Estimating the Natural Rate of Interest in Real Time

Transcription

1 Estimating the Natural Rate of Interest in Real Time (updated version) Sergiy Kasyanenko November 24, 2017 Abstract I construct a new data set of quarterly vintages of real-time estimates of the natural rate of interest and evaluate the reliability of these estimates for a range of empirical models commonly used in the literature. This data, going back to the early 1980s, enables a previously unexplored evaluation of policy rules with time-varying natural rates of interest estimated in real-time. Ex-post revisions of real-time estimates of the natural rate of interest are smaller than revisions of the output gap because revisions of different components of the natural rate of interest partly offset each other. Hence, policy mistakes originating from the use of these estimates in interest rate rules are less severe than miscalculations introduced by realtime estimates of other variables. Interest rate rules that incorporate real-time estimates of the natural rate of interest describe historic policy choices relatively well. Optimized policy rules indicate that, if policymakers had followed a simple interest rate rule that incorporated these estimates, post-crisis macroeconomic performance could have been improved relative to the actual monetary policy stance. JEL classification: E43, E52, C32 Keywords: Natural rate of interest; Real-time data; Monetary policy rule I am thankful to my advisors David Papell, Kei-Mu Yi and German Cubas for their support, patience and guidance. I am also grateful to all participants of the graduate workshop at the Department of Economics at the University of Houston for their valuable comments and suggestions. The usual disclaimer applies. Data, GAUSS code and online appendix can be downloaded here: online materials University of Houston, Department of Economics, phone: (832) , skasyanenko2@uh.edu

2 ... the evidence on balance indicates that the economy s "neutral" real rate - that is, the level of the real federal funds rate that would be neither expansionary nor contractionary if the economy was operating near its potential - is likely now close to zero. Yellen (2016) 1 Introduction The natural rate of interest - the equilibrium short-term real interest rate that would prevail in the long run when inflation is stable and output is at its potential - is emerging as a key metric to assess and guide monetary policy decisions. In principle, macroeconomic stability can be achieved when the short-term policy rate shadows this natural rate of interest. Then a simple interest rate rule that responds to time variation in the natural rate of interest offers a straightforward benchmark for how to adjust short-term policy rates in order to stabilize real activity and inflation. The natural rate of interest is not directly observable and must be inferred at the time of the interest rate setting decision based on available data and imperfect knowledge about the economy. These estimates are retrieved in real time and are updated as more and revised data arrives in later periods. As a result, inaccuracy of real-time estimates can add substantial miscalculations into the policy response to economic fluctuations. For example, Orphanides and van Norden (2002) show that the unreliability of output gaps estimated in real time may become a source of serious policy mistakes. Yet, despite increasing importance of the natural rate of interest for policy conduct, the question of whether it can be accurately measured in real time remains unanswered. In order to explore the implications of measurement errors in the natural rate of interest this paper adopts the baseline empirical model of Laubach and Williams (2003, 2016). This model features regularly and prominently in the Fed outlooks for economic conditions. Hence, it may be a good description of how policymakers perceive the natural rate of interest. In addition, current research on the natural rate of interest is mostly centered around this estimation strategy. However, existing data on real-time estimates of the natural rate of interest is too short to conduct a meaningful policy evaluation analysis. The model jointly identifies the natural rate of interest and the output gap - two unobserved policy variables that have to be estimated in real time to make interest rate rules operational. Thus, it quantifies the extent to which realtime measurement errors in the output gap pass through to real-time misperceptions about the natural rate of interest. The paper contains four principal contributions. First, I construct a new data set on real-time estimates of the natural rate of interest tracing back to the early 1980s and covering empirical specification commonly used in the literature. This data enables previously unexplored evalua- 2

3 tion of interest rate rules that include real-time estimates of the natural rate of interest. Second, I perform a detailed analysis of how accurately the natural rate of interest can be measured in real time. The extent to which past estimates of the natural rate of interest are revised as new data arrives in subsequent quarters and the sources of these revisions are quantified for different empirical models. Third, I evaluate how well real-time estimates of the natural rate of interest perform as policy inputs in a range of simple Taylor (1993) type interest rate rules. In particular, I find that revisions to the natural rate of interest estimated in real time add less to policy miscalculations compared to revisions of the output gap. Fourth, I show that if policymakers had followed a simple interest rate rule that includes a real-time estimate of natural-rate of interest as its intercept, post-crisis macroeconomic performance could have been significantly improved relative to the actual monetary policy stance. Because real-time estimates play a central role in policy guidance (see, for example, Orphanides (2004)), understanding how accurate these estimates are is important for monetary policymakers. However, despite the explosion of interest in real-time data (as discussed by Croushore (2011)), research on real-time estimates of natural rate of interest remains rather sparse. Clark and Kozicki (2005) study 22 annual vintages of real-time estimates of the natural rate of interest for the period from 1982 to 2003 and the earliest quarterly vintage of real-time estimates of the natural rate of interest for the baseline specification of Laubach and Williams (2003) begins in This paper takes advantage of much richer data on real-time estimates of the natural rate of interest - quarterly vintages are constructed starting the first quarter of 1982 and across several specifications commonly used in the literature. As a result, this new data can be used to investigate several important issues that are becoming increasingly relevant for policy conduct and evaluation in the environment when policy decisions are guided by time-varying natural rates of interest. My data offers a more complete view of how estimates of the natural rate of interest retrieved in real time are subsequently revised in the future by providing a detailed description of time-series properties of the natural rate of interest estimated in real time. I find that revisions of real-time estimates of the natural rate of interest are smaller than the revisions of the output gap because revisions of different components that determine the level of the natural rate of interest move in opposite directions. Following the approach proposed by Orphanides and van Norden (2002), I decompose revisions of real-time estimates of the natural rates of interest into various sources. First, real-time estimates may be revised simply because researchers 1 Quarterly vintages of the natural rate of interest are available from 2005 onwards. This real-time data, which matches my estimates for the same period, is accessible from the Federal Reserve Bank of San Francisco at 3

4 have access to revised and more accurate data in later periods. 2 Second, more information in subsequent quarters may provide a better understanding about where the economy is in a business cycle relative to estimates based on real-time data. Third, longer and revised macroeconomic data may lead to revisions of the model describing the economy. This decomposition reveals that revisions of real-time estimates of natural rate of interest are model specific. For example, in models with cost-push shocks, revisions of real-time estimates of the natural rate of interest are mostly driven by revised estimates of the model parameters, while data revisions dominate in more parsimonious specifications. The view that real-time estimates of the natural rate of interest can be informative for policy guidance is firmly gaining currency among monetary policymakers, which warrants a comprehensive inquiry into how real-time accuracy of estimated natural rates of interest affects policy decisions. 3 I investigate this question by analyzing deviations of policy prescriptions from the actual monetary policy stance, where interest rate rules are calculated with real-time estimates of the natural rate of interest and output gap jointly retrieved from the baseline empirical specifications. Unlike a conventional policy evaluation approach, where unobserved policy variables are often drawn from different models, my approach is internally consistent because I use estimated natural rates of interest that are compatible with estimated output gaps. 4 I show that (1) interest rate rules with my real-time estimates of policy variables match the actual policy stance relatively well and (2) ex-post revisions of policy prescriptions obtained with the latest data vintage outperform alternative specification of policy rules commonly studied in the literature. Finally, I investigate what implications data revisions and uncertainty about the true level of the natural rate of interest have for the optimal response to economic slack and deviations of inflation from the target. I show that losses from the 2008 recession could have been significantly lower if the Fed responded more aggressively to real-time estimates of output gap relative to the actual post-crisis monetary policy stance. This result is primarily due to large negative revisions of real-time estimates of the natural rate of interest and the output gap during the initial phase of the Great Recession. I argue that inaccuracy of real-time estimates lead to a systemic underestimation of the economic weakness, constraining the policy response and, hence, creating the environment for a persistent undershooting of the inflation target. The rest of this paper proceeds as follows. Section 2 briefly outlines my estimation strategy 2 See Kozicki (2004) for a brief introduction to the concepts of the real-time data and a survey of the revisions to the key U.S. macroeconomic time series. 3 See, for example, comments by Yellen (2015), Williams (2016, 2017) and Constancio (2016). 4 This makes my policy evaluation analysis immune from criticism by Taylor and Wieland (2016). 4

5 and empirical specifications explored in this paper. In Section 3, I describe ex-post estimates of the natural rate of interest to explain differences across specifications and their implications for the measurement of natural rate of interest in real time. In Section 4, quarterly vintages of the natural rate of interest estimated in real time are presented. This is followed by a discussion of revisions of these estimates within each empirical framework and a decomposition of the sources of these revisions. Finally, Section 5 illustrates how real-time estimates of the natural rate of interest can be used to interpret the conduct of monetary policy, especially during the post-crisis period when nominal policy rate hit its zero lower bound. 2 Empirical Specifications for Estimating the Natural Rate of Interest Following Woodford (2003) and Gali (2008), the natural or equilibrium rate of interest is defined here as the equilibrium real rate of return when prices are fully flexible, markups are constant and aggregate demand is equal to the natural level of output. 5 This definition implies that, absent shocks to the economy, the equilibrium short-run rate of interest is consistent with the long-run price stability. Importantly, this concept of the natural rate of interest seems to offer a relatively good match to the view on the equilibrium real rate held by the policymakers at the Federal Reserve. 6 In particular, it tends to be less sensitive to short-term cyclical shocks to either aggregate demand or supply and captures more persistent medium and long-run structural shifts in the economy. 2.1 Laubach and Williams Model The empirical equivalent of the natural rate of interest explored in this paper is a timevarying estimate of the natural real rate obtained using a procedure suggested by Laubach and Williams (2003). In particular, the real-time natural rates of interest are retrieved by estimating a reduced form New-Keynesian model, where the natural rate of interest, output gap and the trend growth of real output are unobserved state variables. This empirical specification has two blocks: an IS or aggregate demand equation: ỹ t = α 1 ỹ t 1 + α 2 ỹ t 2 + α r 2 2 ( rt j r ) t j + ɛỹ,t (1) j=1 5 For the exposition of the difference between the natural and potential (or efficient) levels of output see Primiceri and Justiniano (2009). The potential level of output is the level of output when products and labor markets are perfectly competitive. 6 See, for example, comments by Fischer (2016). 5

6 where ỹ t is the output gap and r t is the real effective federal funds rate and r t real rate of interest; and a Phillips curve or the aggregate supply equation: is the natural π t = B π (L)π t 1 + b y ỹ t 1 + ɛ π,t (2) The link between the natural real rate of interest and the the trend growth of the natural level of output is established through a log-linear approximation of the consumer s optimality condition (or the Euler equation) around a steady state. In particular, the natural rate of interest is defined by two unobserved components: (1) the growth rate of the natural level of output g t and (2) other determinants of the natural rate of interest z t, for example, shifts in households intertemporal preferences: r t = θg t + z t (3) where θ is an inverse of the intertemporal elasticity of substitution. 7 A relative persistence of the medium to long-run structural shifts in the natural rate of interest is modeled by assuming that all unobserved state variables, such as the natural level of output or y t, its steady-state growth rate, or g t, and other determinants of the natural rate of interest z t, follow a random walk process: y t = y t 1 + g t 1 + ɛ y,t (4) g t = g t 1 + ɛ g,t (5) z t = z t 1 + ɛ z,t (6) where ɛ y,t, ɛ g,t and ɛ z,t are serially and contemporaneously uncorrelated shocks. This specification implies that shocks to consumer preferences as well as the level and growth rate of potential output will have a long lasting impact on the level of the natural rate of interest. In a state-space form, equations 1 and 2 are measurement equations, while equations 4, 5 and 6 are state transition equations. Unobserved state variables as well as model parameters are estimated by maximizing the log-likelihood function following the application of the Kalman filter to this state-space model. 8 Because unobserved state variables are not stationary, maximum likelihood estimates may result in the pile-up problem, as mentioned in Stock (1994). In particular, this pile-up problem may arise when the variability of innovations to the trend growth of potential output and consumer preferences, or z t, is small relative to the overall variability 7 In equation 3, g t is the annualized quarterly growth rate of the natural level of output. 8 See Harvey (1989), chapter 4 for more details. 6

7 of the shocks to potential and actual output. As a result, maximum likelihood estimates may fail to detect time variation in those unobserved states. Therefore, parameter estimates are obtained in a sequential procedure, which allows for median unbiased estimates of the standard deviations of the innovations to the trend growth of output and other determinants of the natural rate of interest, or σ g and σ z, respectively, providing that these processes exhibit small to moderate variability Alternative Specifications of the Natural Rate of Interest This paper investigates several empirical specifications of the time-varying natural rate of interest. The following estimation frameworks are most commonly used in the relevant literature: The baseline LW1 model is the original empirical framework of Laubach and Williams (2003). In particular, a Phillips curve (equation 2) is augmented with two relative price shocks - core import price inflation and lagged crude imported oil price inflation. As discussed below, the inclusion of these relative price shocks will influence the estimates of the variability of the trend growth of output. r t = θg t + z t LW2 model is identical to the LW model with one exception - an inverse of the intertemporal elasticity of substitution, or θ, is restricted to unity implying a one-to-one relationship between the natural rate of interest and the growth rate of potential output. r t = g t + z t HLW1 model is a more parsimonious version of Laubach and Williams (2003) as its Phillips curve contains only lags of the core PCE price inflation and the lagged output gap. As in Holston et al. (2017), this specification imposes a one-to-one feedback from the trend output growth to the natural rate of interest. r t = g t + z t HLW2 model has a similar Phillips curve as the HLW1 model; yet in this specification, the 9 See Stock and Watson (1998) for the discussion on the median unbiased estimation of the models with timevarying parameters. 7

8 inverse of the intertemporal elasticity of substitution is estimated for each data vintage. r t = θg t + z t HLW3 model drops the link between the natural rates of interest and the growth rate of potential output altogether and treats the natural rate of interest as a random walk process: rt = rt 1 + ɛ r,t In all models described above the output gap and the natural rate of interest are jointly estimated. As a result, the accuracy of LW/HLW specifications may be subject to misspecification of the system of reduced-form equations governing the dynamics of output gap and inflation. An alternative approach is to estimate the natural rate of interest when the output gap is determined exogenously. Another reason to investigate real-time estimates of r using this approach is a common practice in the literature to extract output gaps with univariate detrending methods (for example, the Hodrick-Prescott filter) or CBO output gaps for policy evaluation. 10 Thus, this paper will also retrieve real-time estimates of r consistent with real-time CBO output gap for two specifications of the natural rate of interest: CBO1 model includes only an IS curve (equation 1) as its measurement equation in the state-space representation, where the output gap is constructed using the CBO projections of the real potential output. In addition, the intertemporal elasticity of substitution in this specification is restricted to unity. rt = g t + z t CBO2 model estimates an IS curve with the constructed CBO output gaps but models the natural rate of interest as a random walk process. This specification allows to partly mute the impact of frequent and large data revisions in the growth rate of the CBO potential output on the estimates of the natural rate of interest. r t = r t 1 + ɛ r,t Finally, in all empirical specifications considered here unobserved state variables are not stationary, which exacerbates the aforementioned "pile-up" problem. Thus, all models are estimated in a sequential procedure, which, in particular, producers vintage-specific median un- 10 See, for example, Clarida et al. (2000) and Taylor (1999). 8

9 biased estimates of the standard deviation of innovations to the trend output growth as well as median unbiased estimates of the standard deviation of shocks to other determinants of r. This is a departure from the previous studies (for example, Clark and Kozicki (2005)) where variances of innovations in the random-walk components of the natural rate of interest relative to shocks to the potential and actual output where fixed at the baseline values obtained by Laubach and Williams (2003). Vintage-specific estimates from this paper are a more appropriate approach to represent policymakers beliefs about the variability of unobserved non-stationary components of the natural rate of interest in real time. 2.3 Current Estimates of r Existing research shows that estimated trajectories of the natural rate of interest are as diverse as the universe of models, estimation approaches and identifying assumptions underpinning those measurements. Most, studies, however, are built around the baseline specification of Laubach and Williams (2003). For example, Lewis and Vazquez-Grande (2017) study alternative specifications of the original Laubach-Williams model using Bayesian methods and, in contrast to Holston et al. (2017), detect a stronger recovery of the natural rates of interest since the trough of the Great Recession. Kiley (2015) reaches a similar conclusion by including additional controls into the aggregate demand and supply equations, while estimates by Johannsen and Mertens (2016) are either above or below the results reported in Holston et al. (2017) depending on whether the ZLB is explicitly modeled or ignored. A time-varying parameter vector autoregressive model of Lubik and Matthes (2015) produces similar estimates as the model of Laubach and Williams (2003). However, there is a common thread running through most empirical studies of r - a broad agreement that the natural rate of interest is well below its historical level both in the U.S. and worldwide (see, for example, Hamilton et al. (2015) and Rachel and Smith (2015)), with confidence intervals around point estimates being particularly large. One important implication of this low interest rate environment is that conventional monetary policy becomes more frequently constrained by the zero lower bound (ZLB) and, therefore, may fail to deliver desired macroeconomic stabilization. Kiley and Roberts (2017) argue that episodes when the ZLB binds for the Taylor-type policy rules are increasingly likely if r drops to as low as 1 percent - a threshold, which it is unlikely to beat for the foreseeable future. Not surprisingly, policymakers across the globe are paying much more heed to where the natural rate of interest is as well as what its recent decline entails for the conduct of monetary policy. In fact, this stubbornly low natural rate of interest is already prompting a heated debate on how 9

10 to revise and realign the design of monetary policies (see, for example, Bernanke (2017)). At the same time, the inherent lack of precision in the measurement of the natural rate of interest and possible misperceptions about its true level arising from the estimation of different models are behind some lingering skepticism regarding the relevance of estimated time-varying r for policy conduct (see, for example, Taylor and Wieland (2016)). The next section addresses this issue by exploring how different empirical specifications affect the estimated level of r. However, to make policy rules with time-varying natural rate of interest operational a better understanding of how those estimates are revised over time is needed. After all, a consensus in the literature is that inference on unobservables produces large and highly persistent revisions and the accuracy of real-time estimates further deteriorates during the cyclical turns in economic activity. As a result, real-time measurement errors may be a source of substantial policy mistakes. Therefore, this paper also attempts to answer the question of whether revisions to r are model specific and quantifies the size and sources of these revisions across different empirical specifications. 3 Ex-post Revised Estimates of the Natural Rate of Interest All state-space models defined by the measurement equations (1)-(2) and the state transition equations (4)-(6) are estimated using the latest vintage of U.S. data spanning the period from 1961:1 to 2017:1. 11 The one-sided or filtered estimates of the natural rate of interest are shown in Figure 1: panel a) displays estimates for the LW and HLW specifications and panel b) shows estimates obtained using the constructed CBO output gap. All approaches paint a broadly similar picture - the estimated natural rate of interest fell sharply following the Great Recession. As can be seen in Table 1, parameter estimates are reasonably consistent across all empirical models - the sum of autoregressive coefficients α 1 and α 2 in the IS curve ranges from.94 in the LW specification to.91 in the CBO2 model. Meanwhile, a one percentage point widening of the real interest rate gap, or the difference between the real effective federal funds rate and the natural rate of interests r t rt, increases economic slack or output gap by about percentage points in the LW/HLW models and by around 0.09 percentage points in the CBO specifications. However, the estimated Phillips curves differ across the LW and HLW specifications. First, shocks to inflation are more persistent over the short and medium term within the HLW framework as it yields a larger estimate of the responsiveness of the current inflation rate to the 11 That is, the latest data vintage is labeled 2017:2 and contains all information that was available to policymaker in the second quarter of

11 lagged inflation rate, or β 1. Second, the sensitivity of inflation to lagged output gap, or β y, is nearly twice as big in more parsimonious HLW models versus the baseline LW specification. 3.1 Ex-post Estimates of the Output Gap and Trend Output Growth This difference has several implications for the estimated values of the natural rates of interest. First, the degree to which output gap responds to the real interest rate gap, or α r, influences estimates of the natural rate of interest because these estimates are designed to bridge the gap between the realized level of output and some measure of the natural level of output. In particular, when the real interest rate gap converges to zero, output moves closer to its natural level, which stabilizes inflation. This means that, everything else being equal, α r < 0 and a larger negative output gap in these empirical models is associated with a bigger estimate of the real interest rate gap r t rt, or, alternatively, a smaller estimate of the natural rate of interest.12 And because a Phillips curve serves to identify output gap, the degree by which inflation responds to lagged demand conditions will also influence estimates of the natural rate of interest. Figure 2, panel a) displays estimated output gaps from the LW and HLW specifications as well as the output gap calculated using the CBO projections of the potential GDP. A comparison of the LW and HLW models offers some insight on how those estimates are linked to the measurement of other unobserved state variables. Estimated LW output gaps tend to be below the HLW output gaps, which implies that LW models appear to overestimate the natural level of output relative to the HLW specifications. This difference virtually vanished by the beginning of 2000 s indicating that the estimated natural level of output obtained from the HLW models has been catching up with its LW counterpart. This also implies that during the same period, the trend output growth estimated within the HLW empirical framework accelerated and eventually surpassed its LW equivalent. However, starting early 2000s, a difference between estimates of the HLW output gaps and the LW output gaps reappears again pointing to a slowing trend growth of output derived from the HLW empirical specification relative to the LW models. Estimates of the growth rates of the natural level of output shown in Figure 2, panel b) are consistent with this narrative. Much of this divergence in the estimated growth rates of the natural level of output can be traced back to different specifications of the Phillips curves (equation 2) in LW and HLW models. More specifically, in addition to the lagged inflation rates, the LW models allow the current inflation rate to respond to cost-push shocks, namely, relative price shocks from the lagged oil prices and prices of imports. 13 This, in turn, induces a smaller response, or β y, 12 This, in particular, helps explain why estimates of the natural rate of interest exhibit a procyclical pattern. 13 As in Laubach and Williams (2003), measures of the relative price shocks are calculated by subtracting the 11

12 of the current inflation rate to the output gap in the LW models since a bigger fraction of the variation in the current inflation rate is captured by the relative price shocks originating from the oil prices and prices of imports rather than lagged demand conditions. Meanwhile, a Phillips curve without the relative price shocks, as in the HLW specifications, requires a stronger response of the current inflation to the lagged output gap to explain the observed variation in inflation rates. Thus, a positive (or negative) shock to inflation in a Phillips curve without costpush shocks induces overestimation (or underestimation) of the demand conditions (or output gaps) in the HLW models relative to LW specifications. Put differently, inflation surprises and their feedback to estimated output gaps are more pronounced in models, which lack additional controls for cost-push shocks. Broadly speaking, results from models with and without additional control variables in the Phillips or IS curves will depend on the extent to which such variables deviate from their historical norms in the estimation sample. Intuitively, all empirical frameworks discussed here allow for random shocks to affect both the level and the growth rate of the natural level of output. The first step of the sequential estimation approach, which, as in Kuttner (1994), assumes that the growth rate of the natural level of output is constant, attributes time variation in the natural output exclusively to the shocks to its level. Hence, compared to LW models the HLW specifications generate relatively stronger response of inflation to output gaps as well as smaller measured shocks to the level of natural output in the presence of large cost-push shocks to the Phillips curve. And because shocks to the natural output permanently shift its level, subsequent periods with moderate cost-push disturbances to the Phillips curve induce larger adjustments to the natural level of output in HLW models relative to LW specifications. As a result, less variability of estimated natural level of output in the LW models allows to pin down the growth rate of the natural level of output with a better precision. Indeed, as can be seen in Table 1, median unbiased estimates of the ratio of the standard deviation of shocks to the trend growth of output to the standard deviation of innovations to the level of natural output, or λ g, is 2.7 times larger in the HLW1 model compared to the LW1 specification. Thus, estimated variance of the growth rate of potential output, σ g, in the LW models is more than twice as low compared to the HLW specifications. This, in particular, may help improve the precision of estimated natural rates of interest in models, where a Phillips curve is augmented with additional control variables and the natural rate of interest is tied the trend growth of output. overall inflation rate from the core import price inflation and imported oil price inflation. 12

13 3.2 The Effect of Other Determinants on Estimated r Still, the difference in the estimated growth rates of the natural level of output cannot account for the divergence between the estimates of the natural rate of interest retrieved from the LW and HLW models. For example, from the first quarter of 2008 onwards, the estimated trend growth of natural output was on average 0.8 percentage points higher in the LW models versus the HLW specifications. This difference alone should widen the gap between the estimated natural rates of interest within the baseline LW1 specification relative to the HLW2 model (both models do not impose a unity restriction on θ) by nearly two percentage points. However, HLW estimates of the natural rate of interest are, on average, above their LW counterparts - a gap that grows even wider in the aftermath of the Great Recession. This result indicates that the variability of the estimates of other determinants of the natural rate of interest, or z t, is pushing the difference between the estimates of the natural rates of interest obtained from LW and HLW specifications in a different direction relative to the difference between the estimated trend growth of output obtained from those models. In addition, as can be seen in the last row of Table 1, most of the variance in r conditional on the estimates of z t and g t is attributed to other determinants of the natural rate rather than the trend growth of output. 14 Because the real interest rate gap is equal to r t θg t z t, larger estimates of the growth rate of the natural level of output should be accompanied with lower estimates of z t to force this gap converge to zero. In principle, an accommodative monetary policy or low real federal funds rates, r t, during economic downturns, when output gaps are below zero, will cause those estimates of drop even deeper into the negative territory. 15 This also implies that real-time estimates of z t will turn more negative as the estimation sample covers more periods when the economy was in a recession. And since, by assumption, all past shocks to z t are highly persistent, more negative estimates of z t in the LW specifications (which, as discussed above, producers higher estimates of the trend growth of output relative to HLW models) at the start of the sample result in lower values of those estimates for the most resent observations as well, pushing the natural rate of interest below the estimates obtained from the HLW models. 14 Clark and Kozicki (2005) reach a similar conclusion, which related literature often presents as a proof of a weak link between the natural rate of interest and the trend growth of output (see, for example, Kiley (2015), Hamilton et al. (2015) andlewis and Vazquez-Grande (2017)). In addition, Hamilton et al. (2015) argue that the correlation between ex-ante real interest rates and long-run GDP growth across major economies is modest. This finding, however, is sensitive to the choice of the data samples. 15 Because α r is negative, output stabilization implies that the real interest rate gap, r t rt, should be countercyclical, which means that other determinants of the natural rate of interest are procyclical. 13

14 3.3 A Trade-off in the Precision of Estimates of g t and z t Furthermore, because within the LW specifications growth rates of the natural level of output are estimated with better precision, the same empirical framework should allow for a greater variability in z t to capture variability in the output gap. Indeed, as can be seen in Table 1, estimated conditional variance of other determinants of the natural rate of interest, σ z, in the HLW models is over 40% lower versus estimates of σ z from the LW models. Essentially, the median unbiased estimate of λ z, or the ratio of the standard deviation of shocks to other determinants of r to the standard deviation of innovations to the IS curve, obtained from the HLW specifications is lower compared to its counterpart derived from the LW models. This result owes to the previously discussed higher volatility of the trend growth of output in the HLW models determined during the first step of the sequential estimation procedure. In particular, the estimate of λ z is derived from the test of multiple shifts in the intercept, or ϕ 0, of the following regression: ỹ t = ϕ 0 + ϕ 1 ỹ t 1 + ϕ 2 ỹ t 2 + ϕ r t j + ϕ 4 g t + υ t (7) j=1 where ỹ t and g t are the output gap and trend growth of the natural output obtained from the second estimation step, which assumes that z t is time-invariant. 16 The estimated trend growth of output is more volatile in the HLW specifications, therefore, accounts for a larger portion of variation in the estimated output gaps. This allows for more stability in ϕ 0 and, consequently, delivers a lower estimate of λ z Implications for Real-Time Estimates of r These results have several implications for the measurement of the natural rate of interest in real time. First, the LW specifications appear to be more immune to revisions of the growth rate of the natural level of output as this unobserved component of the natural rate of interest is estimated with better accuracy compared to the HLW models. Second, most uncertainty about the estimates of the natural rates of interests originates from the lack of precision in the measurement of the other determinants of rt. This, in particular, is a consequence of the unreliability of the measured output gaps, which, as pointed out by Orphanides and van Norden (2002), are subject to considerable uncertainty at the end-of-sample estimation. Because a sequential estimation procedure introduces a trade-off between how accurately the trend growth of output can be measured and the precision of estimates of other random walk components 16 See, for example, Andrews et al. (1996) for more details on the test of the regression parameter changes at multiple unknown points in time. 14

15 of r, it may be possible to improve the accuracy of the output gap estimates by augmenting an IS curve with the demand-shift variables. Gilchrist and Zakrajsek (2012) argue that credit spreads may provide additional explanatory power for the business cycle fluctuations. Including credit spreads into an IS equation, may, however, exacerbate the "pile-up" problem. Because credit spreads are highly countercyclical, the maximum-likelihood estimation may fail to discern moderate volatility in the unobserved states even when a sequential estimation procedure is used. Intuitively, the inclusion of demand-shift variables improves the identification of the cyclical turns in business activity and, thus, leaves less residual volatility in data to pickup time variation in other unobserved and highly persistent components of the model. Indeed, the LW specifications with lagged credit spreads do not detect time-variation in other determinants of the natural rate of interest. 17 This "pile-up" problem appears to be less sever when additional shift variables are included in an IS curve but not into a Phillips curve. When credit spreads are included in the aggregate demand equation, estimated natural rate of interest is, on average, a full percentage point above estimates obtained from the two baseline specifications of LW1 and HLW1. 18 Meanwhile, estimated variance of the growth rate of the natural level of output is indeed smaller compared to the models discussed above. At the same time, there is no gain in accuracy with which other determinants of the natural rate of interest are measured. Thus, a trade-off between how precisely two unobserved components of the natural rate of interest can be measured remains even when the IS curve is augmented with demand-shift variables. 4 Real-time Estimates and Revisions of the Natural Rate of Interest Ex-post revised estimates of the natural rate of interest obtained with the latest data vintage may be suitable for exploring how those measurements vary across different empirical specifications. They are, however, a poor substitute for real-time estimates when it comes to understanding the historical path of monetary policy and the assessment of future policy prescriptions. Indeed, a reliance on ex post revised data can deliver an incorrect interpretation of the past policy choices, as shown by Orphanides (1998, 2004). Real-time estimates of the natural rate of interest may offer a better insight on how policymakers respond to information that was available to them at the time of the interest rate setting decision. In addition, if revisions of 17 Kiley (2015) applies Bayesian methods to bypass this "pile-up" problem of classical inference. However, he finds that the inclusion of demand-shift variables is relatively uninformative about the data generating process of the natural rate of interest. 18 These results appear to be on par with Kiley (2015) findings, who estimates that the natural rates of interest stood at about 1.25 percent by the end of

16 real-time estimates of the natural rates of interest in subsequent quarters can equip policymakers with the knowledge about the magnitude and direction of the measurement bias, they may be a useful guide to future policy actions as well. 4.1 Real-Time Data Real-time estimates of the natural rate of interest in the LW and HLW specifications are obtained with data vintages of the real GDP and the Core Price Index for Personal Consumption Expenditures (PCE) from the Real-Time Data Research Center of the Federal Reserve Bank of Philadelphia. Since the first available series of the core PCE index is of 1996:2 vintage, the Price Index for Personal Consumption Expenditures is used instead to retrieve real-time estimates for data vintages prior to the first quarter of Data vintages of the CBO output gaps are constructed with the CBO s projections of potential GDP. In particular, the CBO usually revises its projections of potential output twice a year: in January and August (from 1991 to 2001, only January s projections are available). The CBO output gap is calculated using data vintages for real GDP and corresponding vintages of the CBO s projections of potential output. Hence, January s projections of potential output are paired with the real GDP vintages of the first and second quarters of the same year (because GDP data is released with a quarterly lag, the last observations of those vintages are the last quarter of the preceding year and the first quarter of the current year, respectively), while the August s revisions are used to construct output gap vintages of the third and fourth quarters of the same year. The earliest available data on the real GDP series is of 1967:4 vintage. However, this paper, looks at real-time estimates for data vintages beginning in 1982:2. First, short sample sizes of earlier data vintages make the Kalman filter recursion more sensitive to the choice of the initial state vector and its covariance matrix, which greatly weakens the reliability of estimates. Short samples render estimates of λ g and λ z much less reliable as well. In addition, the way monetary policy was conducted during the pre-volcker period may have contributed to macroeconomic instability (see, for example, Clarida et al. (2000)). As a result, estimates from the samples consisting mostly of observations from that period may not be fully comparable to estimates obtained with more recent data vintages. 19 It is a common convention in the real-time literature to mark the data vintage by the date when information became first available to the public. Most macroeconomic time series are released with a quarterly delay. Thus, the last observation on output and prices in every data vintage lags the date of that vintage by one quarter. 16

17 4.2 Real-Time Estimates of r and Vintage-Specific Revisions Figure 3 shows real-time estimates of the natural rate of interest for all seven empirical models. Light-shaded bands indicate the range of updated estimates once new observations and data revisions were introduced in subsequent quarters. These bands are fairly narrow for more recent vintages as successive estimation samples contain fewer new observations as well as minor revisions to data relative to earlier vintages. In fact, with an exception of CBO models, where frequent and large revisions to the level of projected potential output considerably add to the uncertainty of the real time estimates, the LW and HLW specifications produce real-time estimates that are rather close to measured natural rates of interest obtained from the latest data vintage. To explore whether this pattern holds for earlier vintages as well, all vintage-specific estimates of the natural rate of interest were compared to the real-time series of the natural rate of interest. Figure 4 plots an average difference between the vintage-specific estimates of the natural rate of interest and their corresponding real-time estimates obtained with past data vintages. These vintage-specific revisions are averaged for data vintages covering three time period. The first period includes data vintages when the federal funds rate was at its zero lower bound, or from 2009:3 to 2017:2. The second period averages across vintages from earlier 2000 s to the start of the Great Recession, which is roughly equivalent to the time period when the economy went through the phase of productivity slowdown. Finally, the last period looks at data vintages from earlier 1990 s to the beginning of 2000 s. As can be seen from the top panel of Figure 4, on average, real-time estimates of the natural rate of interest retrieved for data vintages following the Great Recession only modestly undershoot their ex-post revised measures even after four years of new observations are added to the estimation sample. The data for earlier vintages tells a different story - real-time estimates obtained in 1990 s tended to overstate the natural rates of interest relative to revised estimates obtained in subsequent quarters (see bottom panel of Figure 4). For example, during that period, an average downward revision of the natural rates of interest within the LW1 specification would amount to about 0.3 percentage points four years after the initial real-time measurement was made. This difference in revisions across vintages can be explained by a bigger impact of new observations on revised estimates for earlier data vintages relative to more recent data vintages. After all, the estimation sample, which starts in 1961:1 for all data vintages, is growing longer with time. As a result, new observations trigger larger revisions to the real-time measurement for earlier data vintages when the initial estimation sample is shorter. For instance, adding 17

18 16 more quarters of data to the 1991:2 vintage increases the sample size by over 13% versus a 9% increase in the sample size if the same number of new observations are added to the 2009:3 vintage. Furthermore, data revisions appear to be more extensive for earlier vintages, which generates larger revisions of measured natural rates of interest in subsequent quarters. Lastly, the federal funds rate stood virtually unchanged since the second quarter of New observations may have contributed little to the revision of the real interest rate gap, which kept ex-post revised natural rates of interest fairly close to their real-time levels. An alternative way to judge how far real-time estimates deviate from their ex-post revised levels of the natural rate of interest is to look at vintage-specific revisions for all vintages for which real-time measures are calculated. Figure 5 displays a quarterly revision of the real-time estimate of r averaged across four quarters preceding each vintage-specific series of ex-post revised estimates. 20 For example, a point on Figure 5 corresponding to the 2009:1 data vintage shows an average quarterly revision of real-time estimates obtained in each quarter of 2008 relative to ex-post estimates retrieved in the first quarter of This comparison is informative because it shows the extent to which estimated natural rates of interest are subject to a revision within a relatively short window after the initial real-time measurement was made. If the size of those revisions is large, real-time estimates may be a source of bigger measurement errors in policy rules that contain time-varying natural rates of interest. As can be seen in Figure 5, the magnitude of revisions for most recent vintages is, in fact, fairly modest. There are, however, dissimilarities between LW and HLW models. Most interestingly, real-time measurements underestimated natural rates of interest compared to ex-post revised estimates in the LW1 specification during the Great Recession; while revisions from the HLW1 specifications tended to move in the opposite direction over the same period. This difference reflects the inclusion of cost-push shocks into the Phillips curve equation in LW models. Since oil prices where particularly volatile from late 2008 through the first half of 2009, each new observation induced a bigger revision of the output gap (and, consequently, the real interest rate gap) relative to the size of output gap revisions in the HLW specifications. 4.3 Revisions of r Estimated During Recessions These short-term revisions should not raise much concern about the measurement error if they offset each other in subsequent quarters. For example, a researcher employing the LW1 model in the first quarter of 2008 would estimate the natural rate of interest for the last quarter of 2007 at 1.95%. This estimate would be revised 37 times and the latest-available measurement 20 Since LW2 and HLW2 specifications deliver revisions that are almost identical to revisions from LW1 and HLW1 models they are not shown on the plot. 18

19 retrieved with the 2017:2 data vintage would put r in 2007:4 at 2.08% - a fairly trivial revision compared to its real-time estimate first obtained at the start of It is, however, important to understand how those revisions evolve over time. Indeed, as was mentioned previously, a common finding in the literature is that the accuracy of real-time estimates of unobserved policy variables tends to deteriorate during the cyclical turns in economic activity. Thus, if real-time measurements of r retrieved during economic downturns are subject to big revisions in the future, they may give rise to more serious policy errors. The magnitude of revisions appears to be contingent on the vintage of the data which is used to generate updated estimates. This is especially relevant for the LW models, where additional control variables in their Phillips curves tend to be more volatile during economic downturn triggering larger revisions of the initial estimates in the future. To answer this question I examine a full history of revisions r for estimates made in real time around the turning points in the U.S. business cycles. Real-time estimates of r retrieved at the start of the 2008 recession undershoot final estimates produced with the 2017:2 data vintage by about percentage points in LW and HLW specifications. However, if revisions remained relatively stable within the HLW models, revisions to the LW estimates of r jumped to as high as a half of a percentage point by mid-2010 or two years after the initial real-time measurements were made and then fell back closer to the size of the revisions in the HLW specifications. Meanwhile, revisions of the real-time measurements of r spanning the period of the 2008 recession are negative, indicating that the initial real-time estimates tended to overstate the natural rate of interest relative to measurements obtained in subsequent quarters. However, this measurement error was smaller for the HLW1 and HLW2 models - less than 0.1 percentage points versus nearly a half of a percentage point overestimation in LW and HLW3 specifications. These results point to several considerations that have to be taken into account when the natural rate of interest is estimated in real time. First, the inclusion of additional control variables, such as oil prices and imports prices shocks in the LW specifications, affects both the estimates of r as well as how real-time measurements from past data vintages are revised over time. In particular, substantial deviations of control variables from their past averages trigger bigger updates of unobserved states in the recursive algorithms, such as the Kalman filter, where a larger weight is placed on more recent observations. Indeed, more parsimonious empirical models, such as, the HLW specifications, carry out the identification and updating of output gaps based on the arrival of new information on inflation rates alone - a relatively inertial macroeconomic time series. As a result, the extent to which initial measurements of the output gap are revised in those models will mostly depend on the size of revisions to the 19