Exploratory Analysis and Modeling of Stock Returns

Abstract

In this article, novel joint semiparametric spline-based modeling of conditional mean and volatility of financial time series is proposed and evaluated on daily stock return data. The modeling includes functions of lagged response variables and time as predictors. The latter can be viewed as a proxy for omitted economic variables contributing to the underlying dynamics. The conditional mean model is additive. The conditional volatility model is multiplicative and linearized with a logarithmic transformation. In addition, a cube-root power transformation is employed to symmetrize the lagged response variables. Using cubic splines, the model can be written as a multiple linear regression, thereby allowing predictions to be obtained in a simple manner. As outliers are often present in financial data, reliable estimation of the model parameters is achieved by trimmed least-square (TLS) estimation for which a reasonable amount of trimming is suggested. To obtain a parsimonious specification of the model, a new model selection criterion corresponding to TLS is derived. Moreover, the (three-parameter) generalized gamma distribution is identified as suitable for the absolute multiplicative errors and shown to work well for predictions and also for the calculation of quantiles, which is important to determine the value at risk. All model choices are motivated by a detailed analysis of IBM, HP, and SAP daily returns. The prediction performance is compared to the classical generalized autoregressive conditional heteroskedasticity (GARCH) and asymmetric power GARCH (APGARCH) models as well as to a nonstationary time-trend volatility model. The results suggest that the proposed model may possess a high predictive power for future conditional volatility. Supplementary materials for this article are available online.

Key Words:

• Conditional mean
• Conditional volatility
• Financial time series
• Prediction

ACKNOWLEDGMENTS

This research was partially supported by NSF grants DMS 09-05400, DMS 09-07622, DMS 12-09226, DMS 13-05858, and DMS 14-07530.

Additional information

Notes on contributors

Kimihiro Noguchi

Kimihiro Noguchi (E-mail: [email protected]) is Assistant Professor, Department of Mathematics, Western Washington University, Bellingham, WA 98225. Alexander Aue (E-mail: [email protected]) is Associate Professor and Prabir Burman (E-mail: [email protected]) is Professor, Department of Statistics, University of California, Davis, CA 95616.

Alexander Aue

Kimihiro Noguchi (E-mail: [email protected]) is Assistant Professor, Department of Mathematics, Western Washington University, Bellingham, WA 98225. Alexander Aue (E-mail: [email protected]) is Associate Professor and Prabir Burman (E-mail: [email protected]) is Professor, Department of Statistics, University of California, Davis, CA 95616.

Prabir Burman

Kimihiro Noguchi (E-mail: [email protected]) is Assistant Professor, Department of Mathematics, Western Washington University, Bellingham, WA 98225. Alexander Aue (E-mail: [email protected]) is Associate Professor and Prabir Burman (E-mail: [email protected]) is Professor, Department of Statistics, University of California, Davis, CA 95616.