Common Persistence in Conditional Variances

Summary

A 20-page theoretical and empirical paper introducing co-persistence in variance — the exact second-moment analogue of cointegration. Even when individual time series exhibit integrated-GARCH (IGARCH) behavior (persistent, non-mean-reverting conditional variances), a linear combination of those series may have a stationary conditional variance. The co-persistent vector plays the same role as the cointegrating vector: it eliminates the unit-eigenvalue subspace of the dynamics. An empirical application to daily DM (Deutsche Mark) and BP (British pound) vs. USD returns finds the bilateral DM/BP cross rate shows no variance persistence — most USD exchange rate volatility is dollar-specific news.

Key Claims

Definitions and Formal Framework

• Persistence in variance (Definition): $\{y_t\}$ is persistent in variance if $\limsup_{t\to\infty} \|H_t^*(s)\| \neq 0$ a.s. for some $s>0$, where $H_t^*(s) = E_t(\text{vech}(H_{t+s})) - E_s(\text{vech}(H_{t+s}))$ measures how current shocks influence future covariance forecasts. Analogue of $I(1)$ (integrated of order one) in the mean.

• Theorem 1 (Covariance stationarity): The vector GARCH($p,q$) process is covariance stationary if and only if all roots of $\det[s^q I - A(s^{-1}) - B(s^{-1})] = 0$ lie strictly inside the unit circle — equivalently, the spectral radius $\rho(A_1 + \cdots + A_q + B_1 + \cdots + B_p) < 1$.

• Co-persistence in variance (Definition): $\{y_t\}$ is co-persistent in variance if there exists a nonzero vector $\gamma$ such that individual variances are persistent but $\limsup_{t\to\infty} |E_t(\gamma'H_{t+s}\gamma) - E_s(\gamma'H_{t+s}\gamma)| = 0$ a.s. — the portfolio $\{\gamma'y_t\}$ has no variance persistence.

• Theorem 2: The vector GARCH($p,q$) process is co-persistent in variance if and only if $\text{vech}(\gamma\gamma')'v_i = 0$ for all right eigenvectors $v_i$ corresponding to eigenvalues $|\lambda_i| \geq 1$ of $(A_1 + B_1)$. The number of linearly independent co-persistent vectors equals $N(N+1)/2 - r$, where $r$ is the number of unit or explosive eigenvalues.

Factor GARCH Connection

• Lemma 1: $\{\gamma'e_t\}$ follows a univariate GARCH($p,q$) if and only if $\text{vec}_2(\gamma)'A_i = \alpha_i\,\text{vec}_2(\gamma)'$ and $\text{vec}_2(\gamma)'B_i = \beta_i\,\text{vec}_2(\gamma)'$ for all $i$ — a set of testable over-identifying restrictions.

• Lemma 2 (Factor GARCH): In a $K$-factor GARCH($p,q$) model, any portfolio $\gamma$ orthogonal to all $r$ integrated factors is co-persistent in variance, with $\gamma'y_t$ following a univariate GARCH($p,q$). Co-persistence exists whenever the number of persistent variance factors $r < N$.

Asset Pricing Motivation

• In a $K$-factor excess return structure, if some factors have persistent conditional variances (IGARCH), long-term contracts bear a risk premium that is itself time-varying and horizon-dependent. A portfolio orthogonal to all persistent factors eliminates variance persistence — its long-run risk premium is time-invariant even when individual assets are IGARCH.

• Optimal portfolio allocation differs radically across investment horizons when persistent variance factors are present: long-horizon investors face a qualitatively different risk landscape than short-horizon investors.

Empirical Example: DM and BP vs. USD (1980–1985)

• Daily data, 1,245 observations (Jan 1980–Feb 1985); martingale in mean; no cointegration in mean.
• Univariate GARCH(1,1): $\hat\alpha+\hat\beta = 0.966$ (DM) and $0.971$ (BP) — neither rejects IGARCH at conventional levels.
• Bivariate GARCH(1,1) estimated by quasi-maximum likelihood estimation (QMLE) with Lemma 1 constraints; 3 over-identifying restrictions: Lagrange multiplier (LM) statistic = 2.488 (not rejected). LM test for 4 ARCH lags: not rejected at conventional levels.
• Largest characteristic root of estimated $A+B$: $|\hat\lambda_1| = 0.982$ — persistent, with half-life $\approx 38$ trading days.
• Estimated co-persistent vector: $\hat\gamma \approx (1, -1.176) \approx (1, -1)$ — the DM/BP bilateral rate (triangular arbitrage).
• Univariate GARCH(1,1) for the DM/BP bilateral rate: $\hat\alpha + \hat\beta = 0.247 + 0.608 = 0.855 < 1$; $t$-statistic for $\hat\alpha+\hat\beta=1$ equals $6.033$ — overwhelmingly rejects persistence.
• Interpretation: persistence in USD exchange rate volatility is dollar-specific news; the bilateral European DM/BP rate exhibits no variance persistence.

Inference Caveat

• Appropriate asymptotic theory for QMLE in multivariate persistent variance processes was not fully worked out at the time. Authors note QMLE provides a generalized least squares (GLS)-type correction that may ensure standard asymptotic properties (citing Lee-Hansen 1994; Lumsdaine 1989 for the univariate IGARCH case).

Concepts Introduced or Extended

• GARCH and BEKK-GARCH — co-persistence in variance; Theorems 1–2; Lemmas 1–2; factor GARCH co-persistence condition; DM/BP empirical application
• Cointegration — variance analogy: co-persistence in variance mirrors co-integration in mean; same algebraic structure (null space of explosive eigenvectors)

Entities Mentioned

• Tim Bollerslev
• Robert Engle

Quotes

"Although many economic or financial time series may exhibit persistence in their conditional variances, a nontrivial linear combination of such variables may have no persistence in variance. In that situation the variables are naturally defined to be co-persistent in variance, and the co-persistent linear combination may be interpreted as a long-run relationship."

My Take

The paper is primarily a theoretical contribution — the formal parallel between I(1)/co-integration in the mean and IGARCH/co-persistence in the variance is clean, elegant, and actionable. The empirical application is compact but convincing: the near-$(1,-1)$ co-persistent vector arising naturally from estimation, matching the no-arbitrage DM/BP cross rate, is a satisfying confirmation. The practical limitation is that inference in IGARCH systems was not fully worked out at the time (the authors note this explicitly), so the empirical section remains somewhat provisional from a statistical inference standpoint. The factor GARCH framing — co-persistence = fewer persistent factors than assets — is the most actionable takeaway for practitioners building multivariate volatility models.