Do high-frequency measures of volatility improveforecasts of return distributions?
John M. Maheu and Thomas H. McCurdyUniversity of TorontoRCEA and CIRANO
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Important Problems in Finance
• Derivative pricing
• Risk measurement and management
• Portfolio choice
• Each requires a full characterization of the multiperiod forecast density of returns
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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
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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)
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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
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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
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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.
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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.
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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)
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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
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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
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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
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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
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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)
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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
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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)
).
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Joint Models of RV and Returns
Component Specifications
rt = µ + εt, εt = σtut, ut ∼ NID(0, 1)
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
σ2t = Et−1RVt = exp
(Et−1 log(RVt) +
1
2Vart−1(log(RVt)
).
- fat tails ut ∼ tν(0, 1)
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Joint Models of RV and Returns
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),
σ2t = Et−1RVt = exp
(Et−1 log(RVt) +
1
2Vart−1(log(RVt)
).
- fat tails ut ∼ tν(0, 1)
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Joint Models of RV and Returns
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
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
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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
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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
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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)
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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)
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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.
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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)
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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
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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).
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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
Diebold-Mariano Test Statistics
2Comp-OSV vs 2Comp-OSV-mixture2Comp-OSV vs 2Comp-OSV-GARCH
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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
Diebold-Mariano Test Statistics
2Comp-OSV vs EGARCH2Comp-OSV vs 2Comp
2Comp-OSV vs HAR
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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
Diebold-Mariano Test Statistics
2Comp-OSV vs 2Comp-OSV-mixture2Comp-OSV vs HAR-OSV
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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
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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
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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
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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.
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Future Work
• Extension to many assets
• Modeling of realized covariance by a time varying Wishart distribution
• Importance of component dynamics in realized covariance
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