Diebold-Inoue (2001) Long Memory and Regime Switching

Summary

Diebold and Inoue argue that long memory and regime switching — usually studied as distinct phenomena — are intimately related. They show analytically that stochastic regime switching is easily confused with long memory, even asymptotically, provided only a "small" amount of switching occurs (a switching probability that shrinks with the sample size at a precise rate). The result holds in a simple mixture model, Engle-Lee's (1999) stochastic permanent break model, and Hamilton's (1989) Markov switching model, and a Monte Carlo study supports its practical relevance.

Key Claims

Small-switching mechanism (mixture model). For a mixture $v_t=w_t$ with probability $p$ (else $0$ ), $w_t\sim N(0,\sigma_w^2)$ : if $p$ is held constant the partial-sum variance grows like $O(T)$ (short memory), but letting $p=O(T^{2d-2})$ for $0<d<1$ makes $\mathrm{var}(\sum_t v_t)=O(T^{2d+1})$ — exactly the growth rate of an $I(d)$ (long-memory) process (Proposition 1).
Robust across models. The same "regime switching mimics long memory" behaviour arises in a mean-plus-noise state-space extension, in Engle-Lee's stochastic permanent break model, and in Hamilton's two-state Markov switching model.
Asymptotic, not just finite-sample. The confusion persists asymptotically under the shrinking-switching parameterization — long-memory estimators (e.g. GPH) return significantly positive $\hat d$ when the true process is pure regime switching.
Monte Carlo confirmation. Simulations show estimated $\hat d>0$ and slowly-decaying sample autocorrelations under regime switching, matching the theory.
Practical warning. A finding of long memory may reflect infrequent regime switches or structural breaks rather than genuine fractional integration; the two are observationally near-equivalent.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"We show analytically that stochastic regime switching is easily confused with long memory, even asymptotically, so long as only a 'small' amount of regime switching occurs, in a sense that we make precise."

My Take

A cautionary classic: it makes rigorous the uneasy suspicion that estimated long memory might be an artifact of rare breaks. The elegant part is the rate result — tie the switching probability to $T^{2d-2}$ and the partial-sum variance replicates $I(d)$ exactly, so no finite-sample hand-waving is needed. Paired with Granger-Hyung (2004) and Hillebrand (2004) on the variance side, it establishes the now-standard hygiene rule the long-memory page states: test for structural breaks before declaring fractional integration. The flip side, worth remembering, is that the equivalence runs both ways — it does not prove long memory is always spurious, only that the data alone cannot easily separate the two.