Trend Inflation, Indexation, and Inflation Persistence in the New Keynesian Phillips Curve

The authors ask whether the persistence long observed in the estimated New Keynesian Phillips Curve (NKPC) reflects a genuine structural feature of price-setting – backward-looking indexation, as assumed with an indexation parameter near one in Christiano-Eichenbaum-Evans (2005) and about 0.25 in Smets-Wouters (2007) – or is instead an artifact of estimating the NKPC around a fixed or de-meaned inflation trend when trend inflation has actually drifted with monetary policy. They argue for the latter: “accounting for trend-inflation drift allows a purely forward-looking model to fit the data well” (Introduction, p. 2101). The distinction matters because backward indexation as a structural mechanism has no clean micro-foundation and is hard to reconcile with survey and micro price evidence, whereas drifting trend inflation has an independent monetary-policy interpretation.

Trend inflation is extracted from a Bayesian time-varying-parameter VAR(2) with stochastic volatility, following Cogley and Sargent (2005a), estimated over four quarterly U.S. variables – output growth, real marginal cost (labor share in the nonfarm business sector), GDP-deflator inflation, and the nominal interest rate – over 1960:I-2003:IV, with a 1954:I-1959:IV training sample used to set priors. VAR coefficients evolve as driftless random walks with reflecting barriers, and the volatilities in the stochastic-volatility block evolve as driftless geometric random walks (Eqs. 20 and 23, p. 2109-2110). Trend inflation pi-bar_t is defined as the long-horizon limit of the VAR’s time-varying conditional mean of inflation, pi-bar_t = e_pi’(I - A_t)^{-1} mu_t (Eq. 24, p. 2110) – so, unlike a fixed sample mean or a low-pass filter, it moves with the estimated drift in the whole VAR system, and the authors interpret its movements “as reflecting changes in this aspect of monetary policy” (p. 2110).

Over the full 1960-2003 sample, raw inflation’s autocorrelation (0.834) is only modestly above the trend-based inflation gap’s autocorrelation (0.769), but the two diverge sharply after the Volcker disinflation: over 1984-2003 the gap’s autocorrelation falls to just 0.305, far below raw inflation’s 0.784 (Table 1, p. 2111). Before 1984 the two measures were closer (0.801 for the gap versus 0.843 for raw inflation). The authors read this as showing that “this ’excess persistence’ [criticized in purely forward-looking Calvo models] reflects an exaggeration of the persistence in mean-based measures of the gap rather than a deficiency of persistence in forward-looking models” (p. 2111) – i.e., once trend inflation is allowed to move, especially post-Volcker, most of what looked like structural persistence in the gap disappears.

The three free structural parameters psi = (alpha, theta, rho) – Calvo stickiness, the elasticity of substitution between goods, and the indexation parameter – are estimated by minimum distance, choosing psi to minimize the distance between the TVP-VAR’s estimated reduced-form forecasting parameters (mu-hat_t, A-hat_t) and the values those parameters would take under the cross-equation restrictions implied by the NKPC (Section III.A, pp. 2107-2109; Section IV, pp. 2112-2114). The strategic-complementarity parameter omega is calibrated rather than estimated, set to delta/(1-delta) = 0.429 given a Cobb-Douglas labor elasticity delta = 0.3. Parameters are constrained to economically meaningful ranges: alpha in (0,1), rho in [0,1], theta in (1, infinity) (Table 2, p. 2113).

The indexation parameter is estimated at essentially zero: median rho = 0, with a 90 percent credible interval of (0, 0.15), and “approximately 78 percent of the estimates lie exactly on the lower bound of zero, and 90 percent are less than 0.15” (Table 3, p. 2113). This contrasts with prior estimates in the literature the authors cite: rho around 0.15 in Sbordone (2006), rho = 1 in Giannoni-Woodford (2003), and rho approximately 0.25 in Smets-Wouters (2007) – all of which, on the authors’ account, are estimated without allowing for drifting trend inflation.

Median Calvo stickiness is alpha = 0.588 (90% CI 0.44-0.70), implying a median price-reset waiting time of about 1.31 quarters, or roughly 3.9 months (90% CI 2.5-5.8 months); the median elasticity of substitution theta = 9.8 (90% CI 7.4-12.1) implies a steady-state markup of about 11 percent, which the authors describe as “in line with other estimates in the literature” (Section IV, p. 2114). These estimates are joint with rho = 0 – i.e., they characterize a purely forward-looking Calvo model once trend inflation is accounted for.

Yes: the correlation between the TVP-VAR’s reduced-form inflation forecasts and the forecasts implied by the NKPC’s cross-equation restrictions is 0.978, which the authors read as providing “little evidence against the cross-equation restrictions” (Section V, pp. 2114-2116). The model is reported to fit the data well throughout the sample, including both the Great Inflation and the Great Moderation episodes, despite the substantial swings in trend inflation across those periods.

Because the NKPC’s coefficients are nonlinear functions of trend inflation and the Calvo primitives (Eq. 8 and Eqs. 31-32, 49, pp. 2107, 2120, 2122), the model implies genuinely time-varying coefficients – the marginal-cost coefficient zeta_t falls as trend inflation rises, and the coefficient on expected future inflation b1_t shifts from about 1 at zero trend inflation to roughly 1.05-1.10 at the estimated trend, while the coefficient on the discount-factor term b3_t stays near zero throughout (Section V, pp. 2115-2117). The authors argue that forward-looking NKPC terms, which are positively correlated with past inflation whenever inflation predicts future inflation more than one quarter ahead, will spuriously mimic a backward-looking coefficient once trend inflation is omitted from the specification (p. 2113 and footnote 17) – this is the mechanism by which a constant-trend NKPC estimation can manufacture apparent indexation that is not really there.

The rho = 0 finding is robust across four alternative specifications reported in Appendix C – omitting the interest-rate and output-growth terms, additionally omitting extra inflation-lead terms, and additionally shutting off the time-varying coefficients – with alpha stable around 0.57-0.59 and theta around 9.8-12.1 throughout (Appendix C, p. 2124). The duration estimate is broadly consistent with Bils and Klenow’s (2004) adjusted median duration of 5.5 months, but Nakamura and Steinsson’s (2007) approximately 8-month estimate lies outside the paper’s 90 percent confidence interval of (2.5, 5.8) months – a tension the authors do not resolve. More fundamentally, they acknowledge, citing Beyer and Farmer (2007), that distinguishing forward- from backward-looking NKPC components rests on auxiliary identifying assumptions, writing that “in a fundamental sense, whether our estimates are supported by convincing economic restrictions is not clear” (p. 2114). The sample ends in 2003:IV and therefore does not address the zero lower bound or the post-2008 low-inflation environment, and the authors note that sensitivity to alternative lag lengths or variable orderings “is left to future research” (footnote 11, p. 2110).