The Use of High-Frequency Data in Financial Econometrics ...statweb.stanford.edu/~ckirby/scqf2010/SCQFslides_Hansen.pdf · The Use of High-Frequency Data in Financial Econometrics:

The Use of High-Frequency Datain Financial Econometrics:Recent Developments

Peter Reinhard Hansen

Department of Economics, Stanford University

Stanford Conference in Quantitative Finance, 2010

Peter Reinhard Hansen (Stanford) Financial Econometrics November 2010 1 / 96

Financial Econometrics

The Society for Financial Econometrics (SoFiE) Founded 2008

Journal of Financial Econometrics (2003).

Oxford-Man Institute @ Oxford University (2007)

Volatility Institute @ NYU Stern (Robert Engle) (2009)

VLAB


Introduction

Part 1: Accurate Measures of VolatilityRealized Measures of Volatility Computed from High-Frequency Data

Ideal Case: Realized VarianceNoisy Data (Market Microstructure)Empirical Properties of NoiseRobust Estimators:Realized Kernel & Markov Chain Estimator

Part 2: ApplicationsUtilizing Realized Measures for Volatility Modeling and Forecasting

Realized GARCH Models


Realized Measures

of

Volatility


High-Frequency Prices


A Measure of Variation of Asset Prices

Suppose that Yt = logPt is a semi-martingale

Yt =

∫ t

0

audu +

∫ t

0

σudBu + Jt ,

where

a is a predictable locally bounded drift,σ is a cadlag volatility process,B is a standard Brownian motion, andJt =

∑Nt

i=1Di is a �nite activity jump process.

Quadratic Variation (over [0, 1]) is

QV =

∫1

0

σ2udu +

N1∑i=1

D2

i .


Intraday Returns

Divide day into n subintervals

0 = T0 < T1 < · · · < Tn−1 < Tn = 1,

(e.g. equidistant Tj − Tj−1 = 1

n).

Intraday returns

yj ,n = YTj− YTj−1 , j = 1, . . . , n.

Daily return is the sum of intraday returns,

Y1 − Y0 = y1,n + · · ·+ yn,n.


Realized Variance (Empirical QV)

Realized variance is de�ned by

RV =n∑

j=1

y2j ,n.

Properties (ideal case)

RVp→ QV as supi=1,...,n |Ti − Ti−1| → 0.

No jumps (i.e. Jt = 0) then

QV = IV :=

∫1

0

σ2udu, (integrated variance)

and √n(RV − IV )

d→ N(0,V ).


Literature

Literature Invigorated byAndersen and Bollerslev (1998)Realized Variance useful for evaluation of GARCH models

Statistical Properties of Realized VarianceAndersen et al. (2001)Barndor�-Nielsen and Shephard (2002)Jacod (1994), Jacod and Protter (1998)


The Great Tragedy of Science:

The Slaying of a Beautiful

Hypothesis by an Ugly FactThomas H. Huxley (1825-1895).


The Great Tragedy of Science:

The Slaying of a Beautiful

Hypothesis by an Ugly FactThomas H. Huxley (1825-1895).


High-Frequency Prices: AA 2007-05-04






High-Frequency Prices: AA 2007-01-26 (zoom)


High-Frequency Returns are Autocorrelated

Autocorrelation causes RV to be biased/inconsistent.

Computing RV with tick-by-tick returns says more about �noise� than�volatility�.

Ad-hoc resolution: Sample sparsely.

Compute RV using 5-minute returns.


Sample Sparsely


Volatility Signature Plot Reveals Bias Problem


Price Without Noise


Prices With Noise


Properties of the Noise

Hansen and Lunde (2006)Journal of Business and Economic StatisticsInvited paper with Comments and Rejoinder.

Noise Ut is...

... serial dependent

... endogenous (not independent of Yt)

... has changed over time (tick size)

... is �small�.


RobustRealized Measures


Realized Kernel

Realized Autocovariances

γh =n∑

i=1

yi ,nyi−h,n.

So RV = γ0.

RK =∞∑

h=−∞k( h

H)γh,

where

k(·) is a kernel function andH is a bandwidth parameter.


Kernel Functions


Realized Kernel: Various Crises

Figure: Realized volatility for the period 1997-2009 (annualized) and the time ofsome of the major crises and events.


Volatility Signature Plot


Refresh Time and Semi-Stale Prices

Multivariate Problem

Asynchronous Trading (quoting)Refresh Time �aligns� observations,and induces additional noise


Realized Beta from Realized Kernel


Literature on Realized Kernel

Univariate Realized KernelBarndor�-Nielsen et al. (2008)

Multivariate Realized KernelBarndor�-Nielsen et al. (2010a)

Realized Kernels in PracticeBarndor�-Nielsen et al. (2009)

Relation to other estimatorsBarndor�-Nielsen et al. (2010b)


Markov Chain Estimator(Personal Favorite)

Hansen and Horel (2009)


Prices On A Grid


The Markov Chain Estimator

Exploits discreteness of price changes.

Simple to compute

Estimate MC model (requires counting)Estimator computed with basic matrix operations

Inference...

Standard errors given in closed-formAsymptotics is reliable


Empirical Example. GE 2004-11-01

x =

−3−2−11

2

3

P =

0.17 0.00 0.33 0.50 0.00 0.000.00 0.06 0.23 0.51 0.17 0.030.00 0.01 0.25 0.71 0.02 0.000.00 0.02 0.72 0.25 0.01 0.000.00 0.14 0.64 0.19 0.03 0.000.00 0.50 0.50 0.00 0.00 0.00

π =

0.000.020.480.480.020.00

.

Compute

Λπ = diag (π) , Z = (I − P + Π)−1 Π = ιπ′.

Markov Chain Estimator is: MC = x ′[

nclog

Λπ(2Z − I )]x

= x ′

0.007 −0.000 −0.001 0.000 −0.000 −0.000−0.000 0.035 −0.013 −0.003 0.010 0.002−0.000 −0.012 0.533 0.256 −0.004 −0.002−0.001 −0.004 0.250 0.532 −0.008 −0.000−0.000 0.008 0.004 −0.014 0.033 0.000−0.000 0.004 −0.001 −0.003 0.000 0.004

x = 0.7469.


Filtering Argument

Observed Prices: XTj, j = 0, 1, . . . , n, with

∆XTj= XTj

− XTj−1 a homogeneous ergodic MC.

Consider �ltered pricesE(XTj+h

|FTj).

Easy to compute within the MC framework because

E(XTj+h|FTj

) = XTj+

h∑i=1

E(∆XTj+i|FTj

).

Even as h→∞!


Martingale Representation

We haveXTj

= µj + YTj+ UTj

,

where

µj = µ · j , with µ = E(∆XTj);

YTj= lim

h→∞E(XTj+h

− µj+h|FTj)

is a Martingale; and

UTjis stationary, ergodic, bounded process.


MC Estimator is RV of Filtered Price

XTj= µj + YTj

+ UTj,

MC estimator is basically the realized variance of YTj,

MC# = nx ′Λπ(2Z − I )x .

Consistent with a Gaussian limit distribution under appropriateassumption.

Log-correction

MC =nx ′Λπ(2Z − I )x

1

n

∑nj=1

X 2

Tj

.


Con�dence Intervals


Financial Crisis:

Markov Chain Estimates

USD/YEN


















Financial Crisis:

Markov Chain Estimates

SPY











High-Frequency Data andForecasting


Volatility Forecasting using High-Frequency Data

Hansen and Lunde (2011) (Oxford Handbook of Economic Forecasting).

HF data improves...

understanding of volatility dynamics � key for forecasting.understanding of the driving forces of volatility.For instance: Study of news announcements and their e�ect on the�nancial markets.evaluation of models/forecast

Realized measures...

good predictors of future volatility.led to new and better volatility models... yield more accurateforecasts.facilitate better estimation of complex volatility models.


Realized GARCH Models

joint with

Albert Huang and Howard Shek


ARCH and GARCH Models

ARCH-type models (Engle, 1982)

µt ≡ Et−1(rt) ht ≡ vart−1(rt).

Studentized returns

zt =rt − µt√

ht∼ (0, 1)

GARCH(1,1) (Bollerslev, 1986)

ht = ω + αr2t−1 + βht−1.

β ≈ 0.95 and α ' 0.05 in practice.


GARCH Model

Squared returns, r2t−1, de�nes the dynamic of the conditional variance

ht = ω + αr2t−1 + βht−1.

r2t can be viewed as a noisy measure of volatility

Realized measures provide more accurate measurements of volatility.


GARCH-X Model

Engle (2002), and many others

ht = ω + αr2t−1 + βht−1 + γxt−1.

xt is a realized measure of volatility (e.g. RV)

Huge improvement in the empirical �t.

Typically

γ ' 0.5.α ' 0. (ARCH parameter becomes insigni�cant)


GARCH Model

Re-write GARCH(1,1) equation:

ht = ω + πht−1 + α(r2t−1 − ht−1)

π = α + β measures how persistent is volatility.

α ' �the strength of the signal r2t−1�

β ≈ 0.95 and α ' 0.05 in practice.


GARCH is Slow


GARCH-X with a Realized Measure is Fast


GARCH-X is Incomplete

Data (rt , xt), but model only speci�es rt |rt−1,xt−1, . . .Simple case

rt =√htzt .

ht = ω + αr2t−1 + βht−1 + γxt−1.

Need a Model for xt .


Realized GARCH


Realized GARCH: Simple Case

GARCH-X structure

rt =√htzt

ht = ω + αr2t−1 + βht−1 + γxt−1

Measurement Equation completes the model

xt = ξ + ϕht + Errort .

xt is noisy measurement of QVt

QVt is ht + volatility shock.


Logarithmic Speci�cation

Logarithmic speci�cation is preferred

log ht = ω + β log ht−1 + γ log xt−1.

log xt = ξ + ϕ log ht + τ(zt) + ut .

Leverage Function:

τ(z) = τ1z + τ2(z2 − 1)

Captures the joint dependence between

return shocks, ztvolatility shocks, τ(zt) + ut .


Key Features of Realized GARCH

Empirical Features

Easy to estimate.Captures return-volatility dependence (leverage e�ect).Properties of multiperiod returns (skewness and kurtosis)Outperforms conventional GARCH

Theoretical Features (elegant mathematical structure)

Parsimonious

Tractable analysis (quasi maximum likelihood).Induced simple ARMA structure for both x and h

Natural extension of conventional GARCH


Data

Dow Jones Industrial Average stocks and SPY (ETF).

2002-01-01 to 2007-12-31 as in-sample data and2008-01-01 to 2008-08-31 as out-of-sample.

For x , we use the realized kernel (RK) by BHLS (2008)

xt ≈ ht with open-to-close returnsxt<ht (on average) with close-to-close returns


Linear Model (SPY Open-to-Close)

GARCH Equation

ht = 0.09(0.05)

+ 0.29(0.16)

ht−1 + 0.63(0.18)

xt−1

Measurement Equation

xt = −0.05(0.09)

+ 1.01(0.19)

ht +−0.02(0.02)

zt + 0.06(0.01)

(z2t − 1)︸︷︷︸τ(z)

+ ut

Standard deviation of ut : σu = 0.51(0.05)

.


Linear Model (SPY Close-to-Close)

GARCH Equation

ht = 0.07(0.04)

+ 0.29(0.15)

ht−1 + 0.87(0.25)

xt−1


xt = +0.00(0.08)

+ 0.74(0.14)

ht +−0.07(0.02)

zt + 0.03(0.01)

(z2t − 1)︸︷︷︸τ(z)

+ ut


.


Log-Linear Model (SPY Open-to-Close)

GARCH Equation

log ht = 0.04(0.02)

+ 0.70(0.05)

log ht−1 + 0.45(0.04)

log xt−1 − 0.18(0.06)

log xt−2


log xt = −0.18(0.05)

+ 1.04(0.07)

log ht +−0.07(0.01)

zt + 0.07(0.01)

(z2t − 1)︸︷︷︸τ(z)

+ ut


.

Persistence Parameter

π = β + (γ1 + γ2)ϕ = 0.986


Log-Linear Model (SPY Close-to-Close)

GARCH Equation

log ht = 0.11(0.02)

+ 0.72(0.05)

log ht−1 + 0.48(0.06)

log xt−1 − 0.21(0.07)

log xt−2


log xt = −0.42(0.06)

+ 1.00(0.10)

log ht +−0.11(0.01)

zt + 0.04(0.01)

(z2t − 1)︸︷︷︸τ(z)

+ ut


.

Persistence Parameter

π = β + (γ1 + γ2)ϕ = 0.987


Estimated News Impact Curve


Skewness and Kurtosis of Cumulative Returns

Figure: Skewness and kurtosis in line with empirical returns


Multi-Period Forecast

Multi-period ahead predictions with the Realized GARCH model isstraightforward.

When p = q = 1. we obtain VARMA(1,1) structure[htxt

]=

[β γϕβ ϕγ

] [ht−1xt−1

]+

[ω

ξ + ϕω

]+

[0

τ(zt) + ut

],

so we can write, Yt = AYt−1 + µ+ εt .


Realized GARCH Volatility

during the

Global Financial Crisis


Realized GARCH: Global Financial Crisis

Figure: Conditional volatility during the global �nancial crisis with some of themajor events .Peter Reinhard Hansen (Stanford) Financial Econometrics November 2010 88 / 96

Extensions

Realized GARCH


Realized EGARCH with Multiple RM (w. Huang)

Multiple Realized Measures (like MEM by Engle & Gallo)

log ht = ω + β log ht−1 + γ ′ logXt−1 + τ(zt−1)

logXt = ξ +ϕ log ht + δ(zt) + Ut .

RK crowds out RV and Range (High minus Low)

var(Ut) yields information about accuracy about RMs.


Multivariate Realized GARCH (w. Lunde & Voev)

Build Realized GARCH for Market Returns, r0,t = µ0 +√h0,tz0,t .

Asset i 's returns, ri ,t = µi +√hi ,tzi ,t , conditional on r0,t , x0,t .

Key in this model:ρt = cov(z0,t , zi,t |Ft−1).

Measurement equation with Fisher transform

z(ρt) = a0i + b0iz(ρt−1) + c0iz(y1,t−1).

1-factor structure where we can extract

βt = ρt

√hi ,t/h0,t .


Correlation Measurement Equation (CVX)


Time Series for ρt and βt (CVX)


Cross Sectional Beta-Quantiles


Conclusion

Realized Measures

Empirical Issues with High-Frequency DataRealized Variance, Realized Kernel, Markov Chain Estimator

Volatility Forecasting

Realized GARCH

Easy to estimate, Tractable, can explain empirical �stylized facts�.Multivariate extension very promising.


Bibliography

Andersen, T. G., Bollerslev, T., 1998. Answering the skeptics: Yes, standard volatility models do provide accurateforecasts. International Economic Review 39 (4), 885�905.

Andersen, T. G., Bollerslev, T., Diebold, F. X., Ebens, H., 2001. The distribution of realized stock return volatility.Journal of Financial Economics 61 (1), 43�76.

Barndor�-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2008. Designing realised kernels to measure theex-post variation of equity prices in the presence of noise. Econometrica 76, 1481�536.

Barndor�-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2009. Realised kernels in practice: Trades andquotes. Econometrics Journal 12, 1�33.

Barndor�-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2010a. Multivariate realised kernels: consistentpositive semi-de�nite estimators of the covariation of equity prices with noise and non-synchronous trading. Jounalof Econometrics forthcoming.

Barndor�-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2010b. Subsampling realised kernels. Journal ofEconometrics forthcoming, forthcoming.

Barndor�-Nielsen, O. E., Shephard, N., 2002. Econometric analysis of realised volatility and its use in estimatingstochastic volatility models. Journal of the Royal Statistical Society B 64, 253�280.

Hansen, P. R., Horel, G., 2009. Quadratic variation by markov chains. working paper

http://www.stanford.edu/people/peter.hansen.Hansen, P. R., Lunde, A., 2006. Realized variance and market microstructure noise. Journal of Business and Economic

Statistics 24, 127�218, the 2005 Invited Address with Comments and Rejoinder.

Hansen, P. R., Lunde, A., 2011. Forecasting volatility using high frequency data. In: Clements, M., Hendry, D. (Eds.),Handbook of Economic Forecasting. Oxford University Press.

Jacod, J., 1994. Limit of random measures associated with the increments of a Brownian semimartingalePreprintnumber 120, Laboratoire de Probabilitiés, Université Pierre et Marie Curie, Paris.

Jacod, J., Protter, P., 1998. Asymptotic error distributions for the Euler method for stochastic di�erential equations.Annals of Probability 26, 267�307.


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The Use of High-Frequency Data in Financial Econometrics ...statweb.stanford.edu/~ckirby/scqf2010/SCQFslides_Hansen.pdf · The Use of High-Frequency Data in Financial Econometrics: