The evaluation and comparison of three benchmark asset pricing models with daily data: supplementary evidence

ABSTRACT

Recent studies advocate two new benchmark models (the Fama-French five-factor model and the Hou, Xue and Zhang four-factor model) with monthly data. Our daily data approach provides considerable supplement to the monthly data approach presented in recent studies. We adopt the advanced bootstrap methodology by replicating the original data sample, and this approach should effectively alleviate the problem of too much noise in the data of daily return. A two-pass cross-sectional regression and GMM with several useful testing statistics are used to more thoroughly diagnose the specifications of the model. The following consistency is observed when using different frequencies of sample data: the evidence indicates that the two newer benchmark models (the Fama-French five-factor model and the Hou, Xue and Zhang four-factor model) outperform the Fama-French three-factor model in estimating a few well-known portfolios (formed on different anomalies). However, several specification tests do not robustly accept the correct specifications of the Fama-French five-factor model and the Hou, Xue and Zhang four-factor model.

KEYWORDS:

• Supplement
• daily data
• bootstrap
• benchmark models

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplementary material

Supplemental data for this article can be accessed here.

Notes

1. For example, Faff (2004) is the only author to collect a sample of Australian daily data to test the performance of the Fama-French three-factor model, and he concludes that the size factor has a negative risk premium, which appears to reveal performance problems regarding the use of the three-factor model as a benchmark. In another strand of literature using daily data of Gençay, Selçuk, and Whitcher (2003), (2005)) address the problem of wavelet beta in asset pricing.

2. Cochrane (2005) and Hayashi (2001) suggest that the efficient weighting matrix appears to have poor small sample properties; for example, the size of the efficient Wald statistic in small samples substantially exceeds the assumed significance value, and a poor spectral density matrix leads to misleading standard errors.

3. The time-series methodology is adequate when all the factors in the model represent excess stock returns (see Cochrane (2005)).

4. Cochrane (2005) explicitly stated that the GMM, time-series, cross-sectional procedures and distribution theory are similar but not identical. Time-series procedures are also the first stage of a two-pass cross-sectional regression. Since GMM and cross-sectional procedures are not directly comparable, most current studies use both methods to circumvent any estimation problems associated with a specific methodology.

5. The different estimations of risk premiums in beta method (the cross-sectional stage) usually can lead to different estimations of pricing errors.

6. For example (in our study), if the risk factor mean is 0.01, but the risk premium is 0.03 from cross-sectional regression. We suggest such difference is somewhat big, as 0.03 of the risk premium is (0.03–0.01)/0.01*100% rate higher than 0.01 of risk factor mean.

7. Monthly data are medium-frequency data, while daily data are high-frequency data.

8. Time-series regression is adequate for these two models, but it does not illustrate the total image of empirical asset pricing. Rather, our study is just interested in other aspect of empirical tests, where the related statistic can in fact be somewhat different from time-series regression.

9. Our study specifically clarifies this issue.

10. However, these two series are highly correlated. We incorporate such a small difference in our empirical work.

11. Lewellen, Nagel, and Shanken (2010) argue that this strategy can relax the tight structure of the SBM25 portfolios to an efficient degree.

12. Lewellen, Nagel, and Shanken (2010) indicate that no model is perfect or will explain all patterns in the data; this truism makes it tempting to view a model as successful if it explains even a few anomalies.

13. We estimate the full-sample betas, following Lettau and Ludvigson (2001), Petkova (2006), and Kan, Robotti, and Shanken (2013), among others.

14. In our replication of the bootstrap simulation, we assume that the (demeaned) asset returns and pricing factors are all derived from normal distributions. Moreover, the bootstrap accounts for the contemporaneous cross-correlation among the test assets that leads to their small factor structure (see Lewellen, Nagel, and Shanken (2010)).

15. Following the previous literature, we also use the identity-weighting weighting matrix in the first stage to obtain the optimal weighting matrix.

16. http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

17. Notably, the high value of the autocorrelation is also consistent with the previous literature (e.g., Maio and Santa-Clara (2012)).

18. In particular, the Fama-French three-factor model only explains 0.02 of Campbell and Vuolteenaho (2004) R² as the FF25 portfolios should, to some extent, be relaxed by industrial portfolios; additionally, it is well-known that the variation in industrial portfolios is difficult for a model to explain.

19. Throughout our empirical study, a sufficient high value of Shanken t-statistic of a factor (exceeding 3.00) mostly implies that the corresponding p-value of the bootstrap t-statistic accepts the significance of that pricing factor.

20. When we refer to a t-statistic as ‘high,’ we refer to its absolute value, thus including cases in which the t-statistic is negative, throughout the study.

21. The negative value of the cma risk premium is not consistent with Fama and French (2015) We explain such variation as following two reasons: first of all, the risk premium may be either positive or negative for a few pricing factors; for example, the premium of inflation could be either positive or negative in pricing different asset sample. Also, the liquidity risk could be either positive or negative (see Martınez et al. (2002)). The negative value of risk premium, such as our case, could usually be explained by the hedge effect. For example, Hsu and Huang (2010) repots the positive risk premium of technology risk factor; however, they also find negative betas (slope) in big value stock and they explain that estimation can be due to the hedge demand. Second, Faff (2004) also reports a negative value of size premium in estimating Fama-French three-factor model with daily data sample. Our estimations of variations are similar as Faff’s (2004) empirical results and we suggest that such variations (the sign of risk premium) mainly due to different features of daily data sample and monthly data sample (see summarized statistics in our on-line Appendix Tables).

22. The percentage decrease in the HJ-distance in total amount is logical in this place.

23. A few statistics are not highly significant, i.e., the t-statistic of the augmented factors in the Fama-French five-factor models do not significantly exceed 3.00.

24. For example, the Fama and French (2015) model should perform better than FF3 model in cross-sectional regression (10 short reversal portfolios) using the daily data sample. But the time-series estimations appear to reject this point.