Meddahi (2002) ARMA Representation of Two-Factor Models

Summary

Derives closed-form expressions for the moving-average roots of a two-factor ARMA model — the analytically hard part of characterising the ARMA(p,p) structure of a variable defined as the sum of p AR(1) processes. Applies the result to characterise the weak GARCH(2,2) representation (Drost-Nijman 1993) of squared returns for two-factor continuous-time stochastic volatility models (affine, GARCH diffusion, constant-elasticity-of-variance (CEV), Ornstein-Uhlenbeck, eigenfunction, square-root stochastic autoregressive volatility (SR-SARV)) at any observation frequency $h$, linking continuous-time parameters directly to discrete-time GARCH coefficients.

Key Claims

• Granger-Morris structure: A sum of $p$ AR(1) processes is ARMA(p,p) (with noise) or ARMA(p,p-1) (without); the AR roots equal the persistence parameters $\gamma_i$. The MA roots are the novel contribution.
• MA roots (Proposition 2.1): Given the first two autocorrelations $\rho_1, \rho_2$ of the MA(2) component $z_t$, closed-form formulas for $\beta_2$ (via a quadratic in $u$ with $\mathrm{sign}(\rho_2)$ determining the root) and $\beta_1 = \rho_1 p_2^{-1}\beta_2(1-\beta_2)$; MA roots $\lambda_{1,2} = (\beta_1 \pm \sqrt{\beta_1^2+4\beta_2})/2$.
• Two routes to ARMA structure: Structural (Proposition 2.2 — in terms of $\gamma_i$ and noise covariance) and reduced-form (Proposition 2.3 — in terms of observable autocovariances of $y_t$).
• Weak GARCH(2,2) (Proposition 3.2): For squared returns from any two-factor SR-SARV continuous-time SV model without leverage, the discrete-time squared returns satisfy $E[(e_t^h)^2 \mid \mathcal{H}_{t-1}] = \omega^h + \alpha_1^h(e_{t-1}^h)^2 + \alpha_2^h(e_{t-2}^h)^2 + \beta_1^h\nu_{t-1} + \beta_2^h\nu_{t-2}$ with all coefficients derived from $(\kappa_i, \theta_i, \xi_i, h)$.
• Temporal aggregation: At weekly frequency and below, the two-factor model reduces empirically to GARCH(1,1) because the fast-reverting factor (large $\kappa$) averages out, explaining GARCH(1,1)'s empirical dominance. At very high frequencies, AR and MA roots both $\to 1$, making parameters near-unidentified and the ARCH effect $\to 0.0488$ in the DM/$ calibration.
• Kalman filter equivalence: The GARCH(2,2) recursive predictor (eq. 2.10) is the analytical steady-state of the Kalman filter for the latent $(f_{1t}, f_{2t})$ state — can replace the KF in quasi-maximum likelihood (QML) estimation (Harvey-Ruiz-Shephard 1994 approach).
• Empirical illustration: Bollerslev-Zhou (2002) two-factor affine calibration (DM/$, 1986–1996): fast factor$\kappa_1=0.571$, slow factor$\kappa_2=0.076$; GARCH(2,2) at daily frequency reduces to near-GARCH(1,1) at weekly.

Concepts Introduced or Extended

• GARCH and BEKK-GARCH
• Stochastic Volatility
• Diffusion Process

Entities Mentioned

• Nour Meddahi

Quotes

"Many financial time series models are specified through a structural representation. Nonetheless, knowing their reduced ARMA form may be useful for impulse response analysis, filtering, forecasting, and for purposes of statistical inference."

"While the model is a (weak) GARCH(2,2) for all frequencies, it appears as a (weak) GARCH(1,1) for the weekly frequency and lower ones. The main reason is that the very volatile but non persistent factor has no impact on volatility clustering. This explains the empirical relevance of the GARCH(1,1) with respect to other ARCH-type models."

My Take

The paper fills a clean gap: Granger-Morris (1976) told us the ARMA order of a sum of AR(1) processes but could not analytically characterise the MA roots for orders > 1. The closed-form in Proposition 2.1 is the main new result, and the SV application in Section 3 is its payoff. The temporal-aggregation result (GARCH(2,2) $\to$ GARCH(1,1) at lower frequencies) provides a satisfying theoretical explanation for a long-standing empirical regularity. The identification difficulty at high frequencies (AR $\approx$ MA roots) is an important caveat for microstructure-heavy intra-daily estimation.