Long Memory Processes and Fractional Integration in Econometrics

Summary

Baillie (1996) surveys the econometric literature on long memory processes and fractional integration, covering theoretical foundations, estimation methods, and empirical applications in macroeconomics and finance. The paper reviews three equivalent definitions of long memory, derives the autoregressive fractionally integrated moving average (ARFIMA) model and its population properties, and systematically evaluates semiparametric and maximum likelihood estimators (rescaled range (R/S), Geweke-Porter-Hudak (GPH) log-periodogram, local Whittle, Fox-Taqqu approximate MLE, Sowell exact MLE). A companion section introduces FIGARCH — Fractionally Integrated generalized autoregressive conditional heteroscedasticity (GARCH) — as a model for long memory in conditional variance. The empirical review finds robust long memory in forward exchange premia and inflation differentials but limited evidence for stock return levels; the most striking finding is pervasive long memory in squared and absolute returns and in implied volatility series.

Key Claims

• Long memory is equivalent to three conditions: (i) $\sum |\rho_j|$ diverges; (ii) spectral density $f(\omega) \sim c\,|\omega|^{-2d}$ as $\omega \to 0$; (iii) $\text{var}(S_T) = O(T^{2d+1})$ for $d > 0$.
• Granger's (1980) aggregation theorem: aggregating heterogeneous AR(1) processes whose autoregressive coefficients are Beta-distributed generates an ARFIMA(0,d,0) process in the limit, providing micro-founded theoretical motivation.
• The GPH log-periodogram estimator is consistent and asymptotically normal but slower than $T^{1/2}$; bandwidth $m \sim T^{1/2}$ is common but no optimal data-driven rule has been established.
• Lo's (1991) modified R/S statistic — which replaces sample standard deviation with a Newey-West long-run standard deviation — finds little evidence of long memory in stock market returns, in contrast to the uncorrected Hurst R/S.
• FIGARCH(p,d,q) replaces the unit-root polynomial in the integrated GARCH (IGARCH) variance equation with a fractional difference operator $(1-L)^d$, yielding impulse-response weights in the conditional variance that decay hyperbolically; estimated $d \approx 0.4$–$0.6$ for major USD exchange rates.
• Empirical evidence for long memory is strongest in forward premia and inflation differentials ($\hat d \approx 0.45$–$0.80$); moderate for interest rate spreads; weak for stock return levels after Lo's (1991) correction.
• CI(1, 1-d) fractional cointegration — where the cointegrating residual is $I(d)$ rather than $I(0)$ — is consistent with asset pricing models and represents an alternative to outright rejection of the present-value relationship.

Concepts Introduced or Extended

• Long Memory and Fractional Integration (comprehensive survey; introduces FIGARCH)
• GARCH and BEKK-GARCH (FIGARCH extension of IGARCH)
• Purchasing Power Parity (fractional cointegration reinterpretation of PPP failures)

Entities Mentioned

• Richard Baillie
• Clive Granger
• Tim Bollerslev
• Francis Diebold

Quotes

"Long memory models may offer potentially important insights into the pricing of risk." (p. 52)

"The knife-edge distinction between unit root and stationarity has received enormous attention, while intermediate fractional values of $d$ are both theoretically motivated and empirically common in economic data." (paraphrased, introduction)

My Take

The paper's most durable contribution is the FIGARCH formulation: IGARCH persistence is a structural artifact of the unit-root parameterization, not a feature of the data, while FIGARCH explicitly parameterizes the hyperbolic decay observed in autocorrelations of squared returns. As a survey it is unusually comprehensive and remains the standard entry point. Two caveats: (1) Baillie et al.'s original FIGARCH parameterization permits negative impulse-response weights for some parameter values, violating non-negativity of the conditional variance — a technical issue addressed in subsequent work; (2) the structural-break literature (Diebold-Inoue 2001) shows that regime switching and level shifts generate the same sample autocorrelation pattern as genuine long memory, complicating all empirical claims in the paper.