**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: [email protected], address: Bank for International Settlements, Centralbahnplatz 2, CH 4051 Basel, Switzerland.
- email: [email protected], 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, n_{t}, 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 n_{t} into trend inflation, T_{t}, and gap inflation, s_{t}, which restricts the impact of permanent and transitory shocks on n_{t}. When permanent shocks dominate movements in n_{t}, the inference is that inflationary expectations are not well anchored. The SW-UC model also imposes stochastic volatility (SV) on the innovations of T_{t} and s_{t}. Trend and gap SV creates nonlinearities in inflation dynamics, which produce bursts of volatility in n_{t}. Persistence is not

often imposed on s_{t} when estimating the SW-UC-SV model. We depart from this assumption by giving s_{t} 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, F_{t}n_{t}+h, h = 1, ..., H. The result is that the SI inflation prediction evolves as a weighted average of the lagged SI forecast, F_{t-}i n_{t}+h, and a RE inflation forecast, E_{t}n_{t}+h, where the weights are A and 1-A. The result is the SI law of motion F_{t}n_{t}+h = AF_{t-1}n_{t}+h + (1 - A)E_{t}n_{t}+h, where F_{t}n_{t}+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 F_{t}n_{t}+h = A_{t}F_{t-1}n_{t}+h + (1 - A_{t})E_{t}n_{t}+h, where the TVP-SI parameter, A_{t}, evolves as an exogenous and bounded random walk (RW), A_{t}+_{1} = A_{t} + a_{K}K_{t}+*1*, and its innovation is drawn from a truncated normal distribution (TN), K_{t}+_{1} ~ TN (0, 1; A_{t}+1 e (0, 1)). The SI forecaster’s information set includes the innovation K_{t} when F_{t-1}n_{t}+h is updated to F_{t}n_{t}+h, which implies that A_{t} 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 F_{t}n_{t},_{t}+h by adding a classical measurement error, Zt,h, to set nl^{p}+_{h} = Ftnt+h + vz,hZh,t, where Zh,t ~ N(0, 1), h = 1, ..., H. The nl^{p}+_{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 T_{t}, s_{t}, and SI state variables into movements in n_{t} and nff/h. Estimates of the joint DGP provide evidence about drift in A_{t} and its co-movement with the SVs of T_{t} and a_{t} and drifting persistence in s_{t}. If A_{t} 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 T_{t}, a_{t}, 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 A_{t}+_{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 n_{t} 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 |n_{t}, nf[+_{1}, ..., nf[f_{5}j _{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 n_{t} 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 n_{t} and nj"^{pF}h, h = 1,..., H. Stock and Watson (2007) is the source of the statistical model to which we add drifting persistence to s_{t}. 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 n_{t}, Zn,_{t}, is uncorrelated with T_{t} and s_{t} and the innovations, n_{t} and u_{t}, 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 9_{t}+_{1}, restricting the RW of 9_{t}+_{1} e (-1, 1) ensures stationarity of £_{t}+1 at each date t + 1, and innovations to the linear state variables, n_{t} and u_{t}, and innovations to nonlinear state variables, %_{n},_{t}+_{1}, 5_{0},_{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, a_{n} = Q_{n},_{t} and = g_{0},_{t}, and eliminate gap inflation persistence, 9_{t} = 0, for all dates t. The result is a fixed coefficient SW-UC model with an IMA 1,1 reduced form, (1 - L)n_{t} = (1 - wL)v_{t}, where the MA1 coefficient w e (-1,1), L is the lag operator, n_{t-1} = Ln_{t}, and the one-step ahead forecast error v_{t} = n_{t} + s_{t} + T_{t} - T_{t-1}|_{t-1}.^{[1]} The IMA(1,1) implies a RE inflation updating equation, E {n_{t}+_{1}1 n^{t}, a_{n}, a_{0}} = (1 - Wn_{t} + wE {n_{t} | n^{t-1}, a_{n}, a_{0}}, where n^{t} is the date t history of inflation, n_{t}, ..., n_{1}.

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 w_{t} 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 n_{Wit} = (1 - w_{t})/w_{t}.

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

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

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, A_{t}, and HA,_{t} = (1 - A_{t})/A_{t}. The SI-EWMA smoother (4) nests the RE forecast, limA_{t-}^_{0}F_{t}n_{t}+h = E_{t}n_{t}+h, and the pure SI update, limA_{t-}^_{1} F_{t}n_{t}+h = j VA,t-i (n^{J}f=1 At-() Et- jn_{t}+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 E_{t}n_{t}+h period by period. At the other extreme, less weight is placed on E_{t}n_{t}+h and more on E_{t-}jn_{t}+h, j > 1, as A_{t} rises to one. Thus, F_{t-1}n_{t}+h summarizes the SI inflation forecast.

Between these polar cases, shocks to A_{t} alter the discount applied to the history of E_{t}n_{t}+h in the SI-EWMA smoother (4). This information aids in identifying movements in with

respect to innovations in A_{t}. The EWMA smoother (2) shows a similar relationship exists between E_{t}n_{t}+h, n_{t}, and the time-varying discount generated by c;_{n},_{t}, ^_{0},_{t}, and d_{t}. This gives us several sources of information to identify movements in n_{t} 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 X_{t} = [r_{t} s_{t}]' and SI analogues F_{t}X_{t} = [F_{t}r_{t} F_{t}s_{t}]'. The problem is the law of iterated expectation (LIE) cannot be employed to create predictions of X_{t}+h or F_{t} X_{t}+h because forecasts of d_{t} 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 n_{t} to T_{t}, s_{t}, and Zn,_{t}, which can be rewritten as

where 5_{X} = [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 d_{t} 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 F_{t}+_{1}X_{t}+_{1}. The SI-EWMA formula (4) depends on the RE inflation forecasts E_{t-J}-n_{t}+h. Since these RE forecasts are E_{t-J}n_{t}+h = 5_{X}©^{h}+^{j}X_{t-}j under the AUM, other RE forecasts are needed to replace E_{t}_jU_{t}+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, E_{t-J}n_{t}+h = 5_{X}@^{h}+^{j}X_{t-}j in the SI-EWMA smoother (4). The result is F_{t}n_{t}+h = 5_{X}®^{h}\_{t} X7=o t-j (n^=_{0} A_{t-}^ \ _{t}X_{t-}j. Next, we conjecture the law of motion of the SI state vector is F_{t}X_{t}+h = (1 - A_{t}) E_{t}X_{t}+h + A_{t}F_{t-1}X_{t}+h. An implication is the SI-EWMA smoother F_{t}X_{t}+h = XJ=_{0} FA,_{t-}j (n^=_{0} A_{t-}^ E_{t-}jX_{t}+h. Condition on the date t drift in inflation persistence on date t information, 0_{t}|_{t}, to find F_{t}X_{t}+h = Xj=o F\,_{t-}j (n^=_{0} X_{t-}^ ©^{h}+^{j}X_{t}

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 F_{At-j} (U^{J}_{£=0} h-l) j X_{t-} j, which by induction gives a law of motion, F_{t}X_{t} = (1 - A_{t})X_{t} + A_{t}©_{t}\_{t}F_{t-1}X_{t-1}. We create state equations for F_{t}+_{1}X_{t}+_{1} by pushing this law of motion forward a period and substituting for X_{t}+_{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 0_{t}+_{1} is dropped. The state equations (8.1) show shocks to A_{t} alter the transition and impulse dynamics only of F_{t}T_{t} and F_{t}s_{t}. Changes in d_{t} shift the transition dynamics of all elements of S_{t} while its impulse dynamics react to £_{n},_{t}, and c,o,_{t}.

We complete the SSM by constructing its observation equations. First, replace F_{t}n_{t}+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

[^{Z}n,^{t} Z*1*,_{t} ... Z_{H},_{t}~\ , and Q_{U} = DD'. The SSM integrates F_{t}n_{t}+h out of the observation equations (8.2) using the SI term structure of inflation forecasts (7). As a result, F_{t}s_{t} produces mean reversion in n^{fpF}h while permanent movements are tied directly to F_{t}T_{t} and 0_{t} and indirectly to g_{n},_{t}, Qu,_{t}, A_{t}, and 0_{t}. The direct response of nfpFh to 0_{t} is produced by the observation equations (8.2). Drift in 0_{t} also alters transition dynamics in the state equations (8.1), which generates movements in F_{t}T_{t} and F_{t}s_{t}, 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 T_{t}, a_{t}, F_{t}T_{t}, F_{t}s_{t}, g_{n},_{t}, Qu,_{t}, A_{t}, and d_{t}. Lindsten, Bunch, Sarkka, Schon, and Godsill (2016) give instructions for a PS that generates smoothed estimates of these state variables.

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

where = [07J t+_{1} =%u,_{t}+_{1} _{t}*+1* K_{t}+1] The RB-APF uses the KF to create an analytic distribution of S_{t} using the SSM (8.1) and (8.2), given simulated values of V_{t}. 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^ a^{2} 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 S_{t} 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 D_{g}. First, we give j^{2} 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 d_{t}+_{1} and A_{t}+_{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 n_{t} 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 t_{0} and F_{0}t_{0} 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 100^{2} indicates a flat prior over a wide range of values for t_{0} and F_{0}t_{0}. The joint prior of f_{0} and F_{0}e_{0} is drawn from N (0_{2x1}, $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 g_{Ut}_{0}, 0_{0}, and A_{0}; see the notes to table 2. We also restrict priors on t_{0}, f_{0}, F_{0}t_{0}, and F_{0}e_{0} 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.

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, S_{t}, 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 [w_{t} ^ _{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 [w^{y}_{t} j _{1} are also resampled generating [w_{t} ^ _{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^{(}_{t}^{i)} are employed in the entire KF to update j and produce new weights w^{(}_{t}^{l)} = exp j^ j jX^{M} exp j l t^{l)}}, 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 S_{t}|_{t}, £_{t}|_{t}, and V_{t}|_{t} rely on the weights w^{(}_{t}^{i)} = w^{(}_{t}^{i)}/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 d_{t} = 0 or estimating a constant SI parameter, X_{t} = 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 S_{t}, E_{t}, V_{t}, 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) model. The 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^{(}j^{i)}, given Y_{1: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 r_{t}^{(}_\ to r_{t}^{(i)}. The transition equations are appended to the process that draws V^ to sample T^{(i)} ~ P (t| r_{t}^{(i)}), which in essence equates P (t | Y _{1:t}, V^{(}^ to P (t| T_{t}^{(i)}). We denote the system of transition equations r^{(i)} = |(r_{t}^{(}_\, Y_{1:t}, V^{(}_{t}^{i)}, V^{(}_{t}^{i}__{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 §t^{i)}, 2^{(i)}, and Vt^{i)}. Thus, T is estimated by the PLE jointly with S_{t}|_{t}, 2_{t}|_{t}, and V_{t}|_{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 d_{t+1} is a£^{(i)} ~IG I -2t, ~*2*^{lt} I, where a_{t} = a_{t}__{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^{(t}_{t}, f^{(t}_{t}, f^{(}_{c}p^{>}_{t}, and f^{(t}_{t} in steps 2 and 3.(a) of the RB-APF algorithm. The algorithm samples a^{2}^, affl, ^{a}*4*>^{(}t_{t}^{)}, and a^{2}^ from particle streams of sufficient statistics. The law of motion of sufficient statistic matches the transition equation P^{(}g t-_{1} , Y_{1:t}, V^{(}_{t}^{t)}, 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 a2_{n} 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.

Lindsten, Bunch, Särkkä, Schön, and Godsill (2016) develop an algorithm to compute smoothed estimates of S_{t} and V_{t}, given Y_{1: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 V_{t} conditional on forward filtered particles. Forward filtering operations are conducted using the SSM (8.1) and (8.2) to produce smoothed estimates of S_{t}, given smoothed estimates of V_{t}. Lindsten, 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 V_{t} can cause its Markov structure to be lost. A reason is that marginalizing the linear states produces a likelihood that depends on V_{1:t} rather than V_{t}.

LBSSG solve this sampling problem by decomposing the target density P (V_{1:T} | Y_{1: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 V_{t}+_{1:T}.

The aforesaid factorization of P (V_{1:r} | Ynr; ^{Y}) is also useful because there is information in P (v_{1:t} |v_{1:t+1}, Y_{1: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 V_{1:t} is not easy to evaluate. LBSSG’s propose simulation methods to perform the backward filtering implicit in P (V_{1:t} |v_{1: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, V_{t}+_{1:r} |v_{1:t}, Y_{1: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 S_{t} and E_{t} by drawing from P [s_{t} | Y_{1:t}, V_{1: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 S_{t} does not enter P (V_{1:t} |v_{1: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, d_{t} = 0 or drift in SI updating A_{t} = A. The goal is to evaluate the impact of inflation gap persistence or SI on the dynamics of n_{t} and n^{jfF}_{h}, 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 T_{t} and F_{t}T_{t} with n_{t}, and nf^+h, (ii) fluctuations in a_{t} and F_{t}s_{t}, (iii) the history of ^_{n>t} and £_{Utt} since the start of the sample, (iv) movements in d_{t} and A_{t} over the business cycle, and (v) the contributions of Y_{t}, n_{t}, and nj^f+h to variation in Tt and FtTt .

Our estimates rest on a sample of real-time realized inflation, n_{t}, 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 Philadelphia. The 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 deflator. These 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, P_{t}. The quarterly price level data are transformed into inflation measured at an annualized rate using n_{t} = 400 [lnP_{t} - lnP_{t-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 n_{t}. We comply with this timing protocol by assuming the average nowcast, 1-quarter, ..., and 4-quarter ahead predictions, which are denoted nf^{p}+_{1}, ^{n}f^{p}+_{2},. ., and nf^{p}+_{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.

The plots reveal several features of n_{t} and the average SPF inflation predictions. First, average SPF inflation predictions exhibit less variation than n_{t} throughout the sample. Next, as h increases, average SPF inflation predictions become smoother and are centered on n_{t}. 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 T_{t}, s_{t}, F_{t}T_{t}, and F_{t}s_{t}. 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 n^{j}f+_{2} to n^{jf}+_{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 recession. Our 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.

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, d_{t} = 0, only SV drives gap inflation, and a fixed parameter, A_{t} = 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 d_{t} and A_{t}. In contrast, a2 and a} are about equal in the DGP with 0_{t} = 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 0_{t} and A_{t}, a^ and a^{2}, 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 n_{t}, a^ _{n}, because it is nearly twice as large in the DGP that estimates 0_{t} and A_{t} 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 | Y_{1: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|y_{1:T}). The estimates of L(y | Y_{1: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 a^{2}, a}, a^, and a^{2} 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 a^{2} changes little from the start to the end of sample. The PLE path of a^{2} 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.

Figure 3 contains n_{t}, the average SPF inflation nowcast and 4-quarter ahead inflation prediction, ^{n}tt-+1 and -nff+5, filtered RE trend inflation, T_{t}|_{t}, filtered SI trend inflation, F_{t}|_{t}T_{t}, filtered RE gap inflation, f_{t}|_{t}, and filtered SI gap inflation, F_{t}|_{t}f_{t}, on the 1968Q4 to 2017Q2 sample. Plots of n^{5}f_{+}^{f}1 , F_{t}|_{t}T_{t}, and its 68 percent uncertainty bands are in figure 3(a). Figure 3(b) is similar, but replaces nf[+_{1} ^{with n}t,t+5. In these figures, solid (blue) lines are average SPF inflation predictions and F_{t}|_{t}T_{t} is the dotted (black) lines. Figure 3(c) displays T_{t}|_{t} with a dash (green) line, F_{t}|_{t}T_{t} with a dotted (black) line, and n_{t} with a dot-dash (red) line. Estimates of RE and SI gap inflation appear in figure 3(d) as a dashed (green) line, f_{t}|_{t}, and dotted (black) line, F_{t}|_{t}f_{t}.

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 F_{t}|_{t}T_{t} is only 3.8 percent. At the same time, nf[+_{5} is 6.1 percent. The peaks in nf[+_{5} and F_{t}|_{t}T_{t}, which occur a year and a half later, are close to 6.5 percent. The next peaks in nf[+_{1} and nff+^{7}_{5} are 9.5 in 1979Q4 and 8.3 percent in 1980Q1. However, only in 1981Q2 does F_{t}|_{t}T_{t} peak at 7.5 percent. After 1983, nf[+_{1}, nf[+_{5}, and F_{t}|_{t}T_{t} fall steadily before leveling off in the late 1990s as figures 3(a) and 3(b) show. However, F_{t}|_{t}T_{t} often deviates from nff+J between 1983 and 2000. As a result, nf[+_{1} often is outside the 68 percent uncertainty bands of F_{t}|_{t}T_{t} during this period while nff+5 falls within the 68 percent uncertainty bands of F_{t}|_{t}T_{t} after the Volcker disinflation in figure 3(b).

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

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 T_{t}\_{t} and F_{t}\_{t}T_{t} 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 s_{t}\_{t} and F_{t}\_{t}f_{t}\_{t} in figure 3(d). These plots show s_{t}\_{t} and F_{t}\_{t}e_{t} 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, s_{t}\_{t} and F_{t}\_{t}e_{t} turn negative before the 1973-75 recession, which coincides with the largest spikes in s_{t}\_{t} and F_{t}\_{t}s_{t} of nearly nine percent. These spikes are followed by s_{t}\_{t} and F_{t}\_{t}e_{t} falling to about -2.5 percent by 1976. From the late 1970s to 1981, s_{t}\_{t} and F_{t}\_{t}s_{t} range from about zero to 3.7 percent.

There are two more aspects of figure 3(d) worth discussing. First, s_{t}\_{t} and F_{t}\_{t}e_{t} are less volatile subsequent to the Volcker disinflation compared with the 1970s. After 1983, (the absolute values of) s_{t}\_{t} and F_{t}\_{t}e_{t} are never larger than three percent. Second, s_{t}\_{t} and F_{t}\_{t}e_{t} 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.

The Volcker disinflation is another example. After 1983, n_{t} and F_{t}\_{t}T_{t} began to fall, but the drop in n_{t} is steeper as figure 3(c) shows. These plots are consistent with mostly negative realizations for F_{t}\_{t}s_{t} from 1983 to 2000 as in figure 3(d). As discussed previously, we assign these movements in F_{t}\_{t}T_{t} and F_{t}\_{t}e_{t} to the average SPF participant expecting a temporary fall in n_{t} 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.

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,_{n}t\_{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 n_{t} and n^{Sp}+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}\_{T} (£_{o},_{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,_{n}t\_{T} while this process starts in 1975 for £_{o},_{t}\_{T}.

Finally, our estimates show Q_{n}tl_{T} is smaller than £u,_{t}iT 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.

Figures 5(a) and 5(b) display filtered and smoothed estimates of drifting inflation gap persistence, 0_{t}|_{t} and 0_{t}|T. Dotted and dot-dash (orange) lines denote 0_{t}|_{t} and 0_{t}|_{T}. Surrounding 0_{t}|_{t} and 0_{t}|_{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, | 0_{t}|_{T} |, and accumulated changes of this absolute value, | 0_{t}|_{T} | - | 0i|_{T} |. These plots depict | 0_{t}|_{T} | and | 0_{t}|_{T} | - | 0_{1}|_{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 0_{t}|_{t} and 0_{t}|_{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 0_{t}|_{t} and 0_{t}|_{T} while there are troughs between these recession. Post- 1981, 0_{t}|_{t} and 0_{t}|_{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 0_{t}|_{t} and 0_{t}|_{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 d_{t}\_{T}, | d_{t}\_{T} |. The plot of | d_{t}\_{T} | gives evidence similar to that found in figure 5(b). There is evidence of a shift in business cycle behavior of | d_{t}\_{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 9_{t}\_{t}, 9_{t}\_{T}, and | 9_{t}\_{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 | 9_{t}\_{T} |, | 9_{t}\_{T} | - | 9_{1}\_{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 | 9_{t}\_{T} | - | 9_{1}\_{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.

Figure 6 presents filtered and smoothed estimates of the time variation in the frequency of SI updating, A_{t}\_{t} and A_{t}\_{T}. These panels plot A_{t}\_{t} and A_{t}\_{T} as dotted (light green) and dot-dashed (brick) lines. The thin solid (brick) lines denote 90 percent uncertainty bands of A_{t}\_{T} and 90 percent uncertainty bands of A_{t}\_{t} are depicted with light gray areas. Figures 6(b) and 6(d) plot accumulated changes in A_{t}\_{T}, A_{t}\_{T} - A_{1}\_{T}. In these panels, dark and light gray areas are 68 and 90 percent uncertainty bands of A_{t}\_{T} - A_{1}\_{T}. The top row of figure 6 has A_{t}\_{t}, A_{t}\_{T}, and A_{t}\_{T} - A_{1}\_{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 9_{t} = 0 for all dates t.

Plots of A_{t}\_{t} and A_{t}\_{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 E_{t}n_{t}+h because A_{t}\_{T} varies between 0.01 and 0.35. However, there is uncertainty about these estimates because the 90 percent confidence bands of A_{t}\_{T} range from 0.01 to 0.60.

Figures 6(a) also shows A_{t}\_{t} and A_{t}\_{T} reach a plateau from 1994 to 2007 before falling during the 2007-09 recession. From 1995 to 2008, A_{t}\_{t} and A_{t}\_{T} range between 0.50 and 0.70. The recession of 2007-2009 sees A_{t}\_{T} (A_{t}\_{t}) dropping to 0.25 (0.35). Subsequently, A_{t}\_{T} (A_{t}\_{t}) recovers to 0.47 (0.60) before 2017Q2. The filtered and smoothed estimates of A_{t} are also associated with substantial uncertainty. For example, when A_{t}\_{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 A_{t}\_{t} and A_{t}\_{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 A_{t}\_{T} - A_{1}\_{T} for the joint DGP in which there is drift in inflation gap persistence. In this case, the path of A_{t}\_{T} - A_{1}\_{T} in figure 6(b) is similar to A_{t}\_{T} displayed in figure 6(a) with respect to level and slope. Another interesting feature of figure 6(b) is the uncertainty bands surrounding A_{t}\_{T} - A_{1}\_{T}. Figure 6(b) displays 90 percent uncertainty bands of A_{t}\_{T} - A_{1}\_{T} that are narrower for the entire sample compared with the analogous confidence bands of A_{t}\_{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 A_{t}\_{T} - A_{1}\_{T} conditional on a joint DGP in which there is no persistence in the inflation gap. Given 9_{t} is zero, SI inflation updating is less frequent quarter by quarter, as depicted by A_{t}\_{t} and A_{t}\_{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 9_{t} = 0, the plot of A_{t}\_{T} - A_{1}\_{T} in figure 6(d) indicates otherwise. Figure 6(d) depicts A_{t}\_{T} - A_{1}\_{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 A_{t}\_{t}, A_{t}\_{T}, and A_{t}\_{T} - A_{1}\_{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.

Figure 7 displays conditional volatilities of RE trend inflation, T_{t}, and SI trend inflation, F_{t}T_{t}. The plots quantify uncertainty over time in T_{t} and F_{t}T_{t} conditional on the history of Y_{t}, or histories of subsets of its elements, smoothed estimates of the nonlinear states, V_{t}|_{T}, and estimates of the static scale volatility coefficients, T. The measure of the volatility of T_{t} is Var(T_{t} | Y _{1:t}, V_{t}|_{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 F_{t}T_{t}. 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 T_{t} and F_{t}T_{t}.

Figure 7(a) plots the conditional volatilities of T_{t}. The conditional volatilities of F_{t}T_{t} 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 | Y_{1:t}, V_{t}|_{T}, T), Var(x | n_{1:t}, V_{t}|_{T}, ^{T}), Var(x | n^{fPF}, V_{t}|_{T}, ^{T}), and Var(x | n_{1:t}, n^{fPF}, V_{t}|_{T}, T), respectively, where x = T_{t}, F_{t}T_{t}.

**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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