Fiorentini-Sentana (1998) Conditional Means of Time Series Processes

Summary

Fiorentini and Sentana (1998) derive the exact time series process followed by the conditional mean $\boldsymbol{\mu}_t = E_{t-1}(\mathbf{x}_t)$ of any vector linear process $[I-A(L)]\mathbf{x}_t = [I-B(L)]\boldsymbol{\epsilon}_t$ . The conditional mean follows a vector linear process one order lower: $[I-A(L)]\boldsymbol{\mu}_t = [A(L)-B(L)]\boldsymbol{\epsilon}_t$ . Corollaries characterise conditional means for autoregressive integrated moving average (ARIMA), autoregressive fractionally integrated moving average (ARFIMA), and multivariate GARCH processes, and establish that GARCH-in-mean (GARCH-M) models can only generate nonnegative autocorrelations in returns — explaining their poor empirical fit. The paper closes with a bivariate application showing that U.S. stock returns are approximately white noise even though their conditional mean follows a highly persistent first-order autoregression (AR(1)), consistent with a large negative contemporaneous correlation ( $\rho \approx -0.95$ ) between innovations to actual and expected returns.

Key Claims

Proposition 1: If $[I-A(L)]\mathbf{x}_t = [I-B(L)]\boldsymbol{\epsilon}_t$ then $[I-A(L)]\boldsymbol{\mu}_t = [A(L)-B(L)]\boldsymbol{\epsilon}_t$ ; the conditional mean is a vector linear process of degree $m-1$ where $m = \max(k,h)$ .
Corollary 1 (ARIMA): An ARIMA $(p,d,q)$ process has conditional mean whose autocorrelation function (ACF) matches an autoregressive-moving-average ARMA $(p, m-1)$ process with the same AR coefficients, $m = \max(p+d, q)$ .
Corollary 2 (Multivariate GARCH): For GARCH $(p,q)$ , $\text{vech}(\Sigma_t)$ displays the ACF of a vector autoregressive-moving-average VARMA $(m, q-1)$ process with the same AR coefficients, $m = \max(p,q)$ ; for GARCH(1,1) this reduces to a vector autoregression VAR(1).
Lemma 3 (GARCH-M constraint): If $x_t = \delta\sigma_t^2 + \epsilon_t$ then $\text{Cov}(x_t, x_{t-k}) = \delta^2 \text{Cov}(\sigma_t^2, \sigma_{t-k}^2) \geq 0$ for all $k \geq 1$ ; GARCH-M can only produce nonnegative autocorrelations.
ARFIMA persistence: For ARFIMA $(0,\gamma,0)$ , the conditional mean is also fractionally integrated of order $\gamma$ but has an infinite moving-average (MA) representation.
Persistence identity: $P_\infty(\mu_{t+1}|\epsilon_t) = P_\infty(x_t|\epsilon_t) - 1$ where $P_\infty(x_t|\epsilon_t) = V(x_t)/V(\epsilon_t) = \sum_{j=0}^\infty \psi_j^2$ ; univariate white noise forces constant conditional mean.
Bivariate result: In a VAR(1) with $|A| = 0$ (reduced rank), returns can be white noise while the conditional mean follows a persistent AR(1); the mechanism is a large negative contemporaneous correlation between the two innovations.
Empirical (U.S. monthly returns, 1952:1–1994:12): $\theta = 0.9916$ (returns nearly white noise); conditional mean AR(1) with coefficient 0.9755; innovation standard deviation (SD) 42× smaller than observed returns; $\rho(u,w) = -0.9466$ ; $P_\infty(r_t|w_t) = 1600$ .

Concepts Introduced or Extended

Entities Mentioned

Quotes

"As a first application of Proposition 1, we derive the time series processes for the conditional mean of integrated and fractionally integrated processes." (p. 1104)

"The fact that the autocorrelation function of $x_t$ can only take nonneg values ... explains why GARCH-M models tend to fit financial time series so poorly." (p. 1107)

My Take

The paper's main contribution is elegant and underappreciated: it shows that the conditional mean inherits a specific, derivable ARMA structure from the original process, a result rarely used in applied work. The GARCH-M nonnegative-ACF constraint (Lemma 3) is the sharpest empirical implication and provides a clean theoretical explanation for an observed stylised fact. The bivariate stock-return application is compelling — the reduced-rank VAR(1) delivers white-noise returns with a highly persistent conditional mean via large negative contemporaneous correlation — but the identification relies on imposing $|A| = 0$ , which is only marginally significant ( $\chi^2_1 = 3.558$ , $p = 0.059$ ). The persistence measure $P_\infty = V(x)/V(\epsilon)$ is a useful complement to unit-root-based measures for stationary processes, though the paper does not develop its inferential properties.