Forecasting Realized Volatility with Changing Average …risk2015.institutlouisbachelier.org/...GO_MSAMEM_Paris_Mar30Slides.… · Forecasting Realized Volatility with Changing Average

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


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


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





misspecification





misspecification





misspecification





misspecification





misspecification


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


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


In This Paper






In This Paper






In This Paper






In This Paper






In This Paper






In This Paper






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


Why a MEM?







Why a MEM?







Why a MEM?







Why a MEM?







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)






























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


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)


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


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 ]


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


Descriptive Statistics


MEM/AMEM Estimation Results


Gamma density for the AMEM


Residual Autocorrelation Results


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.


MS–AMEM Estimation Results

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


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


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




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


Inference on the Regimes: MS(4)-AMEM


Unconditional Volatilities by Regime


Different Classification by Regime


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


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


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.


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




MSE and MAE Results


Diebold Mariano tests


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.


Prediction confidence intervals - March 2005


Prediction confidence intervals - October 2008


Prediction confidence intervals - March 2011


Extending the Information SetRealized Kernel Volatility and VIX


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)


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)


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


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


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


Some limitations of the ST–AMEM


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 •


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).


Realized Traffic and Change of Regimes!


Forecasting Realized Volatility with Changing Average …risk2015.institutlouisbachelier.org/...GO_MSAMEM_Paris_Mar30Slides.… · Forecasting Realized Volatility with Changing Average

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