-time forecasts are straightforward to produce. We simply 13

17 11 Figure 5: KANSM Level KANSM real-time estimates KANSM Slope 10 9 Realtime estimates Percent Percent Years KANSM Bow Years KANSM Phi Percent 2 4 Percent Years Years Note: Realtime estimates of KANSM state variables. The sample starts in December 1985 and the realtime vintages start in December The sample ends in December Figure 6: US Level and inflation Macro-finance linkages US Slope and real activity Level 5 Core CPI (annualised) Slope Capacity utilisation Output gap Note: The correlation between the Level factor and core CPI inflation is The correlation between the Slope factor and capacity utilisation is

18 Figure 7: Accommodating data unavailable in real time Time Variable r L S B π y t-3 t-2 t-1 t O O t+1 x x x x x x t+2 x x x x x x x x x x x x Note: Capacity utilization and inflation data are unavailable at time t, and O denotes the nowcast for those variables using t 1 data. x denotes the forecasts from the resultant balanced panel. use the pseudo real-time series (i.e the final-vintage data truncated at the end of each month) as a complete balanced panel of data to estimate the VAR models outlined in section 2, and then use those estimated models to obtain forecasts for each variable from t + 1 until t + 48 (fours years ahead). Producing genuine real-time forecasts is a little more involved, partly because historical vintages need to be used, but also because the one-month publication lag for capacity utilization and inflation data needs to taken into account. That is, both of those variables for a given month are only released during the following month. Hence, those data need to treated as missing at the end of month, and so we therefore have an unbalanced panel of data when the real-time forecasts are undertaken. We accommodate the missing capacity utilization and inflation observations as illustrated in figure 7. That is, at each point in time t, any VAR model with missing capacity utilization and inflation data is estimated up to t 1, and that t 1 VAR is used to now-cast the missing observations at time t. The VAR model is then re-estimated with the balanced panel, including the now-cast observations, up to time t, and that time t VAR model is used to produce forecasts, again from t + 1 until t A one-step estimation using a state space formulation would provide a more formal resolution of the unbalanced panel. However, as mentioned in footnote 7, the repeated estimations required for the real-time forecasts would be too computationally onerous. The key point for our real-time forecast exercises is that any missing data are treated in the same way in the benchmark and macro-finance models. 15

19 Both the pseudo real-time and real-time forecasts use an expanding sample. This follows the literature, and reflects the relatively short sample sizes involved. 3.5 Evaluating forecast performance For the macroeconomic variables, we use forecasts relative to the final data vintages to evaluate the performance of the model estimates. Given the stability in the real-time estimates, we have also chosen to evaluate the forecasts of yield curve components against their final vintage. An alternative would be to calculate yield forecasts for some specific maturities and compare those to realized yield series, but evaluating the yield curve components directly provides a general test across the entire yield curve. We calculate the root mean squared forecast error (RMSFE) for our forecasts and evaluate those against the RMSFEs from appropriate benchmark models. For macroeconomic variables, the benchmark model is the macro-only model. For yield curve factors, the benchmark model is the yields-only model. We have also calculated single equation AR(1) forecasts for all variables, which is a standard benchmark for forecasts from VAR models. Note that we do not subject forecasts of the Federal Funds Rate to be bounded by zero in any of our models. It would be possible to impose values of zero on any negative forecasts, but the key point in our relative forecasting exercise is that all models have identical treatment. We test the statistical significance of the RMSFE differences relative to benchmark models using the Diebold and Mariano (1995) test for non-nested models, with the Clark and West (2007) correction for nested models. For example, the AR(1), macro-only, and yields-only models are nested within the unrestricted macro-finance VAR model. We use one-sided tests, given we are interested in model outperformance over the benchmark. Note that whenever forecasts of monthly data are made for horizons, h, greater than one month, the time series of forecast errors will overlap, which produces serial correlation. We therefore use a Newey-West estimator with a window length of h 1 to correct for that autocorrelation when calculating the statistical significance. 16

20 4 Results In this section we present the results of our forecasting exercises. Section 4.1 contains the pre-gfc results from December 1986 to December 2007, and section 4.2 contains the full sample results from December 1986 to December We report the results grouped by macroeconomic variables, where the benchmark is the macro-only VAR model, and yield curve components, where the benchmark is the yields-only model. The outright RMSFEs from the benchmark model are reported for each variable, with the units as the variables were used in the VARs, i.e. index points for capacity utilization, and annualized percentage points for CPI inflation, the Federal Funds Rate, and the yield curve components. The remaining entries in each line of the table are the relative RMSFEs for the forecasts from the alternative models. A ratio smaller than one, in bold type, means the alternative model provides a better forecast than the benchmark model. We will typically refer to the percentage improvement, e.g. if the relative RMSFE is 0.80, then the forecast improvement is 20 percent relative to the benchmark model. The indicators *, **, and *** respectively denote the 10, 5, and 1 percent levels of statistical significance. Note that entries of n/a in the table indicate that the forecast errors are zero by definition. This occurs for all of the pseudo real-time forecasts for the h = 0 horizon, where the data is implicitly assumed to be known at the end of the month. The only occurrence in the real-time forecasts is for the Federal Funds Rate rate at the h = 0 horizon, because that data is observed at the end of the month and the final vintage contains no revisions relative to the historical vintages. Conversely, the nowcasts of demeaned capacity utilization and inflation, and the historical vintage estimates of the yield curve components are compared to their final (revised) vintages, and so there will be some forecast error at the h = 0 horizon. 4.1 Pre-GFC forecast results Table 1 contains the forecast performance results for the macroeconomic variables over the pre-gfc sample. The first point of note is that the pseudo real-time results are consistent with those already reported in the literature. That is, including yield curve information improves macroeconomic forecasts relative to the macro-only model. The improvements are more apparent for longer horizons, and are generally better for the restricted versus the unrestricted macro-finance model. Inflation and the FFR are 17

21 the variables with the largest and most consistent forecast improvements, with respective gains of 35 and 43 percent at the four-year horizon (and the nowcast for inflation beats the macro-only benchmark by 10 percent). Those improvements are highly statistically significant, and very material in economic significance. On the latter, for example, using the table entries to calculate 4.25 (1 0.57) gives a 1.83 percentage point forecast improvement for the FFR. The second point of note is that, for capacity utilization, the real-time forecast improvements are markedly smaller than those for pseudo real-time forecasts. Indeed, the macro-finance models underperform the benchmark out to two or three years, respectively for the restricted and unrestricted macro-finance models. The maximum outperformance is at the four year horizon, but only to the extent of a (statistically significant) 11 percent improvement for the restricted model, compared to 18 percent in the same model in pseudo real time. The third point of note is that, while there is some deterioration, the real-time forecasts of inflation and the FFR largely maintain the profile, magnitude, and of the outperformances in the pseudo real-time results. Finally, the macro-finance models outperform the AR benchmark, both in pseudo real time and in real time. Table 2 contains the forecast performance results for the yield curve components over the pre-gfc sample. The pseudo real-time results are again consistent with those already reported in the literature, in that including macroeconomic information improves yield curve forecasts relative to the yields-only model. However, unlike the macroeconomic results, the forecast improvements are not consistent across all horizons, with the Level and Slope forecasts underperforming the yields-only benchmark for shorter horizons. Also, there is no obvious advantage from the restricted versus the unrestricted macro-finance model. In addition, when there are forecast improvements, they are typically less impressive than for the macroeconomic variables. Nevertheless, the forecast improvements for long horizons are statistically significant and economically material, particularly for the Slope variable where the 33 percent improvement translates to a 1.14 percentage point improvement in the RMSFE relative to the yields-only model. The real-time forecasts of the yield curve components largely maintain the mixed results noted for the pseudo real-time results, and there is some deterioration in the Slope and Bow components. Surprisingly, the real-time Level results are better than their pseudo real-time counterparts. 18

22 Table 1: Pre-GFC forecast results for macroeconomic variables Capacity utilization, pseudo real-time Capacity utilization, actual real-time Macro MF MF AR Macro MF MF AR h BM Unres. Res. BM Unres. Res. 0 n/a n/a n/a n/a *** *** *** ** ** * 0.86 ** * 0.81 ** ** * 0.89 ** 1.80 Inflation, pseudo real-time Inflation, actual real-time Macro MF MF AR Macro MF MF AR h BM Unres. Res. BM Unres. Res. 0 n/a n/a n/a n/a *** 0.90 ** *** 0.88 *** *** 0.86 *** *** 0.88 ** *** 0.88 *** *** 0.88 *** *** 0.88 *** *** 0.87 ** *** 0.87 ** *** ** *** 0.83 *** *** 0.82 *** *** 0.75 *** *** 0.75 *** *** 0.65 *** *** 0.69 *** 0.98 Fed. Funds Rate, pseudo real-time Fed. Funds Rate, actual real-time Macro MF MF AR Macro MF MF AR h BM Unres. Res. BM Unres. Res. 0 n/a n/a n/a n/a n/a n/a n/a n/a *** 0.88 *** *** 0.87 *** *** 0.90 ** *** 0.89 ** *** 0.91 ** *** 0.91 ** *** 0.95 * *** * ** ** ** * * * *** 0.67 *** 0.99 * *** 0.79 *** *** 0.57 *** 0.87 *** *** 0.65 *** 0.97 Notes: The Macro Benchmark (BM) results are RMSFEs for each horizon h. The remaining results are RMSFEs relative to the BM results. *, **, and *** are respectively 10, 5, and 1 percent levels of statistical significance based on the one-sided Diebold-Mariano -West test, with the Clark-West correction for the nested models (MF Unres. and AR). 19

23 Table 2: Pre-GFC forecast results for yield curve components Level, pseudo real-time Level, actual real-time Yields MF MF AR Macro MF MF AR h BM Unres. Res. BM Unres. Res. 0 n/a n/a n/a n/a * ** 0.98 ** 0.99 ** * 0.98 ** *** 0.96 ** 0.98 ** * 0.97 ** *** 0.95 ** 0.97 ** ** ** 0.92 * 0.97 ** ** * *** ** 0.90 * ** *** *** 0.84 *** 0.94 *** Slope, pseudo real-time Slope, actual real-time Yields MF MF AR Macro MF MF AR h BM Unres. Res. BM Unres. Res. 0 n/a n/a n/a n/a *** 0.89 * * *** 0.74 *** *** 0.84 *** *** 0.67 *** ** 0.69 *** 0.86 Bow, pseudo real-time Bow, actual real-time Yields MF MF AR Macro MF MF AR h BM Unres. Res. BM Unres. Res. 0 n/a n/a n/a n/a * ** * ** ** * 0.91 ** 0.96 ** ** *** 0.81 *** 0.94 ** ** *** 0.77 *** 0.90 ** * 0.87 ** 0.89 ** *** 0.84 *** 0.92 *** *** 0.83 *** 0.87 *** ** 0.97 *** ** 0.92 *** Notes: The Yields Benchmark (BM) results are RMSFEs by horizon h. The remaining results are RMSFEs relative to the BM results. *, **, and *** are respectively 10, 5, and 1 percent levels of statistical significance based on the one-sided Diebold-Mariano -West test, with the Clark-West correction for the nested models (MF Unres. and AR). 20

24 Table 3: Full-sample forecast results for macroeconomic variables Capacity utilization, pseudo real-time Capacity utilization, actual real-time Macro MF MF AR Macro MF MF AR h BM Unres. Res. BM Unres. Res. 0 n/a n/a n/a n/a *** *** 0.97 * *** 0.96 * 0.99 * *** 0.96 * *** 0.95 ** ** ** 0.88 ** ** ** 0.79 *** * 0.85 ** * 0.74 ** * 0.78 * 1.43 Inflation, pseudo real-time Inflation, actual real-time Macro MF MF AR Macro MF MF AR h BM Unres. Res. BM Unres. Res. 0 n/a n/a n/a n/a *** *** *** *** *** *** *** *** *** *** 0.80 *** *** 0.81 *** *** 0.72 *** *** 0.74 *** 0.90 * *** 0.61 *** 0.83 * *** 0.66 *** 0.89 Fed. Funds Rate, pseudo real-time Fed. Funds Rate, actual real-time Macro MF MF AR Macro MF MF AR h BM Unres. Res. BM Unres. Res. 0 n/a n/a n/a n/a n/a n/a n/a n/a *** 0.86 *** 0.99 *** *** 0.85 *** 0.93 *** *** 0.88 *** 0.97 *** *** 0.86 *** 0.91 *** *** 0.89 *** 0.96 *** *** 0.88 *** 0.89 *** *** 0.92 ** 0.92 ** *** 0.92 ** 0.85 *** *** ** *** *** *** 0.88 * 0.84 ** *** *** *** 0.73 *** 0.78 ** *** 0.80 *** 0.80 ** *** 0.64 *** 0.66 ** *** 0.70 *** 0.74 ** Notes: The Macro Benchmark (BM) results are RMSFEs for each horizon h. The remaining results are RMSFEs relative to the BM results. *, **, and *** are respectively 10, 5, and 1 percent levels of statistical significance based on the one-sided Diebold-Mariano -West test, with the Clark-West correction for the nested models (MF Unres. and AR). 21

25 Table 4: Full-sample forecast results for yield curve components Level, pseudo real-time Level, actual real-time Yields MF MF AR Macro MF MF AR h BM Unres. Res. BM Unres. Res. 0 n/a n/a n/a n/a * * ** ** ** *** ** Slope, pseudo real-time Slope, actual real-time Yields MF MF AR Macro MF MF AR h BM Unres. Res. BM Unres. Res. 0 n/a n/a n/a n/a ** * ** * ** * ** ** * * Bow, pseudo real-time Bow, actual real-time Yields MF MF AR Macro MF MF AR h BM Unres. Res. BM Unres. Res. 0 n/a n/a n/a n/a * * 0.86 * *** 0.87 *** * 0.90 *** * 0.86 *** *** 1.07 Notes: The Yields Benchmark (BM) results are RMSFEs by horizon h. The remaining results are RMSFEs relative to the BM results. *, **, and *** are respectively 10, 5, and 1 percent levels of statistical significance based on the one-sided Diebold-Mariano -West test, with the Clark-West correction for the nested models (MF Unres. and AR). 22

26 4.2 Full-sample forecast results Tables 3 and 4 respectively contain the forecast performance for the macroeconomic variables and the yield curve components for the full sample. The main points of note parallel those already outlined for the pre-gfc results. Specifically, for the macroeconomic variables, both the pseudo real-time and real-time forecast improvements remain very similar to the results for the pre-gfc sample. Indeed, the real-time forecasts of capacity utilization relative to the pseudo real-time forecasts no longer deteriorate as much, with the respective improvements of 27 and 22 percent from the unrestricted and restricted macro-finance models at the four-year horizon comparable to the improvements for inflation and the FFR. Regarding the yield curve components, both the pseudo real-time and realtime results forecast performances remain mixed by horizon and model. In general, there appears to be a larger deterioration for the real-time forecasts relative to the pseudo real-time forecasts in the full sample compared to the pre-gfc sample. 4.3 Discussion of results The results in the previous two sections provide the basis for several implications and related discussion. First, there is a clear benefit from using yield curve information when undertaking forecasts of macroeconomic variables in real time. This is an intuitive result for the FFR, because the yield curve should contain information about the expected path of the policy rate. Similarly, the improvement in the inflation forecasts is consistent with the empirical and theoretical link between inflation forecasts and the inflation component in nominal interest rates. While the capacity utilization results are consistent with literature showing that yield curve information helps to forecast economic downturns, the caveat from our analysis is that the short- and medium-horizon forecast improvements that appear possible with pseudo real-time analysis are not obtained in the real-time setting. Second, the real-time forecast improvements are greatest at longer horizons, even including material gains for capacity utilization. These longer-horizon results are particularly useful for central banks. That is, central banks typically set monetary policy to target macroeconomic outcomes/objectives over longer horizons. So long as the level and shape of the prevailing yield 23