Testing economic signals when extreme values and long memory collide

Two problems that reinforce each other

A predictor can be difficult to analyze for two different reasons. Long memory means that dependence between observations fades slowly. Heavy tails mean that extreme observations occur relatively frequently; in the models studied here, the underlying distribution can have infinite variance, so its spread cannot be summarized by a finite population variance.

Together, these properties complicate regression inference: deciding whether an estimated relationship is distinguishable from chance. A fitted coefficient is not enough. Researchers also need a test whose false-positive rate behaves as advertised.

In an arXiv preprint revised on November 8, 2026, Ioannis Kasparis of the University of Cyprus develops methods intended to recover familiar statistical tests across several classes of persistent, heavy-tailed predictors.

The interaction matters because heavier tails change the boundary between stationary and nonstationary fractional models. Stationarity describes statistical behavior that remains stable over time. In the paper’s framework, the boundary depends jointly on the memory parameter, d, and the tail parameter, α: it occurs at d = 1 − 1/α. Smaller α means heavier tails. A predictor that would fall on the stationary side under finite-variance assumptions can therefore fall on the nonstationary side when tails are sufficiently heavy.

Transform the instrument, not the evidence

The proposed method extends IVX, an instrumental-variable approach that constructs an auxiliary series from changes in the predictor. Its filter reduces the instrument’s persistence relative to a strongly nonstationary predictor.

Kasparis adds weight functions that limit how quickly transformations of this instrument grow. One example is a signed square root: retain an observation’s sign but take the square root of its magnitude. This is not a procedure for deleting extreme observations. It changes how the constructed instrument enters estimation and testing.

Filtering alone is not sufficient in the heavy-tailed setting. The weights help keep the mathematical quantities used in the analysis within manageable growth limits.

Under the stated assumptions, the resulting weighted IVX t-statistic approaches the standard normal distribution as the sample grows. That allows familiar significance thresholds without requiring precise knowledge of the predictor’s persistence.

A separate weighted instrumental-variable estimator, WIV, serves mainly for estimating coefficients and quantities needed to calculate uncertainty. Under nonstationarity it can converge faster than weighted IVX, but it does not generally provide equally dependable conventional tests. The distinction is between estimating a relationship and reliably testing it.

A different route through the mathematics

The theoretical contribution also addresses a missing mathematical tool. Analysts often use functional central limit theorems to describe the behavior of an entire rescaled time series, then derive regression results from that description.

For part of the heavy-tailed fractional setting considered here, the prospective limiting process has paths that are unbounded on every interval. The usual path-based argument cannot apply.

Instead, the paper establishes limiting results directly for suitable transformed sample averages. These results provide a route to inference even where the conventional whole-path approach fails.

What the simulations show

The simulation study uses 20,000 replications with sample sizes of 250, 500 and 1,000, varying persistence, tail behavior and dependence between predictor innovations and regression errors.

For tests intended to reject a true null hypothesis 5% of the time, Kasparis reports weighted IVX rejection rates reaching roughly 8% in some configurations. Corresponding WIV and ordinary least-squares tests exceeded 35% in some nonstationary configurations. Some alternative weights produced weighted IVX rates approaching 9%.

That is evidence of substantially improved false-positive control in these experiments, not perfect calibration or a guarantee for arbitrary data. Some simulated settings also go beyond the proven sufficient conditions.

Financial application and remaining limits

An application uses stock-return predictors from a dataset updated through 2024. Results vary with the return series, sampling frequency and weight function. For example, stock variance shows no significant monthly predictability in the reported tables, while some quarterly specifications using the CRSP_SPvw return series do show significance. These regression tests do not establish causation or demonstrate profitable out-of-sample trading.

The inference theory also assumes finite-variance regression errors with zero conditional mean and constant conditional variance. Allowing infinite-variance predictors does not mean allowing unrestricted infinite-variance errors. Additional restrictions govern predictor models and weight functions.

Finally, the supplied preprint explicitly says that a coefficient-uniform justification of a truncation estimate remains to be checked before submission. This account does not independently verify the proofs. The work offers a proposed route to more reliable testing, supported by simulations, but its preprint status and that unresolved check remain material qualifications.