Inflation and professional forecast dynamics: an evaluation of stickiness, persistence, and volatility

BIS Working Papers

No 713

 

Inflation and professional forecast dynamics: an evaluation of stickiness, persistence, and volatility

by Elmar Mertens and James M. Nason

 

Monetary and Economic Department

April 2018

 

JEL classification: E31, C11, C32

Keywords: inflation; unobserved components; professional forecasts; sticky information; stochastic volatility; time-varying parameters; Bayesian; particle filter.

 

This publication is available on the BIS website (www.bis.org).

 

© Bank for International Settlements 2017. All rights reserved. Brief excerpts may be reproduced or translated provided the source is stated.

ISSN 1020-0959 (print)

ISSN 1682-7678 (online)

 

Inflation and Professional Forecast Dynamics: An Evaluation of Stickiness, Persistence, and Volatility

Elmar Mertens† and James M. Nason

Current draft: February 16, 2018

First Draft: February 15, 2015

 

Abstract

This paper studies the joint dynamics of real-time U.S. inflation and average inflation predictions of the Survey of Professional Forecasters (SPF) based on sample ranging from 1968Q 4 to 2017Q 2. The joint data generating process (DGP) comprises an unobserved components (UC) model of inflation and a sticky information (SI) prediction mechanism for the SPF predictions. We add drifting gap inflation persistence to a UC model in which stochastic volatility (SV) affects trend and gap inflation. Another innovation puts a time-varying frequency of inflation forecast updating into the SI prediction mechanism. The joint DGP is a nonlinear state space model (SSM). We estimate the SSM using Bayesian tools grounded in a Rao-Blackwellized auxiliary particle filter, particle learning, and a particle smoother. The estimates show that (i) longer horizon average SPF inflation predictions inform estimates of trend inflation; (ii) gap inflation persistence is procyclical and SI inflation updating is frequent before the Volcker disinflation; and (iii) subsequently, gap inflation persistence turns countercyclical and SI inflation updating becomes infrequent.

JEL Classification Numbers: E31, C11, C32.

Key Words: inflation; unobserved components; professional forecasts; sticky information; stochastic volatility; time-varying parameters; Bayesian; particle filter.


  • email: elmar.mertens@bis.org, address: Bank for International Settlements, Centralbahnplatz 2, CH 4051 Basel, Switzerland.
  • email: jmnason@ncsu.edu, address: Department of Economics, Campus Box 8110, NC State University, Raleigh, NC 27695–8110 and the Centre for Applied Macroeconomic Analysis.
  • An earlier version of this working paper has been circulated as CAMA Working Paper 6/2015, Centre for Applied Macroeconomic Analysis, Crawford School of Public Policy, The Australian National University.

We thank Gregor Smith for several conversations that motivated this paper. We also received valuable comments from Todd Clark, Patrick Conway, Drew Creal, Bill Dupor, Andrew Filardo, Monica Jain, Alejandro Justiniano, and Wolfgang Lemke and suggestions from colleagues and participants at numerous seminars and conferences. Jim Nason thanks the Jenkins Family Economics Fund at North Carolina State University for financial support. The views herein are those of the authors and do not represent the views of the Bank for International Settlements.

 

1 Introduction

Central banks pay particular attention to inflation expectations. A good reason for this focus is that inflation expectations contain information about private agents’ beliefs about the under­lying factors driving observed inflation dynamics. We label these factors the inflation regime. For example, Bernanke (2007) argues that well anchored inflation expectations are necessary for a central bank to be able to stabilize inflation. However, since monetary policy makers lack direct knowledge of inflation expectations, they must infer such expectations from estimates of the inflation regime. These estimates often rely on realized inflation and combinations of financial market data, statistical and economic models, and forecast surveys.

This paper estimates inflation regimes from the joint data generating process (DGP) of realized inflation and the inflation predictions of professional forecasters grounded in a non­linear state space model (SSM). We tap a sample of inflation predictions from the Survey of Professional Forecasters (SPF) to extract the “beliefs” held by the average respondent about the (in)stability of the persistence, volatility, and stickiness of inflation. Average SPF inflation predictions are attractive for evaluating the SSM because, as Faust and Wright (2013), and Ang, Bekaert, and Wei (2007) observe, SPF inflation predictions often dominate model-based out of sample forecasts. This forecasting performance suggests that average SPF inflation predictions coupled with realized inflation harbor useful information to measure inflation expectations.

We study the joint DGP of realized inflation, nt, and average SPF inflation predictions by linking a Stock and Watson (2007) unobserved components (SW-UC) model of inflation to a version of the Mankiw and Reis (2002) sticky information (SI) model. The SW-UC model is useful for evaluating the impact of different types of shocks on inflation and inflation expectations. First, it decomposes nt into trend inflation, Tt, and gap inflation, st, which restricts the impact of permanent and transitory shocks on nt. When permanent shocks dominate movements in nt, the inference is that inflationary expectations are not well anchored. The SW-UC model also imposes stochastic volatility (SV) on the innovations of Tt and st. Trend and gap SV creates nonlinearities in inflation dynamics, which produce bursts of volatility in nt. Persistence is not

often imposed on st when estimating the SW-UC-SV model. We depart from this assumption by giving st drifting persistence in the form of a time-varying parameter first-order autoregression, or a TVP-AR(1). Drifting gap persistence is another source of nonlinearity in a SW-UC model, which can exhibit pro- or countercyclical changes. We label the extended version of the DGP of nt as the SW-UC-SV-TVP-AR(1) model.

Coibion and Gorodnichenko (2015) adapt a SI model to a setup in which forecasters update their rational expectations (RE) information set with a fixed probability 1-A. Averaging across forecasters defines the h-step ahead SI inflation prediction, Ftnt+h, h = 1, ..., H. The result is that the SI inflation prediction evolves as a weighted average of the lagged SI forecast, Ft-i nt+h, and a RE inflation forecast, Etnt+h, where the weights are A and 1-A. The result is the SI law of motion Ftnt+h = AFt-1nt+h + (1 - A)Etnt+h, where Ftnt+h updates at the frequency 1/(1 - A) on average. In this reading, A reflects the average forecaster’s beliefs about the persistence or stickiness of the inflation regime.

We innovate on the Coibion-Gorodnichenko static coefficient SI-law of motion by investing A with drift. The result is a nonlinear SI-law of motion Ftnt+h = AtFt-1nt+h + (1 - At)Etnt+h, where the TVP-SI parameter, At, evolves as an exogenous and bounded random walk (RW), At+1 = At + aKKt+1, and its innovation is drawn from a truncated normal distribution (TN), Kt+1 ~ TN (0, 1; At+1 e (0, 1)). The SI forecaster’s information set includes the innovation Kt when Ft-1nt+h is updated to Ftnt+h, which implies that At is also part of this information set.

A motivation for placing At in the SI-law of motion is to uncover evidence about changes in the beliefs that the average SPF participant holds about the inflation regime. Changes in these beliefs are embedded in observed movements of the average SPF participant’s h-step ahead inflation prediction, nff^h. We relate nfp^h to Ftnt,t+h by adding a classical measurement error, Zt,h, to set nlp+h = Ftnt+h + vz,hZh,t, where Zh,t ~ N(0, 1), h = 1, ..., H. The nlp+h observation equation, SI-law of motion, and RW of At form the SI-prediction mechanism.

The joint DGP of the SI-prediction mechanism and SW-UC-SV-TVP-AR(1) model maps shocks to Tt, st, and SI state variables into movements in nt and nff/h. Estimates of the joint DGP provide evidence about drift in At and its co-movement with the SVs of Tt and at and drifting persistence in st. If At exhibits meaningful statistical and economic time variation and it moves with the SVs or drifting inflation gap persistence, we find evidence that shifts in SI inflation updating are attuned to the hidden factors driving the inflation regime.

Another contribution is the sequential Monte Carlo (SMC) methods that we use to estimate the joint DGP of the SI-prediction mechanism and SW-UC-SV-TVP-AR(1) model. These methods consist of the particle learning estimator (PLE) of Storvik (2002) and the particle smoother (PS) of Lindsten, Bunch, Sarkka, Schon, and Godsill (2016). The PLE and PS rely on a Rao-Blackwellized auxiliary particle filter (RB-APF). Our joint DGP is susceptible to Rao-Blackwellization because Tt, at, and the SI state variables form a linear SSM for given realizations of the nonlinear state variables — which are trend and gap inflation SVs, drifting inflation persistence, and At+1, and estimates of the static coefficients of the SI-prediction mechanism and SW-UC-SV-TVP- AR(1) model. Applying the Kalman filter (KF) produces estimates of the distribution of the conditionally linear states that are integrated analytically, which increases the efficiency of the RB-APF. The RB-APF estimates the nonlinear states by simulation.

We estimate the joint DGP of the SI-prediction mechanism and SW-UC-SV-TVP-AR(1) model on a quarterly sample from 1968Q4 to 2017Q2. The sample matches nt with the GNP or GDP deflator inflation available to the SPF in real time at date t. The average SPF inflation prediction is denoted nf^^h, where H = 5 or h = 1, ..., 1- to 5-quarter ahead forecast horizons.

Given only a sample of |nt, nf[+1, ..., nf[f5j 1 and our priors, the SSM yields posterior estimates of the beliefs that the average SPF participant has about the hidden factors underlying

the inflation regime. Our estimates of trend inflation are aligned with average SPF inflation predictions, especially at longer horizons. Gap inflation is more volatile before the Volcker disinflation than afterwards. There is a spike in gap inflation SV during the 1973-75 recession while trend inflation SV displays peaks during the 1981-82 and 2007-09 recessions. The drift in gap inflation persistence is procyclical before the Volcker disinflation, turns countercyclical afterwards, disappears by the 2007-09 recession, and returns to pre-2000 rates by 2014. The average SPF participant updates SI inflation forecasts frequently from the late 1960s to 1988. The frequency of SI inflation updating falls from 1990 to 1995 and then remains steady until 2007. During the 2007-09 recession, SI inflation updating occurs more frequently and drops slowly afterwards. Thus, movements in the frequency of SI inflation updating displays co­movement with trend inflation, its SV, and drifting inflation persistence. We conclude that the beliefs of the average SPF respondent are sensitive to the impact of permanent shocks on the conditional mean of inflation and that the Volcker disinflation marks the moment at which the behavior of trend inflation, its SV, and the cyclicality of the drift in inflation gap persistence changed.

The structure of the paper is as follows. In section 2, we build a SSM of the joint DGP of nt and nSpFh, h = 1, . ., H. Section 3 discusses the SMC methods used to estimate the SSM. Results appear in section 4. Section 5 offers our conclusions.

2 Statistical and Econometric Models

This section describes the statistical and economic models used to estimate the joint dynamics of nt and nj"pFh, h = 1,..., H. Stock and Watson (2007) is the source of the statistical model to which we add drifting persistence to st. The economic model is a SI-prediction mechanism that has a SI-TVP parameter. Drift in inflation persistence and the frequency of SI inflation updating create nonlinearities in the state transition dynamics of the SSM. The SI-TVP also interacts with trend and gap inflation SVs to produce nonlinearites in the impulse structure of the SSM.

2.1 The SW-UC Model

The SW-UC model generates nt. Stock and Watson (2010), Creal(2012), Shephard (2013), Cogley and Sargent (2015), and Mertens (2016) have estimated versions of the model in which SV in innovations to Tt and st is the source of nonlinearity in nt. We add an additional nonlinearity to the SW-UC-SV model in the form of drift in the persistence of st created by a TVP-AR(1). We collect these features into the SW-UC-SV-TVP-AR(1) model

where measurement error on nt, Zn,t, is uncorrelated with Tt and st and the innovations, nt and ut, these innovations are afflicted by SV, which evolve as RWs in ln g^,t+1 and ln qU,t+1, drifting persistence in £t+1 is tied to 9t+1, restricting the RW of 9t+1 e (-1, 1) ensures stationarity of £t+1 at each date t + 1, and innovations to the linear state variables, nt and ut, and innovations to nonlinear state variables, %n,t+1, 50,t+1, and &t+1, are uncorrelated.

A special case of the SW-UC-SV-TVP-AR(1) model gives a result about forecasting traced to Muth (1960). Shut down SV, an = Qn,t and = g0,t, and eliminate gap inflation persistence, 9t = 0, for all dates t. The result is a fixed coefficient SW-UC model with an IMA 1,1 reduced form, (1 - L)nt = (1 - wL)vt, where the MA1 coefficient w e (-1,1), L is the lag operator, nt-1 = Lnt, and the one-step ahead forecast error vt = nt + st + Tt - Tt-1|t-1.[1] The IMA(1,1) implies a RE inflation updating equation, E {nt+11 nt, an, a0} = (1 - Wnt + wE {nt | nt-1, an, a0}, where nt is the date t history of inflation, nt, ..., n1.

Stock and Watson   (2007),   Grassi and Prioietti (2010) and Shephard (2013) note the SW-UC- SV model replaces w with the      time-varying local weight wt in the reduced form IMA(1, 1). The result is a exponentially weighted moving average (EWMA) updating recursion or smoother

in which the discount wt adjusts to changes in the latest data, where nWit = (1 - wt)/wt.

2.2 The SI-Prediction Mechanism

This section begins by reproducing the SPF observation equation, the nonlinear Sl-law of motion, and the random walk law of motion of At. These elements form the system of equations

where Etnt+h is conditional on the       average SPF participant’s statistical model of inflation and At e (0, 1) for all dates t. Equations (3.1)-(3.3) define the Sl-prediction mechanism through which shocks to At and movements in other state variables generate fluctuations in nj*pFh.

The Sl-law of motion (3.2) implies a EWMA smoother. Iterate (3.2) backwards, substitute the result into (3.2), and repeat the process many times to produce the SI-EWMA smoother

where the discount rate is the SI-TVP, At, and HA,t = (1 - At)/At. The SI-EWMA smoother (4) nests the RE forecast, limAt-^0Ftnt+h = Etnt+h, and the pure SI update, limAt-^1 Ftnt+h = j VA,t-i (nJf=1 At-() Et- jnt+h. The former limit shuts down SI as At falls to zero because the discount on Et-jnt+h increases with j. In this case, SI inflation forecast updates rely only on Etnt+h period by period. At the other extreme, less weight is placed on Etnt+h and more on Et-jnt+h, j > 1, as At rises to one. Thus, Ft-1nt+h summarizes the SI inflation forecast.

Between these polar cases, shocks to At alter the discount applied to the history of Etnt+h in the SI-EWMA smoother (4). This information aids in identifying movements in with

respect to innovations in At. The EWMA smoother (2) shows a similar relationship exists be­tween Etnt+h, nt, and the time-varying discount generated by c;n,t, ^0,t, and dt. This gives us several sources of information to identify movements in nt and rf^+h within the joint DGP of the Sl-prediction mechanism and the SW-UC-SV-TVP-AR(l) model.

2.3 The State Space Model of the Joint DGP

Drift in inflation gap persistence complicates building a SSM for the joint DGP of the Sl-prediction mechanism and SW-UC-SV-TVP-AR(l) model. The SSM rests on the RE and SI term structures of inflation forecasts for which the latent factors are the RE state variables Xt = [rt st]' and SI analogues FtXt = [Ftrt Ftst]'. The problem is the law of iterated expectation (LIE) cannot be employed to create predictions of Xt+h or Ft Xt+h because forecasts of dt are needed. Instead, we construct RE and SI term structures of inflation forecasts in the presence of drifting gap inflation persistence by invoking the anticipated utility model (AUM).

The RE term structure of inflation forecasts is based on the observation and state equations of the SW-UC-SV-TVP-AR(l) model. The observation equation (1.1) of the SW-UC-SV-TVP-AR(l) model links nt to Tt, st, and Zn,t, which can be rewritten as

 

 

where 5X = [1 1. The state equations of the SW-UC-SV-TVP-AR(l) model are created by placing the random walk (1.3) below the TVP-AR(l) (1.3)

We appeal to two aspects of the AUM to solve the problem. The AUM resurrects the LIE by (i) assuming agents are ignorant of the true DGP and (ii) treating the TVPs of the joint DGP as fixed (locally) at each date. These assumptions are instructions to hold TVPs at their date t values within RE and SI forecasts that condition on date t information.

The AUM restricts the impact of drifting inflation gap persistence on these RE forecasts by conditioning dt on date t information.

Next, we show the SI term structure of inflation forecasts is built on the SI-EWMA formula (4), RE term structure of inflation forecasts (6), and a conjecture about the law of motion of the SI vector Ft+1Xt+1. The SI-EWMA formula (4) depends on the RE inflation forecasts Et-J-nt+h. Since these RE forecasts are Et-Jnt+h = 5X©h+jXt-j under the AUM, other RE forecasts are needed to replace Et_jUt+h in the SI-EWMA smoother (4). Our solution assumes the average member of the SPF fixes drift in inflation gap persistence at its current value when iterating SI recursions back­wards. Under this assumption, Et-Jnt+h = 5X@h+jXt-j in the SI-EWMA smoother (4). The result is Ftnt+h = 5X®h\t X7=o t-j (n^=0 At-^ \ tXt-j. Next, we conjecture the law of motion of the SI state vector is FtXt+h = (1 - At) EtXt+h + AtFt-1Xt+h. An implication is the SI-EWMA smoother FtXt+h = XJ=0 FA,t-j (n^=0 At-^ Et-jXt+h. Condition on the date t drift in inflation persistence on date t information, 0t|t, to find FtXt+h = Xj=o F\,t-j (n^=0 Xt-^ ©h+jXt

The online appendix has details about the SI term structure of inflation forecasts (7).

The online appendix also develops state equations for Ft+iXt+i. Remember the SI-EWMA smoother of Ft Xt is Xj=o FAt-j (UJ£=0 h-l) j Xt- j, which by induction gives a law of motion, FtXt = (1 - At)Xt + At©t\tFt-1Xt-1. We create state equations for Ft+1Xt+1 by pushing this law of motion forward a period and substituting for Xt+1 with the state equations (5.2) of the SW-UC-SV-TVP-AR(i) model. Stack the latter equations on top of the former to obtain the state equations of the SSM of the joint DGP

the conditioning time subscript on 0t+1 is dropped. The state equations (8.1) show shocks to At alter the transition and impulse dynamics only of FtTt and Ftst. Changes in dt shift the transition dynamics of all elements of St while its impulse dynamics react to £n,t, and c,o,t.

We complete the SSM by constructing its observation equations. First, replace Ftnt+h in the SPF measurement equation (3.1) with the SI term structure of inflation forecasts (7) for h = 1, ..., H. Place the results below the observation equation (5.1) of the SW-UC-SV-TVP-AR(1) model to form the SSM’s observation equations

[Zn,t Z1,t ... ZH,t~\ , and QU = DD'. The SSM integrates Ftnt+h out of the observation equations (8.2) using the SI term structure of inflation forecasts (7). As a result, Ftst produces mean reversion in nfpFh while permanent movements are tied directly to FtTt and 0t and indirectly to gn,t, Qu,t, At, and 0t. The direct response of nfpFh to 0t is produced by the observation equations (8.2). Drift in 0t also alters transition dynamics in the state equations (8.1), which generates movements in FtTt and Ftst, and hence, nf^^h.

3 Econometric Methods

We combine a RB-APF algorithm adapted from Lopes and Tsay (2011) with the PLE of Storvik (2002) to estimate the SSM (8.1) and (8.2); also see Carvalho, Johannes, Lopes, and Polson (2010), Creal(2012), andHerbst and Schorfheide (2016). The RB-APF and PLE produce filtered estimates of Tt, at, FtTt, Ftst, gn,t, Qu,t, At, and dt. Lindsten, Bunch, Sarkka, Schon, and Godsill (2016) give instructions for a PS that generates smoothed estimates of these state variables.

3.1 Rao-Blackwellization of a Nonlinear State Space Model

 We generate updates of the nonlinear states by simulating the multivariate RW process

 

 

where = [07J t+1 =%u,t+1 t+1  Kt+1]  The RB-APF uses the KF to create an analytic distribution of St using the SSM (8.1) and (8.2), given simulated values of Vt. Analytic integration endows the RB-APF estimator of the linear state variables with greater numerical efficiency.

3.2 Priors and Initial Conditions

We posit priors for the static volatility parameters and initial conditions to generate synthetic samples of linear and nonlinear states using the SSM (8.1) and (8.2) and multivariate RW (9). The static scale volatility parameters are collected in T = [ajj aU a^ a2 a^ n a^ 1 ... a^ 5] . Priors on T are grounded in restrictions of the joint DGP of the Sl-prediction mechanism and SW-UC-SV-TVP-AR(l) model while remaining consistent with the PLE of Storvik (2002). The PLE requires priors for T to have analytic posterior distributions. The posterior distributions serve as transition equations to update or “learn” about the joint distribution of St and T.

Table 1 lists our priors for the static volatility parameters found in T. We endow these parameters with inverse gamma (JS) priors. Columns labeled a£ and $£ denote the scale and shape parameters of the JS priors of the elements of T, the mean is 0.5^/(0.5 a£ - 1), and the two right most columns display the associated 2.5 and 97.5 percent quantiles, where £ = n, u, 4>, k, Zn, Zh, and h = 1, ..., 5.

Priors on the static volatility coefficients are o£ ~ JS ^y-g,where  a£ and Pr are scale and shape parameters, £ = n, u, $, k, Zn, Zh, and h = 1, ..., 5

Two features are worth discussing about our priors on the scale volatility coefficients of Dg. First, we give j2 and aU prior means equal to 0.04. These prior means are larger than the prior mean of 0.01 placed on jJ and a^. Second, our priors on a,, aU, a,, and aK deliver 2.5 and 97.5 percent quantiles that exhibit greater variation in innovations to ln G,t+1 and ln gU t+1 compared with variation in innovations to dt+1 and At+1. Nonetheless, the 2.5 and 97.5 percent quantiles of a, n, a, 1, ..., a, 5 reveal our belief that volatility in the measurement errors of nt and nfpFh dominate shock volatility in the joint DGP of the SI-prediction mechanism and SW-UC-SV-TVP-AR(1) model.

Priors on initial conditions of the linear state variables appear in the left two columns of table 2. We draw t0 and F0t0 from normal priors with a mean of two percent, which is about the mean of GNP deflator inflation on a 1958Q1 to 1967Q4 training sample. A variance of 1002 indicates a flat prior over a wide range of values for t0 and F0t0. The joint prior of f0 and F0e0 is drawn from N (02x1, $0), which equates the prior means to zero (i.e., unconditional means). Prior variances are produced by the ergodic bivariate normal distribution of particle draws of gUt0, 00, and A0; see the notes to table 2. We also restrict priors on t0, f0, F0t0, and F0e0 by splitting the training sample variance of the first difference of GNP deflator inflation between trend (one-third) and gap (two-thirds) shocks.

The last two columns of table 2 lists priors on initial conditions of the nonlinear state variables. We endow priors of ln gU,0 and ln g,,0 with normal distributions. Prior means are calibrated to pre-1968 inflation data similar to Stock and Watson (2007). Uncertainty about ln gU,0 and ln g,,0 is reflected in prior variances of ten. Table 2 shows that d0 is drawn from a standard normal, subject to truncation at (-1, 1, and another truncated normal bounds A0 e 0, 1 with (untruncated) mean of 0.5 and a unit variance. These priors are in essence uninformative about values inside the bounds.

3.4 The Particle Learning Estimator

Section 3.1 applies the RB process to the SSM (8.1) and (8.2). This process increases the numer­ical efficiency of the estimator of the linear states, St, by shrinking Monte Carlo error. Another method to improve the efficiency of this estimator is the APF of Pitt and Shephard (1999, 2001). In this section, we sketch a RB-APF to estimate the linear and nonlinear states that begins from algorithm 2 of Lopes and Tsay (2011, p.173); also see Creal(2012, section 2.5.7).[1] The online appendix provides a complete exposition of our implementation of the RB-APF.

A RB-APF obtains estimates of the likelihood by running the prediction step of the KF on the SSM (8.1) and (8.2) particle by particle. At date t, the KF predictive step produces the log likelihood, 1(i), and particle weights, = expj/Xj expjl , i = 1, ..., M. Stratified resampling of [wt ^ 1 yields indexes that are used to regroup St-1 |t-1, its mean square error (MSE), ■E(-1|t_1, and V(i); see steps 3(a) and 3(b) of section A3.1 of the online appendix and Hol, Schon, and Gustafsson (2006). This step aims to prevent a particle from receiving all the

probability mass as M becomes large. The ensemble of weights [wyt j 1 are also resampled generating [wt ^ 1; see step 3(d) of section A3.1 of the online appendix. The resampled particles S(_)1|t_1, Z[_)1|t-1, and V(ti) are employed in the entire KF to update j and produce new weights w(tl) = exp j^ j jXM exp j l tl)}, i = 1, ...,M; see step 3(e) of section A3.1 of the online appendix. By simulating the multivariate RW (9), the nonlinear states are updated to V(+)1 across the M particles. Estimates of St|t, £t|t, and Vt|t rely on the weights w(ti) = w(ti)/w^; see step 4 of section A3.1 of the online appendix.

Section 4reports estimates of the joint DGP of the Sl-prediction mechanism and SW-UC-SV-TVP-AR(1) model. Its estimated log likelihood is compared with the log likelihood of a joint DGP estimated conditional on setting dt = 0 or estimating a constant SI parameter, Xt = A. Thus, we use log likelihood (11) to evaluate competing joint DGPs, but only after marginalizing T. The next section discusses the PLE used to estimate T. We estimate the joint posterior distribution of St, Et, Vt, and T by embedding the RB-APF in the PLE of Storvik (2002), given priors on the joint DGP of the Sl-prediction mechanism and SW- UC-SV-TVP-AR(1) modelThe PLE rests on two insights. First, choosing conjugate priors for T yields an analytic solution of its posterior distributions. The posterior distribution is recovered conditional on the states and sample data. The idea is to draw T from particle streams of a vector of sufficient statistics, r(i) that depend on V(ji), given Y1:t. Since the sufficient statistics are grounded in the IG priors of T, the mapping to the analytic posterior distributions is a system of transition equations that simulate M particles to learn about or update from rt(_\ to rt(i). The transition equations are appended to the process that draws V^ to sample T(i) ~ P (t| rt(i)), which in essence equates P (t | Y 1:t, V(^ to P (t| Tt(i)). We denote the system of transition equations r(i) = |(rt(_\, Y1:t, V(ti), V(ti_1), i = 1, ..., M.

Second, the PLE marginalizes T out of the posterior of the states produced by the RB-APF. The idea is to update r(i) at the same time the RB-APF generates §ti), 2(i), and Vti). Thus, T is estimated by the PLE jointly with St|t, 2t|t, and Vt|t.

As noted, we place IG priors on T to expedite Storvik’s PLE. The priors, which are reviewed in section 3.2 and table 1, are a£ ~ IG ^, where £ indexes the elements of T. The IG priors are useful because the associated posterior distributions are solved analytically. For example, the posterior distribution of the static volatility coefficient of the RW of dt+1 is a£(i) ~IG I  -2t, ~2lt I, where    at   =  at_1    + t_1 and        = X£=1 [_ ^£i_^ . The process generating I, where the shapeparameter is a sufficient statistic for a^We extend the idea of identifying Pj t as sufficient statistics to the entire collection of static volatility parameters in T.

The online appendix gives procedures to simulate and update f(tt, f(tt, f(cp>t, and f(tt in steps 2 and 3.(a) of the RB-APF algorithm. The algorithm samples a2^, affl, a4>(tt), and a2^ from particle streams of sufficient statistics. The law of motion of sufficient statistic matches the transition equation P(g t-1 , Y1:t, V(tt), Vt—i), for P = n, u, F, and k.

This leaves us to describe the routines that sample the measurement error scale volatil­ity parameters, a^ n and a^ h, h = 1, ..., 5. Since these variances lack laws of motion that can be employed to build transition equations, the relevant shape parameters are updated on information obtained from KF operations of the RB-APF. For example, we sample a^h | Y 1:t ~IG I at, I, where updates of p^h t are calculated using information from step 3.(b) of the RB-APF; see the online appendix. Thus, updates of the shape parameters of the posterior distributions of a2n and a^ h, which are the sufficient statistics P^n t and Px^ht, are driven by the KF prediction error of Y t weighted by the “gain” of these innovations.

3.5 A Rao-Blackwellized Particle Smoother

Lindsten, Bunch, Särkkä, Schön, and Godsill (2016) develop an algorithm to compute smoothed estimates of St and Vt, given Y1:T and Y. The algorithm is a forward filter-backward smoother (FFBS) for SSMs amenable to Rao-Blackwellization. The forward filter is the RB-APF described in section 3.3 and online appendix. The FFBS applies Rao-Blackwellization methods moving from date T to date 1 to generate smoothed estimates of Vt conditional on forward filtered particles. Forward filtering operations are conducted using the SSM (8.1) and (8.2) to produce smoothed estimates of St, given smoothed estimates of VtLindsten, Bunch, Särkkä, Schön, and Godsill (LBSSG) refer to the entire process as a forward-backward-forward smoother.

The RB-PS operates only on the nonlinear states of the joint DGP of the SI-prediction mechanism and SW-UC-SV-TVP-AR(1) model. The problem is, when moving backwards from date t to date t-1, smoothing Vt can cause its Markov structure to be lost. A reason is that marginalizing the linear states produces a likelihood that depends on V1:t rather than Vt.

LBSSG solve this sampling problem by decomposing the target density P (V1:T | Y1:T; y) into p(^1:t IV 1:t+1, Y 1:T; ’y) P (Vt+1:T | Y1:T; y) . Drawing from p(Vt+1:T I Y 1:T; y) yields an in­complete path of the approximate smoothed nonlinear states from date t +1 to date T, which is denoted Vt+1:T

The aforesaid factorization of P (V1:r | Ynr; Y) is also useful because there is information in P (v1:t |v1:t+1, Y1:r; t) about the probabilities (i.e., normalized weights) needed to draw smoothed nonlinear states. Gaining access to this information is difficult because the condi­tional density of V1:t is not easy to evaluateLBSSG’s propose simulation methods to perform the backward filtering implicit in P (V1:t |v1:t+1, Ynr; Y). This density can be decomposed into P (V 1:t |V1:t + 1, Y 1:r; Y) « P (Yt + 1:r, Vt + 1:r |V1:t, Y1:t; Y) P (V1:t | Y1:t; Y) , where the object of interest is the predictive density P^+nr, Vt+1:r |v1:t, Y1:t; t). LBSSG P (Yt+1:r, Vt+1:r | St, Vt; Y) P (St | Y1:t, V1:t; y) dSt. Hence, run the KF forward to obtain estimates of St and Et by drawing from P [st | Y1:t, V1:t; Y). The mean and MSE of St are employed in simulations to generate sufficient statistics that approximate the density of the SSM (8.1) and (8.2), which when normalized are the probabilities of drawing a pathof V 1:r. The upshot is, although St does not enter P (V1:t |v1:t+1, Ynr; Y), the conditionally linear states are relevant for estimating the probability of sampling V 1:r. In a final step that is conditional on the path of V 1:t, the linear states are smoothed by iterating the KF forward.

4 The Data and Estimates​​​​​​​

We present estimates of the joint DGP of the Sl-prediction mechanism and SW-UC-SV-TVP-AR(1) model in this section. These estimates are compared with ones gleaned from joint DGPs that lack inflation gap persistence, dt = 0 or drift in SI updating At = AThe goal is to evaluate the impact of inflation gap persistence or SI on the dynamics of nt and njfFh, h = 1, ..., 5. The joint DGPs are estimated using a RB-APF, PLE, and PS that engage M = 100,000 particles. These estimates are used to study (i) comovement of Tt and FtTt with nt, and nf^+h, (ii) fluctuations in at and Ftst, (iii) the history of ^n>t and £Utt since the start of the sample, (iv) movements in dt and At over the business cycle, and (v) the contributions of Yt, nt, and nj^f+h to variation in Tt and FtTt .

4.1 The Data​​​​​​​

Our estimates rest on a sample of real-time realized inflation, nt, and h-step ahead average

SPF inflation prediction, nff^. We obtain the data from the Real-Time Data Set for Macroe­conomists (RTDSM), which is compiled by the Federal Reserve Bank (FRB) of PhiladelphiaThe data consist of observations from 1968Q4 through 2017Q2 for real-time realized inflation and average SPF inflation predictions.

Realized inflation is the RTDSM’s quarterly real-time vintages of the GNP and GDP de­flatorThese vintages reflect data releases that were publicly available around the middle of quarter t and most often the publicly available information contains observations through quarter t-1. We employ these vintages to compute the quarterly difference in the log levels of real-time observations on the GNP or GDP deflator, Pt. The quarterly price level data are transformed into inflation measured at an annualized rate using nt = 400 [lnPt - lnPt-1].

Average SPF inflation predictions include a nowcast of the GNP or GDP deflator’s level and forecasts of these price levels 1-, 2-, 3-, and 4-quarters ahead. These surveys are collected at quarter t without full knowledge of nt. We comply with this timing protocol by assuming the average nowcast, 1-quarter, ..., and 4-quarter ahead predictions, which are denoted nfp+1, nfp+2,. ., and nfp+5, are conditional on data available at the end of quarter t-1. These inflation predictions are the annualized log differences of the average SPF prediction of the deflator’s level and one lag of the real-time realized price level supplied by the RTDSM.

Figure 1 plots nt and four different average SPF inflation predictions. Plots of nt and the average SPF inflation nowcast, nfp+1, appear in figure 1(a). Realized inflation is also found in figure 1(b), but the 1-quarter ahead average SPF inflation prediction, nSp+2, replaces nfp+1. Figure 1(c) displays nt and the 3-quarter ahead average SPF inflation prediction, nSp+4, and figure 1(d) has nt and the 4-quarter ahead average SPF inflation prediction, nfp+75. The panels depict nt with a dot-dash (red) line and average SPF inflation predictions with a solid (blue) line. Vertical gray shaded bars denote NBER recession dates.

 

The plots reveal several features of nt and the average SPF inflation predictions. First, average SPF inflation predictions exhibit less variation than nt throughout the sample. Next, as h increases, average SPF inflation predictions become smoother and are centered on nt. All this suggests the average SPF surveys provide useful forecasts of inflation, which is a point made by Ang, Bekaert, and Wei (2007), Faust and Wright (2013), Mertens (2016), and Nason and Smith (2016a), among others.

Differences between the average SPF nowcast are 4-quarter ahead prediction contain infor­mation to identify Tt, st, FtTt, and Ftst. For example, the average SPF inflation nowcast peaks close to 10 percent during the 1973-75 recession and around the double dip recessions of the early 1980s as Figure 1(a) shows. The former peak in inflation falls moving from njf+2 to njf+5 in figures 1(b), 1(c), and 1(d). At a 4-quarter ahead horizon, the average SPF inflation prediction rises steadily from about three percent in the early 1970s to a peak greater than eight percent around the 1980 recessionOur estimates rely on this information, which is a function of the SPF inflation prediction horizon, to identify persistence, stickiness, and volatility in RE and SI trend and gap inflation.

4.2 Posterior Estimates of Ψ and Fit of the Joint DGPs​​​​​​​

Table 3 lists full sample estimates of T, T, for three joint DGPs. The DGPs combine the SI- prediction mechanism and SW-UC-SV-TVP-AR(1) model, SI-prediction mechanism and a SW-UC model in which no persistence, dt = 0, only SV drives gap inflation, and a fixed parameter, At = A, SI-prediction mechanism and the SW-UC-SV-TVP-AR(1) model.

The restrictions on inflation gap persistence and the frequency of SI inflation updating affect T in several ways. First, innovations to the RW of trend inflation SV are more volatile than innovations to the RW of gap inflation SV in the DGPs with drifting gap persistence because an > a}. However, an is larger while a} is smaller in the DGP that estimates dt and At. In contrast, a2 and a} are about equal in the DGP with 0t = 0 and close to the calibrated values Stock and Watson (2007) and Creal (2012) use to estimate the state of the SW-UC-SV model. Next, there is little variation in estimates of the scale volatility on innovations to the RWs of 0t and At, a^ and a2, across the DGPs in which these parameters appear. The DGPs with drifting gap persistence produce estimates of the scale volatility on the measurement errors of SPF inflation predictions, a^ h, h = 1, ..., 5, that are quantitatively similar. The converse is true for estimates of the scale volatility on the measurement errors of nt, a^ n, because it is nearly twice as large in the DGP that estimates 0t and At compared with the other two DGPs.

Estimates of log marginal data densities (MDDs) appear at the bottom of table 3 for the three joint DGPs. Equation (11) is used to calculate L(y | Y1:T), which is the log MDD for a joint DGP tied to Y. Standard errors of the log MDDs are beneath estimates of L(y|y1:T). The estimates of L(y | Y1:T), indicate the data have, at a minimum, a very strong preference for the joint DGP of the SI prediction mechanism and SW-UC-SV-TVP-AR(1) model. Hence, the rest of the paper reports evidence this joint DGP has for the stickiness, persistence, and volatility of Tt, Ft|tTt, Ft, and Ft\t£t.

Figure 2 plots the PLE paths of a2, a}, a^, and a2 consistent with the joint DGP favored by the data. The scale volatility parameters are plotted with solid (navy blue) lines and 68 and 90 percent uncertainty bands appear as dark and light shading in figures 2(a)-2(d). These figures show a2 more than doubles, a} falls by about a third, a2 rises by about a quarter, and a2 changes little from the start to the end of sample. The PLE path of a2 drifts up for much of the sample as seen in figure 2(a). However, the PLE paths of these parameters are smooth from the 2001 recession to the end of the sample. Also, the 68 percent uncertainty bands are tight for the most part in figure 2, but the 90 percent uncertainty bands are wider and on occasion display substantial variation.

The table presents posterior means of the elements of Y, which are calculated using the full sample at date T = 2017Q2. The values in brackets below the posterior means are 5 and 95 percent quantiles. The model in which the SI parameter is fixed yields the posterior mean A = 0.304 with 5 and 95 percent quantiles of 0.250 and 0.360 conditional on the data and priors. The log MDDs are computed using the formula for L(y | Y1:T) described by equation (11) in section 3.3. Volatility over the log MDDs are measured by standard errors that appear in parentheses. The estimates of the static scale volatility parameters and log marginal data densities are created using M = 100,000 particles.

4.3 Trend and Gap Inflation​​​​​​​

Figure 3 contains nt, the average SPF inflation nowcast and 4-quarter ahead inflation prediction, ntt-+1 and -nff+5, filtered RE trend inflation, Tt|t, filtered SI trend inflation, Ft|tTt, filtered RE gap inflation, ft|t, and filtered SI gap inflation, Ft|tft, on the 1968Q4 to 2017Q2 sample. Plots of n5f+f1 , Ft|tTt, and its 68 percent uncertainty bands are in figure 3(a). Figure 3(b) is similar, but replaces nf[+1 with nt,t+5. In these figures, solid (blue) lines are average SPF inflation predictions and Ft|tTt is the dotted (black) lines. Figure 3(c) displays Tt|t with a dash (green) line, Ft|tTt with a dotted (black) line, and nt with a dot-dash (red) line. Estimates of RE and SI gap inflation appear in figure 3(d) as a dashed (green) line, ft|t, and dotted (black) line, Ft|tft.

Estimates of SI trend inflation are informed by the 1973-75 recession, inflation surge of the late 1970s and early 1980s, and Volcker disinflation.[1] In 1974Q4, figure 3(a) displays a spike in nf[+1 of nearly 10 percent, but Ft|tTt is only 3.8 percent. At the same time, nf[+5 is 6.1 percent. The peaks in nf[+5 and Ft|tTt, which occur a year and a half later, are close to 6.5 percent. The next peaks in nf[+1 and nff+75 are 9.5 in 1979Q4 and 8.3 percent in 1980Q1. However, only in 1981Q2 does Ft|tTt peak at 7.5 percent. After 1983, nf[+1, nf[+5, and Ft|tTt fall steadily before leveling off in the late 1990s as figures 3(a) and 3(b) show. However, Ft|tTt often deviates from nff+J between 1983 and 2000. As a result, nf[+1 often is outside the 68 percent uncertainty bands of Ft|tTt during this period while nff+5 falls within the 68 percent uncertainty bands of Ft|tTt after the Volcker disinflation in figure 3(b).

Figure 3(c) has several interesting features. First, nt is volatile compared with Tt|t and Ft|tTt. Another striking aspect of figure 3(c) is Tt|t and Ft|tTt are nearly identical for much of the sample. This is not true for nt and Ft|tTt (or Tt|T) from 1968Q4 to 2000. For example, Tt|t and Ft|tTt are less than a third of nt during the first oil price shock. However, Ft|tTt explains much of the increases in nt and nf[+1 by the late 1970s and early 1980s. Hence, Tt|t and Ft|tTt respond slowly to the first oil price shock, but the inflation shock of the late 1970s and early 1980s produces quicker responses in Tt|t and Ft|tTt. Subsequently, nt is often less than Tt|t and Ft\tTt from 1983 to 2000. Beginning in 2003, Tt\t and Ft\tTt are often centered on nt.

The estimates RE and SI trend inflation are a counterpoint to studies in which gap inflation dominates movements in inflation; see Cogley and Sbordone (2008) among others. One reason is Tt\t and Ft\tTt condition on nf^, h = 1, ..., 5. This differs from studies that rely on univariate SW-UC-SV models; for example see Grassi and Prioietti (2010), Creal (2012), and Shephard (2013).

We plot st\t and Ft\tft\t in figure 3(d). These plots show st\t and Ft\tet are nearly identical for the 1968Q4-2017Q2 sample. These estimates of gap inflation rise from less than one percent in 1968Q4 to about 3.5 percent in 1970. Thereafter, st\t and Ft\tet turn negative before the 1973-75 recession, which coincides with the largest spikes in st\t and Ft\tst of nearly nine percent. These spikes are followed by st\t and Ft\tet falling to about -2.5 percent by 1976. From the late 1970s to 1981, st\t and Ft\tst range from about zero to 3.7 percent.

There are two more aspects of figure 3(d) worth discussing. First, st\t and Ft\tet are less volatile subsequent to the Volcker disinflation compared with the 1970s. After 1983, (the absolute values of) st\t and Ft\tet are never larger than three percent. Second, st\t and Ft\tet are often negative from 1983 to 2000, which leads the average SPF participant to expect an increase in future growth in realized inflation. Nelson (2008) explains this prediction is an implication of the Beveridge and Nelson (1981) decomposition, which is built into the SW-UC-SV-TVP-AR(1) model of the joint DGP. Hence, the average SPF participant believes the Volcker disinflation produced only a transitory drop in realized inflation.

Movements in Ft\tet have parallels in monetary policy. Remember the average SPF partic­ipant expects mean reversion in nt during the 1973-75 recession. However, in the late 1970s the average SPF participant believe unit root dynamics dominates nt. An explanation for this shift in the average SPF participant’s beliefs about the inflation regime is discussed by Meltzer (2014, pp. 1006-1007). He notes that in the 1970s U.S. monetary policy makers did not dis­tinguish permanent from transitory shocks. As a result, their responses to the first oil price shock contributed to unanchored inflation expectations by the late 1970s.

 

The Volcker disinflation is another example. After 1983, nt and Ft\tTt began to fall, but the drop in nt is steeper as figure 3(c) shows. These plots are consistent with mostly negative realizations for Ft\tst from 1983 to 2000 as in figure 3(d). As discussed previously, we assign these movements in Ft\tTt and Ft\tet to the average SPF participant expecting a temporary fall in nt during and after the Volcker disinflation. The assessment agrees with Goodfriend and King (2005) and Meltzer (2014, p. 1131). They argue households, firms, and investors expected only a transitory drop in inflation after 1983.

4.4 Trend and Gap Inflation Volatilities​​​​​​​

Estimates of filtered and smoothed trend and gap inflation SVs appear in figure 4. Figures 4(a) and 4(c) contain dotted lines, which are c;n,t\t (purple) and c;o,t\t (teal). Dot-dashed (purple and teal) lines are £n,t\T and £o,t\T in figures 4(b) and 4(d). These figures also include 90 percent uncertainty bands, which are thinner solid (black) lines.

Figure 4 makes several points about c;n,t\t, £o,t\t, c,nt\T, and £o,t\T. Figures 4(a) shows the largest peaks in £n,t\t occur in 1977, 1983, and 2009 while £o,t\t is dominated by a spike in 1975 in figure 4(c). Figures 4(b) and 4(d) display peaks in £n,t\T and c;o,t\T during the 1981-82 recession and in 1975, respectively. Hence, these plots are more evidence shocks to gap infla­tion dominate movements in nt and nSp+h during the 1973-75 recession, but in the inflation surge of the late 1970s and early 1980s permanent shocks are more important.

Another revealing feature of figures 4(a) and 4(c) is the behavior of SV around NBER dated recessions. The filtered SVs, £n,t\t and £o,t\t, often rise during or after a NBER recessions as depicted in figures 4(a) and 4(c). There are peaks c;n,t\To,t\T) during the 1990-91 and 2007-09 (1981-82, 1990-91, 2001, and 2007-09) recessions.

Figure 4(b) and 4(d) are also informative about the long run behavior of £n,t\T and c;o,t\T. These SVs display steady declines for extended periods during the sample. The descent starts in 1983 for c,nt\T while this process starts in 1975 for £o,t\T.

 

Finally, our estimates show QntlT is smaller than £u,tiT for the entire sample. These es­timates differ from Grassi and Prioietti (2010), Stock and Watson (2010), Creal (2012), and Shephard (2013). These authors report trend SV dominates inflation gap SV from the 1970s well into the late 1990s. However, Creal and Shephard find that gap inflation SV is greater than trend SV after 2000.

4.5 Drifting Inflation Gap Persistence​​​​​​​

Figures 5(a) and 5(b) display filtered and smoothed estimates of drifting inflation gap persis­tence, 0t|t and 0t|T. Dotted and dot-dash (orange) lines denote 0t|t and 0t|T. Surrounding 0t|t and 0t|T are 68 and 90 percent uncertainty bands in the dark and light gray shaded areas. Figures 5(c) and 5(d) plot the absolute value of smoothed inflation gap persistence, | 0t|T |, and accumulated changes of this absolute value, | 0t|T | - | 0i|T |. These plots depict | 0t|T | and | 0t|T | - | 01|T | with dot-dashed (orange) lines, where the dark and light gray shaded areas are 68 and 90 percent uncertainty bands.

There is co-movement between 0t|t and 0t|T with NBER dated cycles in figures 5(a) and 5(b). The co-movement is procyclical during the 1969-70,1973-75, and 1980 recessions. These recessions see peaks in 0t|t and 0t|T while there are troughs between these recession. Post- 1981, 0t|t and 0t|T turn countercyclical. Filtered and smoothed estimates of drifting inflation gap persistence peak between the recessions of 1981-82, 1990-91, 2001, and 2007-09 while these recessions see troughs in 0t|t and 0t|T.

Uncertainty bands of 0t|t and 0t|T also appear in figures 5(a) and 5(b). The 90 percent quantiles of 0t|T (0t|t) cover zero in 1971-72, 1990-91, and 2006-14 (1968-69, 1972-73, 1975, 1976-78, 1983, 1990-93, and 2003-14). Hence, we infer there are episodes in which inflation gap persistence is zero. These results are similar to evidence presented by Cogley, Primiceri, and Sargent (2010). They find inflation gap persistence drops after 1983. However, our evidence is tied to procyclical troughs in 0t|T before 1983 and to the 2007-09 recession and its aftermath, which occurs more than 20 years after the Volcker disinflation.

 

Another take on the statistical and economic significance of drifting gap inflation persis­tence appears in figure 5(c). This figure displays the absolute value of dt\T, | dt\T |. The plot of | dt\T | gives evidence similar to that found in figure 5(b). There is evidence of a shift in business cycle behavior of | dt\T | around the Volcker disinflation. Drift in the absolute value of inflation gap persistence also declines steadily from the late 1990s to 2013.

There remains the inference problem that 9t\t, 9t\T, and | 9t\T | are not necessarily infor­mative about the statistical and economic content of changes in drifting inflation gap persis­tence during the sample. We address this problem by plotting accumulating changes in | 9t\T |, | 9t\T | - | 91\T |, in figure 5(d). Figure 5(d) shows these changes have tighter uncertainty bands compared with the plots in figures 5(a), 5(b), and 5(c). Nonetheless, the path of | 9t\T | - | 91\T | continues to show peaks that coincide with pre-1981 recessions and troughs occurs between these recessions. The opposite is observed post-1981.

Hence, figure 5 gives evidence that dates a switch from procyclical to countercyclical drift in inflation gap persistence to 1981. This break is consistent with an argument made by Meltzer (2014, p. 1006 and p. 1207). He contends there was a shift in the pattern of U.S. inflation persistence because of changes to the way the Fed operated monetary policy in the 1980s and 1990s compared with the 1970s.

4.6 Time Variation in the Frequency of SI Updating​​​​​​​

Figure 6 presents filtered and smoothed estimates of the time variation in the frequency of SI updating, At\t and At\T. These panels plot At\t and At\T as dotted (light green) and dot-dashed (brick) lines. The thin solid (brick) lines denote 90 percent uncertainty bands of At\T and 90 percent uncertainty bands of At\t are depicted with light gray areas. Figures 6(b) and 6(d) plot accumulated changes in At\T, At\T - A1\T. In these panels, dark and light gray areas are 68 and 90 percent uncertainty bands of At\T - A1\T. The top row of figure 6 has At\t, At\T, and At\T - A1\T estimated using the joint DGP of the SI-prediction mechanism and the SW-UC-SV- TVP-AR(1) model. Figures 6(c) and 6(d) report similar estimates, but the SW-UC-SV model lacks persistence in gap inflation, or 9t = 0 for all dates t.

Plots of At\t and At\T display a decade long swing from more frequent to less frequent updating beginning in the late 1980s in figure 6(a). From the late 1960s to the 1988, the average SPF inflation respondent is estimated to update almost every quarter to changes in Etnt+h because At\T varies between 0.01 and 0.35. However, there is uncertainty about these estimates because the 90 percent confidence bands of At\T range from 0.01 to 0.60.

Figures 6(a) also shows At\t and At\T reach a plateau from 1994 to 2007 before falling during the 2007-09 recession. From 1995 to 2008, At\t and At\T range between 0.50 and 0.70. The recession of 2007-2009 sees At\T (At\t) dropping to 0.25 (0.35). Subsequently, At\T (At\t) recovers to 0.47 (0.60) before 2017Q2. The filtered and smoothed estimates of At are also associated with substantial uncertainty. For example, when At\T plateaus in the late 1990s, the five percent quantile is as low as 0.20 and the 95 percent quantile is as high as 0.95. Fur­thermore, the 90 percent uncertainty bands of At\t and At\T remain wide in figure 6(a) as the sample moves past the recession of 2001, the “considerable” and “extended” period policy regimes of the Greenspan and Bernanke Feds of the early 2000s, the 2007-09 recession, and unconventional policy regimes of the Bernanke and Yellen Feds.

There are useful inferences to draw from At\t and At\T, even with the uncertainty sur­rounding these estimates. For example, infrequent SI inflation updating by the average member of the SPF lets the Fed engage in a policy of “opportunistic disinflation” during the 1990s as described by Meyer (1996) and Orphanides and Wilcox (2002). Orphanides and Wilcox argue that in the mid 1990s Fed policy makers advocated to wait for a state of the world in which there is little cost to monetary policy lowering inflationary expectations rather than to take actions during periods when the potential for a costly disinflation are large. However, since the joint DGP of the SI-prediction mechanism and SW-UC-SV-TVP-AR(1) model is the source of At\t and At\T, we only have estimates of the average SPF respondent’s beliefs about changes in the inflation regime and not evidence about shifts in the monetary policy regime

There is greater support for statistically and economically important time variation in the frequency of SI inflation updating in figure 6(b). This figure plots At\T - A1\T for the joint DGP in which there is drift in inflation gap persistence. In this case, the path of At\T - A1\T in figure 6(b) is similar to At\T displayed in figure 6(a) with respect to level and slope. Another interesting feature of figure 6(b) is the uncertainty bands surrounding At\T - A1\T. Figure 6(b) displays 90 percent uncertainty bands of At\T - A1\T that are narrower for the entire sample compared with the analogous confidence bands of At\T in figure 6(a). These estimates strengthen the case that changes in the frequency of SI inflation updating by the average member of the SPF are statistical and economic important.

This message is reinforced by figure 6(d). This figure presents estimates of At\T - A1\T conditional on a joint DGP in which there is no persistence in the inflation gap. Given 9t is zero, SI inflation updating is less frequent quarter by quarter, as depicted by At\t and At\T in figure 6(c) compared with the estimates found in figure 6(a). Although figure 6(c) suggests that there is useful information about the frequency of SI inflation updating conditional on 9t = 0, the plot of At\T - A1\T in figure 6(d) indicates otherwise. Figure 6(d) depicts At\T - A1\T as fluctuating around zero with 90 percent uncertainty bands that often contain zero under the joint DGP in which inflation gap has no persistence.

This section reports estimates of At\t, At\T, and At\T - A1\T shows that SI inflation updating by the average SPF respondent is statistically and economically significant for the last 48 years. These results agree with Coibion and Gorodnichenko (2015). Nonetheless, our estimates also reveal shifts in SI inflation updating during the sample. From the 1969 to 1988, the frequency of SI inflation updating occurred almost every quarter. The frequency declines to about once every two to three quarter until 2007, followed by a sharp increase during the 2007-09 recession. Afterwards, the frequency drops by 2017Q2. These shifts in estimates of SI inflation updating indicate the average SPF participant’s beliefs about the inflation regime changed within a few years of the end of the Volcker disinflation. The average SPF participant’s beliefs about the inflation regime also appear to have been altered by the recession of 2007-09.

4.7 SPF Inflation Predictions and Trend Inflation Uncertainty​​​​​​​

Figure 7 displays conditional volatilities of RE trend inflation, Tt, and SI trend inflation, FtTt. The plots quantify uncertainty over time in Tt and FtTt conditional on the history of Yt, or histories of subsets of its elements, smoothed estimates of the nonlinear states, Vt|T, and estimates of the static scale volatility coefficients, T. The measure of the volatility of Tt is Var(Tt | Y 1:t, Vt|T, T), where the entire information set runs from the first observation to quarter t, the smoothed nonlinear states begin at quarter t and end with quarter T, and estimates of the static scale volatility parameters are full sample. Similar computations are used to produce the conditional volatility of FtTt. Thus, the paths of the nonlinear states and parameter estimates are held fixed across changes in the sample data fed into the KF to produce estimates of the conditional volatilities of Tt and FtTt.

Figure 7(a) plots the conditional volatilities of Tt. The conditional volatilities of FtTt are found in figure 7(b). In these figures, the solid (black) line, dashed (blue) line, dotted (red) line, and dot-dashed (green) line areVar(x | Y1:t, Vt|T, T), Var(x | n1:t, Vt|T, T), Var(x | nfPF, Vt|T, T), and Var(x | n1:t, nfPF, Vt|T, T), respectively, where x = Tt, FtTt.

Figures 7(a) and 7(b) reveal nt and nff+5 jointly contribute the bulk of the informa­tion pertinent to estimate Tt and FtTt. The reason is the dot-dashed (green) lines of fig­ures 7(a) and 7(b) are always near the solid (black) lines. Hence, given only nt and nff+5, Var(Tt | nt, nS:ppFt+5, V)t|T, T) and Var(FtTt | nt, nf:f^+5, V)t|T, T) are close to the estimates con­ditioned on the entire information set, Var(Tt |Yt, V)t|T, T) and Var(FtTt |Yt, V)t|T, T). In con­trast, the dotted (blue) lines are far from the solid (black) and large dot-dashed (green) lines in the first half of the sample. Hence, prior to the Volcker disinflation, there is insufficient information in nt alone to estimate Tt and FtTt without also generating more variation in these estimates compared with estimates conditioning on either Yt or nt and nff+5. How­ever, conditioning only on nf f+5 produces substantial variation around FtTt that is manifested as large differences between plots of Var(FtTt | nf:p^+5, V}t|T, T) and Var(FtTt |Yt, V}t|T, T) or Var(FtTt| nt, n fPF+5, Vt|T, Ÿ) in figure 7(b).

​​​​​​​

5 Conclusions

This paper studies the joint dynamics of realized inflation and inflation predictions of the Sur­vey of Professional Forecasters (SPF). The joint data generating process (DGP) mixes a Stock and Watson (2007) unobserved components (SW-UC) model with the Coibion and Gorodnichenko (2015) version of the Mankiw and Reis (2002) sticky information (SI) model. The SW-UC model with stochastic volatility (SV) in trend and gap inflation is extended to include drift in inflation gap persistence. The SI law of motion is endowed with drift in the SI inflation updating pa­rameter. We estimate the joint DGP on a sample of real-time realized inflation and averages of SPF inflation predictions from 1968Q4 to 2017Q2. The estimator embeds a Rao-Blackwellized auxiliary particle filter into the particle learning estimator of Storvik (2002). Smoothed esti­mates of the state variables are constructed using an algorithm developed by Lindsten, Bunch, Sarkka, Schon, and Godsill (2016).

There are five key results to draw from our estimates. First, longer horizon average SPF inflation predictions provide useful information for estimating rational expectations (RE) and SI trend inflation and reducing uncertainty around these estimates. Second, RE and SI inflation gaps dominate inflation fluctuations during the first oil price shock. This is reversed during the late 1970s and early 1980s. Third, trend (gap) inflation SV falls steadily after 1983 (1975). We also find that inflation gap persistence is procyclical before 1981 and turns countercycli­cal afterwards. Fifth, changes in the frequency of SI inflation updating are statistically and economically important. The average SPF participant is updating SI inflation predictions often from the late 1960s through the late 1980s. Subsequently, the frequency of SI inflation updat­ing falls to levels associated with estimates reported by Coibion and Gorodnichenko (2015), among others, and remains low until the 2007-09 recession.

Our results fit into a literature represented by, among others, Krane (2011), and Nason and Smith (2016a, b). These authors find that the responses of professional forecasters to per­manent shocks are greater than for those to transitory shocks when revising their predictions. In the same way that this research inspired us, we hope that this paper stimulates further work on the ways in which professional forecasters and other economic agents process information to form beliefs and predictions about future economic outcomes and events.

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