How much data turns a tiny edge significant

Question

A strategy has a small but fixed per-day signal-to-noise ratio: its average daily return divided by the daily standard deviation is $0.1$ . The z-statistic for "mean return $> 0$ " grows with the number of days $n$ as $z = 0.1\sqrt{n}$ .

About how many days of data are needed for this fixed edge to just reach two-sided significance at the 5% level, and what does that dependence on $n$ tell you about reading p-values?

Accepted Answer

The computation. Two-sided significance at 5% needs z \geq 1.96. Setting z = 0.1\sqrt{n} = 1.96: \sqrt{n} = \frac{1.96}{0.1} = 19.6, \qquad n = 19.6^2 \approx 384 \approx 385 	ext{ days}. So the very same daily edge is nowhere near significant at n = 100 (there z = 0.1 	imes 10 = 1.0, p \approx 0.32) but clears the bar at about 385 days, and is comfortably significant at n = 400 (where z = 2.0, p \approx 0.046). What this means. The p-value is not a measure of how big or real the edge is. The edge here never changed. All that changed was the amount of data, which shrinks the noise in the es…

How much data turns a tiny edge significant

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