Delgado-Velasco (2005) Sign Tests for Long-Memory Time Series

Summary

Delgado and Velasco (2005) develop sign-based tests for the long-memory parameter $d$ of ARFIMA(p,d,q) (autoregressive fractionally integrated moving average) processes. The tests use only the signs of (pre-whitened) residuals, making them exact (finite-sample distribution-free) for simple hypotheses and locally most powerful (LMP) among sign tests. The key appeal is robustness to fat-tailed and infinite-variance innovations — precisely the cases where standard tests (Dickey-Fuller, Tanaka score test) collapse.

Key Claims

• The sign test statistic $T_n(d_0) = n\sum_{j=1}^{n-1} j^{-1}\tilde\gamma_{nj}(d_0)$ (sum of weighted sign-autocorrelations) is LMP among sign tests and has an exact distribution under $H_0$ that is independent of the error distribution.
• Asymptotically $n^{-1/2}T_n \to N(0, \pi^2/6)$, but convergence is very slow — exact Monte Carlo critical values are necessary for $n < 2000$.
• For composite ARFIMA hypotheses, plugging in $\sqrt{n}$-consistent robust autoregressive-moving-average (ARMA) estimates for nuisance parameters yields a $\chi^2(1)$ (or $N(0,1)$) limiting statistic for $d$ alone.
• Monte Carlo: Under Cauchy ($t_1$) errors, the sign test has 16% power at $n=400$, $a=0.5$; Dickey-Fuller has 3.2% and Tanaka has 1.6% — near or below nominal size.
• Sign test power increases with fatter tails: heavier tails make sign changes more informative about $d$.

Concepts Introduced or Extended

• Long Memory and Fractional Integration — sign-based robust tests for $d$; exact finite-sample inference under infinite variance

Entities Mentioned

• Miguel A. Delgado
• Carlos Velasco

Quotes

"This methodology is especially appealing when applied to time series exhibiting infinite variance, which are likely in many fields, such as finance, economics, telecommunications and hydrology."

"Tanaka's test performs very poorly in the presence of errors with fat distribution tails."

My Take

The exactness result is elegant: because $S_t(d_0) = \text{sign}(\varepsilon_t(d_0))$ are i.i.d. Rademacher under $H_0$ for any symmetric continuous innovation distribution, the test requires only the median-zero assumption. The practical caveat is that the asymptotic normal approximation is useless for economically relevant sample sizes — the paper itself shows severe under-rejection even at $n=2000$. The test is most useful as a robustness check when there is concern about fat-tailed or heavy-tailed innovations.