In this paper, we propose a semiparametric method for modeling the volatility in financial time series. The aim is to improve the forecasting capabilities of the most popular parametric volatility models and we also compare our approach to two recent semiparametric models in the literature. Our method is based on the bivariate Bernstein basis polynomials and the functional gradient descent (FGD) algorithm. We evaluate our method through simulated and real datasets. The results demonstrate its good predictive potential for financial volatility.
AudrinoF. (2005). Local likelihood for nonparametric ARCH(1) models. Journal of Time Series Analysis, 26, 251-278.
2.
AudrinoF., & BühlmannP. (2009). Splines for financial volatility. Journal of Royal Statistical Society B, 71, 655-670.
3.
BabuG. J.CantyA. J., & ChaubeyY. P. (2002). Application of Bernstein polynomials for smooth estimation of a distribution and density functon. Journal of Statistical Planning and Inference, 105, 377-392.
4.
BaillieR. T.BollerslevT., & MikkelsenH. O. (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 74, 3-30.
ChangI. S.ChienL. C.HsiungC. A.WenC. C., & WuY. J. (2007). Shape restricted regression with random Bernstein polynomials. Lecture Notes-Monograph Series, 54, 187-202.
10.
ChangI. S.HsiungC., & WuY. (2005). Bayesian survival analysis using Bernstein polynomials. Scandinavian Journal of Statistics, 32, 447-466.
11.
DieboldF. X., & MarianoR. S. (1995). Comparing predictive accuracy. Journal of Business and Economic Statistics, 13(3), 253-263.
12.
EngleR. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflations. Econometrica, 50, 987-1007.
13.
EngleR. F., & RangelJ. G. (2008). The spline-GARCH model for low frequency volatility and its global macroeconomic causes. The Review of Financial Studies, 21, 1187-1222.
14.
FellerW. (1965). An Introduction to probability theory and its applications volume I. Wiley, New York.
15.
FriedmanJ. H. (2001). Greedy function approximation: A gradient boosting machine. Annals of Statistics, 29, 1189-1232.
16.
GlostenL. R.JagannathanR., & RunkleD. E. (1993). On the relation between the expected value and the volatility of nominal excess return on stocks. Journal of Finance, 48, 1779-1801.
17.
GokcanS. (2000). Forecasting volatility of emerging stock markets: Linear versus non-linear GARCH models. Journal of Forecasting, 19(6), 499-504.
18.
GrenanderU. (1981). Abstract inference. Wiley, New York.
19.
HansenP. R., & LundeA. (2005). A forecast comparison of volatility models: Does anything beat a GARCH (1, 1)?Journal of Applied Econometrics, 20, 873-889.
20.
KimW., & LintonO. B. (2004). The live method for generalized additive volatility models. Econometric Theory, 20, 1094-1139.
21.
KooB., & LintonO. (2013). Robust estimation of semi-multiplicative volatility models. Working paper, The Institute for Fiscal Studies, Department of Economics, UCL.
22.
LintonO. B. (2009). Semiparametric and nonparametric ARCH modeling, in Handbook of Financial Time Series, Anderson et al. Springer-Verlag, Berlin.
LintonO. B., & YanY. (2011). Semi- and nonparametric ARCH process. Journal of Probability and Statistics, 1-17.
25.
LiuR., & YangL. (2016). Spline estimation of a semiparametric GARCH model. Econometric Theory, 32, 1023-1054.
26.
MalmstenH., & TerasvirtaT. (2004). Stylized facts of financial time series and three popular models of volatility. SSE/EFI Working Paper Series in Economics and Finance at Stockholm School of Economics, 563.
27.
McAleerM., & MedeirosM. C. (2008). Realized volatility: A review. Econometric Reviews, 27, 10-45.
28.
NelsonD. B. (1990). Stationarity and persistence in the GARCH (1, 1) model. Econometric Theory, 6, 318-334.
29.
NelsonD. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrika, 59, 347-370.
30.
PetroneS. (1999). Random bernstein polynomials. Scaninavian Journal of Statistics, 26, 373-393.
31.
RuppertR. (2012). Selecting the number of knots for penalized splines. Journal of Computational and Graphical Statistics, 4, 735-757.
32.
RydbergT. H. (2000). Realistic statistical modeling of financial data. International Statistical Review, 68, 233-258.
33.
SpiritiS.EubankR.SmithP. W., & YoungD. (2013). Knot selection for least-squares and penalized splines. Journal of Statistical Computation and Simulation, 83, 1020-1036.
34.
TsayR. S. (2010). Analysis of financial time series. Wiley, New Jersey, 3rd edition.
35.
WandM. P., & JonesM. C. (1995). Kernel Smoothing. Chapman and Hall, London.
36.
WangJ., & GhoshS. K. (2012). Shape-restricted nonparametric regression with Bernstein polynomials. Computational Statististics and Data Analysis, 56, 2729-2741.
37.
WangJ., & GhoshS. K. (2013). Nonparametric models for longitudinal data using Bernstein polynomial sieve. North Carolina State University Technical Report, 1.
ZhangX.FengY., & PeitzC. (2017). A general class of semigarch models baed on the box-cox transformation. Center for International Economics Working Paper Series, (No. 6).
40.
ZhouS., & ShenX. (2001). Spatially adaptive regression splines and accurate knot selection shemes. Journal of the American Statistical Association, 96, 247-259.
41.
ZiegelmannF. A. (2011). Semiparametric estimation of volatility: Some models and complexity choice in the adaptive functional-coefficient class. Journal of Statistical Computation and Simulation, 81(6), 707-728.
