Volatility Models: Early and New ApproachesVolatility Models and their Applications, John Wiley and...

Basics of Statisticsand Econometrics

Main Stylized Features of . . .

Univariate GARCH Models

Multivariate GARCH Models

Realized Volatility

Multiplicative Error Models

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Volatility Models: Early and NewApproaches

Module 1 of the Course ofEconometrics

Edoardo Otranto(Universita di Messina)

e-mail: [email protected]

International Doctoral Program inEconomics

Scuola Superiore Sant’Anna

November 2012- January 2013





Realized Volatility



Contents

• Main Stylized Features of financial returns

• Univariate GARCH Models

• Multivariate GARCH Models

• Realized Volatility

• Multiplicative Error Models





Realized Volatility



References:

• Bauwens L., Hafner C., Laurent S. (2012): Handbook ofVolatility Models and their Applications, John Wiley andSons

• Bollerslev, T. (2008): Glossary to ARCH (GARCH).Research Paper 2008-49, CREATES

• Francq, C., Zakoian, J.-M. (2010): GARCH Models:Structure, Statistical Inference and FinancialApplications. Wiley

• Hamilton, J.D. (1994): Time Series Analysis. Princeton

• Bauwens, L., Laurent, S., Rombouts, V.K. (2006):Multivariate GARCH Models: a Survey, Journal ofApplied Econometrics 21, 79-109.

• Silvennoinen, A., Terasvirta, T. (2009): MultivariateGARCH Models. In Andersen, Davis, Kreiss, Mikosch(Eds.): Handbook of Financial Time Series. pp. 201-228.Springer

• Engle, R. (2009): Anticipating Correlations, Princeton.





Realized Volatility



In finance, volatility is a measure for variation of price of afinancial instrument over time. It is unobservable and linkedto the standard deviation of returns.

1. Main Stylized Features offinancial returns

• Volatility clustering: “Large changes tend to be followedby large changes, of either sign, and small changes tend tobe followed by small changes” (Mandelbrot, Journal ofBusiness 1963). This means that volatility measured bysquared returns is persistent.

• The unconditional probability distributions areleptokurtic (fatter tails and more mass around their centerthan the Gaussian distribution)

• The returns are not independent





Realized Volatility



• Asymmetry: Large negative returns are more probablethan large positive returns (widespread but not universalfeature)

• Leverage effect: negative news tend to increase volatilitystronger than positive news.

• Co-movements in volatilities: when stock volatilitieschange, they all tend to change in the same direction.





Realized Volatility







Realized Volatility



2. Univariate GARCH ModelsLet yt an asset return at time t

yt − µt = εt = σtzt

zt are i.i.d. with E(zt) = 0 and V ar(zt) = 1 ∀ tµt and σt can depend on past information It−1E(yt|It−1) = µt and V ar(yt|It−1) = σ2

t .

so that:

E(εt|It−1) = 0 and V ar(εt|It−1) = σ2t .

The i.i.d. hypothesis for the zt process can be replaced by theassumption that the process is a martingale differencesequence (m.d.s) such that

E(zt|It−1) = 0 and V ar(zt|It−1) = 1.





Realized Volatility



GARCH(1,1) (Engle, Econometrica 1982; Bollerslev, J. ofEcon. 1986):

σ2t = ω + αε2t−1 + βσ2

t−1

Positivity constraints:

ω > 0, α ≥ 0, β ≥ 0

Using p lags of εt and q lags of σ2t we obtain the

GARCH(p,q) model.Autocorrelation coefficients:

ρ1 = α(1−β2−αβ)1−β2−2αβ

ρj = (α + β)ρj−1 for j > 1 (α + β < 1)

σ2 = V ar(εt) = ω1−α−β exists and εt is covariance stationary.

(α + β) is the persistence of the conditional variance.

K =E(ε4t )

V ar(εt)2= λ

1− α2 − β2 − αβ1− λα2 − β2 − 2αβ

λ = E(z4t ) and λα2 − β2 − 2αβ < 1. As a consequenceK > λ.





Realized Volatility



GJR-GARCH (Glosten, Jagannathan, Runkle, J. of Finance1993)

σ2t = ω + αε2t−1 + βσ2

t−1 + γε2t−1I(εt−1 < 0)

If γ > 0 the response is stronger for a past negative shockthan for a positive one (leverage effect).

This effect for a particular firm says that a negative shock (areturn below its expected value) implies that the firm is moreleveraged, i.e. has a higher ratio of debt to stock value, and istherefore more risky, so that the volatility should increase.





Realized Volatility



Probability distributions for ztThe Gaussian distribution was the first used for the MLE. Inthis case it has a QML interpretation, providing consistent andasymptotically Normal estimators of the conditional mean andGARCH parameters (provided that they are correctlyspecified).

The Normal assumption implies a conditional kurtosis equalto 3 for yt and unconditional leptokurtosis, but the degree ofleptokurtosis may be too small to fit the kurtosis of the data.

Alternatives:t-distribution: in this case we have another parameter to beestimated (the dof ν > 2). When ν > 4 the fourth momentexists and the conditional kurtosis λ is 3 + 6(ν − 4)−1.

GE distribution: it is another distribution symmetric around 0,which adds a shape parameter to the Normal distribution.





Realized Volatility







Realized Volatility



Note: The Normal, t and GE distributions are symmetricaround 0. The symmetry of the conditional distribution doesnot necessarily imply the same property for the unconditionalone.

He, Silvennoinen and Terasvirta (J. of Financial Econ., 2008)show that conditional symmetry combined with a constantconditional mean implies unconditional symmetry, whateverthe GARCH equation.

They also show that conditional symmetry combined with atime-varying conditional mean is sufficient for creatingunconditional asymmetry, but the conditional mean dynamicshas to be very strong to induce non-negligible unconditionalasymmetry.

On the other side the conditional asymmetry impliesunconditional asymmetry

Other Alternatives:skewed-t (Hansen, Int. Ec. Rev., 1994) or estimatenonparametrically the conditional distribution (Engle andGonzalez-Rivera, JBES 1991)





Realized Volatility



Drawbacks of early GARCH modelsThe models may be too rigid for fitting return series,especially over a long span.

• often the estimated persistence of conditional variance ishigh. Engle and Bollerslev (Econ. rev., 1986) propose theIGARCH model imposing α + β = 1, but this impliesthat the unconditional variance does not exist.

• Diebold (Ec. Rev., 1986) mentions that the highpersistence of conditional variances may be provoked byoverlooking changes in the conditional variance interceptω (or in the unconditional variance). In other wordschanges in ω induce non-stationarity, which is capturedby high persistence.

• the GJR-GARCH model implies that that conditionalvariances persist more strongly after a large negativeshock than after a large positive shock, but this is notrealistic in correspondence of abrupt changes. Forexample, after the October 87 crash, the volatility in USstock market reverted swiftly to its pre-crash normal level.

These considerations allowed the development of changingparameter models.





Realized Volatility



Component models (Engle and Lee, 1999)These models hypothesize that there is a long-run component(qt) in volatilities, which changes smoothly, and a short-runone (σ2

t ), changing more quickly and fluctuating around qt

σ2t = qt + α(ε2t−1 − qt−1) + β(σ2

t−1 − qt−1)qt = σ2 + φ(ε2t−1 − σ2

t−1) + ρqt−1

If ρ > (α + β), qt evolves more smoothly than σ2t .

If ρ < 1, the forecasts of both qt and σ2t converge to σ2

1−ρ as theforecast horizon tends to infinity. In other words, qt reverts toa constant level. This feature does not fit to the viewpoint thatthe level of unconditional volatility can itself evolve throughtime.





Realized Volatility



Smooth Transition models (Lanne and Saikkonen, Econom.J. 2005)

In these models the parameters of the GARCH equationchange more or less quickly through time, according to asmooth function (Terasvirta, JASA 1994).

σ2t = ω1 + ω2G(εt−1) + αε2t−1 + βσ2

t−1G(εt−1) = {1 + exp [−γ(εt−1 − k)]}−1

where γ > 0 represents the speed of the transition. If it islarge, the transition function is close to a step functionjumping at the value k, which represents the location of thetransition





Realized Volatility



Mixture models (Haas, Mittnik and Paolella, J. of FinancialEconom. 2004)

The idea is that the model is based on two (or more) variancecomponents (regimes):

σ2i,t = ωi + αiε

2t−1 + βiσ

2i,t−1 i = 1, 2

where (εt|It−1) ∼[wN(µ1, σ

21,t) + (1− w)N(µ2, σ

22,t)],

with wµ1 + (1− w)µ2 = 0.

One regime could feature a low mean with high variance (bullmarket), and the other a high mean with low variance (bearmarket) (µ1 < µ2 and ω1

1−β1−α1> ω2

1−β2−α2).

The model is useful also to capture unconditional skewnessand kurtosis.

It is based on the (not realistic) idea that the regime indicatorvariables are independent through time.





Realized Volatility



Markov Switching models (Haas, Mittnik and Paolella, J. ofFinancial Econom. 2004)

Following the idea of Hamilton (Econometrica, 1989), inthese models it is assumed that the regime indicator variablesare dependent, in the form of a Markov process of order 1.

The parameters of the GARCH model change according tothis Markov process. Let st denote a discrete random variabletaking the values 1 and 2.

εt(st) = σt(st)ztσ2t (st) = ωst + αstε

2t−1(st−1) + βstσ

2t−1(st−1)





Realized Volatility



Path dependence problem: to compute the value of theconditional variance at date t, one must know the realizedvalues of all sτ for τ ≤ t (2t possible paths). This renders MLestimation infeasible.

Approximated solutions: Gray (J. of Financial ec. 1996),Dueker (JBES 1997), Klaassen (Empirical Ec., 2002), Haas,Mittnik and Paolella (J. of Financial Econom. 2004)

Exact Solutions: GMM (Francq and Zakoian, CSDA 2008),Bayesian MCMC (Bauwens, Preminger and Rombouts,Econom. J. 2010), MCML (Augustyniak, CSDAforthcoming).

This problem does not occur if βst = 0 ∀st value.





Realized Volatility



3. Multivariate GARCH ModelsFirst attempt: Kraft and Engle (Mimeo, UCSD 1982);

First bivariate ARCH application: Engle, Granger and Kraft(J. of Ec. Dynamics and Control, 1984)

Extension to GARCH (VEC model): Bollerslev, Engle andWooldridge (J. of Political Ec. 1988)

BEKK Model: Engle and Kroner (Econom. Theory, 1995)

Factor Model: Engle, Ng and Rothschild (J. of Econom.1990)

Constant Conditional Correlation (CCC) Model: Bollerslev(Rev. of Ec. and Stat. 1990)

Time-Varying Conditional Correlation Model: Tse and Tsui(JBES 2002); Engle (JBES 2002)

Main Problems: dimensionality, positive definiteness





Realized Volatility



VEC

ht = c + Fηt−1 +Ght−1,

where n is the number of assets and vech(·) the operator thatstacks the lower triangular part of a (n× n) matrix as avector (n(n + 1)/2× 1).

Ht is the conditional covariance matrix and ht = vech(Ht).

ηt = vech(εtε′

t)

c is a (n(n + 1)/2× 1) vector of parameters and F andGare (n(n + 1)/2× n(n + 1)/2) matrices of parameters.





Realized Volatility



BEKK

Ht = CC ′ + Fεt−1ε′

t−1F′ +GHt−1G

′

whereC , F andG are n× n matrices of parameters (C isupper triangular to ensure positive definiteness ofHt).





Realized Volatility



Factor ModelThe idea is that εt is a linear function of p factors (p < n)collected in the vector ft:

εt = Bft + νt

whereB is the (n× p) factor loading matrix (rank p) and νtis a white noise vector (idiosyncratic component).

Assuming V ar(νt|It−1) = Ψ (full rank),V ar(ft|It−1) = Φt and Cov(νt,ft) = 0, then:

Ht = BΦtB′ + Ψ

which is positive definite.

Φt follows an MGARCH process (in general it is supposedthat it is a diagonal matrix of univariate GARCH).

Alternatively ft can be chosen as a linear combination of theelements of εt





Realized Volatility



Conditional Correlations

Ht = StRtSt,

where St is a diagonal matrix containing the conditionalstandard deviations andRt is a time-varying positive definitematrix of correlations.

Estimation: 2-step procedure of Engle (JBES 2002): in thefirst step we estimate the parameters of St (call them θV )using n univariate models for the conditional variances (forexample, simple GARCH models); in the second step weestimate the parameters present inRt (call them θR),conditioning on the estimate of θV . This is possible becausethe full log-likelihood function can be split into the sum of thefollowing components:

L(θV ) = −∑T

t=1 [log(|St|) + 0.5u′tut] ,

L(θR|θV ) = −∑T

t=1 [log(|Rt|) + 0.5u′tR−1t ut] ,

where ut = S−1t εt





Realized Volatility



Constant Conditional Correlations

Rt = R

Using the 2-step estimation we obtain the consistentestimator:

R =1

T

T∑t=1

utu′t

In finite samples, the diagonal elements of R are not exactlyequal to 1, so that it should be transformed to a correlationmatrix:

R = (In � R)−1/2R(In � R)−1/2

where� is the Hadamard product (element by elementmultiplication).





Realized Volatility



Time-Varying Conditional Correlations (TVCC)

Rt = (1− α− β)R + αΨt−1 + βRt−1

where Ψt−1 is the correlation matrix calculated on(ut−1,ut−1, . . . ,ut−M), with M > n to ensure that Ψt bepositive definite.

If the starting matrixR0 is a positive-definite correlationmatrix, α ≥ 0, β ≥ 0, (α + β) < 1, thenRt is a positivedefinite correlation matrix andR is its expected value(correlation targeting).

R can be replaced by R.





Realized Volatility



Dynamic Conditional Correlations (DCC)Engle (JBES 2002) proposes to model the dynamic process ofthe covariance matrixQt and transform it to the correlationmatrixRt:

Rt = Q−1t QtQ−1t ,

Qt = (1− a− b)R + aut−1u′

t−1 + bQt−1,Qt = diag(

√q11,t√q22,t, . . . ,

√qnn,t),

IfQ0 is symmetric and positive-definite and α and β satisfythe same restrictions as in TVCC model, thenQt is symmetricand positive-definite andRt is a correlation matrix. R isestimated by R but it is inconsistent (Aielli, WP University ofFirenze 2009). A consistent specification ofQt is given by:

Qt = (1− a− b)R + aQt−1ut−1u′

t−1Qt−1 + bQt−1,





Realized Volatility



4. Realized VolatilityThe model described are parametric and designed to estimatethe daily, weekly or monthly volatility using data sampled atthe same frequency.

The trading and pricing of securities in many of today’s liquidfinancial asset markets is evolving in a near continuousfashion throughout the trading day. The price and the returnseries of financial assets are discrete observations from acontinuous time-process.

dp(t) = µ(t)dt + σ(t)dW (t) t ≥ 0

where dp(t) is the logarithmic price increment, µ(t) is acontinuous locally bounded variation process (drift), σ(t) is astrictly positive and right-continuous with left limits volatilityprocess (spot volatility) and W (t) is a standard Brownianmotion.





Realized Volatility



Assuming that the time length of one day is one, theone-period daily return is:

rt = p(t)− p(t− 1) =

∫ t

t−1µ(s)ds +

∫ t

t−1σ(s)dW (s)

and (conditional on the sample path of the drift and the spotvolatility):

rt ∼ N(

∫ t

t−1µ(s)ds, IVt)

where IVt =∫ tt−1 σ

2(s)ds is the so-called integrated variance(volatility).





Realized Volatility



Mincer-Zarnowitz (MZ) regression:

σ2t = a0 + a1σ

2t + εt

where σ2t is the ex-post volatility, σ2

t is the forecastedvolatility.

To judge the quality of the GARCH forecasts econometriciansfirst used σ2

t = r2t .

Andersen and Bollerslev (Int. Ec. Rev. 1998) have shownthat, if rt follows a GARCH(1,1) process, the R2 isnecessarily lower than 1

3.





Realized Volatility



Simulation Experiment (Bauwens, Hafner and Laurent,2012)





Realized Volatility



• daily squared return is an extremely noisy estimator

• the conditional variance of the GARCH is much lessnoisy than squared returns

• if σ2t = r2t the R2 of the MZ regression is 0.07

• if σ2t = IVt the R2 of the MZ regression is more than 0.50

• IVt is not computable in practical applicationsAndersen and Bollerslev (1998) suggests the realizedvolatility estimator, obtained by simply summing up intradaysquared returns:

RVt =M∑i=1

r2t,i

In the simulation experiment using 5-minute returns(M = 288), the correlation between RVt and IVt is 0.989.

It can be shown that RVt is consistent for IVt.





Realized Volatility







Realized Volatility



At very high frequencies, returns are polluted bymicrostructure noise (bid-ask bounce, unevenly spacedobservations, discreteness,...). This problem causes thehigh-frequency returns to be autocorrelated.

A robust estimator is the realized kernel volatility(Barndorff-Nielsen, Hansen, Lunde and Shephard;Econometrica 2008)∑H

h=−H k( hH+1

)γh γh =∑h

j=|h|+1 rj,trj−|h|,t

k(r) =

1− 6r2 + 6r3 0 ≤ r ≤ 0.52(1− r)3 0.5 < r ≤ 10 r > 1

where k(r) is the Parzen kernel function and H its bandwidth.





Realized Volatility



5. Multiplicative Error ModelsThe volatility is the evolution of a non-negative valuedprocess.

Engle (J. App. Econom. 2002) proposes to model it as theproduct of a time varying scale factor (which depends uponthe recent past of the series) and a standard positive valuedrandom variable.

Such a specification was adopted in Engle and Russell(Econometrica 1998) for durations, Manganelli (WP ECB2002) for volume transaction data and Chou (MimeoAcademia Sinica 2001) for highlow range.





Realized Volatility



Extended Model(Engle, J. App. Econom. 2002; Engle and Gallo, J. Econom.2006)

µt = ω +p∑i=1

αixt−i +q∑j=1

βjµt−j +m∑h=1

γhzh,t

where zh,t are (non negative) covariance stationary weaklyexogenous variables.

Asymmetric MEM (AMEM)

(Engle and Gallo, J. Econom. 2006; Gallo and Otranto, WPUniv. FI 2012)

µt = ω + αxt−1 + βµt−1 + γDt−1xt−1

Dt =

1 if rt < 0

0 if rt ≥ 0





Realized Volatility



Mixture MEM(Lanne, J. Financial Econom. 2006)

εt = π1ε1,t + π2ε2,t π2 = (1− π1)εi,t ∼ Gamma(ai, 1/ai) i = 1, 2

andµi,t = ωi + αixt−1 + βiµi,t−1 i = 1, 2

E(εt|It−1) = 1 and V ar(εt|It−1) = π1

1

a1+ π2

1

a2

⇓

E(xt|It−1) = π1µ1,t+π2µ2,t andV ar(xt|It−1) = π1

µ21,t

a1+π1

µ22,t

a2





Realized Volatility



MS-AMEM(Gallo and Otranto, WP Univ. FI 2012)

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

µt,st = ω +∑n

i=1 kiIst + αstxt−1 + βstµt−1,st−1 + γstDt−1xt−1

st ∈ [1, . . . , n] is the regime at time t;

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

Ist is an indicator equal to 1 when st ≤ i and 0 otherwise;ki ≥ 0 and k1 = 0.

The positiveness and stationary constraints of the AMEMhold within each regime





Realized Volatility



Asymmetry in Probability MS-AMEM(AsyP-MS-AMEM)

(Gallo and Otranto, WP Univ. FI 2012)The asymmetry deriving from the sign of the returns mayaffect not only the average level within a certain regime, butalso the transition probabilities:

Pr(st|st−1, rt−1) = pij,t =

{p−ij if rt−1 < 0p+ij if rt−1 ≥ 0

A possible reparameterization of the transition probabilitiescould be made using a multinomial logit (i = 1, ..., n;j = 1, ...n− 1):

pij,t = exp(φij+ϑijDt−1)

1+∑n−1

h=1 exp(φih+ϑihDt−1)

pin,t = 1−∑n−1

j=1 pij,t

with pij,t = p−ij when Dt−1 = 1 and pij,t = p+ij whenDt−1 = 0.

Note: MS-AMEM is nested in AsyP-MS-AMEM





Realized Volatility



Smooth Transition AMEM (ST-AMEM)(Otranto J. Applied Stat., forthcoming)

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

µt = ω1 + ω2gt + αxt−1 + βµt−1 + γDt−1xt−1

gt = (1 + exp(−δ(xt−1 − c)))−1





Realized Volatility



Composite AMEM(Brownlees, Cipollini and Gallo 2012)

They decompose µt in the sum of a short (ξt) and a long (χt)time dynamics:

µt = ξt + χtξt = αξ(xt−1 − µt−1) + β∗ξξt−1 + γξ(xt−1 − µt/2)χt = ωχ + αχ(xt−1 − µt−1) + β∗χχt−1β∗ξ = βξ + αξ + γξ/2β∗χ = βχ + αχ





Realized Volatility



Component AMEM(Brownlees, Cipollini and Gallo, J. Financial Econom. 2011)

xtj is the volatility of day t at instant j (j = 1, . . . J areequally spaced intervals). They propose to distinguish a dailycompoent ηt, a periodic intra-daily componet sj and anon-periodic intra-daily dynamic component µt,j:

xt,j = ηtsjµt,jεt,jηt = ωη + αηx

(η)t−1 + βηηt−1 + γηx

−(η)t−1

ln(sj) =∑J

k=1 δkI(k = j)

µt,j = ωµ + αµx(µ)t,j−1 + βµµt,j−1 + γµx

−(µ)t,j−1

x(η)t = 1

J

∑Jj=1

xt,jsjµt,j

x(µ)t = xt,j

sjηt

x−(η)t = xt(η)I(rt < 0)

x−(µ)t = xt,j(µ)I(rt < 0)





Realized Volatility



AMEM with Spillover Effects (SAMEM)(Otranto, CRENoS WP 2012)

He proposes to capture the spillover effects in volatility in aunivariate factorial framework:Let yt the volatility of the market of interest and zt = (yt,xt)

′

a (n + 1)× 1 a vector of variables, each one representing thevolatility relative to a certain financial market.

yt = µtεt εt|I t−1 ∼ Gamma(a, 1/a) for each tµt = ζt +

∑ni=1 ξi,t

ζt = ω +∑p0

h=1 α0,hyt−h +∑q0

j=1 β0,jζt−j + γ0D0,t−1yt−1ξi,t =

∑pih=1 αi,hxi,t−h +

∑qij=1 βi,jξi,t−j + γiDi,t−1xi,t−1





Realized Volatility



Vector MEM (VMEM)(Cipollini, Engle and Gallo, NBER WP 2006)

zt = µt � εt (εt|I t−1) ∼ D(1,Σ)µt = ω +Azt−1 +Bµt−1 + Γdt−1 � zt−1

where dt−1 contains the n dummies and εt contains ndisturbances; D is a multivariate distribution (multivariateGamma or copula-based distribution).

E(zt|I t−1) = µt V ar(zt|I t−1) = µtµ′t �Σ

Volatility Models: Early and New ApproachesVolatility Models and their Applications, John Wiley and...

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