forecasting realized volatility with changing average...

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Forecasting Realized Volatility with Changing Average Levels Giampiero M. Gallo * Edoardo Otranto ** * Dipartimento di Statistica, Informatica, Applicazioni (DiSIA) G. Parenti – Università di Firenze ** Dipartimento di Scienze Cognitive e della Formazione and CRENoS – Università di Messina Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 1 / 49

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Forecasting Realized Volatilitywith Changing Average Levels

Giampiero M. Gallo∗ Edoardo Otranto∗∗∗Dipartimento di Statistica, Informatica, Applicazioni (DiSIA) G.Parenti – Università di Firenze∗∗Dipartimento di Scienze Cognitive e della Formazione andCRENoS – Università di Messina

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 1 / 49

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 2 / 49

Outline

Introduction

On Today’s Menu

MEM and MS–MEM

Regimes in the Volatility of S&P500ModelingInference on RegimesIn– and Out–of–sample Forecasting

Extensions

Conclusions

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 3 / 49

Persistence as a Challenge to Volatility Modeling

BackgroundI Direct measurement of volatility: ideal properties of the

estimators ex postI Modeling for forecasting purposes: dynamic modelsI Clustering features are presentI Choice of modeling volatility or log–volatilityI Residual diagnostics as a guideline to specificationI Possible recalcitrant residual correlation as a guide to

misspecification

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 4 / 49

Persistence as a Challenge to Volatility Modeling

BackgroundI Direct measurement of volatility: ideal properties of the

estimators ex postI Modeling for forecasting purposes: dynamic modelsI Clustering features are presentI Choice of modeling volatility or log–volatilityI Residual diagnostics as a guideline to specificationI Possible recalcitrant residual correlation as a guide to

misspecification

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 4 / 49

Persistence as a Challenge to Volatility Modeling

BackgroundI Direct measurement of volatility: ideal properties of the

estimators ex postI Modeling for forecasting purposes: dynamic modelsI Clustering features are presentI Choice of modeling volatility or log–volatilityI Residual diagnostics as a guideline to specificationI Possible recalcitrant residual correlation as a guide to

misspecification

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 4 / 49

Persistence as a Challenge to Volatility Modeling

BackgroundI Direct measurement of volatility: ideal properties of the

estimators ex postI Modeling for forecasting purposes: dynamic modelsI Clustering features are presentI Choice of modeling volatility or log–volatilityI Residual diagnostics as a guideline to specificationI Possible recalcitrant residual correlation as a guide to

misspecification

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 4 / 49

Persistence as a Challenge to Volatility Modeling

BackgroundI Direct measurement of volatility: ideal properties of the

estimators ex postI Modeling for forecasting purposes: dynamic modelsI Clustering features are presentI Choice of modeling volatility or log–volatilityI Residual diagnostics as a guideline to specificationI Possible recalcitrant residual correlation as a guide to

misspecification

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 4 / 49

Persistence as a Challenge to Volatility Modeling

BackgroundI Direct measurement of volatility: ideal properties of the

estimators ex postI Modeling for forecasting purposes: dynamic modelsI Clustering features are presentI Choice of modeling volatility or log–volatilityI Residual diagnostics as a guideline to specificationI Possible recalcitrant residual correlation as a guide to

misspecification

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 4 / 49

S&P500 volatility (Jan. 3, 2000 to Oct. 26, 2012)

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 5 / 49

Nonlinear Effects in VolatilityHow to model such a series

I Underlying movements at low frequencyI A variety of interpretations/choices

I Levels vs logs; Variance vs VolatilityI Long memory (ARFIMA on log–vol; Andersen et al. 2003;)I Quasi long memory (HAR on log-vol: Corsi, 2009;

extensions: McAleer and Medeiros, 2008)I Spline fitting (van Bellegem and von Sachs, 2004; Engle

and Rangel, 2008; Brownlees and Gallo, 2010)I Markov Switching linear model on vol (Maheu and

McCurdy, 2002)I Markov Switching and fractionally integrated dynamics

(Bordignon and Raggi, 2010)I Smooth multiplicative component in a GARCH framework

(Amado and Teräsvirta, 2012)I Interpretation of the time-varying unconditional volatility

(Engle and Rangel, 2008)Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 6 / 49

Nonlinear Effects in VolatilityHow to model such a series

I Underlying movements at low frequencyI A variety of interpretations/choices

I Levels vs logs; Variance vs VolatilityI Long memory (ARFIMA on log–vol; Andersen et al. 2003;)I Quasi long memory (HAR on log-vol: Corsi, 2009;

extensions: McAleer and Medeiros, 2008)I Spline fitting (van Bellegem and von Sachs, 2004; Engle

and Rangel, 2008; Brownlees and Gallo, 2010)I Markov Switching linear model on vol (Maheu and

McCurdy, 2002)I Markov Switching and fractionally integrated dynamics

(Bordignon and Raggi, 2010)I Smooth multiplicative component in a GARCH framework

(Amado and Teräsvirta, 2012)I Interpretation of the time-varying unconditional volatility

(Engle and Rangel, 2008)Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 6 / 49

Nonlinear Effects in VolatilityHow to model such a series

I Underlying movements at low frequencyI A variety of interpretations/choices

I Levels vs logs; Variance vs VolatilityI Long memory (ARFIMA on log–vol; Andersen et al. 2003;)I Quasi long memory (HAR on log-vol: Corsi, 2009;

extensions: McAleer and Medeiros, 2008)I Spline fitting (van Bellegem and von Sachs, 2004; Engle

and Rangel, 2008; Brownlees and Gallo, 2010)I Markov Switching linear model on vol (Maheu and

McCurdy, 2002)I Markov Switching and fractionally integrated dynamics

(Bordignon and Raggi, 2010)I Smooth multiplicative component in a GARCH framework

(Amado and Teräsvirta, 2012)I Interpretation of the time-varying unconditional volatility

(Engle and Rangel, 2008)Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 6 / 49

Nonlinear Effects in VolatilityHow to model such a series

I Underlying movements at low frequencyI A variety of interpretations/choices

I Levels vs logs; Variance vs VolatilityI Long memory (ARFIMA on log–vol; Andersen et al. 2003;)I Quasi long memory (HAR on log-vol: Corsi, 2009;

extensions: McAleer and Medeiros, 2008)I Spline fitting (van Bellegem and von Sachs, 2004; Engle

and Rangel, 2008; Brownlees and Gallo, 2010)I Markov Switching linear model on vol (Maheu and

McCurdy, 2002)I Markov Switching and fractionally integrated dynamics

(Bordignon and Raggi, 2010)I Smooth multiplicative component in a GARCH framework

(Amado and Teräsvirta, 2012)I Interpretation of the time-varying unconditional volatility

(Engle and Rangel, 2008)Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 6 / 49

Nonlinear Effects in VolatilityHow to model such a series

I Underlying movements at low frequencyI A variety of interpretations/choices

I Levels vs logs; Variance vs VolatilityI Long memory (ARFIMA on log–vol; Andersen et al. 2003;)I Quasi long memory (HAR on log-vol: Corsi, 2009;

extensions: McAleer and Medeiros, 2008)I Spline fitting (van Bellegem and von Sachs, 2004; Engle

and Rangel, 2008; Brownlees and Gallo, 2010)I Markov Switching linear model on vol (Maheu and

McCurdy, 2002)I Markov Switching and fractionally integrated dynamics

(Bordignon and Raggi, 2010)I Smooth multiplicative component in a GARCH framework

(Amado and Teräsvirta, 2012)I Interpretation of the time-varying unconditional volatility

(Engle and Rangel, 2008)Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 6 / 49

Nonlinear Effects in VolatilityHow to model such a series

I Underlying movements at low frequencyI A variety of interpretations/choices

I Levels vs logs; Variance vs VolatilityI Long memory (ARFIMA on log–vol; Andersen et al. 2003;)I Quasi long memory (HAR on log-vol: Corsi, 2009;

extensions: McAleer and Medeiros, 2008)I Spline fitting (van Bellegem and von Sachs, 2004; Engle

and Rangel, 2008; Brownlees and Gallo, 2010)I Markov Switching linear model on vol (Maheu and

McCurdy, 2002)I Markov Switching and fractionally integrated dynamics

(Bordignon and Raggi, 2010)I Smooth multiplicative component in a GARCH framework

(Amado and Teräsvirta, 2012)I Interpretation of the time-varying unconditional volatility

(Engle and Rangel, 2008)Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 6 / 49

Nonlinear Effects in VolatilityHow to model such a series

I Underlying movements at low frequencyI A variety of interpretations/choices

I Levels vs logs; Variance vs VolatilityI Long memory (ARFIMA on log–vol; Andersen et al. 2003;)I Quasi long memory (HAR on log-vol: Corsi, 2009;

extensions: McAleer and Medeiros, 2008)I Spline fitting (van Bellegem and von Sachs, 2004; Engle

and Rangel, 2008; Brownlees and Gallo, 2010)I Markov Switching linear model on vol (Maheu and

McCurdy, 2002)I Markov Switching and fractionally integrated dynamics

(Bordignon and Raggi, 2010)I Smooth multiplicative component in a GARCH framework

(Amado and Teräsvirta, 2012)I Interpretation of the time-varying unconditional volatility

(Engle and Rangel, 2008)Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 6 / 49

Nonlinear Effects in VolatilityHow to model such a series

I Underlying movements at low frequencyI A variety of interpretations/choices

I Levels vs logs; Variance vs VolatilityI Long memory (ARFIMA on log–vol; Andersen et al. 2003;)I Quasi long memory (HAR on log-vol: Corsi, 2009;

extensions: McAleer and Medeiros, 2008)I Spline fitting (van Bellegem and von Sachs, 2004; Engle

and Rangel, 2008; Brownlees and Gallo, 2010)I Markov Switching linear model on vol (Maheu and

McCurdy, 2002)I Markov Switching and fractionally integrated dynamics

(Bordignon and Raggi, 2010)I Smooth multiplicative component in a GARCH framework

(Amado and Teräsvirta, 2012)I Interpretation of the time-varying unconditional volatility

(Engle and Rangel, 2008)Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 6 / 49

Nonlinear Effects in VolatilityHow to model such a series

I Underlying movements at low frequencyI A variety of interpretations/choices

I Levels vs logs; Variance vs VolatilityI Long memory (ARFIMA on log–vol; Andersen et al. 2003;)I Quasi long memory (HAR on log-vol: Corsi, 2009;

extensions: McAleer and Medeiros, 2008)I Spline fitting (van Bellegem and von Sachs, 2004; Engle

and Rangel, 2008; Brownlees and Gallo, 2010)I Markov Switching linear model on vol (Maheu and

McCurdy, 2002)I Markov Switching and fractionally integrated dynamics

(Bordignon and Raggi, 2010)I Smooth multiplicative component in a GARCH framework

(Amado and Teräsvirta, 2012)I Interpretation of the time-varying unconditional volatility

(Engle and Rangel, 2008)Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 6 / 49

Nonlinear Effects in VolatilityHow to model such a series

I Underlying movements at low frequencyI A variety of interpretations/choices

I Levels vs logs; Variance vs VolatilityI Long memory (ARFIMA on log–vol; Andersen et al. 2003;)I Quasi long memory (HAR on log-vol: Corsi, 2009;

extensions: McAleer and Medeiros, 2008)I Spline fitting (van Bellegem and von Sachs, 2004; Engle

and Rangel, 2008; Brownlees and Gallo, 2010)I Markov Switching linear model on vol (Maheu and

McCurdy, 2002)I Markov Switching and fractionally integrated dynamics

(Bordignon and Raggi, 2010)I Smooth multiplicative component in a GARCH framework

(Amado and Teräsvirta, 2012)I Interpretation of the time-varying unconditional volatility

(Engle and Rangel, 2008)Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 6 / 49

In This Paper

Two strong choicesI Concentrate on Multiplicative Error Models

I (over GARCH) Use volatility rather than squared returnsI (over linear models) Directly applied to volatility (and not

logs)I QMLE interpretation ensures consistency

I Adopt a Markov Switching approachI Exploit well established propertiesI Classification of periods by regime

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 7 / 49

In This Paper

Two strong choicesI Concentrate on Multiplicative Error Models

I (over GARCH) Use volatility rather than squared returnsI (over linear models) Directly applied to volatility (and not

logs)I QMLE interpretation ensures consistency

I Adopt a Markov Switching approachI Exploit well established propertiesI Classification of periods by regime

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 7 / 49

In This Paper

Two strong choicesI Concentrate on Multiplicative Error Models

I (over GARCH) Use volatility rather than squared returnsI (over linear models) Directly applied to volatility (and not

logs)I QMLE interpretation ensures consistency

I Adopt a Markov Switching approachI Exploit well established propertiesI Classification of periods by regime

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 7 / 49

In This Paper

Two strong choicesI Concentrate on Multiplicative Error Models

I (over GARCH) Use volatility rather than squared returnsI (over linear models) Directly applied to volatility (and not

logs)I QMLE interpretation ensures consistency

I Adopt a Markov Switching approachI Exploit well established propertiesI Classification of periods by regime

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 7 / 49

In This Paper

Two strong choicesI Concentrate on Multiplicative Error Models

I (over GARCH) Use volatility rather than squared returnsI (over linear models) Directly applied to volatility (and not

logs)I QMLE interpretation ensures consistency

I Adopt a Markov Switching approachI Exploit well established propertiesI Classification of periods by regime

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 7 / 49

In This Paper

Two strong choicesI Concentrate on Multiplicative Error Models

I (over GARCH) Use volatility rather than squared returnsI (over linear models) Directly applied to volatility (and not

logs)I QMLE interpretation ensures consistency

I Adopt a Markov Switching approachI Exploit well established propertiesI Classification of periods by regime

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 7 / 49

In This Paper

Two strong choicesI Concentrate on Multiplicative Error Models

I (over GARCH) Use volatility rather than squared returnsI (over linear models) Directly applied to volatility (and not

logs)I QMLE interpretation ensures consistency

I Adopt a Markov Switching approachI Exploit well established propertiesI Classification of periods by regime

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 7 / 49

Why a MEM?

Modeling non-negative time seriesGARCH conditional variance is the expectation of squaredreturns but volatility is measured directly

I A lot of information available in financial markets is positivevalued:

I ultra-high frequency data (within a time interval: range,volume, number of trades, number of buys/sells, durations)

I daily volatility estimators (realized volatility, daily range,absolute returns)

I Time series exhibit persistence which can be modeled à laGARCH

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 8 / 49

Why a MEM?

Modeling non-negative time seriesGARCH conditional variance is the expectation of squaredreturns but volatility is measured directly

I A lot of information available in financial markets is positivevalued:

I ultra-high frequency data (within a time interval: range,volume, number of trades, number of buys/sells, durations)

I daily volatility estimators (realized volatility, daily range,absolute returns)

I Time series exhibit persistence which can be modeled à laGARCH

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 8 / 49

Why a MEM?

Modeling non-negative time seriesGARCH conditional variance is the expectation of squaredreturns but volatility is measured directly

I A lot of information available in financial markets is positivevalued:

I ultra-high frequency data (within a time interval: range,volume, number of trades, number of buys/sells, durations)

I daily volatility estimators (realized volatility, daily range,absolute returns)

I Time series exhibit persistence which can be modeled à laGARCH

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 8 / 49

Why a MEM?

Modeling non-negative time seriesGARCH conditional variance is the expectation of squaredreturns but volatility is measured directly

I A lot of information available in financial markets is positivevalued:

I ultra-high frequency data (within a time interval: range,volume, number of trades, number of buys/sells, durations)

I daily volatility estimators (realized volatility, daily range,absolute returns)

I Time series exhibit persistence which can be modeled à laGARCH

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 8 / 49

Why a MEM?

Modeling non-negative time seriesGARCH conditional variance is the expectation of squaredreturns but volatility is measured directly

I A lot of information available in financial markets is positivevalued:

I ultra-high frequency data (within a time interval: range,volume, number of trades, number of buys/sells, durations)

I daily volatility estimators (realized volatility, daily range,absolute returns)

I Time series exhibit persistence which can be modeled à laGARCH

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 8 / 49

Multiplicative Error Models

I Extension of GARCH approach to modeling the expectedvalue of processes with positive support (Engle, 2002;Engle and Gallo, 2006)

I Autoregressive Conditional Duration is a special case.Absolute returns, high-low, number of trades in a certaininterval, volume, realized volatility can be modeled with thesame structure

I Rather than calling the models Autoregressive ConditionalVolatility, Autoregressive Conditional Volume, etc. call themMEM

I Ease of estimationI Possibility of expanding the information set (main

interesting results)

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 9 / 49

Multiplicative Error Models

I Extension of GARCH approach to modeling the expectedvalue of processes with positive support (Engle, 2002;Engle and Gallo, 2006)

I Autoregressive Conditional Duration is a special case.Absolute returns, high-low, number of trades in a certaininterval, volume, realized volatility can be modeled with thesame structure

I Rather than calling the models Autoregressive ConditionalVolatility, Autoregressive Conditional Volume, etc. call themMEM

I Ease of estimationI Possibility of expanding the information set (main

interesting results)

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 9 / 49

Multiplicative Error Models

I Extension of GARCH approach to modeling the expectedvalue of processes with positive support (Engle, 2002;Engle and Gallo, 2006)

I Autoregressive Conditional Duration is a special case.Absolute returns, high-low, number of trades in a certaininterval, volume, realized volatility can be modeled with thesame structure

I Rather than calling the models Autoregressive ConditionalVolatility, Autoregressive Conditional Volume, etc. call themMEM

I Ease of estimationI Possibility of expanding the information set (main

interesting results)

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 9 / 49

Multiplicative Error Models

I Extension of GARCH approach to modeling the expectedvalue of processes with positive support (Engle, 2002;Engle and Gallo, 2006)

I Autoregressive Conditional Duration is a special case.Absolute returns, high-low, number of trades in a certaininterval, volume, realized volatility can be modeled with thesame structure

I Rather than calling the models Autoregressive ConditionalVolatility, Autoregressive Conditional Volume, etc. call themMEM

I Ease of estimationI Possibility of expanding the information set (main

interesting results)

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 9 / 49

Multiplicative Error Models

I Extension of GARCH approach to modeling the expectedvalue of processes with positive support (Engle, 2002;Engle and Gallo, 2006)

I Autoregressive Conditional Duration is a special case.Absolute returns, high-low, number of trades in a certaininterval, volume, realized volatility can be modeled with thesame structure

I Rather than calling the models Autoregressive ConditionalVolatility, Autoregressive Conditional Volume, etc. call themMEM

I Ease of estimationI Possibility of expanding the information set (main

interesting results)

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 9 / 49

The Asymmetric MEM

xt = µtεt , εt ∼ Gamma(a,1/a) for each t

µt = ω + αxt−1 + βµt−1 + γDt−1xt−1

Dt =

1 if rt < 0

0 if rt ≥ 0

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 10 / 49

The Markov Switching–AMEM

xt = µt ,st εt , εt |st ∼ Gamma(ast ,1/ast ) for each t

µt ,st = ω +∑n

i=1 ki Ist + αst (xt−1 − µt−1,st−1)++β∗st

µt−1,st−1 + γst Dt−1(xt−1 − µt−1,st−1)

st is a discrete latent variable ranging in [1, . . . ,n] (regime attime t). Ist is an indicator equal to 1 when st ≤ i and 0otherwise; ki ≥ 0 and k1 = 0 (non decreasing levels of volatilitypassing to higher regimes)

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 11 / 49

MS–AMEM

Pr(st = j |st−1 = i , st−2, . . . ) = Pr(st = j |st−1 = i) = pij

The unconditional expected value within state j , j = 1, . . . ,n, isequal to:

µj =ω +

∑ji=1 ki

1− αj − βj − γj/2

where βj = β∗j − αj − γj/2. Under this reparameterization, themean equation of the MS-AMEM is:

µt ,st = ω +n∑

i=1

ki Ist + αst xt−1 + βstµt−1,st−1 + γst Dt−1xt−1

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 12 / 49

Estimation Issues

I Hamilton filter and smoother with Kim (1994)approximation to avoid path dependence

I After each step of the Hamilton filter, at time t we collapsethe n2 possible values of µt into n values, by an averageover the probabilities at time t − 1:

µ̂t ,st =

∑ni=1 Pr [st−1 = i , st = j |Ψt ]µ̂t ,st−1,st

Pr [st = j |Ψt ]

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 13 / 49

S&P500 volatility (Jan. 3, 2000 to Oct. 26, 2012)

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 14 / 49

Descriptive Statistics

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 15 / 49

MEM/AMEM Estimation Results

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 16 / 49

Gamma density for the AMEM

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 17 / 49

Residual Autocorrelation Results

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 18 / 49

Nonparametric Bayesian Regime IdentificationOtranto and Gallo (2002) identify the number of regimes inMarkov switching models, based on the detection of theempirical posterior distribution of the number of regimes, viaGibbs sampling, using the nonparametric Bayesian techniquesderived from the Dirichlet process theory.

Strong indication of the presence of regimes and three is thefavored number of regimes.

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 19 / 49

MS–AMEM Estimation Results

Common dynamics detected between regimes 1 and 3 inMS(3). Numerical difficulties encountered with MS(4)

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 20 / 49

Gamma density by regime for the MS(3)–AMEM

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 21 / 49

Gamma density by regime for the MS(4)-AMEM

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 22 / 49

Residual Autocorrelation Results

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 23 / 49

Discussion on Transition Probabilities

I Duration in a certain regime i : ( 11−pii

)I MS(3)–AMEM: average 87 days in Regime 1; 28 days in

Regime 2; 13 days in Regime 3. MS(4)–AMEM: expectedduration equal to 77, 18, 15 and 5 days respectively.

I Off–diagonal elements: strong interaction betweenregimes 2 and 3 while the period of low volatility is a sort ofself standing regime. From Regime 2 higher probability tomove to Regime 3 (and vice versa) than to revert toRegime 1.

I MS(4)–AMEM. Period classification is similar but withfrequent changes in regime

I many isolated dots in the figure (largest smoothedprobability not always close to 1. Regime 3 and 4 in theMS(4)-AMEM capture most of Regime 3 in theMS(3)–AMEM, with a higher interaction between Regimes1 and 2.Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 24 / 49

Inference on the Regimes: MS(3)-AMEM

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 25 / 49

Inference on the Regimes: MS(4)-AMEM

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 26 / 49

Unconditional Volatilities by Regime

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 27 / 49

Different Classification by Regime

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 28 / 49

Volatility relative to regime–specific averages -MS(3)–AMEM

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 29 / 49

Volatility relative to regime–specific averages -MS(4)–AMEM

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 30 / 49

The MS(3)–AMEM with a DummyThe behavior of the MS(4)–AMEM points to some instability inthe estimation and in the interpretation of the results, withundesirable autocorrelation generated. Capture the burst with adummy in the MS(3)–AMEM.

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 31 / 49

Volatility relative to regime–specific averages -MS(3)–AMEM(d)

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 32 / 49

Residual Autocorrelation Results

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 33 / 49

MSE and MAE Results

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 34 / 49

Diebold Mariano tests

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 35 / 49

Results of the Model Confidence SetModel Confidence Set procedure (Hansen, Lunde and Nason,2011): set of models containing the best model at 95%confidence

Hansen (2010): theoretical inverse relationship between in–and out–of–sample performance. Our results (better in–samplefit and equivalent, if not slightly better performanceout–of–sample) as a support for our Markov Switching MEMs.

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 36 / 49

Prediction confidence intervals - March 2005

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 37 / 49

Prediction confidence intervals - October 2008

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 38 / 49

Prediction confidence intervals - March 2011

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 39 / 49

Extending the Information SetRealized Kernel Volatility and VIX

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 40 / 49

Time–varying Transition ProbabilityTVTP as a function of the volatility index vt .TVTP–MS(3)–AMEM

xt = µt ,st εt , εt |st ∼ Gamma(ast ,1/ast ) for each t

µt ,st = ω +∑n

i=1 ki Ist + αst (xt−1 − µt−1,st−1) + β∗stµt−1,st−1 + γst Dt−1(xt−1 − µt−1,st−1)

Dt =

1 if rt < 0

0 if rt ≥ 0

pij,t =exp(θij+φi,j vt−1)

1+exp(θi,j+φi,j vt−1)

(1)

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 41 / 49

Rough or Smooth?Smooth transition driven by vt ST–AMEM

xt = µtεt , εt ∼ Gamma(a,1/a) for each t

µt = ω + (α0 + α1F (vt−1))xt−1 + (β0 − β1F (vt−1))µt−1 + γDt−1xt−1

F (vt ) = (1 + exp(−g(vt − c)))−1

(2)

Gallo & Otranto Changing Average Volatility Paris, Mar 30, 2015 42 / 49

Mixing Frequencies - Quasi Long MemoryBorrow Corsi’s (2009) HAR (heterogeneous dynamics)HMEM Errors enter multiplicatively into HAR

xt = µtεt , εt ∼ Gamma(a,1/a)

µt = ω + αDxt−1 + αW x̄ (5)t−1 + αM x̄ (22)

t−1

MS(3)–HMEM Insert regimes

xt = µt ,st εt , εt |st ∼ Gamma(ast ,1/ast )

µt ,st = ω +∑n

i=1 ki Ist + αD,st xt−1 + αW ,st x̄(5)t−1 + αM,st x̄

(22)t−1

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Autocorrelation PropertiesTVTP–MS(3)–AMEM ST–AMEM HMEM MS(3)–HMEM HAR

lag Q p-value Q p-value Q p-value Q p-value Q p-value1 52.05 0.00 0.04 0.84 1.59 0.21 3.23 0.07 6.12 0.012 76.10 0.00 0.90 0.64 38.25 0.00 54.87 0.00 48.49 0.003 99.15 0.00 1.76 0.62 40.74 0.00 71.89 0.00 59.25 0.004 132.53 0.00 3.29 0.51 46.21 0.00 92.96 0.00 60.21 0.005 177.49 0.00 5.45 0.36 46.24 0.00 98.49 0.00 66.01 0.006 207.78 0.00 6.02 0.42 47.98 0.00 108.47 0.00 68.82 0.007 234.29 0.00 6.39 0.50 52.32 0.00 119.01 0.00 69.16 0.008 264.40 0.00 7.72 0.46 56.87 0.00 127.54 0.00 69.18 0.009 293.85 0.00 10.39 0.32 64.57 0.00 136.43 0.00 86.96 0.00

10 332.39 0.00 15.80 0.11 74.82 0.00 148.65 0.00 88.92 0.0011 363.26 0.00 20.36 0.04 89.91 0.00 162.64 0.00 88.93 0.0012 394.20 0.00 22.55 0.03 97.59 0.00 171.92 0.00 88.93 0.00

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Estimated unconditional volatility

No regime Regime 1 Regime 2 Regime 3 Regime 4MEM 14.25

AMEM 14.39MS(3)–AMEM 9.21 14.39 28.80MS(4)–AMEM 9.28 13.74 26.76 56.97

MS(3)–AMEM(d) 9.20 14.35 28.37TVTP–MS(3)–AMEM 8.69 14.37 55.99

ST-AMEM 7.20 27.40HMEM 14.06

MS(3)–HMEM 9.13 7.30 67.33

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Some limitations of the ST–AMEM

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Best Forecasting Sets - MCS

Model Absolute errors Squared errorsMEM � �AMEM � • � •MS(3)–AMEM � • � •MS(4)–AMEM � � •MS(3)–AMEM(d) � • � •TVTP–MS(3)–AMEM ? � • ? � •ST–AMEM � • � •HMEMMS(3)–HMEM � �HAR •

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Conclusions

I Evidence of “excessive” persistence in realized volatilityI Issue of time–varying unconditional volatilityI Address nonlinearity versus long memory

I statistical fit of a flexible function of time (spline)I smooth transition (multiplicative structure needed cf.

Amado and Teräsvirta, 2012)I Markov Switching

I In Markov Switching abrupt changes in line with marketepisodes

I Transition probabilities could be made dependent on someinterpretable forcing event (some results with lagged VIX).

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Realized Traffic and Change of Regimes!

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