ESSAYS ON VOLATILITY MODELING AND FORECASTING ZHANG SHEN (B.A. 2003, M.A. 2006, NanKai University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ECONOMICS NATIONAL UNIVERSITY OF SINGAPORE 2011 ACKNOWLEDGEMENTS I have benefited greatly from the guidance and support of many people over the past five years. In the first place, I owe an enormous debt of gratitude to my main supervisor, Dr. Han Heejoon, for his supervision from the very early stage of this research. I believe his passion and perseverance in pursuit of the truth in science encourage me a lot on doing research. His extraordinary patience, integrity and wisdom in guiding students will leave me a life-long influence. I am always feeling lucky and honorable to be supervised by him. I would also like to thank Professor Tilak Abeysinghe, whose positive attitudes to research aroused my interest in studying econometrics when I took his module of econometric modeling and applications. I gratefully acknowledge Dr. Park Myung, for his help and constructive suggestions in the third chapter of this research. Along with these professors, I also wish to thank my friends and colleagues at the department of Economics for their helpful comments, especially to Wan Jing and Li Bei. Finally, to my parents, all I can say is that it is your unconditional love that gives me the courage and strength to face the difficulties in pursuing my dreams. Thanks for your acceptance and endless support to the choices I make all the time. i TABLE OF CONTENTS Acknowledgements i Table of Contents ii Summary iv List of Tables v List of Figures vi Chapter 1: Nonstationary Nonparametric Volatility Model Introduction The Model Asymptotic Distribution Theory Simulation 14 Empirical Application 16 5.1 The Data, Models and Estimation Methods 16 5.2 Evaluation Criterion 19 5.3 Estimation and Forecast Evaluation Results 22 Conclusion 25 Chapter 2: Semiparametric ARCH Model for the Long Memory in Volatility 32 Introduction 32 ii Models and Estimation Method 35 2.1 The Model 35 2.2 Estimation Method 37 Empirical Application 40 3.1 The Data, Models and Estimation Methods 40 3.2 Evaluation Criterion 43 3.3 Estimation and Forecast Evaluation Results 44 Conclusion 47 Chapter 3: Multi-step Forecasting of Realized Volatility Measure 53 Introduction 53 Models 55 Empirical Analysis 58 3.1 The Data, Models and Estimation Methods 58 3.2 Out-of-sample Forecasting Methodology 60 3.2.1 Iterated Forecasting 61 3.2.2 Direct Forecasting 61 3.3 Evaluation Criterion 3.3.1 Estimation and Forecast Evaluation Results 62 63 Conclusion 67 Appendix 79 iii SUMMARY This thesis is composed of three essays on the modeling and forecasting of return volatility. The first chapter investigates a new nonstationary nonparametric volatility model, in which the conditional variance of time series is modeled as a nonparametric function of an integrated or near-integrated covariate. This model can generate the long memory property in volatility and allow the nonstationarity in return series. We establish the asymptotic distribution theory for this model and show that it performs reasonably well in the empirical application. The second chapter proposes a semiparametric volatility model which combines the nonparametric ARCH function with a persistent covariate. This new model applies the GARCH-X structure under the semiparametric framework, it can produce long-memory in volatility given the persistent property in the covariate. We show that it provides a better explanation of volatility in the empirical analysis. The last chapter suggests a parametric volatility model and mainly focuses on the multistep forecasting of volatility. We introduce a long-term dynamic component to the HEAVY models to capture the long-memory in volatility. We apply the high-frequency database to our model and the other benchmark models and show that our model outperforms the other models. iv List of Tables Chapter Table Unit root test results for the VIX index 26 Table Comparison of within-sample predictive power for the stock return volatility 26 Table Comparison of out-of-sample predictive power for the stock return volatility 27 Chapter Table Bandwidth Selection 50 Table Within-sample estimation result for parametric models 51 Table Comparison of within-sample predictive power for the stock return volatility 51 Table Comparison of out-of-sample predictive power for the stock return volatility 52 Chapter Table Estimation results for models Table DMW statistic based on QLIKE for within-sample forecasts 68-69 70 Table DMW statistic based on QLIKE for out-of-sample iterative forecasts 71-72 Table DMW statistic based on QLIKE for out-of-sample direct forecasts 73-74 Table MSE result for out-of-sample direct forecasts 75-78 v List of Figures Chapter Figure Graphs for the Monte Carlo simulation 28 Figure Estimate of model for the daily S&P 500 index returns from Jan. 1996 to 27 Feb. 2009 29 Figure Within-sample fitted values of volatility models for Jan. 1996 to 27 Feb. 2009 30 Figure Out-of-sample fitted values of volatility models for 18 Mar. 2004 to 27 Feb. 2009 31 Chapter Figure Estimation result for the nonparametric ARCH component the local exponential method using Figure Estimation result for the nonparametric ARCH component the local log-likelihood method using Figure Plot for and 49 49 50 vi Chapter Nonstationary Nonparametric Volatility Model 1.Introduction ARCH type models have been widely used to model the volatility of economic and …nancial time series since the seminal work by Engle (1982) and the extension made by Bollerslev (1986). Recently there has been active research on nonparametric or semiparametric volatility models. See Linton (2009) for an excellent review. The nonparametric ARCH literature begins with Pagan and Schwert (1990a) and Pagan and Hong (1991). In the nonparametric ARCH model they considered, the conditional variance t of a martingale di¤erence sequence (yt ) is given as t = m (yt 1) ; (1) where m ( ) is a smooth but unknown function, and the multilag version is t = m (yt ; yt ; ; yt d) : They proposed these models to allow for a general shape to the news impact curve and their models can nest all the parametric ARCH processes. However, their models cannot capture adequately the time series properties of many actual …nancial time series, in particular volatility persistence, and the statistical properties of the estimators can be poor, due to curse of dimensionality. See Masry and Tj stheim (1995), Härdle and Tsybakov (1997) for the related literature. To overcome these problems, additive models have been proposed as a ‡exible but parsimonious alternative to nonparametric models. See Engle and Ng (1993), Yang, Härdle and Nielsen (1999), Kim and Linton (2004), Linton and Mammen (2005) and Yang (2006) for the related literature. To capture volatility persistence, some proposed models are intended to nest the GARCH(1,1) model. Among many nonparametric or semiparametric ARCH models, only the models proposed by Audrino and Bühlmann (2001), Linton and Mammen (2005) and Yang (2006) can nest the GARCH(1,1) model. However, it is well known that even the GARCH(1,1) model is inadequate to capture volatility persistence observed in many …nancial time series. While the autocorrelation of squared series of the GARCH(1,1) process decays exponentially and converges to zero very quickly, stock return or exchange rate return series commonly exhibit the long memory property in volatility; the autocorrelation of squared return series decays very slowly. Ding et al. (1993) found earlier that it is possible to characterize the power transformation of stock return series to be long memory. In the literature of parametric ARCH type models, there has been active research on this issue and several models have been proposed to capture the long memory property in volatility.1 These models accommodate fractional integration, structural changes or a persistent covariate in ARCH type models. For the related literature on the long memory property in volatility, see Baillie et al. (1996), Ding and Granger (1996), Bollerslev and Mikkelsen (1996) (fractionality of the order of integration), Engle and Lee (1999) (two This is also an important issue in the literature of stochastic volatility models. See Hurvich and Soulier (2009) for stochastic volatility models with long memory property. But we not consider stochastic volatility models. We focus only on ARCH type models that are parametric, nonparametric or semiparametric. components), Diebold and Inoue (2001) (switching regime), Mikosch and Starica (2004) (structural change), Granger and Hyung (2004) (occasional break) and Park (2002) and Han and Park (2008) (persistent covariate). On the other hand, there has been less attention on the long memory property in volatility in the literature of nonparametric or semiparametric ARCH models. Even if it has been an important issue for nonparametric or semiparametric ARCH models to capture adequately volatility persistence, there has been no attempt to explain the long memory property in volatility in the framework of nonparametric or semiparametric ARCH models. This is the …rst limitation of existing nonparametric or semiparametric ARCH models that we focus on. Moreover, most nonparametric or semiparametric ARCH models assume the covariance stationarity of (yt ) : Hence, these models are valid only for stationary time series, which is the second limitation of existing models that we focus on. Among nonparametric or semiparametric ARCH models, the only exception without this limitation is the splineGARCH model proposed by Engle and Rangel (2008) that allows the unconditional variance of (yt ) to be time-varying. If we model the volatility of …nancial return series, it is quite restrictive to assume that the unconditional variance of …nancial return series is constant for a long time span, in particular, considering that fundamental features of the …nancial markets are continuously and signi…cantly changing.2 The aim of this paper is to develop and investigate a new nonparametric volatility model Starica and Granger (2005) investigated a nonstationary unconditional variance model of stock return series. They discovered that most of the dynamics of stock return series are concentrated in shifts of the unconditional variance. notation of Assumption 2.3 in WP. Following the way to prove (14) in WP, we have n X 1n;a E m2 (dn xt;n ) E dn xt;n h m2 (x) K t=1 = n Z X t=1 n h X (dt;0;n ) dn h dn t=1 n X nh1+2 (dt;0;n ) n X n X n +2m(x) Z Z (dt;0;n ) t=1 dn A t=1 h +2 dn m2 (x) K m2 (dn dt;0;n y) t=1 nh1+ dn nh1+ dn dt;0;n y h x h ht;0;n (y)dy m2 (x) K (y) dy 1 Z n X m(x)j h m1 (y; x)K (y) dy jm(hy + x) (dt;0;n ) n m2 (hy + x) x h 1 Z jm(hy + x) 1 m(x)j m(x)K (y) dy m21 (s; x)K (s) ds (dt;0;n ) t=1 Z m1 (s; x)K (s) ds dn by Assumption 3.2. Since m ( ) is positive, m(hy + x) m(x) + h m1 (y; x): The fourth line follows from this. This completes the proof of (8). We next prove (7). From (11), we have h n X t=1 Kh (xt !1=2 x) (m(x) ^ m(x)) = n X "2t Znt + t=1 dn h n 1=2 2n = 3n ; where Znt = dn nh 1=2 m(xt )K dn xt;n h x h / 3n 81 with n 3n dn X = K nh t=1 dn xt;n h x h : Note that Znt includes m(xt ); which is di¤erent from the case in WP. Hence, the limit of n = Pn t=1 Znt is also di¤erent and given by the following; n n = 3n dn X m (xt )K nh t=1 = n X dn m2 (x)K 3n nh + t=1 n X dn 3n nh Z 1t=1 !p m2 (x) m2 (xt ) dn xt;n h dn xt;n h x h x h m2 (x) K dn xt;n h x h K (s) ds by (11) and (15). As in WP, (dn h=n)1=2 2n = 3n = op (1) because nh1+2 =dn ! 0: Note that "2t is a martingale di¤erence sequence because (xt ) is adapted to (Ft Vn n X n t=1 "2t Znt !d N (0; E("4t ) ). Znt Hence, we can deduce ); (16) for any h satisfying nh=dn ! and nh1+2 =dn ! 0; as in WP. (16) corresponds to the equation (5.21) in WP. They assume that (xt ) is independent of the error term (ut ) so that ut Znt becomes a martingale di¤erence sequence. However, note that we not require the independence between (xt ) and ("t ). Even if (xt ) and ("t ) are dependent, "2t Znt is a 82 martingale di¤erence sequence because (xt ) is adapted to (Ft 1) and, therefore, (16) holds. 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[...]... assumptions Section 3 provides the asymptotic distribution theory of the kernel estimate of the model, and a simulation experiment is conducted in Section 4 Section 5 provides an empirical application of the model, which includes data description, evaluation criterion, and within-sample and out-of sample forecast evaluation results of the model Section 6 concludes the paper, and Appendix contains mathematical... research on loss functions that are 8 See http://realized.oxford-man.ox.ac.uk/ 20 robust to the use of a noisy volatility proxy See Hansen and Lunde (2006), Patton (2010) and Patton and Sheppard (2009) Patton (2010) provides necessary and su¢ cient conditions on the functional form of the loss function to ensure the ranking of various forecasts is preserved when using a noisy volatility proxy, and he... m(x)j < 1 and h m1 (y; x) for all y 2 R and R1 1K 2 (s) m2 (s; x) ds 1 1 such that, when h R1 1K (s) m1 (s; x) ds < < 1: < 1 a.s for some q > 4: Assumptions 3.1 and 3.2(a) are the same as Assumptions 3.1 and 3.2 in Wang and Phillips (2009a) As mentioned in Wang and Phillips (2009a), the conditions in Assumption 10 3.1 and 3.2(a) are quite weak and simply veri…ed for various kernels K(x) and functions m(x)... (2002) that the nonstationary covariate (xt ) plays a crucial role in generating volatility persistence He showed that a nonlinear function of a stationary process, on the other hand, cannot generate the long memory property in volatility 3 Asymptotic Distribution Theory We establish the asymptotic distribution theory for the kernel estimate of our model The nonstationary nonparametric volatility model... However, our nonstationary nonparametric volatility model allows the unconditional variance of (yt ) to be time-varying and, therefore, it could be better to use our model for the stock return series As the covariate (xt ) for our nonstationary nonparametric volatility model, we use the VIX index by the Chicago Board Options Exchange The VIX index is the implied volatility calculated from options on the S&P... next section, it is unnecessary to assume that (xt ) is independent of ("t ) for the kernel estimation of our model: Assumptions 2.1 and 2.2 de…ne the nonstationary nonparametric volatility model The parametric counterpart to this model is the nonstationary nonlinear heteroskedasticity (NNH) model by Park (2002) given as 2 t = f (xt 1 ); (4) where f ( ) is a parametric nonlinear function and (xt )... covariate (xt 1) is nonstationary and, not only in the mean equation, the nonstationary covariate is also included in the conditional variance of the error term (ut ) : Recently, Wang and Phillips (2009a, 2009b) investigated the nonparametric cointegrating regression yt = m(xt ) + ut ; where (xt ) is an integrated or fractionally integrated process The model (5) is as an extended case of the nonparametric... model; we can use an economic or …nancial indicator that contains useful information on the volatility of time series If the chosen covariate xt 1 contains 3 The reader is referred to Park and Phillips (1999, 2001) for more details on these function classes The classes I and H include a wide class, if not all, of transformations de…ned on R The bounded functions with compact supports and more generally... in (7) contains the square of the volatility function m2 (x): This is because the estimation is based on the model (5) in which the error term contains the volatility function Similarly, in the semiparametric GARCH model by Yang (2006), the limiting variance of the estimator also contains the square of the volatility function If one adopts an alternative estimation method that is not 2 based on a rearranged... sample standard deviation of (xt ) We use the cross validation bandwidth for the empirical application in the next section, and it is shown that, for our data, the result using the Silverman’ bandwidth is very similar to the one using the cross validation bandwidth We s also tried ^ x n 1=6 that is a possible optimal bandwidth suggested in the previous section, and the simulation results are still similar . composed of three essays on the modeling and forecasting of return volatility. The first chapter investigates a new nonstationary nonparametric volatility model, in which the conditional variance. Application 40 3.1 The Data, Models and Estimation Methods 40 3.2 Evaluation Criterion 43 3.3 Estimation and Forecast Evaluation Results 44 4 Conclusion 47 Chapter 3: Multi-step Forecasting. nonstationary nonparametric volatility model. The model can generate the long memory property in volatility if the unknown function belongs to the function classes considered by Park (2002), and