Cogley-Sargent (2001) Evolving Post-WWII U.S. Inflation Dynamics

Summary

Cogley and Sargent (2001) estimate a three-variable random-coefficients vector autoregression (VAR) for U.S. quarterly data 1948–2000 in which the coefficient vector follows a driftless random walk, casting the model in state-space form and inferring the full trajectory of time-varying parameters via a Gibbs sampler (Carter-Kohn forward-filter backward-smoother). Using posteriors over coefficient paths, they construct time-varying impulse responses and spectral densities to document that U.S. inflation persistence and the core rate of inflation co-moved strongly over the postwar period, track the natural-rate hypothesis, and measure whether Federal Reserve policy was passive or activist.

Key Claims

• Core inflation and persistence are highly correlated (posterior correlation ≈ 0.915): periods of high trend inflation coincide with periods of high inflation persistence, and vice versa.
• Core inflation and the natural rate of unemployment co-move positively (posterior correlation ≈ 0.63): the natural rate was elevated when core inflation was high (1970s), and both fell after Volcker's disinflation.
• Policy passivity in the 1970s, activism under Volcker/Greenspan: the posterior activism coefficient $A = \beta(1)/(1-\rho(1))$ from a time-varying Taylor rule is negative in 1970s (passive/accommodative) and strongly positive after 1980, with posterior correlation −0.69 (full sample) and −0.87 (Volcker–Greenspan era) with core inflation.
• Natural Rate Hypothesis (NRH) dynamics: the NRH test $\beta_1(1)=1$ is valid only when inflation is sufficiently persistent. It was strongly rejected through 1973, accepted from mid-1970s to mid-1980s, and shows marginal evidence against by 2000.
• Taylor (1999) warning confirmed: given the low persistence as of 2000, there is a 50% probability of a successful NRH rejection within 9 years, and an 85% probability within 20 years.
• No stochastic volatility: the observation covariance $V$ (and hence $R$) is held constant across the full sample — a limitation subsequently corrected in Cogley-Sargent (2002/2005).

Model

Observation equation: $$y_t = X_t'\theta_t + \varepsilon_t, \qquad \varepsilon_t \sim \mathcal{N}(0, R)$$

where $y_t$ is a 3-vector (consumer price index (CPI) inflation, logit-unemployment, ex post real interest rate), $X_t' = I_3 \otimes x_t'$ with $x_t = (y_{t-1}', y_{t-2}', 1)'$, and $\theta_t = \mathrm{vec}[(A_{1t}, A_{2t}, c_t)']$ is the stacked $(K \times 1)$ coefficient vector ($K = 21$ for $n=3$, $p=2$, plus intercepts).

Transition equation (driftless random walk): $$\theta_t = \theta_{t-1} + v_t, \qquad v_t \sim \mathcal{N}(0, Q)$$

Stability is enforced by a reflecting barrier: draws violating VAR stability (any eigenvalue outside the unit circle) are rejected and resampled within the Carter-Kohn block.

Joint covariance prior: $V = \begin{bmatrix} R & C \\ C' & Q \end{bmatrix}$ with inverse-Wishart prior $V^{-1} \sim \mathrm{Wishart}(\Psi_0, m_0)$; the off-diagonal block $C \neq 0$ (Model M1 in Amisano-Federico (2004) taxonomy).

Data and estimation:

• Quarterly 1948:1–2000:4; prior calibrated on 1948:1–1958:4 subsample.
• $Q_{\text{prior}} = (0.01)^2 \cdot \hat{P}$ (very conservative; coefficients barely drift).
• Posterior computations initialized at 1965:4; 10,000 Gibbs iterations (2,000 burn-in); 8,000 predictive trajectories each 120 quarters long.

Two-block Gibbs sampler:

1. Draw $\theta^T = (\theta_1, \ldots, \theta_T)$ given $V, Y^T$: Carter-Kohn forward-filter backward-sampling (FFBS) with stability rejection.
2. Draw $V$ given $\theta^T, Y^T$: inverse-Wishart conjugate.

Four sources of forecast uncertainty (distinguished from CMS (2003)'s three-source decomposition):

1. Unknown current state $\theta_T$
2. Unknown covariance $V$
3. Future state drift $\{v_t\}_{t>T}$
4. Future measurement shocks $\{\varepsilon_t\}_{t>T}$

Concepts Introduced or Extended

• Time-Varying Parameter VAR — first complete application to U.S. inflation with time-varying persistence measurement
• Phillips Curve — time-varying natural-rate hypothesis test via spectral analysis of VAR coefficients
• Gibbs Sampler — Carter-Kohn FFBS with stability constraint via rejection sampling
• State-Space Representation — random walk coefficient time-varying-parameter (TVP) model

Entities Mentioned

• Timothy Cogley
• Thomas Sargent

Quotes

"The correlation between core inflation and the persistence of inflation is 0.915, suggesting that at times of high inflation, inflation was highly persistent, while at times of low inflation, it was not."

"We find that policy was passive, i.e., accommodative of inflation, in the 1970s and activist and aggressive against inflation under Volcker and Greenspan."

"Taylor warns that because inflation is no longer very persistent, the NRH may be rejected in new data."

My Take

This paper is the founding document of the TVP-VAR applied to U.S. inflation. Its central insight — that persistence and trend co-move, that activism and core inflation are negatively correlated — is empirically compelling and methodologically clean. The key limitation (constant $R$) was identified immediately by commentators (Bernanke, Mihov, Sims) and addressed two years later in Cogley-Sargent (2002/2005). The joint inverse-Wishart (IW) prior on $V$ (off-diagonal block non-zero) is model M1 in the Amisano-Federico (2004) taxonomy and has the largest parameter count of the four Ω specifications — the Kronecker M3 structure later shown to be both more parsimonious and better out-of-sample.