Forecasting realized covariance in an adaptive framework

FORECASTING REALIZED COVARIANCE IN AN ADAPTIVE FRAMEWORK LU JUN NATIONAL UNIVERSITY OF SINGAPORE 2012 FORECASTING REALIZED COVARIANCE IN AN ADAPTIVE FRAMEWORK LU JUN (B.Sc. National University of Singapore) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE 2012 ii ACKNOWLEDGEMENTS I am so grateful that I have Assistant Professor Chen Ying as my main supervisor and Professor Xia Yingcun as my co-supervisor. Dr. Chen is truly a great mentor not only in statistics but also in daily life. I would like to thank her for her guidance, encouragement, time, and endless patience. Next, I would like to thank all the PhD and master fellow students for discussion on various topics in research. I also thank all my friends who helped me to make life easier as a graduate student. I wish to express my gratitude to the university and the department for supporting me through NUS Graduate Research Scholarship. Finally, I will thank my family for their love and support. iii CONTENTS Acknowledgements ii Summary vi List of Notations ix List of Tables xi List of Figures Chapter Introduction 1.1 xiii Overview of multivariate volatility models . . . . . . . . . . . . . . 1.1.1 Vector volatility model . . . . . . . . . . . . . . . . . . . . . 1.1.2 Factor GARCH (F-GARCH) model . . . . . . . . . . . . . . 11 CONTENTS 1.2 iv 1.1.3 Conditional correlation multivariate volatility model . . . . . 22 1.1.4 Alternative multivariate volatility models . . . . . . . . . . . 28 The realized covariance matrix model and the new techniques applied 33 1.2.1 Dimension reduction . . . . . . . . . . . . . . . . . . . . . . 35 1.2.2 Structural break and the adaptive procedure . . . . . . . . . 37 Chapter Methods 52 2.1 Realized covariance matrix . . . . . . . . . . . . . . . . . . . . . . . 52 2.2 Common principal components analysis . . . . . . . . . . . . . . . . 56 2.3 Adaptive vector autoregressive model . . . . . . . . . . . . . . . . . 59 2.3.1 Estimation and test of homogeneity . . . . . . . . . . . . . . 60 2.3.2 Adaptive identification of the interval of homogeneity . . . . 66 2.3.3 Choice of parameters and implementation details . . . . . . 70 2.3.4 Theoretical properties . . . . . . . . . . . . . . . . . . . . . 79 Chapter Simulation and Real Data Results 3.1 3.2 84 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.1.1 Adaptive VAR approach: sudden change . . . . . . . . . . . 85 3.1.2 Adaptive CPC VAR approach: smooth change . . . . . . . . 93 Real data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.2.1 Factor realized covariance model . . . . . . . . . . . . . . . . 110 3.2.2 Forecasting ability comparison: the CPC + adaptive VAR approach v.s. non-adaptive approach and other multivariate volatility models . . . . . . . . . . . . . . . . . . . . . . . . 113 3.2.3 Value at risk (VaR) comparison: the CPC + adaptive VAR approach v.s. non-adaptive approach and other multivariate volatility models . . . . . . . . . . . . . . . . . . . . . . . . 116 CONTENTS v Chapter Conclusion 121 Chapter Appendix 123 Bibliography 135 vi SUMMARY Non-stationary financial time series with regime-switch-point inside are frequently encountered by researchers, which pose difficulty in accurately estimating the parameters when some parametric models are applied. This inaccuracy would create non-ignorable risk in financial risk management when financial data are in concern, especially at the time when market volatility is high. To deal with such feature, the adaptive regime-switch-point detection technique is developed. However, the existing approach is largely based on univariate time series and the model set-up is rather restrictive. In this thesis, a new adaptive local model selection technique is developed which is based on multidimensional time series, namely, the realized covariance matrix process is modeled under the framework of adaptive local model selection. Concurrently with the local model selection procedure, Summary the dimension reduction technique (using the common principal component analysis) and a local adaptive VAR model are applied to the modeling of the realized covariance matrix process. The combined approach generates satisfying model parameter estimation results and good financial risk metric measures when applied to financial data set. The work in this thesis contains two parts. First part (Chapter and Chapter 2) discusses the non-stationary feature embedded in the financial time series and establishes the dimension reduction and the adaptive VAR approaches under the framework of the realized covariance matrix process modeling. For the dimension reduction technique, first a vector series containing the driving factor of the realized covariance matrix process is extracted from the realized covariance matrix process. Through this procedure, effectively the dimension is reduced from d(d + 1)/2 to just d (d is the dimension of the realized covariance matrix). Then the adaptive local VAR model is applied to the vector series to dynamically determine the past homogeneous time interval which contains no regime-switch-point. The aim of the adaptive parameter estimation procedure is to achieve an optimal tradeoff between small modeling bias and small estimate variability at the same time, which is a key advantage of this approach over other convectional parameter estimation methodologies. The second part of the thesis deals with simulated data and real data set when applied with the aforementioned dimension reduction and the adaptive local VAR model. By comparing the parameter estimation and time series forecasting accuracy results of the proposed method to other conventional multivariate volatility process models, both simulation and real data analysis unambiguously show the superior performance of the adaptive local model selection technique over the rest vii Summary models. In addition, a financial risk measure (VaR) commonly adopted by financial institutes is calculated using real financial data set (Dow Jones 30 Industrial Average Index). The adaptive procedure clearly outperforms its non-adaptive counterpart. All the data analysis results show that the proposed multidimensional adaptive parameter estimation methodology is a promising aspect for future development in other areas (such as bioinformatics, signal processing, etc) other than its financial application exploited in this thesis. viii ix LIST Of NOTATIONS Rd d-dimensional real space M the transpose of a matrix or a vector M || · || the Euclidean norm i.e. ||x|| = (x21 + · · · + x2d )1/2 for x = (x1 , · · · , xd ) ∈ Rd . If x is a matrix, ||x|| represents the square root of the largest eigenvalue of xx I{·} the indicator function In n × n identity matrix 127 It holds LIt (ΣIt , Σ∗ ) > z = (1 − C · |Σ∗ |/|Σ|) sup Σ ⊆ εε> ∪ − inf εε> inf |Σ|C·|Σ∗ | It ε ε − NIt log(|Σ|/|Σ∗ |) > z − C · |Σ∗ |/|Σ| 2z + NIt log(|Σ|/|Σ∗ |) C · |Σ∗ |/|Σ| − Denote v = − C · |Σ∗ |/|Σ|, then the function f (v) = [2z − NIt log ((1 − v)/C))] /v attains its minimum at some point v † = − C · |Σ∗ |/|Σ† | satisfying the equation: 2z/NI + v† − log v† − − v† C =0 = εε> Therefore εε> inf 2z + NIt log(|Σ|/|Σ∗ |) |Σ|>C·|Σ∗ | It − C · |Σ∗ |/|Σ| It 2z + NIt log(|Σ† |/|Σ∗ |) − C · |Σ∗ |/|Σ† | LIt (Σ† , Σ∗ ) > z ⊆ with |Σ† | = (1 − v † )/ (C · |Σ∗ |). Similarly − εε> It inf 2z + NIt log(|Σ|/|Σ∗ |) |Σ| It 2z + NIt log(|Σ† |/|Σ∗ |) C · |Σ∗ |/|Σ† | − ⊆ {LIt (Σ† , Σ∗ ) > z} for some |Σ† | < C · |Σ∗ |. Hence Lemma is proven. Now we are to establish the following bound EΣ∗ exp µ0 LIt (ΣIt , Σ∗ ) ≡ EΣ∗ exp µ0 NI K (ΣIt , Σ∗ ) ≤ (5.4) 128 As shown in the proof of Lemma 1: 2LIt (ΣIt , Σ∗ ) = NIt log(|Σ∗ |/|Σ|) − (|Σ∗ |/|Σ| · C − 1) εε It = NIt log((1 + u)/C) − u εε It (||ε||2 − 1) = NIt log((1 + u)/C) − NIt u − u It with u = |Σ∗ |/|Σ| · C − 1. For any µ such that maxIt | − uµ| ≤ λ this yields by independence of the ε’s: log EΣ∗ exp 2µLIt (ΣIt , Σ∗ ) = µNIt log((1 + u)/C) − µNIt u −uµ(||ε||2 − 1) + log EΣ∗ exp It There is a well-known result as the following: Lemma If there exists a constant H > and a random variable X such that E exp {tX} < ∞ ∀|t| < H, then there exist a constant g > and a positive T > 0, such that E exp {t(X − EX)} ≤ exp gt2 ∀|t| ≤ T Since the proof is obvious, we ignore the detail proof. Apply the Lemma 2, utilizing the assumptions that E exp {λ||εt ||2 } < ∞ for some positive λ > and E||εt ||2 = 1, then for all | − uµ| ≤ λ, we have: −uµ(||ε||2 − 1) ≤ log EΣ∗ exp It κ0 u2 µ2 It for some κ0 > 0. This yields: ˜ It , Σ∗ ) log EΣ∗ exp 2µLIt (Σ κ0 u2 µ2 ≤ µNIt log((1 + u)/C) − µNIt u + It 129 ≤ µNIt log((1 + u∗ )/C) − u∗ + κ0 µu∗2 = µNIt log(1 + u∗ ) − u∗ + κ0 µu∗2 − µNIt log C Where u∗ is the upper bound for |u|. The condition (Θ) ensures that u = u(|Σ|) = |Σ∗ |/|Σ| · C − is bounded by some constant u∗ for all Σ ∈ Θ. The expression log(1 + u) − u + κ0 µu2 is negative for all |u| ≤ u∗ and sufficiently small µ yielding 5.2. Here is a simple proof for f (u) = log(1 + u) − u + κ0 µu2 is negative for all |u| ≤ u∗ . Proof Take the derivative of f (u): f (u) = − + 2κ0 µu 1+u Equal the above expression to 0, there are two solutions: u = 0, u= − 2κ0 µ 2κ0 µ It can be seen that the derivative of f (u) is negative for all values u ∈ [0, (1 − 2κ0 µ)/(2κ0 µ)]. In addition, since f (0) = and for sufficient small µ, the interval in which the function f (u) is decreasing can be arbitrarily large. Hence as long as there is an upper bound u∗ ≥ u ∀u = |Σ∗ |/|Σ|·C−1, the function f (u) is decreasing for all |u| ≤ u∗ and the function value is negative. The proof is completed. From log EΣ∗ exp 2µLIt (ΣIt , Σ∗ ) < for all |u| ≤ u∗ , we have: EΣ∗ exp µLIt (ΣIt , Σ∗ ) = EΣ∗ exp µ0 NIt K (ΣIt , Σ∗ ) ≤ 130 The inequality 5.2 is established. The Tchebychev’s exponential inequality states that for a random variable w > and a constant v > 0, for any value z, the following relationship holds: P(w > z) = P {exp(vw) > exp(vz)} ≤ E {exp [v(w − z)]} Apply the Tchebychev’ exponential inequality here, for some µ0 > 0, we have: PΣ∗ (LIt (ΣIt , Σ∗ ) > z) ≤ EΣ∗ exp µ0 (LIt (ΣIt , Σ∗ ) − z) = exp {−µ0 z} EΣ∗ exp µ0 LIt (ΣIt , Σ∗ ) Since it has been shown that: EΣ∗ exp µ0 LIt (ΣIt , Σ∗ ) ≤ Then PΣ∗ (LIt (ΣIt , Σ∗ ) > z) ≤ e−µ0 z For all ΣIt . By Lemma 1, we have: LIt (ΣIt , Σ∗ ) > z ⊆ LIt (Σ† , Σ∗ ) > z ∪ {LIt (Σ† , Σ∗ ) > z} for some fixed Σ† , Σ† depending on z. Finally we have: PΣ∗ (LIt (ΣIt , Σ∗ ) > z) ≤ 2e−µ0 z (5.5) 131 Hence upper bound of the probability 5.3 is proven. Theorem 2.3 Assume (Θ) and condition 5.1. Then for any r > 0, there is a constant µ0 such that: ∗ EΣ∗ |LIt (ΣIt , Σ∗ )|r ≡ EΣ∗ |Nt K (ΣIt , Σ∗ )|r ≤ τr µ−r ≡ Rr (Σ ). Where K (ΣIt , Σ∗ ) is the Kullback-Leibler information. τr = 2r z≥0 (5.6) zr−1 e−z dz = 2rΓ(r) and Rr (Σ∗ ) is a constant. Proof of Theorem 2.3 EΣ∗ |(LIt (ΣIt , Σ∗ )|r ≤ − zr dPΣ∗ (LIt (ΣIt , Σ∗ ) > z) z≥0 rzr−1 PΣ∗ (LIt (ΣIt , Σ∗ ) > z)dz = −zr PΣ∗ (LIt (ΣIt , Σ∗ ) > z) + z≥0 zr−1 PΣ∗ (LIt (ΣIt , Σ∗ ) > z)dz ≤ r z≥0 Since PΣ∗ (LIt (ΣIt , Σ∗ ) > z) ≤ 2e−µ0 z Hence EΣ∗ |(LIt (ΣIt , Σ∗ )|r ≤ r zr−1 PΣ∗ (LIt (ΣIt , Σ∗ ) > z)dz z≥0 zr−1 e−µ0 z dz ≤ 2r z≥0 (µ0 z)r−1 e−µ0 z µ−r d (µ0 z) = 2r z≥0 = τr µ−r 132 Hence Theorem 2.3 is proven. The upper bound has been established for the log-likelihood ratio for multivariate case as shown in 5.6. In the following proofs the risk bound of parametric case is extended to the general ’near parametric’ situation under the small modeling bias condition. Theorem 2.4 Let for some θ ∈ Θ and some ∆ ≥ E∆I k (θ) ≤ ∆ where ∆I k (θ) = t∈I k (5.7) K {Xt , Xt (θ)}, Xt is the observable multidimensional time series and Xt (θ) is some local parametric approximation of Xt . It holds for an estimate θ constructed from the observations {Xt }t∈I k that E log + (θ, θ)/Eθ (θ, θ) ≤ + ∆ Where (θ, θ) is some loss function. Proof of Theorem 2.4 Lemma Let P and P0 be two measures such that the Kullback-Leibler divergence E log(dP/dP0 ) satisfies E log(dP/dP0 ) ≤ ∆ < ∞ for some ∆. Then for any random variable ζ with E0 ζ < ∞, it holds that E log(1 + ζ) ≤ ∆ + E0 ζ Proof of Lemma By simple algebra one can check that for any fixed y the maximum of the function f (x) = xy − x log x + x is attained at x = ey leading to the inequality xy ≤ x log x − x + ey . Using this inequality and the representation E log(1 + ζ) = E0 {Z log(1 + ζ)} with Z = dP/dP0 , we obtain: E log(1 + ζ) = E0 {Z log(1 + ζ)} 133 ≤ E0 (Z log Z − Z) + E0 (1 + ζ) = E0 (Z log Z) + E0 ζ − E0 Z + It remains to note that E0 Z = and E0 (Z log Z) = E log Z. Lemma is applied with ζ = (θ, θ)/E (θ, θ) yields the result of the Theorem 2.4 in view of Eθ (ZI,θ log ZI,θ ) = E log ZI,θ = E log t∈I = E E log t∈I p[Xt ] p[Xt (θ)] p[Xt ] |Ft−1 p[Xt (θ)] = E∆Ik (θ) Hence Theorem 2.4 is proven. Theorem 2.4 extends the parametric risk bound in 5.6 to the ’near parametric’ situation under the SMB condition. Corollary 2.5 Let the SMB condition 2.9 hold for some interval I k and θ ∈ Θ. 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Journal of the American Statistical Association 100 1394-1411. 142 [...]... good performance in estimating the VAR values for risk management purpose Overall, the new adaptive technique is a powerful tool in model parameter estimation and forecasting for covariance matrix process under non-stationary circumstance In the following the conventional multivariate volatility models are discussed in the first half of the introduction chapter and the rational of the proposed adaptive. .. important for both economic and econometric reason Knowledge of correlation structures is vital in many financial applications, such as asset pricing, optimal portfolio risk management, and asset allocation Generally speaking, multivariate volatility modeling improves decision tools in various financial areas Moreover, as financial volatility change in similar way across different assets and markets, modeling... change begins 95 Figure 3.8 Scenario 2 change of B The solid line is the mean value of the length of selected homogeneous interval and the vertical lines represent the position of parameter change begins 96 Figure 3.9 Scenario 1 change of Σ The solid line is the mean value of the lengths of selected homogeneous interval and the vertical lines represent the position of parameter change... solid line is the mean value of the length of selected homogenous interval and the vertical lines represent the position of change points 88 List of Figures xvi Figure 3.4 Scenario 3: change b The solid line is the mean value of the lengths of selected homogenous interval and the vertical lines represent the position of change points 89 Figure 3.5 Scenario 4: change all... rather restrictive and generally univariate-oriented In this thesis, a new multivariate adaptive regimeswitch-point-detection technique is developed and applied to multivariate volatility process (realized covariance matrix process), which generates good parameters estimation and multi-step-ahead condition covariance matrix forecasting results using both simulated and real data analysis, as compared... solid line is the mean value of the lengths of selected homogenous interval and the vertical lines represent the position of change points 89 Figure 3.6 Smooth change scenario parameter value change scheme 94 Figure 3.7 Scenario 1 change of B The solid line is the mean value of the length of selected homogeneous interval and the vertical lines represent the position of parameter change... fitting of the real data set The set of candidate homogenous interval lengths is given on the X-axis 78 Figure 3.1 Sudden change scenario parameter value change scheme 86 Figure 3.2 Scenario 1: change B The solid line is the mean value of the length of selected homogenous interval and the vertical lines represent the position of change points 88 Figure 3.3 Scenario 2: change... conditional covariance between any pair of factors is time invariant Another way to express the two-factor model described is as the follows: εt = λ1 f1t + λ2 f2t + et where et denotes the innovations with constant variance matrix and uncorrelated with either of the two factors Each factor fkt has zero conditional mean, and its conditional variance resembles a GARCH(1, 1) process Hence, the K-factor model can... before and after log-transformation, with the standard normal distribution density plot as the comparison benchmark In the upper panel is the eigenvalue series with the largest kurtosis before logtransformation and the one in the lower panel is the one with average kurtosis before transformation The dashed line represents the kernel density of the eigenvalue series without log-transformation and the... position of parameter change begins 96 List of Figures xvii Figure 3.10 Scenario 2 change of Σ The solid line is the mean value of the lengths of selected homogeneous interval and the vertical lines represent the position of parameter change begins 97 Figure 3.11 No change scenario: the solid line is the mean value of the lengths of selected homogeneous interval 99 Figure . FORECASTING REALIZED COVARIANCE IN AN ADAPTIVE FRAMEWORK LU JUN NATIONAL UNIVERSITY OF SINGAPORE 2012 FORECASTING REALIZED COVARIANCE IN AN ADAPTIVE FRAMEWORK LU JUN (B.Sc (using the common principal component anal- ysis) and a local adaptive VAR model are applied to the modeling of the realized covariance matrix process. The combined approach generates satisfying. feature embedded in the financial time series and establishes the dimension reduction and the adaptive VAR approaches under the framework of the realized covariance matrix process modeling. For the