How to Use Paired Comparison Tests for Evaluating Investment Strategies

If two samples are not independent, a paired comparison test should be used for mean testing. The null and alternative hypotheses for two‑tailed and one‑tailed tests are presented, where μd denotes the difference of the two sample means, usually set to 0.

The test statistic follows a t‑distribution with ν = n‑1 degrees of freedom, calculated as t = (d̄) / (sd/√n), where d̄ is the mean of the paired differences and sd is the standard error of the differences.

Example 1: The “D‑10” investment strategy. Mequeen, Shields and Thorley (1997) compared a popular strategy that invests in the ten Dow Jones stocks with the highest returns against a buy‑and‑hold strategy covering all thirty Dow Jones stocks over 1946‑1995. Questions include: (1) state the two‑tailed null and alternative hypotheses that the mean return difference equals zero; (2) identify the test statistic; (3) find the critical value at a given significance level; (4) decide whether to reject the null; (5) explain why a paired test is appropriate.

Solution: (1) μd = 0 is the null hypothesis. (2) Because the population variance is unknown, the test statistic is a t‑test with ν = 49 degrees of freedom. (3) From the t‑distribution table, the critical value at the chosen significance level is 2.68; if the computed t exceeds ±2.68, reject H0. (4) The computed t exceeds the critical value, so H0 is rejected, indicating a statistically significant difference in average returns. (5) The D‑10 and D‑30 strategies share the same stocks, making the samples dependent; the correlation of returns is positive, thus a paired comparison test is appropriate.

Reference: Zhu Shunquan, Economic and Financial Data Analysis and Its Python Application.