ABSTRACT This paper employs a new approach to analyze potentially omitted non-diversifiable factors in the idiosyncratic risks from multi-factor asset pricing models. It is shown that if there is an omitted non-diversifiable hidden factor, the idiosyncratic risks will contain persistent cross-sectional memory. An extended Rescaled Variance test generalized from L. Giraitis, P. Kokoszaka, R. Leipus, and G. Teyssiere  with finite forecast horizon is provided to investigate the cross-sectional memory of forecast errors in multifactor pricing models. Under the null hypothesis that idiosyncratic risks contain only short memory when there is no hidden non-diversifiable factor, we demonstrate that the extendedT-sample Rescaled Variance test statistic approximates a functional of weighted Brownian Bridge, which is distributed asymptotically as the T-sample Watson’s statistic presented by Maag . Using this approach, our empirical tests that compare forecast errors from the CAPM and Fama-French  model with the excess returns of 1391 firms indicate that there is a strong likelihood that the CAPM may require further identification of hidden non-diversifiable factor(s). Yet, there lacks convincing evidence that the Fama-French  model has an omitted non-diversifiable factor in idiosyncratic risks.
Cite this paper
J. Jeng and Q. Liu, "Do Idiosyncratic Risks in Multi-Factor Asset Pricing Models Really Contain a Hidden Non-Diversifiable Factor? A Diagnostic Testing Approach," Journal of Mathematical Finance, Vol. 2 No. 3, 2012, pp. 251-263. doi: 10.4236/jmf.2012.23028.
 L. Giraitis, P. Kokoszaka, R. Leipus and G. Teyssiere, “Rescaled Variance and Related Tests for Long Memory in Volatility and Levels,” Journal of Econometrics, Vol. 112, No. 2, 2003, pp. 265-294.
 U. R. Maag, “A k-Sample Analogue of Watson’s Statistic,” Biometrika, Vol. 53, No. 3-4, 1966, pp. 579-583.
 E. F. Fama and K. R. French, “Common Risk Factors in the Returns on Stocks and Bonds,” Journal of Financial Economics, Vol. 25, No. 1, 1993, pp. 23-49.
 A. Goyal and P. Santa-Clara, “Idiosyncratic Risk Matters,” Journal of Finance, Vol. 58, No. 3, 2003, pp. 975- 1008. doi:10.1111/1540-6261.00555
 D. Mayers, “Nonmarketable Assets, Market Segmentation, and the Level of Asset Prices,” Journal of Financial and Quantitative Analysis, Vol. 11, No. 1, 1976, pp. 1-12.
 B. G. Malkiel and Y. Xu, “Idiosyncratic Risk and Security Returns,” Working Paper, University of Texas, Dallas, 2006.
 H. Guo and R. Savickas, “Does Idiosyncratic Risk Matter: Another Look,” Working Paper 2003-025A, Federal Reserve Bank of St. Louis, 2003.
 T. Bali, N. Cakici, X. Yan and Z. Zhang, “Does Idiosyncratic Risk Really Matter?” Journal of Finance, Vol. 60, No. 2, 2005, pp. 905-929.
 A. Ang, R. J. Hodrick, Y. Xing and X. Zhang, “The Cross-Section of Volatility and Expected Returns,” Journal of Finance, Vol. 61, No.1 , 2006, pp. 259-298.
 F. Fu, “Idiosyncratic Risk and the Cross-Section of Ex- pected Stock Returns,” Journal of Financial Economics, Vol. 91, No. 1, 2009, pp. 24-37.
 H. Guo and R. Savickas, “Average Idiosyncratic Volatility in G7 Countries,” Review of Financial Studies, Vol. 21, No. 3, 2008, pp. 1259-1296. doi:10.1093/rfs/hhn043
 G. Chamberlain and M. Rothschild, “Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets,” Econometrica, Vol. 51, No. 5, 1983, pp. 1281-1304. doi:10.2307/1912275
 F. Lavancier, “Invariance Principles for Non-Isotropic Long Memory Random Fields,” Statistical Inferences on Stochastic Processes, Vol. 10, No. 3, 2007, pp. 255-282.
 H. White, “Asymptotic Theory for Econometricians,” Academic Press, Oxford, 2001, pp. 10-12.
 Y. Zhou and M. S. Taqqu, “How Complete Random Permutations Affect the Dependence Structure of Stationary Sequences with Long-Range Dependence,” Fractals, Vol. 14, No. 3, 2006, pp. 205-222.
 P. Embrechts and M. Maejima, “Self-Similar Processes,” Princeton Uniersity Press, 2002.
 H. Li and Y. Xu, “Survival Bias and the Equity Premium Puzzle,” Journal of Finance, Vol. 57, No. 5, 2002, pp. 1981-1995. doi:10.1111/0022-1082.00486
 B. M. Barber and J. D. Lyon, “Firm Size, Book-to-Market Ratio, and Security Returns: A Hold-out Sample of Financial Firms,” Journal of Finance, Vol. 52, No. 2, 1997, pp. 875-883. doi:10.1111/j.1540-6261.1997.tb04826.x
 Y. Li and W. J. Mayer, “Impact of Corrections for Dynamic Selection Bias on Forecasts of Retention Behav- ior,” Journal of Forecasting, Vol. 26, No. 8, 2007, pp. 571-582. doi:10.1002/for.1028
 G. S. Watson, “Goodness-of-Fit Tests on a Circle,” Biometrika, Vol. 48, No. 1-2, 1961, pp. 109-112.
 B. M. Brown, “Grouping Corrections for Circular Goodness-of-Fit Tests,” Journal of Royal Statistical Society, Serial B, Vol. 56, 1994, pp. 275-283.