Cogley-Sargent (2002): Drifts and Volatilities: Monetary Policies and Outcomes in the Post WWII U.S.

Summary

Cogley and Sargent (2002) extend their earlier time-varying-parameter vector autoregression (TVP-VAR) (2001) by adding multivariate stochastic volatility, addressing the Sims–Stock criticism that constant innovation variance could be inflating evidence of drifting coefficients. Using a three-variable VAR for inflation, unemployment, and the nominal interest rate over 1948Q1–2000Q4, they show that even after accounting for substantial time variation in $R_t$, evidence for drifting VAR coefficients survives — inflation persistence increased in the 1970s and fell under Volcker, core inflation and the natural rate co-moved strongly (correlation 0.748), and the monetary policy activism coefficient shifted from passive (P(A>1) ≈ 0.21 in 1975) to activist (P(A>1) ≈ 0.92 in 1985). Classical stability tests (Andrews sup-Lagrange multiplier (LM), Nyblom-Hansen) have only 11–25% power against this form of drift, which reconciles the failure-to-reject results of Sims (1999) and Bernanke-Mihov (1998) with the TVP evidence.

Key Claims

• Stochastic volatility is real and large. Innovation variances spiked sharply between 1979 and 1981 and fell from the mid-1980s onward (Great Moderation pattern), with a long-term downward trend in unemployment variance since the late 1950s.
• Drift survives stochastic volatility (SV). After adjusting the persistence measure to the normalized spectrum $g(0,t) = f(0,t)/\int f(\omega,t)d\omega$ (which controls for changing innovation variance), inflation persistence still increases in the 1960s–70s and falls post-Volcker. Full-chain correlation with core inflation is 0.92.
• Activism shifted from passive to activist. Point estimates of the Clarida-Gali-Gertler activism coefficient $A = \alpha_1(1-\alpha_3)^{-1}$ were below 1 from the late 1960s through the mid-1970s; P(A>1) rose from 0.208 (1975) to 0.919 (1985) to 0.941 (1995), with 0.923 probability of an increase between 1975 and 1985 along the same posterior paths.
• Classical tests have low power. Andrews sup-LM and Nyblom-Hansen tests fail to reject constant $\theta$ in actual data and have only 11–25% detection probability when the true data-generating process (DGP) is the authors' TVP model. The sup-Wald test for the inflation equation achieves 71% power and does reject time invariance at the 1% level.
• Drift is highly structured. Only two or three principal components of $Q$ (the state innovation covariance) explain 83–95% of variation in $\theta_t$; these components load primarily on lags of nominal interest and unemployment in the inflation equation.
• Q is small, not exaggerated. Imposing the stability prior (all VAR eigenvalues inside the unit circle) actually reduces posterior mean tr(Q) rather than inflating it. The conservative ordering [i, u, π] is used throughout.

How It Works

Model

A trivariate VAR(2) in nominal interest $i_t$, civilian unemployment $u_t$, and consumer price index (CPI) inflation $\pi_t$ (logit-transformed unemployment):

$$y_t = X_t \theta_t + \varepsilon_t, \qquad \varepsilon_t = R_t^{1/2} \xi_t, \quad \xi_t \sim \mathcal{N}(0, I)$$

Drifting coefficients: $$\theta_t = \theta_{t-1} + v_t, \qquad v_t \sim \mathcal{N}(0, Q), \quad \theta_t \text{ subject to stability prior}$$

The stability prior (VAR polynomial roots inside the unit circle) is encoded via a rejection-sampling indicator; proposed draws violating stability are rejected.

Multivariate stochastic volatility (Jacquier-Polson-Rossi 1994): $$R_t = B^{-1} H_t B'^{-1}$$ where $B$ is lower-triangular with unit diagonal (captures contemporaneous correlations) and $H_t = \mathrm{diag}(h_{1t}, h_{2t}, h_{3t})$ with each $h_{it}$ evolving as a driftless geometric random walk: $$\ln h_{it} = \ln h_{i,t-1} + \sigma_i \eta_{it}, \qquad \eta_{it} \sim \mathcal{N}(0,1)$$

This specification permits permanent, recurrent shifts in variance — unlike Markov-switching models, which either cannot recur (absorbing state) or forever cycle between the same configurations.

Markov Chain Monte Carlo (MCMC) Estimation

100,000 Metropolis-within-Gibbs draws; first 50,000 discarded; every 10th saved (5,000 effective draws). The sampler cycles through:

1. $\theta_T$ (full coefficient path) — Carter-Kohn (1994) forward filter / backward simulation smoother with rejection sampling for the stability condition.
2. $Q$ (state innovation covariance) — Inverse-Wishart conjugate draw.
3. $\beta$ (free parameters of $B$) — Normal conjugate from the orthogonalized system.
4. $\sigma^2 = (\sigma_1^2, \sigma_2^2, \sigma_3^2)$ — Inverse-Gamma conjugate draw.
5. $\{h_{it}\}$ (log-volatility paths) — Single-move Metropolis with log-normal proposal (JPR 1994); acceptance rates ≈ 70-80%.

Independence of $\varepsilon_t$ and $v_t$ (eq. 6) is assumed to economize on parameters after introducing the SV extension; this is the key difference from Cogley-Sargent (2001), which allowed $C = \mathrm{cov}(v_t, \varepsilon_t) \neq 0$.

Persistence and Uncertainty Measurement

Inflation persistence is measured by the normalized spectrum at zero frequency: $$g(0, t) = \frac{f(0,t)}{\int_{-\pi}^{\pi} f(\omega,t) d\omega}$$ where $f(\omega, t) = \frac{1}{2\pi} s(I - A_{t|T} e^{-i\omega})^{-1} E(R_t|T)(I - A_{t|T} e^{i\omega})^{-1} s'$ and $s$ selects inflation. The normalization removes the influence of changing $R_t$, making $g(0,t)$ a pure measure of autocorrelation structure. $g(0,t) = 1$ corresponds to white noise.

Uncertainty quantification follows Sims-Zha (1999): Sims-Zha error bands based on the principal components of the posterior covariance matrix $V_{\bar\pi}$, estimated via the delta method from the MCMC ensemble.

Power Analysis

Monte Carlo power study using the estimated TVP model as the DGP, 10,000 artificial samples:

Test	Rejection rate (5% level)
Andrews sup-LM (VAR)	0.252
Andrews sup-LM (inflation eq.)	0.112
Nyblom-Hansen (VAR)	0.234
Andrews sup-Wald (inflation eq.)	0.711
Andrews sup-Wald (VAR)	0.296

The sup-Wald test for inflation has sufficient power and rejects time invariance in the actual data at the 1% level.

Concepts Introduced or Extended

• Time-Varying Parameter VAR
• Stochastic Volatility
• Vector Autoregression
• Metropolis-Hastings
• Structural Breaks
• MCMC Convergence Diagnostics

Entities Mentioned

• Timothy Cogley
• Thomas Sargent

Quotes

"We continue to find evidence that the VAR coefficients have drifted, mainly along one important direction."

"Most of our tests fail to reject time invariance of θ, but most also have low power to detect the patterns of drift we describe above. In the one case where a test has a better-than-even chance of detecting drift in θ, for the data time invariance is rejected at better than the one-percent level."

"One respectable view is that either an erroneous model, insufficient patience, or his inability to commit to a better policy made Arthur Burns respond to the end of Bretton Woods by administering monetary policy in a way that produced the greatest peace time inflation in U.S. history."

My Take

This is the foundational TVP-SV VAR paper (published as Cogley-Sargent 2005 in Review of Economic Dynamics). The core methodological contribution — the $R_t = B^{-1}H_tB'^{-1}$ factorization with geometric-random-walk log-volatilities — became the standard template for Bayesian macroeconometrics and was later generalized by Primiceri (2005) to allow simultaneous drift in the $B$ matrix.

The paper's rhetorical strategy is effective: it directly engages the Sims-Stock criticism, shows the criticism is valid in principle (stochastic volatility is real), but demonstrates it does not undermine the main result. The power analysis is a genuine contribution — the failure to reject time invariance in Bernanke-Mihov and Sims turns out to reflect test weakness rather than evidence against drift.

Two limitations stand out. First, the independence assumption $E(\varepsilon_t v_s') = 0$ is a simplification relative to the 2001 paper; Primiceri (2005) allows the $B$ matrix (the contemporaneous impact matrix) to also drift, which is more general. Second, the activism coefficient estimation uses lagged-variable instruments and assumes the Fed lacks current-quarter data — this is a reasonable approximation but the weak-instruments problem at low-persistence dates is acknowledged and not fully resolved.