On tests of the conditional relationship between beta and returns

Abstract

The Pettengill et al. (1995) test of the conditional relationship between beta and returns has recently become widely used. This article shows that there is a large bias in that test. The test is almost guaranteed to be satisfied, regardless of the model that generates expected returns. In particular, even if the Capital Asset Pricing Model (CAPM) is not true and expected returns and beta are unrelated, the test will detect statistically significant results of the size that they report in line with their hypothesis. The reason for the bias is that the ex post selection criterion used to partition data automatically generates coefficient values that the test interprets as being evidence in favour of the CAPM.

Acknowledgements

The author is grateful to two anonymous referees for helpful comments.

Notes

¹ Pettengill and co-workers (1995), p. 107.

² Pettengill and co-workers (1995), Table 5.

³ Table 6 of the article gives the average return on different portfolios. Averaging these gives an average market return of 15.79% per annum, or 1.316% per month. Subtracting the average risk premium of 0.91% per month gives a riskless rate of 0.406% per month.

⁴ See, for instance, the large literature on beta stability starting with Blume (1971).

⁵ The differences are negligible. The calculations are available from the author.