Statistical Data Analysis and Inference › Limitations and Practical Considerations

Historical-return inference depends on a sampling model. The following limitations affect how we interpret its estimates.

A major issue is non-stationarity. Financial markets evolve over time. When you use a historical mean estimate to forecast returns, you assume that the fitted historical process is informative about the forecast horizon. The strength of that assumption depends on the parameter, asset, horizon, and regime, and should be evaluated with structural-break diagnostics and out-of-sample performance. Regime changes and evolving market microstructure can violate the identical-distribution assumption underlying classical statistics.

A second challenge is the low signal-to-noise ratio in many financial return series. The ratio of daily volatility to mean return depends on the asset, period, and estimator, but mean-return uncertainty often remains large even with long samples. A useful analysis should report the sample-specific standard error or confidence interval rather than rely on a universal volatility-to-mean ratio.

Serial dependence and volatility clustering can further complicate inference. Return autocorrelation varies by asset, frequency, and market microstructure, while squared or absolute returns often show persistence. Standard errors computed under independence can be biased in either direction depending on the dependence pattern; heteroskedasticity- and autocorrelation-consistent estimators or suitable bootstraps estimate the relevant long-run variance.

These limitations shape how you use the results:

• Treat point estimates with skepticism.
• Report intervals from a valid procedure, even when they are wide, rather than presenting uncertain estimates with false precision.
• Consider robust statistics that are less sensitive to outliers.
• Distinguish statistical significance from economic significance and practical tradability.