With fast evolving econometric techniques being adopted in asset pricing, traditional linear asset pricing models have been criticized by their limited function on capturing the time-varying nature of data and risk, especially the absence of data smoothing is of concern. In this paper, the impact of data smoothing is explored by applying two asset pricing models with non-linear feature: cubic piecewise polynomial function (CPPF) model and the Fourier Flexible Form (FFF) model are performed on US stock returns as an experiment. The traditional beta coefficient is treated asymmetrically as downside beta and upside beta in order to capture corresponding risk, and further, to explore the risk premia attached in a cross-sectional context. It is found that both models show better goodness of fit comparing to classic linear asset pricing model cross-sectionally. When appropriate knots and orders are determined by Akaike Information Criteria (AIC), the goodness of fit is further improved, and the model with both CPPF and FFF betas employed showed the best fit among other models. The findings fill the gap in literature, specifically on both investigating and pricing the time variation and asymmetric nature of systematic risk. The methods and models proposed in this paper embed advanced mathematical techniques of data smoothing and widen the options of asset pricing models. The application of proposed models is proven to superiorly provide high degree of explanatory power to capture and price time-varying risk in stock market.
Citation: Fangzhou Huang, Jiao Song, Nick J. Taylor. The impact of time-varying risk on stock returns: an experiment of cubic piecewise polynomial function model and the Fourier Flexible Form model[J]. Data Science in Finance and Economics, 2021, 1(2): 141-164. doi: 10.3934/DSFE.2021008
With fast evolving econometric techniques being adopted in asset pricing, traditional linear asset pricing models have been criticized by their limited function on capturing the time-varying nature of data and risk, especially the absence of data smoothing is of concern. In this paper, the impact of data smoothing is explored by applying two asset pricing models with non-linear feature: cubic piecewise polynomial function (CPPF) model and the Fourier Flexible Form (FFF) model are performed on US stock returns as an experiment. The traditional beta coefficient is treated asymmetrically as downside beta and upside beta in order to capture corresponding risk, and further, to explore the risk premia attached in a cross-sectional context. It is found that both models show better goodness of fit comparing to classic linear asset pricing model cross-sectionally. When appropriate knots and orders are determined by Akaike Information Criteria (AIC), the goodness of fit is further improved, and the model with both CPPF and FFF betas employed showed the best fit among other models. The findings fill the gap in literature, specifically on both investigating and pricing the time variation and asymmetric nature of systematic risk. The methods and models proposed in this paper embed advanced mathematical techniques of data smoothing and widen the options of asset pricing models. The application of proposed models is proven to superiorly provide high degree of explanatory power to capture and price time-varying risk in stock market.
| [1] |
Akaike H (1974) A New Look at the Statistical Model Identification. IEEE Trans Autom Control 19: 716-723. doi: 10.1109/TAC.1974.1100705
|
| [2] |
Andersen TG, Bollerslev T (1998) DM-Dollar Volatility: Intraday Activity Patterns, Macroeconomic Announcements, and Longer Run Dependencies. J Financ 53: 219-265. doi: 10.1111/0022-1082.85732
|
| [3] |
Andersen TG, Bollerslev T, Cai J (2000) Intraday and Interday Volatility in The Japanese Stock Market. J Int Financ Mark Inst Money 10: 107-130. doi: 10.1016/S1042-4431(99)00029-3
|
| [4] | Andersen TG, Bollerslev Z, Diebold FX (2002) Parametric and Nonparametric Volatility Measurement. Center for Financial Institutions Working Papers. Wharton School Center for Financial Institutions, University of Pennsylvania. |
| [5] |
Ang A, Chen J, Xing YH (2006) Downside Risk. Rev Financ Stud 19: 1191-1239. doi: 10.1093/rfs/hhj035
|
| [6] |
Ang A, Kristensen D (2012) Testing Conditional Factor Models. J Financ Econ 106: 132-156. doi: 10.1016/j.jfineco.2012.04.008
|
| [7] |
Bollerslev T, Cai J, Song FM (2000) Intraday Periodicity, Long Memory Volatility, And Macroeconomic Announcement Effects In The US Treasury Bond Market. J Empir Financ 7: 37-55. doi: 10.1016/S0927-5398(00)00002-5
|
| [8] |
Chakrabarti G, Das R (2021) Time-varying beta, market volatility and stress: A comparison between the United States and India. IIMB Manage Rev 33: 50-63. doi: 10.1016/j.iimb.2021.03.003
|
| [9] | Chung U, Jung J, Shee CH, et al. (2001) Economies of Scale and Scope in Korea's Banking Industry: Evidence from The Fourier Flexible Form. J Korean Econ 2: 87-111. |
| [10] | Dobrynskaya V (2020) Is Downside Risk Priced In Cryptocurrency Market? HSE Working papers WP BRP 79/FE/2020, National Research University Higher School of Economics. |
| [11] |
Eilers P, Marx B (1996) Flexible Smoothing with B-Splines and Penalties. Stat Sci 11: 89-121. doi: 10.1214/ss/1038425655
|
| [12] | Eilers P, Marx B (2004) Splines, Knots and Penalties. Comput Stat Sci 2: 637-653. |
| [13] |
Engle RF (2002) Dynamic Conditional Correlation - A Simple Class of Multivariate GARCH Models. J Bus Econ Stat 20: 339-350. doi: 10.1198/073500102288618487
|
| [14] | Engle RF, Gonzalez-Rivera G (1991) Semiparametric ARCH Models. J Bus Econ Stat 9: 345-359. |
| [15] |
Engle RF, Russell JR (1998) Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data. Econometrica 66: 1127-1162. doi: 10.2307/2999632
|
| [16] |
Evans KP, Speight AEH (2010) Intraday Periodicity, Calendar and Announcement Effects In Euro Exchange Rate Volatility. Res Int Bus Financ 24: 82-101. doi: 10.1016/j.ribaf.2009.04.001
|
| [17] |
Fama E, Macbeth J (1973) Risk, Return, and Equilibrium: Empirical Tests. J Political Econ 81: 607-636. doi: 10.1086/260061
|
| [18] | Featherstone AM, Cader HA (2005) Bayesian Inferences on Fourier Flexible Functional Form in Agricultural Production. American Agricultural Economics Association, 2005 Annual Meeting, July 24-27, Providence, RI: 25. |
| [19] | Ferguson JC, Company B (1963) Multi-Variable Curve Interpolation, Boeing Company. |
| [20] |
Gallant AR (1981) On the Bias in Flexible Functional Forms and an Essentially Unbiased Form: The Fourier Flexible Form. J Econometrics 15: 211-245. doi: 10.1016/0304-4076(81)90115-9
|
| [21] |
Gallant AR (1982) Unbiased Determination of Production Technologies. J Econometrics 20: 285-323. doi: 10.1016/0304-4076(82)90022-7
|
| [22] | Gallant AR (1984) The Fourier Flexible Form. Am J Agri Econ 66: 204. |
| [23] |
Giglio S, Kelly B, et al. (2016) Systemic risk and the macroeconomy: An empirical evaluation. J Financ Econ 119: 457-471. doi: 10.1016/j.jfineco.2016.01.010
|
| [24] | Härdle W (1992) Applied Nonparametric Regression, Cambridge University Press. |
| [25] | Härdle W, Hall P, Marron JS (1988) How Far Are Automatically Chosen Regression Smoothing Parameters From Their Optimum? J Am Stat Assoc 83: 86-95. |
| [26] |
Huang F (2019) The impact of downside risk on UK stock returns. Rev Account Financ 18: 53-70. doi: 10.1108/RAF-07-2017-0139
|
| [27] |
Huang TH, Wang MH (2004) Estimation of Scale and Scope Economies in Multiproduct Banking: Evidence from the Fourier Flexible Functional Form with Panel Data. Appl Econ 36: 1245-1253. doi: 10.1080/0003684042000247415
|
| [28] |
Huang TH, Wang MH (2001) Estimating Scale And Scope Economies with Fourier Flexible Functional Form—Evidence from Taiwan's Banking Industry. Aus Econ Papers 40: 213-231. doi: 10.1111/1467-8454.00123
|
| [29] | Horváth L, Li B, Li H, et al. (2020) Time-varying beta in functional factor models: Evidence from China. North Am J Econ Financ 54: 101283. |
| [30] |
Jarrow R, Ruppert D, Tu Y (2004) Estimating the Interest Rate Term Structure of Corporate Debt with a Semiparametric Penalized Spline Model. J Am Stat Assoc 99: 57-66. doi: 10.1198/016214504000000070
|
| [31] | Li W (2021) COVID-19 and asymmetric volatility spillovers across global stock markets. North Am J Econ Financ 58: 101474. |
| [32] |
Li Y, Yang L (2011) Testing Conditional Factor Models: A Nonparametric Approach. J Empir Financ 18: 972-992. doi: 10.1016/j.jempfin.2011.07.004
|
| [33] | Lintner J (1965) Security Prices, Risk, and Maximal Gains from Diversification*. J Financ 20: 587-615. |
| [34] |
Lintner J (1965) The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. Rev Econ Stat 47: 13-37. doi: 10.2307/1924119
|
| [35] | Min BK, Kim TS (2016) Momentum and downside risk. J Bank Financ 72: S104-S118. |
| [36] | Rorres C, Anton H (1984) Applications of Linear Algebra New York, John Wiley and Sons. |
| [37] | Sharpe WF (1964) Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk. J Financ 19: 425-442. |
| [38] | Stone CJ (1986) The Dimensionality Reduction Principle for Generalized Additive Models. Ann Stat 14: 590-606. |
| [39] |
Taylor N (2004a) Modeling Discontinuous Periodic Conditional Volatility: Evidence from The Commodity Futures Market. J Futures Mark 24: 805-834. doi: 10.1002/fut.20114
|
| [40] |
Taylor N (2004b) Trading Intensity, Volatility, and Arbitrage Activity. J Bank Financ 28: 1137-1162. doi: 10.1016/S0378-4266(03)00116-X
|
| [41] |
Vasicek OA, Fong HG (1982) Term Structure Modeling Using Exponential Splines. J Financ 37: 339-348. doi: 10.1111/j.1540-6261.1982.tb03555.x
|
| [42] | Wand P, Jones C (1995) Kernel Smoothing, Chapman & Hall. |
| [43] | Yu Y, Escalante C, Deng X (2007) Evaluating Agricultural Banking Efficiency Using The Fourier Flexible Functional Form. Selected Paper Prepared for Presentation at The Southern Agricultural Economics Association Annual Meetings Mobile, Alabama, February 3-6, 2007: 36. |
| [44] |
Yu Y, Ruppert D (2002) Penalized Spline Estimation for Partially Linear Single-Index Models. J Am Stat Assoc 97: 1042-1054. doi: 10.1198/016214502388618861
|
| [45] |
Zhang MY, Russell JR, Tsay RS (2001) A Nonlinear Autoregressive Conditional Duration Model With Applications To Financial Transaction Data. J Econometrics 104: 179-207. doi: 10.1016/S0304-4076(01)00063-X
|
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Fangzhou Huang, Jiao Song, Nick J. Taylor. The impact of time-varying risk on stock returns: an experiment of cubic piecewise polynomial function model and the Fourier Flexible Form model[J]. Data Science in Finance and Economics, 2021, 1(2): 141-164. doi: 10.3934/DSFE.2021008
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