Asymmetric Effects and Long Memory in the Volatility of DJIA Stocks

Transcription

1 Asymmetric Effects and Long Memory in the Volatility of DJIA Stocks Pontifical Catholic University of Rio de Janeiro November 7, 2006

2 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some stylized facts of financial time series The distribution of financial returns has fat tails: Kurtosis much larger than 3. Volatility clustering. The volatility of returns displays long-range dependence: Autocorrelations decay very slowly. Asymmetries Modeling the dynamics of financial time series Latent volatility models: GARCH family, stochastic volatility, EWMA, etc. Estimate volatility with intraday data (realized volatility) and use standard time series techniques Solution adopted in this paper! Important for risk management and asset pricing!

3 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some stylized facts of financial time series The distribution of financial returns has fat tails: Kurtosis much larger than 3. Volatility clustering. The volatility of returns displays long-range dependence: Autocorrelations decay very slowly. Asymmetries Modeling the dynamics of financial time series Latent volatility models: GARCH family, stochastic volatility, EWMA, etc. Estimate volatility with intraday data (realized volatility) and use standard time series techniques Solution adopted in this paper! Important for risk management and asset pricing!

4 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some stylized facts of financial time series The distribution of financial returns has fat tails: Kurtosis much larger than 3. Volatility clustering. The volatility of returns displays long-range dependence: Autocorrelations decay very slowly. Asymmetries Modeling the dynamics of financial time series Latent volatility models: GARCH family, stochastic volatility, EWMA, etc. Estimate volatility with intraday data (realized volatility) and use standard time series techniques Solution adopted in this paper! Important for risk management and asset pricing!

5 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some stylized facts of financial time series The distribution of financial returns has fat tails: Kurtosis much larger than 3. Volatility clustering. The volatility of returns displays long-range dependence: Autocorrelations decay very slowly. Asymmetries Modeling the dynamics of financial time series Latent volatility models: GARCH family, stochastic volatility, EWMA, etc. Estimate volatility with intraday data (realized volatility) and use standard time series techniques Solution adopted in this paper! Important for risk management and asset pricing!

6 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some stylized facts of financial time series The distribution of financial returns has fat tails: Kurtosis much larger than 3. Volatility clustering. The volatility of returns displays long-range dependence: Autocorrelations decay very slowly. Asymmetries Modeling the dynamics of financial time series Latent volatility models: GARCH family, stochastic volatility, EWMA, etc. Estimate volatility with intraday data (realized volatility) and use standard time series techniques Solution adopted in this paper! Important for risk management and asset pricing!

7 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some stylized facts of financial time series The distribution of financial returns has fat tails: Kurtosis much larger than 3. Volatility clustering. The volatility of returns displays long-range dependence: Autocorrelations decay very slowly. Asymmetries Modeling the dynamics of financial time series Latent volatility models: GARCH family, stochastic volatility, EWMA, etc. Estimate volatility with intraday data (realized volatility) and use standard time series techniques Solution adopted in this paper! Important for risk management and asset pricing!

8 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some stylized facts of financial time series The distribution of financial returns has fat tails: Kurtosis much larger than 3. Volatility clustering. The volatility of returns displays long-range dependence: Autocorrelations decay very slowly. Asymmetries Modeling the dynamics of financial time series Latent volatility models: GARCH family, stochastic volatility, EWMA, etc. Estimate volatility with intraday data (realized volatility) and use standard time series techniques Solution adopted in this paper! Important for risk management and asset pricing!

9 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some stylized facts of financial time series The distribution of financial returns has fat tails: Kurtosis much larger than 3. Volatility clustering. The volatility of returns displays long-range dependence: Autocorrelations decay very slowly. Asymmetries Modeling the dynamics of financial time series Latent volatility models: GARCH family, stochastic volatility, EWMA, etc. Estimate volatility with intraday data (realized volatility) and use standard time series techniques Solution adopted in this paper! Important for risk management and asset pricing!

10 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Example Daily returns of IBM: January 3, 1994 December 31, 2003

11 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Example Daily log realized volatility of IBM: January 3, 1994 December 31, 2003

12 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Example Daily log realized volatility of IBM: January 3, 1994 December 31, 2003

13 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Modeling long-range dependence Fractionally integrated (ARFIMA) models? Regime-switching? Structural breaks? Modeling asymmetries Regime-switching How many regimes?

14 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Modeling long-range dependence Fractionally integrated (ARFIMA) models? Regime-switching? Structural breaks? Modeling asymmetries Regime-switching How many regimes?

15 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Modeling long-range dependence Fractionally integrated (ARFIMA) models? Regime-switching? Structural breaks? Modeling asymmetries Regime-switching How many regimes?

16 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Modeling long-range dependence Fractionally integrated (ARFIMA) models? Regime-switching? Structural breaks? Modeling asymmetries Regime-switching How many regimes?

17 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Modeling long-range dependence Fractionally integrated (ARFIMA) models? Regime-switching? Structural breaks? Modeling asymmetries Regime-switching How many regimes?

18 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Modeling long-range dependence Fractionally integrated (ARFIMA) models? Regime-switching? Structural breaks? Modeling asymmetries Regime-switching How many regimes?

19 Introduction and Motivation Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Modeling long-range dependence Fractionally integrated (ARFIMA) models? Regime-switching? Structural breaks? Modeling asymmetries Regime-switching How many regimes?

20 Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Realized Volatility: Definition and Main Results Definition Realized variance is defined as the sum of all available intraday high frequency squared returns. Realized volatility is the square root of the realized variance. Under the assumption of uncorrelated intraday returns, the realized variance is a consistent estimator of the integrated variance in a continuous-time diffusion model Andersen et al. (Econometrica, 2003) and Barndorff-Nielsen and Shephard (JRRS-B, 2002).

21 Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Realized Volatility: Definition and Main Results Definition Realized variance is defined as the sum of all available intraday high frequency squared returns. Realized volatility is the square root of the realized variance. Under the assumption of uncorrelated intraday returns, the realized variance is a consistent estimator of the integrated variance in a continuous-time diffusion model Andersen et al. (Econometrica, 2003) and Barndorff-Nielsen and Shephard (JRRS-B, 2002).

22 Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Realized Volatility: Definition and Main Results Definition Realized variance is defined as the sum of all available intraday high frequency squared returns. Realized volatility is the square root of the realized variance. Under the assumption of uncorrelated intraday returns, the realized variance is a consistent estimator of the integrated variance in a continuous-time diffusion model Andersen et al. (Econometrica, 2003) and Barndorff-Nielsen and Shephard (JRRS-B, 2002).

23 Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Realized Volatility: Definition and Main Results Definition Realized variance is defined as the sum of all available intraday high frequency squared returns. Realized volatility is the square root of the realized variance. Under the assumption of uncorrelated intraday returns, the realized variance is a consistent estimator of the integrated variance in a continuous-time diffusion model Andersen et al. (Econometrica, 2003) and Barndorff-Nielsen and Shephard (JRRS-B, 2002).

24 Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Realized Volatility: Definition and Main Results Problem Due to microstructure noise, the intraday returns are autocorrelated. Thus, the realized variance is not a consistent estimator of the integrated variance. Solution Use the two time scales estimator (TTSE) put forward by Zhang, Mykland, and Aït-Sahalia (JASA, 2005).

25 Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Realized Volatility: Definition and Main Results Problem Due to microstructure noise, the intraday returns are autocorrelated. Thus, the realized variance is not a consistent estimator of the integrated variance. Solution Use the two time scales estimator (TTSE) put forward by Zhang, Mykland, and Aït-Sahalia (JASA, 2005).

26 Main Contribution Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Key Idea Use an unified tree-structured framework (model) to deal with structural breaks and regime-shifts. Combination of regression trees and smooth transition models. Main Advantages Nests several nonlinear models previously proposed. Genuinely different regimes. Multiple transition variables. Long-range dependence and intermittent dynamics.

27 Main Contribution Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Key Idea Use an unified tree-structured framework (model) to deal with structural breaks and regime-shifts. Combination of regression trees and smooth transition models. Main Advantages Nests several nonlinear models previously proposed. Genuinely different regimes. Multiple transition variables. Long-range dependence and intermittent dynamics.

28 Main Contribution Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Key Idea Use an unified tree-structured framework (model) to deal with structural breaks and regime-shifts. Combination of regression trees and smooth transition models. Main Advantages Nests several nonlinear models previously proposed. Genuinely different regimes. Multiple transition variables. Long-range dependence and intermittent dynamics.

29 Main Contribution Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Key Idea Use an unified tree-structured framework (model) to deal with structural breaks and regime-shifts. Combination of regression trees and smooth transition models. Main Advantages Nests several nonlinear models previously proposed. Genuinely different regimes. Multiple transition variables. Long-range dependence and intermittent dynamics.

30 Main Contribution Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Key Idea Use an unified tree-structured framework (model) to deal with structural breaks and regime-shifts. Combination of regression trees and smooth transition models. Main Advantages Nests several nonlinear models previously proposed. Genuinely different regimes. Multiple transition variables. Long-range dependence and intermittent dynamics.

31 Main Contribution Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Key Idea Use an unified tree-structured framework (model) to deal with structural breaks and regime-shifts. Combination of regression trees and smooth transition models. Main Advantages Nests several nonlinear models previously proposed. Genuinely different regimes. Multiple transition variables. Long-range dependence and intermittent dynamics.

32 Main Contribution Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Key Idea Use an unified tree-structured framework (model) to deal with structural breaks and regime-shifts. Combination of regression trees and smooth transition models. Main Advantages Nests several nonlinear models previously proposed. Genuinely different regimes. Multiple transition variables. Long-range dependence and intermittent dynamics.

33 Main Results Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some questions Do volatility levels change in periods of significant losses or gains (cumulated returns)? Can negative returns over some horizon be associated with regimes of higher volatility? Main findings New transition variable: Past cumulated returns. The effects of macroeconomic announcements and weekdays are also significant. Extremely good forecasts! Jumps do not improve the forecasts.

34 Main Results Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some questions Do volatility levels change in periods of significant losses or gains (cumulated returns)? Can negative returns over some horizon be associated with regimes of higher volatility? Main findings New transition variable: Past cumulated returns. The effects of macroeconomic announcements and weekdays are also significant. Extremely good forecasts! Jumps do not improve the forecasts.

35 Main Results Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some questions Do volatility levels change in periods of significant losses or gains (cumulated returns)? Can negative returns over some horizon be associated with regimes of higher volatility? Main findings New transition variable: Past cumulated returns. The effects of macroeconomic announcements and weekdays are also significant. Extremely good forecasts! Jumps do not improve the forecasts.

36 Main Results Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some questions Do volatility levels change in periods of significant losses or gains (cumulated returns)? Can negative returns over some horizon be associated with regimes of higher volatility? Main findings New transition variable: Past cumulated returns. The effects of macroeconomic announcements and weekdays are also significant. Extremely good forecasts! Jumps do not improve the forecasts.

37 Main Results Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some questions Do volatility levels change in periods of significant losses or gains (cumulated returns)? Can negative returns over some horizon be associated with regimes of higher volatility? Main findings New transition variable: Past cumulated returns. The effects of macroeconomic announcements and weekdays are also significant. Extremely good forecasts! Jumps do not improve the forecasts.

38 Main Results Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some questions Do volatility levels change in periods of significant losses or gains (cumulated returns)? Can negative returns over some horizon be associated with regimes of higher volatility? Main findings New transition variable: Past cumulated returns. The effects of macroeconomic announcements and weekdays are also significant. Extremely good forecasts! Jumps do not improve the forecasts.

39 Main Results Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Some questions Do volatility levels change in periods of significant losses or gains (cumulated returns)? Can negative returns over some horizon be associated with regimes of higher volatility? Main findings New transition variable: Past cumulated returns. The effects of macroeconomic announcements and weekdays are also significant. Extremely good forecasts! Jumps do not improve the forecasts.

40 Main Results Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Example IBM data: Monthly returns x Daily volatility

41 Related literature Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Persistence and regime switching Diebold and Inoue (Journal of Econometrics, 2001) Mikosch and Stǎricǎ (REStat, 2004) Hyung, Poon, and Granger (WP, 2005) Davidson and Sibbertsen (Journal of Econometrics, 2005) Hillebrand (Journal of Econometrics, 2005) Nonlinear models and realized volatility Martens, van Dijk, and de Pooter (WP, 2004) Long memory and nonlinearity

42 Related literature Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper Persistence and regime switching Diebold and Inoue (Journal of Econometrics, 2001) Mikosch and Stǎricǎ (REStat, 2004) Hyung, Poon, and Granger (WP, 2005) Davidson and Sibbertsen (Journal of Econometrics, 2005) Hillebrand (Journal of Econometrics, 2005) Nonlinear models and realized volatility Martens, van Dijk, and de Pooter (WP, 2004) Long memory and nonlinearity

43 Outline Introduction Some Stylized Facts of Financial Time Series Realized Volatility Goals of the Paper 1 The model Overview and motivation Mathematical definition Modeling cycle 2 Estimation results Forecasting results 3

44 Model Setup Model Definition Modeling Cycle Some Notation A parametric model based on the recursive partitioning of the covariate space X. A local model is determined in each of the K N different regions (partitions) of X. The model is displayed in a graph which has the format of a decision tree with N N parent (or split) nodes and K N terminal nodes (or leaves). Usually, the partitions are defined by a set of hyperplanes, each of which is orthogonal to the axis of a given predictor variable, called the split variable. Smooth transition between regimes.

45 Model Setup Model Definition Modeling Cycle Some Notation A parametric model based on the recursive partitioning of the covariate space X. A local model is determined in each of the K N different regions (partitions) of X. The model is displayed in a graph which has the format of a decision tree with N N parent (or split) nodes and K N terminal nodes (or leaves). Usually, the partitions are defined by a set of hyperplanes, each of which is orthogonal to the axis of a given predictor variable, called the split variable. Smooth transition between regimes.

46 Model Setup Model Definition Modeling Cycle Some Notation A parametric model based on the recursive partitioning of the covariate space X. A local model is determined in each of the K N different regions (partitions) of X. The model is displayed in a graph which has the format of a decision tree with N N parent (or split) nodes and K N terminal nodes (or leaves). Usually, the partitions are defined by a set of hyperplanes, each of which is orthogonal to the axis of a given predictor variable, called the split variable. Smooth transition between regimes.

47 Model Setup Model Definition Modeling Cycle Some Notation A parametric model based on the recursive partitioning of the covariate space X. A local model is determined in each of the K N different regions (partitions) of X. The model is displayed in a graph which has the format of a decision tree with N N parent (or split) nodes and K N terminal nodes (or leaves). Usually, the partitions are defined by a set of hyperplanes, each of which is orthogonal to the axis of a given predictor variable, called the split variable. Smooth transition between regimes.

48 Model Setup Model Definition Modeling Cycle Some Notation A parametric model based on the recursive partitioning of the covariate space X. A local model is determined in each of the K N different regions (partitions) of X. The model is displayed in a graph which has the format of a decision tree with N N parent (or split) nodes and K N terminal nodes (or leaves). Usually, the partitions are defined by a set of hyperplanes, each of which is orthogonal to the axis of a given predictor variable, called the split variable. Smooth transition between regimes.

49 Model Setup Model Definition Modeling Cycle Some Notation (cont.) The root node is at position 0 and a parent node at position j generates left- and right-child nodes at positions 2j + 1 and 2j + 2, respectively. Every parent node has an associated split variable x sj t x t, where s j S = {1, 2,..., q}. J and T are the sets of indexes of the parent and terminal nodes, respectively. A tree architecture can be fully determined by J and T.

50 Model Setup Model Definition Modeling Cycle Some Notation (cont.) The root node is at position 0 and a parent node at position j generates left- and right-child nodes at positions 2j + 1 and 2j + 2, respectively. Every parent node has an associated split variable x sj t x t, where s j S = {1, 2,..., q}. J and T are the sets of indexes of the parent and terminal nodes, respectively. A tree architecture can be fully determined by J and T.

51 Model Setup Model Definition Modeling Cycle Some Notation (cont.) The root node is at position 0 and a parent node at position j generates left- and right-child nodes at positions 2j + 1 and 2j + 2, respectively. Every parent node has an associated split variable x sj t x t, where s j S = {1, 2,..., q}. J and T are the sets of indexes of the parent and terminal nodes, respectively. A tree architecture can be fully determined by J and T.

52 Model Setup Model Definition Modeling Cycle Some Notation (cont.) The root node is at position 0 and a parent node at position j generates left- and right-child nodes at positions 2j + 1 and 2j + 2, respectively. Every parent node has an associated split variable x sj t x t, where s j S = {1, 2,..., q}. J and T are the sets of indexes of the parent and terminal nodes, respectively. A tree architecture can be fully determined by J and T.

53 Model Setup Model Definition Modeling Cycle Some Notation (cont.) The root node is at position 0 and a parent node at position j generates left- and right-child nodes at positions 2j + 1 and 2j + 2, respectively. Every parent node has an associated split variable x sj t x t, where s j S = {1, 2,..., q}. J and T are the sets of indexes of the parent and terminal nodes, respectively. A tree architecture can be fully determined by J and T.

54 Model Setup Model Definition Modeling Cycle Example

55 Model Setup Model Definition Modeling Cycle Mathematical Definition Let z t x t such that z t R p, p q. Set ez t = (1, z t). w t R d is a vector of linear regressors. The Smooth Transition Regression Tree (STR-Tree) model (da Rosa, Veiga and Medeiros (WP, 2003)): log(rv t) = H JT (x t, w t; ψ) + ε t = α w t + X i T β iez tb Ji (x t; θ i ) + ε t where B Ji (x t; θ i ) = Y j J G(x sj,t; γ j, c j ) n i,j (1+n i,j ) 2 1 G(xsj,t; γ j, c j ) (1 n i,j )(1+n i,j ) and G( ) is the logistic function.

56 Model Setup Model Definition Modeling Cycle Mathematical Definition Let z t x t such that z t R p, p q. Set ez t = (1, z t). w t R d is a vector of linear regressors. The Smooth Transition Regression Tree (STR-Tree) model (da Rosa, Veiga and Medeiros (WP, 2003)): log(rv t) = H JT (x t, w t; ψ) + ε t = α w t + X i T β iez tb Ji (x t; θ i ) + ε t where B Ji (x t; θ i ) = Y j J G(x sj,t; γ j, c j ) n i,j (1+n i,j ) 2 1 G(xsj,t; γ j, c j ) (1 n i,j )(1+n i,j ) and G( ) is the logistic function.

57 Model Setup Model Definition Modeling Cycle Mathematical Definition Let z t x t such that z t R p, p q. Set ez t = (1, z t). w t R d is a vector of linear regressors. The Smooth Transition Regression Tree (STR-Tree) model (da Rosa, Veiga and Medeiros (WP, 2003)): log(rv t) = H JT (x t, w t; ψ) + ε t = α w t + X i T β iez tb Ji (x t; θ i ) + ε t where B Ji (x t; θ i ) = Y j J G(x sj,t; γ j, c j ) n i,j (1+n i,j ) 2 1 G(xsj,t; γ j, c j ) (1 n i,j )(1+n i,j ) and G( ) is the logistic function.

58 Model Setup Model Definition Modeling Cycle Mathematical Definition (cont.) The variable n i,j is defined as 1 if the path to leaf i does not include the parent node j; 0 if the path to leaf i includes the right-child node n i,j = of the parent node j; 1 if the path to leaf i includes the left-child node of the parent node j. Let J i be the subset of J containing the indexes of the parent nodes that form the path to leaf i. Then, θ i is the vector containing all the parameters (γ k, c k ) such that k J i, i T.

59 Model Setup Model Definition Modeling Cycle Mathematical Definition (cont.) The variable n i,j is defined as 1 if the path to leaf i does not include the parent node j; 0 if the path to leaf i includes the right-child node n i,j = of the parent node j; 1 if the path to leaf i includes the left-child node of the parent node j. Let J i be the subset of J containing the indexes of the parent nodes that form the path to leaf i. Then, θ i is the vector containing all the parameters (γ k, c k ) such that k J i, i T.

60 Model Setup Model Definition Modeling Cycle Mathematical Definition (cont.) The variable n i,j is defined as 1 if the path to leaf i does not include the parent node j; 0 if the path to leaf i includes the right-child node n i,j = of the parent node j; 1 if the path to leaf i includes the left-child node of the parent node j. Let J i be the subset of J containing the indexes of the parent nodes that form the path to leaf i. Then, θ i is the vector containing all the parameters (γ k, c k ) such that k J i, i T.

61 STR-Tree Model Specification Model Definition Modeling Cycle Main Steps Following the specific-to-general principle, we start the cycle from the root node (depth 0). The general steps are: 1 Selection of the relevant variables. 2 Specification of the model by selecting in the depth d, using the LM test, a node to be split (if not in the root node) and a splitting variable. 3 Estimation of the parameters. 4 Evaluation of the estimated model by checking if it is necessary to: 1 Change the node to be split. 2 Change the splitting variable. 3 Remove the split.

62 STR-Tree Model Specification Model Definition Modeling Cycle Main Steps Following the specific-to-general principle, we start the cycle from the root node (depth 0). The general steps are: 1 Selection of the relevant variables. 2 Specification of the model by selecting in the depth d, using the LM test, a node to be split (if not in the root node) and a splitting variable. 3 Estimation of the parameters. 4 Evaluation of the estimated model by checking if it is necessary to: 1 Change the node to be split. 2 Change the splitting variable. 3 Remove the split.

63 STR-Tree Model Specification Model Definition Modeling Cycle Main Steps Following the specific-to-general principle, we start the cycle from the root node (depth 0). The general steps are: 1 Selection of the relevant variables. 2 Specification of the model by selecting in the depth d, using the LM test, a node to be split (if not in the root node) and a splitting variable. 3 Estimation of the parameters. 4 Evaluation of the estimated model by checking if it is necessary to: 1 Change the node to be split. 2 Change the splitting variable. 3 Remove the split.

64 STR-Tree Model Specification Model Definition Modeling Cycle Main Steps Following the specific-to-general principle, we start the cycle from the root node (depth 0). The general steps are: 1 Selection of the relevant variables. 2 Specification of the model by selecting in the depth d, using the LM test, a node to be split (if not in the root node) and a splitting variable. 3 Estimation of the parameters. 4 Evaluation of the estimated model by checking if it is necessary to: 1 Change the node to be split. 2 Change the splitting variable. 3 Remove the split.

65 STR-Tree Model Specification Model Definition Modeling Cycle Main Steps Following the specific-to-general principle, we start the cycle from the root node (depth 0). The general steps are: 1 Selection of the relevant variables. 2 Specification of the model by selecting in the depth d, using the LM test, a node to be split (if not in the root node) and a splitting variable. 3 Estimation of the parameters. 4 Evaluation of the estimated model by checking if it is necessary to: 1 Change the node to be split. 2 Change the splitting variable. 3 Remove the split.

66 Growing the Tree Model Definition Modeling Cycle Testing for an additional split Consider a STR-Tree model with K leaves. We want to test if the terminal node i T should be split or not. Write the model as log(rv t) =α w t + where X i T {i } β iez tb Ji (x t; θ i ) + β 2i +1ez tb J2i +1 (x t; θ 2i +1) + β 2i +2ez tb J2i +2 (x t; θ 2i +2) + ε t, B J2i +1 (x t; θ 2i +1) = B Ji (x t; θ i ) G(x i t; γ i, c i ) B J2i +2 (x t; θ 2i +2) = B Ji (x t; θ i ) [1 G(x i t; γ i, c i )].

67 Growing the Tree Model Definition Modeling Cycle Testing for an additional split Consider a STR-Tree model with K leaves. We want to test if the terminal node i T should be split or not. Write the model as log(rv t) =α w t + where X i T {i } β iez tb Ji (x t; θ i ) + β 2i +1ez tb J2i +1 (x t; θ 2i +1) + β 2i +2ez tb J2i +2 (x t; θ 2i +2) + ε t, B J2i +1 (x t; θ 2i +1) = B Ji (x t; θ i ) G(x i t; γ i, c i ) B J2i +2 (x t; θ 2i +2) = B Ji (x t; θ i ) [1 G(x i t; γ i, c i )].

68 Growing the Tree Model Definition Modeling Cycle Testing for an additional split Consider a STR-Tree model with K leaves. We want to test if the terminal node i T should be split or not. Write the model as log(rv t) =α w t + where X i T {i } β iez tb Ji (x t; θ i ) + β 2i +1ez tb J2i +1 (x t; θ 2i +1) + β 2i +2ez tb J2i +2 (x t; θ 2i +2) + ε t, B J2i +1 (x t; θ 2i +1) = B Ji (x t; θ i ) G(x i t; γ i, c i ) B J2i +2 (x t; θ 2i +2) = B Ji (x t; θ i ) [1 G(x i t; γ i, c i )].

69 Growing the tree Model Definition Modeling Cycle Testing for an additional split (cont.) In a more compact form log(rv t) =α w t + X i T {i } β iez tb Ji (x t; θ i ) + φ ez tb Ji (x t; θ i ) + λ ez tb Ji (x t; θ i ) G(x i t; γ i, c i ) + ε t, where φ = β 2i +2 and λ = β 2i +1 β 2i +2. In order to test the statistical significance of the split, a convenient null hypothesis is H 0 : γ i = 0 against the alternative H a : γ i > 0.

70 Growing the tree Model Definition Modeling Cycle Testing for an additional split (cont.) In a more compact form log(rv t) =α w t + X i T {i } β iez tb Ji (x t; θ i ) + φ ez tb Ji (x t; θ i ) + λ ez tb Ji (x t; θ i ) G(x i t; γ i, c i ) + ε t, where φ = β 2i +2 and λ = β 2i +1 β 2i +2. In order to test the statistical significance of the split, a convenient null hypothesis is H 0 : γ i = 0 against the alternative H a : γ i > 0.

71 Growing the tree Model Definition Modeling Cycle Testing for an additional split (cont.) In a more compact form log(rv t) =α w t + X i T {i } β iez tb Ji (x t; θ i ) + φ ez tb Ji (x t; θ i ) + λ ez tb Ji (x t; θ i ) G(x i t; γ i, c i ) + ε t, where φ = β 2i +2 and λ = β 2i +1 β 2i +2. In order to test the statistical significance of the split, a convenient null hypothesis is H 0 : γ i = 0 against the alternative H a : γ i > 0.

72 Growing the Tree Model Definition Modeling Cycle Identification Problem Under H 0, the parameters λ and c i can assume different values without changing the quasi-loglikelihood function. Solution Third-order Taylor expansion around γ i = 0. log(rv t) =α w t + X i T {i } β iez tb Ji (x t; θ i ) + α 0ez tb Ji (x t; θ i ) + α 1ez tb Ji (x t; θ i ) x i t + α 2ez tb Ji (x t; θ i ) x 2 i t+ α 3ez tb Ji (x t; θ i ) x 3 i t + e t, where e t = ε t + λ ez tb Ji (x t; θ i ) Remainder. Thus the null hypothesis becomes H 0 : α 1 = α 2 = α 3 = 0.

73 Growing the Tree Model Definition Modeling Cycle Identification Problem Under H 0, the parameters λ and c i can assume different values without changing the quasi-loglikelihood function. Solution Third-order Taylor expansion around γ i = 0. log(rv t) =α w t + X i T {i } β iez tb Ji (x t; θ i ) + α 0ez tb Ji (x t; θ i ) + α 1ez tb Ji (x t; θ i ) x i t + α 2ez tb Ji (x t; θ i ) x 2 i t+ α 3ez tb Ji (x t; θ i ) x 3 i t + e t, where e t = ε t + λ ez tb Ji (x t; θ i ) Remainder. Thus the null hypothesis becomes H 0 : α 1 = α 2 = α 3 = 0.

74 Growing the Tree Model Definition Modeling Cycle Identification Problem Under H 0, the parameters λ and c i can assume different values without changing the quasi-loglikelihood function. Solution Third-order Taylor expansion around γ i = 0. log(rv t) =α w t + X i T {i } β iez tb Ji (x t; θ i ) + α 0ez tb Ji (x t; θ i ) + α 1ez tb Ji (x t; θ i ) x i t + α 2ez tb Ji (x t; θ i ) x 2 i t+ α 3ez tb Ji (x t; θ i ) x 3 i t + e t, where e t = ε t + λ ez tb Ji (x t; θ i ) Remainder. Thus the null hypothesis becomes H 0 : α 1 = α 2 = α 3 = 0.

75 Growing the tree Model Definition Modeling Cycle Testing for an additional split (cont.) A robust version of the test can be carried out in stages as follows: 1 Estimate the STR-Tree model with K regimes and save the residuals bε t. 2 Regress bν t on b h t and estimate the residuals br t, where b h t = H JT(x t; ψ) ψ h i and bν t = ψ= ψ b ez b t B Ji x i t,ez b t B Ji xi 2 t, ezt B b Ji xi 3 t 3 Regress a vector of ones on bε t br t and compute the sum of the squared residuals SSR. Compute the LM statistic LM = T SSR d χ 2 (dim(ν t))

76 Growing the tree Model Definition Modeling Cycle Testing for an additional split (cont.) A robust version of the test can be carried out in stages as follows: 1 Estimate the STR-Tree model with K regimes and save the residuals bε t. 2 Regress bν t on b h t and estimate the residuals br t, where b h t = H JT(x t; ψ) ψ h i and bν t = ψ= ψ b ez b t B Ji x i t,ez b t B Ji xi 2 t, ezt B b Ji xi 3 t 3 Regress a vector of ones on bε t br t and compute the sum of the squared residuals SSR. Compute the LM statistic LM = T SSR d χ 2 (dim(ν t))

77 Growing the tree Model Definition Modeling Cycle Testing for an additional split (cont.) A robust version of the test can be carried out in stages as follows: 1 Estimate the STR-Tree model with K regimes and save the residuals bε t. 2 Regress bν t on b h t and estimate the residuals br t, where b h t = H JT(x t; ψ) ψ h i and bν t = ψ= ψ b ez b t B Ji x i t,ez b t B Ji xi 2 t, ezt B b Ji xi 3 t 3 Regress a vector of ones on bε t br t and compute the sum of the squared residuals SSR. Compute the LM statistic LM = T SSR d χ 2 (dim(ν t))

78 Growing the tree Model Definition Modeling Cycle Testing for an additional split (cont.) A robust version of the test can be carried out in stages as follows: 1 Estimate the STR-Tree model with K regimes and save the residuals bε t. 2 Regress bν t on b h t and estimate the residuals br t, where b h t = H JT(x t; ψ) ψ h i and bν t = ψ= ψ b ez b t B Ji x i t,ez b t B Ji xi 2 t, ezt B b Ji xi 3 t 3 Regress a vector of ones on bε t br t and compute the sum of the squared residuals SSR. Compute the LM statistic LM = T SSR d χ 2 (dim(ν t))

79 Data Introduction Estimation Results Forecasting Results Description 16 DJIA stocks: Alcoa (AA), American International Group (AIG), Boeing (BA), Caterpillar (CAT), General Electric (GE), General Motors (GM), Hewlett Packard (HPQ), IBM (IBM), Intel (INTC), Johnson and Johnson (JNJ), Coca-Cola (KO), Microsoft (MSFT), Merck (MRK), Pfizer (PFE), Wal-Mart (WMT) and Exxon (XON). Source and sample: Tick-by-tick quotes extracted from the NYSE Trade and Quote (TAQ ) database for the period of January 3, 1994, through December 31, Days with abnormally small trading volume are excluded.

80 Data Introduction Estimation Results Forecasting Results Description 16 DJIA stocks: Alcoa (AA), American International Group (AIG), Boeing (BA), Caterpillar (CAT), General Electric (GE), General Motors (GM), Hewlett Packard (HPQ), IBM (IBM), Intel (INTC), Johnson and Johnson (JNJ), Coca-Cola (KO), Microsoft (MSFT), Merck (MRK), Pfizer (PFE), Wal-Mart (WMT) and Exxon (XON). Source and sample: Tick-by-tick quotes extracted from the NYSE Trade and Quote (TAQ ) database for the period of January 3, 1994, through December 31, Days with abnormally small trading volume are excluded.

81 Data Introduction Estimation Results Forecasting Results Description 16 DJIA stocks: Alcoa (AA), American International Group (AIG), Boeing (BA), Caterpillar (CAT), General Electric (GE), General Motors (GM), Hewlett Packard (HPQ), IBM (IBM), Intel (INTC), Johnson and Johnson (JNJ), Coca-Cola (KO), Microsoft (MSFT), Merck (MRK), Pfizer (PFE), Wal-Mart (WMT) and Exxon (XON). Source and sample: Tick-by-tick quotes extracted from the NYSE Trade and Quote (TAQ ) database for the period of January 3, 1994, through December 31, Days with abnormally small trading volume are excluded.

82 Data Introduction Estimation Results Forecasting Results Description 16 DJIA stocks: Alcoa (AA), American International Group (AIG), Boeing (BA), Caterpillar (CAT), General Electric (GE), General Motors (GM), Hewlett Packard (HPQ), IBM (IBM), Intel (INTC), Johnson and Johnson (JNJ), Coca-Cola (KO), Microsoft (MSFT), Merck (MRK), Pfizer (PFE), Wal-Mart (WMT) and Exxon (XON). Source and sample: Tick-by-tick quotes extracted from the NYSE Trade and Quote (TAQ ) database for the period of January 3, 1994, through December 31, Days with abnormally small trading volume are excluded.

83 Data Introduction Estimation Results Forecasting Results Pre-processing and volatility estimation Non-standard quotes removal and computation of mid-quote prices one second returns. Following Hansen and Lunde (2006), the previous tick method is used for determining prices at precise time marks. Realized volatility is constructed with the two time scales estimator with five-minute grids.

84 Data Introduction Estimation Results Forecasting Results Pre-processing and volatility estimation Non-standard quotes removal and computation of mid-quote prices one second returns. Following Hansen and Lunde (2006), the previous tick method is used for determining prices at precise time marks. Realized volatility is constructed with the two time scales estimator with five-minute grids.

85 Data Introduction Estimation Results Forecasting Results Pre-processing and volatility estimation Non-standard quotes removal and computation of mid-quote prices one second returns. Following Hansen and Lunde (2006), the previous tick method is used for determining prices at precise time marks. Realized volatility is constructed with the two time scales estimator with five-minute grids.

86 Data Introduction Estimation Results Forecasting Results Pre-processing and volatility estimation Non-standard quotes removal and computation of mid-quote prices one second returns. Following Hansen and Lunde (2006), the previous tick method is used for determining prices at precise time marks. Realized volatility is constructed with the two time scales estimator with five-minute grids.

87 Estimated Models Estimation Results Forecasting Results The STR-Tree model The estimated STR-Tree model has the following structure log(rv t) = X i T β i B Ji (x t; θ i ) + α 1 log(rv t 1) + + α p log(rv t p)+ δ 1I [Mon] t + δ 2I [Tue] t + δ 3I [Wed] t + δ 4I [Thu] t+ δ 5I [FOMC] t + δ 6I [EMP] t + δ 7I [CPI ] t + δ 8I [PPI ] t + ε t I [Mon] t, I [Tue] t, I [Wed] t, and I [Thu] t are dummy variables for the weekdays. I [FOMC] t, I [EMP] t, I [CPI ] t, and I [PPI ] t are dummy variables for the announcement dates. x t contains lagged cumulated returns over the one to 120 days.

88 Estimated Models Estimation Results Forecasting Results The STR-Tree model The estimated STR-Tree model has the following structure log(rv t) = X i T β i B Ji (x t; θ i ) + α 1 log(rv t 1) + + α p log(rv t p)+ δ 1I [Mon] t + δ 2I [Tue] t + δ 3I [Wed] t + δ 4I [Thu] t+ δ 5I [FOMC] t + δ 6I [EMP] t + δ 7I [CPI ] t + δ 8I [PPI ] t + ε t I [Mon] t, I [Tue] t, I [Wed] t, and I [Thu] t are dummy variables for the weekdays. I [FOMC] t, I [EMP] t, I [CPI ] t, and I [PPI ] t are dummy variables for the announcement dates. x t contains lagged cumulated returns over the one to 120 days.

89 Estimated Models Estimation Results Forecasting Results The STR-Tree model The estimated STR-Tree model has the following structure log(rv t) = X i T β i B Ji (x t; θ i ) + α 1 log(rv t 1) + + α p log(rv t p)+ δ 1I [Mon] t + δ 2I [Tue] t + δ 3I [Wed] t + δ 4I [Thu] t+ δ 5I [FOMC] t + δ 6I [EMP] t + δ 7I [CPI ] t + δ 8I [PPI ] t + ε t I [Mon] t, I [Tue] t, I [Wed] t, and I [Thu] t are dummy variables for the weekdays. I [FOMC] t, I [EMP] t, I [CPI ] t, and I [PPI ] t are dummy variables for the announcement dates. x t contains lagged cumulated returns over the one to 120 days.

90 Estimated Models Estimation Results Forecasting Results The STR-Tree model The estimated STR-Tree model has the following structure log(rv t) = X i T β i B Ji (x t; θ i ) + α 1 log(rv t 1) + + α p log(rv t p)+ δ 1I [Mon] t + δ 2I [Tue] t + δ 3I [Wed] t + δ 4I [Thu] t+ δ 5I [FOMC] t + δ 6I [EMP] t + δ 7I [CPI ] t + δ 8I [PPI ] t + ε t I [Mon] t, I [Tue] t, I [Wed] t, and I [Thu] t are dummy variables for the weekdays. I [FOMC] t, I [EMP] t, I [CPI ] t, and I [PPI ] t are dummy variables for the announcement dates. x t contains lagged cumulated returns over the one to 120 days.

91 Estimated Models Estimation Results Forecasting Results The STR-Tree model The estimated STR-Tree model has the following structure log(rv t) = X i T β i B Ji (x t; θ i ) + α 1 log(rv t 1) + + α p log(rv t p)+ δ 1I [Mon] t + δ 2I [Tue] t + δ 3I [Wed] t + δ 4I [Thu] t+ δ 5I [FOMC] t + δ 6I [EMP] t + δ 7I [CPI ] t + δ 8I [PPI ] t + ε t I [Mon] t, I [Tue] t, I [Wed] t, and I [Thu] t are dummy variables for the weekdays. I [FOMC] t, I [EMP] t, I [CPI ] t, and I [PPI ] t are dummy variables for the announcement dates. x t contains lagged cumulated returns over the one to 120 days.

92 Estimated Models Estimation Results Forecasting Results Example The estimated STR-Tree model: The case of IBM

93 Estimated Models Estimation Results Forecasting Results Alternative models Apart from the STR-Tree model, the following models are also estimated: A structural break model (a STR-Tree specification with time as the only transition variable). Linear AR and ARFIMA models. The Heterogeneous Autoregressive (HAR) model put forward by Corsi (2003) The GARCH(1,1) model The Exponentially Weighted Moving Average (EWMA) Note that the EWMA is computed on the log realized volatility series with decay parameter 0.8.

94 Estimated Models Estimation Results Forecasting Results Alternative models Apart from the STR-Tree model, the following models are also estimated: A structural break model (a STR-Tree specification with time as the only transition variable). Linear AR and ARFIMA models. The Heterogeneous Autoregressive (HAR) model put forward by Corsi (2003) The GARCH(1,1) model The Exponentially Weighted Moving Average (EWMA) Note that the EWMA is computed on the log realized volatility series with decay parameter 0.8.

95 Estimated Models Estimation Results Forecasting Results Alternative models Apart from the STR-Tree model, the following models are also estimated: A structural break model (a STR-Tree specification with time as the only transition variable). Linear AR and ARFIMA models. The Heterogeneous Autoregressive (HAR) model put forward by Corsi (2003) The GARCH(1,1) model The Exponentially Weighted Moving Average (EWMA) Note that the EWMA is computed on the log realized volatility series with decay parameter 0.8.

96 Estimated Models Estimation Results Forecasting Results Alternative models Apart from the STR-Tree model, the following models are also estimated: A structural break model (a STR-Tree specification with time as the only transition variable). Linear AR and ARFIMA models. The Heterogeneous Autoregressive (HAR) model put forward by Corsi (2003) The GARCH(1,1) model The Exponentially Weighted Moving Average (EWMA) Note that the EWMA is computed on the log realized volatility series with decay parameter 0.8.

97 Estimated Models Estimation Results Forecasting Results Alternative models Apart from the STR-Tree model, the following models are also estimated: A structural break model (a STR-Tree specification with time as the only transition variable). Linear AR and ARFIMA models. The Heterogeneous Autoregressive (HAR) model put forward by Corsi (2003) The GARCH(1,1) model The Exponentially Weighted Moving Average (EWMA) Note that the EWMA is computed on the log realized volatility series with decay parameter 0.8.

98 Estimated Models Estimation Results Forecasting Results Alternative models Apart from the STR-Tree model, the following models are also estimated: A structural break model (a STR-Tree specification with time as the only transition variable). Linear AR and ARFIMA models. The Heterogeneous Autoregressive (HAR) model put forward by Corsi (2003) The GARCH(1,1) model The Exponentially Weighted Moving Average (EWMA) Note that the EWMA is computed on the log realized volatility series with decay parameter 0.8.

99 Estimated Models Estimation Results Forecasting Results Alternative models Apart from the STR-Tree model, the following models are also estimated: A structural break model (a STR-Tree specification with time as the only transition variable). Linear AR and ARFIMA models. The Heterogeneous Autoregressive (HAR) model put forward by Corsi (2003) The GARCH(1,1) model The Exponentially Weighted Moving Average (EWMA) Note that the EWMA is computed on the log realized volatility series with decay parameter 0.8.

100 Estimated Models Estimation Results Forecasting Results Alternative models Apart from the STR-Tree model, the following models are also estimated: A structural break model (a STR-Tree specification with time as the only transition variable). Linear AR and ARFIMA models. The Heterogeneous Autoregressive (HAR) model put forward by Corsi (2003) The GARCH(1,1) model The Exponentially Weighted Moving Average (EWMA) Note that the EWMA is computed on the log realized volatility series with decay parameter 0.8.

101 Main Setup Estimation Results Forecasting Results Out-of-sample period: 01-Jan-2000 to 31-Dec-2003 (983 observations) Each model is re-estimated daily and then used for point and value at risk forecasting for the horizons of one, five, ten and twenty days ahead. The specification of the STR-Tree model is revised monthly. Point forecasts for the STR-Tree model are calculated through conditional simulation, as well as interval forecasts for all models.

102 Main Setup Estimation Results Forecasting Results Out-of-sample period: 01-Jan-2000 to 31-Dec-2003 (983 observations) Each model is re-estimated daily and then used for point and value at risk forecasting for the horizons of one, five, ten and twenty days ahead. The specification of the STR-Tree model is revised monthly. Point forecasts for the STR-Tree model are calculated through conditional simulation, as well as interval forecasts for all models.

103 Main Setup Estimation Results Forecasting Results Out-of-sample period: 01-Jan-2000 to 31-Dec-2003 (983 observations) Each model is re-estimated daily and then used for point and value at risk forecasting for the horizons of one, five, ten and twenty days ahead. The specification of the STR-Tree model is revised monthly. Point forecasts for the STR-Tree model are calculated through conditional simulation, as well as interval forecasts for all models.

104 Main Setup Estimation Results Forecasting Results Out-of-sample period: 01-Jan-2000 to 31-Dec-2003 (983 observations) Each model is re-estimated daily and then used for point and value at risk forecasting for the horizons of one, five, ten and twenty days ahead. The specification of the STR-Tree model is revised monthly. Point forecasts for the STR-Tree model are calculated through conditional simulation, as well as interval forecasts for all models.

105 Main Setup Estimation Results Forecasting Results Out-of-sample period: 01-Jan-2000 to 31-Dec-2003 (983 observations) Each model is re-estimated daily and then used for point and value at risk forecasting for the horizons of one, five, ten and twenty days ahead. The specification of the STR-Tree model is revised monthly. Point forecasts for the STR-Tree model are calculated through conditional simulation, as well as interval forecasts for all models.

106 Forecasting Results: IBM Estimation Results Forecasting Results MAE and Forecasting Accuracy Tests 1 day 5 days MAE HLN SPA MAE HLN SPA STR-Tree/AE STR-Tree/SB STR-Tree/DJIA STR-Tree/SB+AE HAR ARFIMA AR EWMA GARCH days 20 days MAE HLN SPA MAE HLN SPA STR-Tree/AE STR-Tree/SB STR-Tree/DJIA STR-Tree/SB+AE HAR ARFIMA AR EWMA GARCH