Abstract
The capital asset pricing model (CAPM), Fama-French (FF), and Pástor-Stambaugh (PS) factor models are examined using a new dynamic rolling regression version of the generalized method of moments (GMM) method. This rolling regression framework not only allows us to investigate phases of the business cycle, but also permits regression estimates to vary through time due to changes in the development and efficiency of the sectors. The principal reasons for using the dynamic GMM with robust instruments is that some of these factors are measured with errors and the disturbances may be non-spherical. The CAPM appears as the most parsimonious model to explain the FF sector returns. Furthermore, the rolling GMM approach is clearly more sensitive to dynamic financial episodes than the ordinary least squares approach. In particular, liquidity has some anticipatory power, as it is able to forecast the 2007–2009 crises with heightened volatility starting in late 2005.
Similar content being viewed by others
Notes
Our selection of 60 months follows the convention of Reilly and Brown (2009, p. 219). They note that there is no theoretically correct time interval for estimating returns. They conclude that the 60-month period is widely used by Morningstar and others, for example, and seems to be neither too long nor too short.
For more detail, see Pástor-Stambaugh (2003). For a discussion of liquidity measures, see Johann and Theissen (2013).
HAC is the heteroscedasticity and autocorrelation consistent estimator. We used the “Iterate to Convergence” Newey and West (1987) methodology of EViews 8.1.
The assumption of a normally distributed matrix of errors is used to simplify the mathematical proof of the consistency of the estimators in this paper. This assumption is in no way a limitation in the modeling process of the time series used in this paper. Our proposed GMMd estimator is based on the higher moments of the observed financial data and is thus able to capture the data’s non-linearity, which is one of the important goals of this estimator.
See Benninga (2014), p. 276.
See Roll (1977) for a discussion of the problems in testing the CAPM theory.
French’s website is http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
Pástor’s website is http://faculty.chicagobooth.edu/lubos.pastor/research/liq_data_1962_2013.txt
Markowitz (2012) noted that the mean-variance model still works well in the presence of moderate amounts of skewness and kurtosis.
The LIQ variable is really a measure of illiquidity, not liquidity, as correctly noted by Bodie et al. (2015, p. 406).
For an introduction to non-linear models including multivariate GARCH processes with financial applications, see Racicot (2012).
We conducted other experiments where the dynamic conditional correlation (DCC) between the average of the returns of the FF 12 sector and SMB and the average and LIQ is computed. We find that the DCC correlation between the average and SMB is much higher than the one obtain for LIQ. This is further evidence that SMB could be a good proxy for LIQ.
Although not shown, similar results were obtained for the FF model.
We have conducted further experiments using OLS to benchmark our results for the dynamic GMMd. The results are quite similar although GMMd is more sensitive to the financial crises observed in our sample, which is the virtues the dynamic estimator proposed in this article.
References
Baba, Y., Engle, R., Kraft, D., & Kroner, K. (1990). Multivariate simultaneous generalized ARCH. Department of Economics, University of California, San Diego, CA: Unpublished Paper.
Benninga, S. (2014). Financial Modeling (4th ed.). Cambridge, MA: MIT Press.
Black, F. (1976). Studies in stock price volatility changes. Proceedings of the 1976 Business Meeting of the Business and Economic Statistics Section. American Statistical Association, 177–181.
Bodie, Z., Kane, A., Marcus, A. J., Perrakis, S., & Ryan, P. J. (2015). Investments (8th ed.). Whitby, ON: McGraw-Hill Ryerson.
Campbell, J., Lo, A., & MacKinlay, A. (1997). The Econometrics of Financial Markets. Princeton, NJ: Princeton University Press.
Carhart, M. M. (1997). On persistence in mutual fund performance. Journal of Finance, 52(1), 57–82.
Cochrane, J. (2005). Asset Pricing (revised ed.). Princeton, NJ: Princeton University Press.
Dagenais, M. G., & Dagenais, D. L. (1994). GMM estimators for linear regression models with errors in the variables. In Centre de recherche et développement en économique (CRDE), Working Paper 0594. Montreal, QC: University of Montreal.
Durbin, J. (1954). Errors in variables. International Statistical Review, 22(1/3), 23–32.
Engle, R. F., & Kroner, K. (1995). Multiplicative simultaneous generalized ARCH. Econometric Theory, 11(1), 122–150.
Fama, E. F. (1963). Mandelbrot and the stable Paretian hypothesis. Journal of Business, 36(4), 420–429.
Fama, E. F. (1965). Portfolio analysis in a stable Paretian market. Management Science, 11(3), 404–419.
Fama, E. F., & French, K. R. (1992). The cross-section of expected stock returns. Journal of Finance, 47(2), 427–465.
Fama, E. F., & French, K. R. (1993). Common risk factors in the returns of stocks and bonds. Journal of Financial Economics, 33(1), 3–56.
Fama, E. F., & MacBeth, J. (1973). Risk, return, and equilibrium: empirical tests. Journal of Political Economy, 81(3), 607–636.
Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50(4), 1029–1054.
Jarque, C. M., & Bera, A. K. (1980). Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Economics Letters, 6, 255–259.
Jensen, M. C. (1968). The performance of mutual funds in the period 1945–64. Journal of Finance, 23(2), 389–416.
Johann, T., & Theissen, E. (2013). Liquidity measures. In A. R. Bell, C. Brooks, & M. Prokopczuik (Eds.), Handbook of Research Methods and Applications in Empirical Finance (pp. 238–255). Cheltenham, U.K.: Edward Elgar.
Jurczenko, E., & Maillet, B. (2006). The four-moment capital asset pricing model: Between asset pricing and asset allocation. In E. Jurczenko & B. Maillet (Eds.), Multi-Moment Asset Allocation and Pricing Models (ch. 6). Chichester, England: John Wiley & Sons..
Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business, 36(4), 394–419.
Mandelbrot, B. (1972). Correction of an error in “The variation of certain speculative prices”. Journal of Business, 45(4), 542–543.
Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investments, Cowles Foundation Monograph 16. New York, NY: John Wiley & Sons.
Markowitz, H. (2012). The “Great Confusion” concerning MPT. Aestimatio, The IEB International Journal of Finance, 4, 8–27.
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach. Econometrica, 59(2), 347–370.
Newey, W. K., & West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703–708.
Pagan, A. R. (1984). Econometric issues in the analysis of regressions with generated regressors. International Economic Review, 25(1), 221–247.
Pagan, A. R. (1986). Two stage and related estimators and their applications. Review of Economic Studies, 53(4), 517–538.
Pagan, A. R., & Ullah, A. (1988). The econometric analysis of models with risk terms. Journal of Applied Econometrics, 3(2), 87–105.
Pal, M. (1980). Consistent moment estimators of regression coefficients in the presence of errors in variables. Journal of Econometrics, 14(3), 349–364.
Pástor, L., & Stambaugh, R. F. (2003). Liquidity risk and expected stock returns. Journal of Political Economy, 111(3), 642–685.
Racicot, F. E. (2012). Notes on nonlinear dynamics. Aestimatio, The IEB International Journal of Finance, 5, 162–221.
Racicot, F. E. (2015). Engineering robust instruments for GMM estimation of panel data regression models with errors in variables: a note. Applied Economics, 47(10), 981–989.
Racicot, F. E., & Rentz, W. F. (2015). The Pástor-Stambaugh empirical model revisited: evidence from robust instruments. Journal of Asset Management, 16(5), 329–341.
Racicot, F. E., & Théoret, R. (2009). On optimal instrumental variables venerators, with an application to hedge fund returns. International Advances in Economic Research, 15(1), 30–43.
Reilly, F., & Brown, K. (2009). Investment Analysis and Portfolio Management, 9e. Mason, OH: South-Western Cengage Learning.
Roll, R. (1977). A critique of the asset pricing theory’s tests. Journal of Financial Economics, 4(2), 129–176.
Rubinstein, M. (1973). The fundamental theorem of parameter-preference security valuation. Journal of Financial and Quantitative Analysis, 8(1), 61–69.
Sharpe, W. F. (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance, 19(3), 425–442.
Theil, H., & Goldberger, A. (1961). On pure and mixed estimation in economics. International Economic Review, 2(2), 65–78.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Racicot, FÉ., Rentz, W.F. & Kahl, A.L. Rolling Regression Analysis of the Pástor-Stambaugh Model: Evidence from Robust Instrumental Variables. Int Adv Econ Res 23, 75–90 (2017). https://doi.org/10.1007/s11294-016-9620-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11294-016-9620-x