do high-frequency measures of volatility improve forecasts ... · do high-frequency measures of...

Do high-frequency measures of volatility improveforecasts of return distributions?

John M. Maheu and Thomas H. McCurdyUniversity of TorontoRCEA and CIRANO

Do high-frequency measures of volatility improve forecasts of return distributions? Nr. 1

1

Important Problems in Finance

• Derivative pricing

• Risk measurement and management

• Portfolio choice

• Each requires a full characterization of the multiperiod forecast density of returns


2

Contribution

• Propose integrated modeling approach for returns and observable realized volatility

(RV)

– Extension of Maheu and McCurdy (2007)

• Link dynamics of realized volatility to conditional variance of returns

• Assess value of RV for multiperiod density forecasts

– Common benchmark of return density allows comparison with GARCH models

• Assess importance of the following for density forecasts

– Market microstructure adjustment to RV

– Value of RV versus daily squared returns. Can we do better?

– Time-series information over short and long forecast horizons

– Modeling of persistence and variance targeting

– Fat-tailed innovations vs Normal innovations


3

Realized Volatility – Literature

• Measurement, statistical features

– Andersen and Bollerslev (1998), ABDL(2001), ABDE(2001)

• Theory

– ABDL(2003), Barndorff-Nielsen and Shephard (2002a,2002b), Meddahi

(2002b)

• Modeling

– ADBL(2003), ABD(2007), Ghysels et al(2006), Maheu and McCurdy (2002),

Martens et al (2003), Koopman et al(2005), Taylor and Xu (1997).

• Variance of returns and RV

– ABDL(2003), Goit and Laurent(2004), 2-step approach,

– Bollerslev et al (2007) (in-sample, σ2t ≡ RVt)

– This paper (out-of-sample density forecasts and using σ2t = Et−1RVt, versus

σ2t ≡ RVt)


4

Realized Volatility

• Andersen et al (2003) argue that quadratic variation, QVt, is the relevant ex post

measure of volatility for daily returns.

• Let rt,i, i = 1, ...,M be equally-spaced grid of intraday returns.

• Daily return rt =∑M

i=1 rt,i.

• Realized Volatility: RVt =∑M

i=1 r2t,i, consistent (M → ∞) for QVt

• More accurate than daily squared returns


5

Intuition – Discrete time context

• Let rt,i = σt,izt,i, zt,i ∼ iid(0, 1) i = 1, ...,M be the intraday returns.

• Daily return rt =∑M

i=1 rt,i.

• Given Ft = {σ2t,i}M

i=1, Var(rt|Ft) =∑M

i=1 σ2t,i. This is what we want to estimate!

• E[RVt|Ft] =∑M

i=1 σ2t,i. Unbiased.

• If zt,i ∼ NID(0, 1), daily return rt|Ft ∼ N(0,∑M

i=1 σ2t,i).

• More efficient than daily squared returns.

Var(r2t |Ft) > Var(RVt|Ft)

since

r2t = (

∑Mi=1 rt,i)

2 = RVt + 2∑

i<j rt,irt,j


6

Market Microstructure

• Bid-ask bounce, random arrival of trades etc

• RVt is biased and inconsistent. Ex pt,i = p̃t,i + ηt,i, p̃t,i = fundamental log-price,

then rt,i = r̃t,i + εt,i, εt,i = ηt,i − ηt,i−1

=⇒ E[RVt|Ft] =∑M

i=1 σ2t,i + Mσ2

ε

• Solutions

– Sample at a lower frequency where no market microstructure effects are present

– Adjust RVt

• Follow bias correction of Hansen and Lunde (2006)

RVt,AC1 =

M∑i=1

r2t,i + 2

M−1∑i=1

rt,irt,i+1. (1)

This estimator suffers from the possibility of a negative value for RV.


7

Market Microstructure

• To rule out possible negative RVt

RVt,ACqb = ω0γ̂0 + 2

q∑j=1

ωjγ̂j (2)

γ̂j =

M−j∑i=1

rt,irt,i+j,

in which the weights follow a Bartlett scheme ωj = 1 − jq+1, j = 0, 1, ..., q.


8

Data

• S&P 500 equity index using the Standard & Poor’s Depository Receipt (Spyder)

which is a tradeable security (Exchange Traded Fund).

• IBM

• Compute daily RVt and rt

• Daily return is open to close and matches RVt.

– TAQ database. Corrections for errors

– Construct 5 minute grid using nearest previous transaction, 9:30 – 16:00 EST

– 228,394 5 minute returns for S&P 500, 286,988 5 minute returns for IBM

– S&P 500 January 2, 1996 to August 29, 2007 (2936 daily observations)

– IBM January 3, 1993 to August 29, 2007 (3693 daily observations)


9

ACF of 5-Minute Return Data

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0 2 4 6 8 10 12 14 16 18 20

S&P500

-0.08-0.07-0.06

-0.05-0.04-0.03-0.02

-0.01 0

0.01

0 2 4 6 8 10 12 14 16 18 20

IBM


10

Summary Statistics: Daily Returns and Realized Volatility

Mean Variance Skewness Kurtosis Min Max

SPY

rt -0.018 0.967 0.080 6.180 -7.504 8.236

RVu 1.210 2.640 6.932 84.936 0.055 33.217

RVAC1b 1.079 2.373 7.670 96.439 0.047 30.789

RVAC2b 1.013 2.115 7.530 88.588 0.043 25.227

RVAC3b 0.978 2.054 8.071 102.635 0.036 26.329

IBM

rt -0.037 2.602 0.074 3.898 -11.699 11.310

RVu 2.825 9.161 5.145 54.879 0.150 58.270

RVAC1b 2.623 9.433 6.051 75.409 0.132 65.069

RVAC2b 2.558 9.875 6.377 82.091 0.114 66.594

RVAC3b 2.531 10.095 6.362 81.024 0.010 65.235

rt are daily returns constructed from open and close prices. RVu is constructed from raw 5-minute returns withno adjustment. RVACqb, q = 1, 2, 3 are computed as:

RVACqb = γ̂0 + 2

qX

j=1

ωj γ̂j , γ̂j =

M−jX

i=1

rt,irt,i+j , ωj = 1 −j

q + 1, j = 0, 1, ..., q


11

S&P500

-8-6-4

-2 0 2 4

6 8

10

1996 1998 2000 2002 2004 2006

A. Returns

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

5 5.5

1996 1998 2000 2002 2004 2006

B. Square-root RVAC3b


12

IBM Daily Returns and RV

-15

-10

-5

0

5

10

15

1994 1996 1998 2000 2002 2004 2006

A. Returns

0 1 2

3 4 5 6

7 8 9

1994 1996 1998 2000 2002 2004 2006

B. Square-root RVAC1b


13

RV Models

Component Specifications, Maheu and McCurdy (2007)

log(RVt) = ω +

2∑i=1

φisi,t + γut−1 + ηvt, vt ∼ NID(0, 1)

si,t = (1 − αi) log(RVt−1) + αisi,t−1, 0 < αi < 1, i = 1, 2

ut−1 is a return innovation

1. αi close to 1 puts less weight on recent observations and more weight on past si,t−1

2. coefficient on log(RVt−j) is φ1(1 − α1)αj−11 + φ2(1 − α2)α

j−12

3. asymmetry term γut−1

4. mean reversion if 0 < φi < 1 which allows for variance targeting,

E[log(RVt)] = ω/(1 − φ1 − φ2),

set ω = mean(log(RVt))(1 − φ1 − φ2)


14

RV Models

Heterogeneous AutoRegressive (HAR) Specifications, Corsi(2003),

ABD(2007)

log(RVt) = ω + φ1 log(RVt−1) + φ2 log(RVt−5,5) + φ3 log(RVt−22,22)

+ γut−1 + ηvt, vt ∼ NID(0, 1),

where log(RVt−h,h) =1

h

h−1∑i=0

log(RVt−h+i), log(RVt−1,1) = log(RVt−1).

ut−1 is a return innovation

1. good approximation to long memory

2. linear – easy to estimate

3. asymmetry terms γut−1

4. mean reversion if stationary, E[log(RV )] = ω/(1−φ1−φ2−φ3) variance targeting

possible


15

Joint Models of RV and Returns

Linking Conditional Variance of Returns to RV

1. ABDL (2003) Et−1(QVt) = Vart−1(rt) ≡ σ2t

2. log-normality

σ2t = Et−1RVt = exp

(Et−1 log(RVt) +

1

2Vart−1(log(RVt)

).


16


Component Specifications

rt = µ + εt, εt = σtut, ut ∼ NID(0, 1)

log(RVt) = ω +

2∑i=1




(Et−1 log(RVt) +

1

2Vart−1(log(RVt)

).

- fat tails ut ∼ tν(0, 1)


17


HAR Specifications

rt = µ + εt, εt = σtut, ut ∼ NID(0, 1)

log(RVt) = ω + φ1 log(RVt−1) + φ2 log(RVt−5,5) + φ3 log(RVt−22,22)

+ γut−1 + ηvt, vt ∼ NID(0, 1),


(Et−1 log(RVt) +

1

2Vart−1(log(RVt)

).

- fat tails ut ∼ tν(0, 1)


18


2 Component Observable SV: (2Comp-OSV)

• Assume σ2t ≡ RVt

• This is like an observable stochastic volatility model

rt = µ + εt, εt =√

RVtut, ut ∼ NID(0, 1)

log(RVt) = ω +

2∑i=1




19

Joint Model Summary

• Component or HAR specification for log(RVt)

• With and without variance targeting

• With and without asymmetry terms

• Normal return innovations vs t-innovations.

• Link RV to variance by σ2t = Et−1RVt or σ2

t ≡ RVt


20

Benchmark Model

EGARCH model on daily returns

rt = µ + εt, εt = σtut ut ∼ NID(0, 1),

log(σ2t ) = ω + β log(σ2

t−1) + γut−1 + α|ut−1|.

• Others: GARCH, GJR-GARCH, with and without t-innovations


21

Density Forecast Evaluation

Predictive Likelihood

Average predictive likelihood over the out-of-sample observations

t = τ + kmax, ..., T − k,

DM,k =1

T − τ − kmax + 1

T−k∑t=τ+kmax−k

log fM,k(rt+k|Φt, θ), k ≥ 1,

• fM,k(x|Φt, θ) is the k-period ahead predictive density for model M , given Φt eval-

uated at the realized return x = rt+k.

• Models that better account for the data produce larger DM,k. A measure of

accuracy.

• Term structure of predictive likelihoods: k vs DM,k

• T = 2936, τ = 1200, kmax = 60 so that τ + kmax − 1 = 1259

• DM,k is computed for each k using the out-of-sample returns r1260, ..., r2936

• DM,k is comparable across k (time series information) and across different models

(both EGARCH and joint RV-return models)


22

Forecast Density Evaluation

Computations

For k > 1, fM,k(rt+k|Φt, θ) is unknown. Approximate using

fM,k(rt+k|Φt, θ) =

∫f (rt+k|µ, σ2

t+k)p(σ2t+k|Φt)dσ2

t+k

≈ 1

N

N∑i=1

f (rt+k|µ, σ2(i)t+k), σ

2(i)t+k ∼ p(σ2

t+k|Φt)

• f (rt+k|µ, σ2(i)t+k) is the data density with mean µ and variance σ2

t+k,

• Draws from p(σ2t+k|Φt) come from simulating σ

2(i)t+k out N times according to the

model dynamics.

• θ = θ̂, maximum likelihood estimate

• Models re-estimated every 50 observations

• S&P 500 out-of-sample 2000/12/26 – 2007/8/29 (1677 density forecasts)

• IBM out-of-sample 1997/12/24 – 2007/8/29 (2434 density forecasts)


23

Log-Forecast Densities for Different Forecast Horizons

-14

-12

-10

-8

-6

-4

-2

0

-6 -4 -2 0 2 4 6

k=1k=20k=60

Forecast densities k-periods ahead for the one-component model with variance targetingand Normal innovations to returns. The forecast densities are based on information upto and including observation 1999/6/1.


24

Statistical Significance

• Is the average predictive likelihood for two models statistically different?

• Use Diebold Mariano (1995) tests adjusted by Amisano and Giacomoni (2007)

• Null hypothesis of equal forecast performance.

• Models A and B define

tkA,B = (DA,k − DB,k)/(σ̂AB,k/√

T − τ − kmax + 1)a∼ N(0, 1)

• σ̂AB,k is the Newey-West long-run sample variance (HAC) estimate for dt =

log fA,k(rt+k|Φt, θ̂) − log fB,k(rt+k|Φt, θ̂)


25

S&P 500, Joint Models versus EGARCH

-1.3

-1.28

-1.26

-1.24

-1.22

-1.2

-1.18

-1.16

0 10 20 30 40 50 60

Aver

age

Pred

ictiv

e Li

kelih

ood

Forecast Horizon k

2Comp-OSV2Comp1Comp

HAREGARCH

-1

0

1

2

3

4

5

6

0 10 20 30 40 50 60

Forecast Horizon k

Diebold-Mariano Test Statistics

2Comp-OSV vs EGARCH2Comp-OSV vs 2Comp

2Comp-OSV vs HAR


26

Robustness

• log(RVt) to have a fat-tailed distribution with a mixture of two normals

vt ∼{

N(0, σ2v,1) with probability π

N(0, σ2v,2) with probability 1 − π

with η = 1.

• log(RVt) has GARCH(1,1) conditional variance dynamics

η2t = κ0 + κ1[log(RVt−1) − Et−2 log(RVt−1)]

2 + κ2η2t−1.

where log(RVt−1) − Et−2 log(RVt−1) denotes the innovation to log(RV ) at time

(t − 1).


27

S&P 500, Robustness, non-Normal Innovations log(RV )

-1.25

-1.24

-1.23

-1.22

-1.21

-1.2

-1.19

-1.18

-1.17

0 10 20 30 40 50 60

Aver

age

Pred

ictiv

e Li

kelih

ood

Forecast Horizon k

2Comp-OSV2Comp-OSV-mixture

2Comp-OSV-GARCH

-4

-3

-2

-1

0

1

2

0 10 20 30 40 50 60

Forecast Horizon k


2Comp-OSV vs 2Comp-OSV-mixture2Comp-OSV vs 2Comp-OSV-GARCH


28

IBM, Joint Models versus EGARCH

-1.8

-1.78

-1.76

-1.74

-1.72

-1.7

-1.68

-1.66

0 10 20 30 40 50 60

Aver

age

Pred

ictiv

e Li

kelih

ood

Forecast Horizon k

2Comp-OSV2Comp

HAREGARCH

0 1 2 3 4 5 6 7 8 9

10

0 10 20 30 40 50 60

Forecast Horizon k


2Comp-OSV vs EGARCH2Comp-OSV vs 2Comp

2Comp-OSV vs HAR


29

IBM, Robustness to non-Normal Innovations to log(RV )

-1.76

-1.75

-1.74

-1.73

-1.72

-1.71

-1.7

-1.69

-1.68

-1.67

0 10 20 30 40 50 60

Aver

age

Pred

ictiv

e Li

kelih

ood

Forecast Horizon k

2Comp-OSV2Comp-OSV-mixture

HAR-OSV

-6

-4

-2

0

2

4

6

8

10

0 10 20 30 40 50 60

Forecast Horizon k


2Comp-OSV vs 2Comp-OSV-mixture2Comp-OSV vs HAR-OSV


30

Summary of Forecast Density Results

• Quality of density forecasts deteriorate at longer horizons

• Adjusting RV for market microstructure improves density forecasts

• EGARCH is the best model based on only daily return data

• The best joint return-RV models are significantly better than EGARCH

• Variance targeting dominated by unrestricted models

• 2 components always better than 1 component

– the advantage of the joint models improve with the forecast horizon

• 2 Component models better than HAR versions.

• Best models: 2 Component with t-density and σ2t = Et−1RVt and 2 Component-

Observable SV

• S&P 500: mixture of normals or GARCH not important

• IBM: mixture of normals improves density forecasts, GARCH not important


31

S&P 500 Model Estimates2Comp-OSV Model

rt = µ + εt, εt =p

RVtut, ut ∼ D(0, 1)

log(RVt) = ω +2

X

i=1

φisi,t + γut−1 + ηvt, vt ∼ NID(0, 1),

si,t = (1 − αi) log(RVt−1) + α1si,t−1, i = 1, 2.

2Comp Model

rt = µ + εt, εt = σtut, ut ∼ tv(0, 1), σ2t = exp

„

Et−1 log(RVt) +1

2V art−1(log(RVt))

«

Parameter ut ∼ N(0, 1) ut ∼ tν(0, 1)2Comp-OSV 2Comp

µ0.038

(0.011)-0.018(0.014)

ω-0.026(0.012)

-0.025(0.013)

φ10.476

(0.007)0.402

(0.147)

φ20.476

()0.543

(0.154)

α10.888

( 0.017)0.911

(0.045)

α20.435

(0.037)0.508

(0.105)

γ-0.129(0.010)

-0.141(0.011)

η0.531

(0.009)0.528

(0.009)

1/ν0.089

(0.016)lgl -5646.725 -5916.342


32

Summary of Full Sample Estimates

• Evidence of high and low persistent component in log(RV )

• fat-tails

• asymetric effect of returns on log(RV )

• according to lgl, observable SV is much better – but about equal in forecast quality


33

Conclusion

• Yes, high-frequency measures of volatility provide significant improvements in den-

sity forecasts

• Improvements are greater for longer forecast horizons

• Ignoring market micorstructure results in a deterioration of forecast quality

• A bivariate, 2 component model with observable SV is a good overall model.


34

Future Work

• Extension to many assets

• Modeling of realized covariance by a time varying Wishart distribution

• Importance of component dynamics in realized covariance


35

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