One-sided or two-sided, and when halving is fair

Question

A strategy's mean daily return is tested against zero and yields $z = 1.8$ . Before running the backtest, the researcher had committed to a directional hypothesis: the strategy makes money (mean return $> 0$ ).

Compute the one-sided p-value for $z = 1.8$ , state how it compares to the two-sided p-value, and explain the rule for when you are allowed to use the one-sided version.

Accepted Answer

The one-sided computation. A one-sided (upper-tail) test only counts evidence in the predicted direction. The p-value is the area in the upper tail beyond the observed z: p{	ext{one}} = 1 - \Phi(1.8) = 1 - 0.9641 = 0.0359 \approx 0.036. Compared to two-sided. The two-sided test counts extremes in both directions, so it doubles the tail: p{	ext{two}} = 2\,[1 - \Phi(1.8)] = 0.072. Here the one-sided p is significant at 5% while the two-sided p is not. That is exactly why the choice is tempting, and why it must be made honestly. The rule for halving. You may use the one-sided p-value only if y…

One-sided or two-sided, and when halving is fair

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