Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Do high-frequency measures of volatility improve forecasts of return distributions?
John M. Maheu and Thomas H. McCurdy
January 31, 2012
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Contents 1
Introduction
2
Data & RV estimation Effects of MSN on RV estimation
3
Return-RV models
4
Density Forecasts
5
Results
6
Conclusion John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Introduction
Predict distribution of returns rather than point estimates Option pricing Value-at-Risk Portfolio choice without quadratic utility or normality Can realized volatility (RV) estimates improve forecasts? Comparative study Joint return-RV models Longer forecast horizon
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
Data
1 equity index (S&P500), 1 individual stock (IBM) 02/01/1996-29/08/2007, 04/01/1993-29/08/2007 5 minute prices Cont. compounded log returns Daily: rt , t = 1, ..., T Intraday: ri,t , i = 1, ..., I , t = 1, ..., T
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
RV estimation
Each day RV is estimated: P 2 Na¨ıve estimator: RVt,u = Ii=1 rt,i Et−1 (RVt ) = Vart−1 (rt ) = σt2 Adjust for Microstructure P Noise (MSN) RVt,ACq = ω0 γˆ0 + 2 qj=1 ωj γˆj P −j γˆj = Ii=1 rt,i rt,i+j Bartlett weights: ωj = 1 −
j q+1
Consider q = 1, 2, 3
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
Bai, Russel & Tiao (2004)
Exchange rate data 87-96 15-minute returns Dollar/Deutsche Mark, Dollar/Yen, Dollar/French Franc Mondays excluded Mean zero, left-skewed Adjusted for time-of-day effects
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
Sample Autocorrelation function (a) Dollar / Deutschmark 0.1 0.0 -0.1 -0.2 1
12
24
48
Autocorrelation
(b) Dollar / French Franc 0.1 0.0 -0.1 -0.2 1
12
24
48
(c) Dollar / Japanese Yen 0.1 0.0 -0.1 -0.2 1
12
John M. Maheu and Thomas H. McCurdy
24
48
DoLag high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
(a) Dollar / Deutchmark 0.3 0.2 0.1 0.0 1
12
24
48
Autocorrelation
(b) Dollar / French Franc 0.3 0.2 0.1 0.0 1
12
24
48
(c) Dollar / Japanese Yen 0.3 0.2 0.1 0.0 1
12
24
48
Lag Figure 4: Sample ACF of Squared MA(1) innovations
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
Findings Evidence of MA(1) structure Residual ACF suggests clustering They find: Higher frequency leads to worse estimates True price process pt not observed Due to Market Microstructure Noise Bid-Ask spread Rounding Noisy quotes Lagged reporting Trades on different markets Human error Information effects John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
The Problem
Suppose we observe Xt = pt + Ut , Ut ∼ (0, σu2 ), Ut iid for now) ∗ +U −U rt,i = rt,i t,i t,i−1
`
pt (&
Bias in RV h i Pnt 2 |r ∗ = E [RVt,u |r ∗ ] = i=1 E rt,i Pnt Pnt 2 = RV ∗ + 2I σ 2 ∗ 2 t,u u i=1 E (Ut,i + Ut,i−1t ) i=1 (rt,i ) + Increasing in sampling frequency RVt,u → ∞ as I → ∞
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
Sparse Sampling Andersen, Bollerslev, Diebold & Labys (1999) propose Volatility Signature Plots Efficient Sampling Tradeoff: observations vs. bias Volatility signature plots Figure 6 Representative Volatility Signature Plots Observations ”thrown away” Liquid and Illiquid Assets Average Realized Volatility
0.0008
Liquid
0.0007 0.0006 0.0005 0.0004 0.0003 0
100
200
300
400
500
k
John M. Maheu and Thomas H. McCurdy 1.3
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
Multiple Scales Zhang, Mykland & A¨ıt-Sahalia (2005) Optimal sample size Sub-grids and averaging 2 Determined by IQ and σu2 Two Scale bias correction (TSRV): Highest frequency to estimate 2σu2 (I −1 RVt,u ) Low frequency ¯I to estimate σt2 + 2¯I σu2 (RVt,sparse ) ¯ TSRVt = RVt,sparse − II RVt,u
Converges to σt2 at rate I 1/6
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
Extensions Bandi & Russel (2005) Pre-filtering Optimal sampling with dependent noise Using RVt,sparse Zhang (2006): More than two scales (MSRV) Fastest possible rate I 1/4 when all possible scales used Optimal choice of weights
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
Dependent Noise Autocorrelogram from AIG Transactions
Autocorrelogram from MMM Transactions
1
1
0.75
0.75
0.5
0.5
0.25
0.25
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
- 0.25
- 0.25
- 0.5
- 0.5
Autocorrelogram from INTC Transactions
Autocorrelogram from MSFT Transactions
1
1
0.75
0.75
0.5
0.5
0.25
0.25
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
- 0.25
- 0.25
- 0.5
- 0.5
1.0
1.0
0.5
0.5
0.0
0.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
- 0.5
- 0.5
Fig. 4. Top and middle panels: Log-return autocorrelograms from transactions for American International Group, Inc. (AIG), 3M Co. (MMM), Intel (INTC) and Microsoft (MSFT), last ten trading days in April 2004. Bottom panel: log-return autocorrelogram from the same transactions for Intel and Microsoft, superimposed M. Maheu and McCurdy Do high-frequency measures of volatility improve forecasts of retu with theJohn autocorrelogram fittedThomas from theH.basic i.i.d. plus AR(1) model for the noise.
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
Extensions II
A¨ıt-Sahalia, Mykland & Zhang (2010) Evidence for dependence in Ut in NYSE & NASDAQ stocks More likely in liquid stocks TSRV and MSRV robust Empirical findings: Largely unaffected by choice of I and ¯I MSRV very close to TSRV Both improvement over RV
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
Realized Kernels
Barndorff-Nielsen, Hansen, Lunde & Shepherd (2006) Kernel estimation Define: Realized autocorrelation: P t γh (Xδ ) = nj=1 (Xjδ − X(j−1)δ )(X(j−h)δ − X(j−h−1)δ ) γ˜h (Xδ ) = γh (Xδ ) + γ−h (Xδ ) P ˜ (Xδ ) = γ0 (Xδ ) + H k h−1 γ˜ (Xδ ) Realized Kernel: K h=1 H
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Effects of MSN on RV estimation
Results ˜ (Xδ ) consistent estimator of K H, I → ∞
R1 0
σ(t − 1 + τ )2 dτ for
Mixed normal, complicated variance Rate of convergence depends on H(I ) I 1/6 if H = cI 2/3 I 1/4 if H = cI 1/2 (if k 0 (0) = k 0 (1) = 0) c may be chosen to minimize the variance
Flat top kernels robust to dependent, endogenous noise or endogenous timing Estimation more accurate than RVt,sparse
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
6.377 6.362
82.091 Introduction Data & RV estimation 81.024 Return-RV models Density Forecasts
0.114 0.010
Effects of MSN on RV estimation
66.594 65.235
adjustment, and RVACq , q = 1, 2, 3, are constructed Results as in Eq. (2.2). Conclusion
ation , we their
ights ch, a
(2.2) j , q +1
et al. ype. , are d the plays mates ance
Fig. 1. ACF of 5-minute return data.
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
ARTICLE
Effects of MSN on RV estimation
J.M. Maheu, T.H. McCurdy / J
Table 1 Summary statistics: daily returns and realized volatility. Mean SPY rt RVu RVAC 1 RVAC 2 RVAC 3 IBM rt RVu RVAC 1 RVAC 2 RVAC 3
Variance
S
0.018 1.210 1.079 1.013 0.978
0.967 2.640 2.373 2.115 2.054
0 6 7 7 8
0.037 2.825 2.623 2.558 2.531
2.602 9.161 9.433 9.875 10.095
0 5 6 6 6
rt are daily returns, RVu are constructed from raw 5-minute returns with no adju John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Benchmark model
EGARCH: rt 2 log (σt )
= µ + εt , εt = σt ut ut ∼ NID(0, 1) 2 = ω + βlog (σt−1 ) + γut−1 + α|ut−1 |
Allows for leverage effect Estimated from daily returns
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
RV-based models
Joint modeling of returns and RV allows multi horizon forecasts Component-RV (2Comp) Heterogeneous Autoregressive-RV (HAR) Extensions: t-distributed noise in rt Mixed normal noise in log (RVt ) GARCH dynamics in log (RVt ) Observed stochastic volatility (OSV)
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
2Comp-RV
2 component model with different decay rates: = µ + εt , εt = σt ut ut ∼ NID(0, 1) 2 X log (RVt ) = ω + φi si,t + γut−1 + ηvt , vt ∼ NID(0, 1) rt
i=1
si,t
= (1 − αi )log (RVt−1 ) + αi si,t−1 , 0 < αi < 1, i = 1, 2
σt2 = Et−1 (RVt ) = exp (Et−1 log (RVt ) + .5Vart−1 (log (RVt )))
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
2Comp-RV Due to Maheu & McCurdy (2007) Estimated by max likelihood RVt estimated from intra day data No infinite exponential smoothing No mean reversion in volatility Degenerate in asymptotic limits Weighting parameters φi ∈ (0, 1) and mean ω
Variance targeting sets ω = mean(log (RV ))(1 − φ1 − φ2 ) No leverage term
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
HAR-RV
Define RVt−h,h = rt
1 h
Ph−1 j=0
log (RVt−h+j ).
= µ + εt , εt = σt ut 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) σt2
= Et−1 (RVt ) = exp (Et−1 log (RVt ) + .5Vart−1 (log (RVt )))
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
HAR-RV
Due to Corsi (2009), Andersen et al. (2007) Captures long memory dependence No leverage term
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Extensions
Robustness is tested w.r.t. certain alternatives: t-distributed noise (ut ∼ tυ (0, 1)) Mixed normality: η = 1 vt ∼ (0, σv2,1 ) w/ prob π; vt ∼ (0, σv2,2 ) w/ prob 1 − π η follows GARCH(1,1) 2 ηt2 = κ0 + κ1 [log (RVt−1 ) − Et−2 log (RVt−1 )]2 + κ2 ηt−1 Set σt2 = RVt (OSV model)
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
How to evaluate density forecasts? Predictive Likelihood Evaluate density forecast at realized return Usually 1-step ahead Φt is the information set at time t, θ a set of parameters Consider out of sample observations t = τ + kmax , ..., T − k. fM,k (x|Φt , θ) denotes k-step ahead forecasted likelihood of model M For S&P, T = 2936, τ = 1200 kmax = 60
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Average predictive likelihood: DM,k
1 = T − τ − kmax + 1
T −k X
log (fM,k (rt+k |Φt , θ))
t=τ +kmax −k
Compare models A and B with test stat due to Diebold & Mariano (1995) and Amisano & Giacomini (2007): √ (DA,k − DB,k ) T − τ − kmax + 1 k tA,B = σ ˆAB,k Asymptotically N(0, 1).
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Computation
If k > 1, fM,k (rt+k |Φt , θ) is unknown. Can be estimated by monte carlo (MC), i = 1, .., N = 10000 Distribution know conditional on variance Rao-Blackwellization Reduces variance of MC estimates
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Computation Consider the EGARCH: Z 2 2 2 fM,k (rt+k |Φt , θ) = f (rt+k |µ, σt+k )p(σt+k |Φt )dσt+k ≈
N 1 X 2(i) f (rt+k |µ, σt+k ), N
2(i)
2 σt+k ∼ p(σt+k |Φt )
i=1
Other models require simulation of RV N times 2(i)
2(i)
σt+k = Et+k−1 RVt+k
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
General Results
Unrestricted models > variance targeting When σt2 = Et−1 RVt , t-innovations > N 2 components > 1 Intraday info, timing and non-normality are important Focus on best models
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
ARTICLE IN PRESS
S&P 6
J.M. Maheu, T.H. McCurdy / Journal of Econometrics (
)
–
Fig. 4. IBM, robustne Fig. 2.H.S&P 500, joint models EGARCH. measures of volatility improve John M. Maheu and Thomas McCurdy Doversus high-frequency forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
IBM
Fig. 2. S&P 500, joint models versus EGARCH.
Fig. 4. IBM, robustnes
functional forms (either 2out we forecast. The three best bivaria 2Comp and HAR. For the S for IBM forecasts the 2Co The additional informatio volatility (OSV) assumptio to in-sample fit as shown b density forecasts for long 500. The OSV assumption the IBM case, as shown b Fig. 3 for ‘2Comp-OSV vs 2 Fig. 4 evaluates the robu for IBM to a generalizatio log(RV). In particular, as Eq. (3.10) to allow eithe parameterization of the co neither of these generaliz sample density forecasts that a mixture-of-Norma log(RV) improves densit distributed alternative for Table 2 provides full-s bestimprove bivariate specificatio Fig.H. 3. McCurdy IBM, joint modelsDo versus EGARCH. John M. Maheu and Thomas high-frequency measures of volatility forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Results
Forecast performance deteriorates as k increases 2COMP, 2COMP-OSV and HAR perform best Intraday data improves forecasts significantly Advantage of flexible models increase in k
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Robustness ARTICLE IN PRESS
.M. Maheu, T.H. McCurdy / Journal of Econometrics (
s EGARCH.
)
–
John M. Maheu and Thomas H. McCurdy Do high-frequency measures of volatility improve forecasts of retu Fig. 4. IBM, robustness to non-normal innovations to log (RV).
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
S&P Estimation Results
ARTICLE IN PRES J.M. Maheu, T.H. McCurdy / Journal of Econometrics (
Table 2 S&P 500 model estimates. 2Comp-OSV model
p
rt = µ + ✏t , ✏t = RVt ut , ut ⇠ D(0, 1) P2 log(RVt ) = ! + i=1 i si,t + ut 1 + ⌘vt , vt ⇠ NID(0, 1), si,t = (1 ↵i ) log(RVt 1 ) + ↵i si,t 1 , i = 1, 2. 2Comp model
rt = µ + ✏t , ✏t = t ut , ut ⇠ tv (0, 1) 1 2 t = exp Et 1 log(RVt ) + 2 Vart 1 (log(RVt ))
P2
log(RVt ) = ! + i=1 si,t = (1 ↵i ) log(RVt Parameter
µ ! 1 2
↵1 ↵2 ⌘ 1/⌫ lgl
i si,t
1
+ ut 1 + ⌘vt , vt ⇠ NID(0, 1), ) + ↵i si,t 1 , i = 1, 2.
2Comp-OSV: ut ⇠ N (0, 1) 0.038 (0.011) 0.026 (0.012) 0.476 (0.007) 0.476 0.888 (0.017) 0.435 (0.037) 0.129 (0.010) 0.531 (0.009)
5646.725
2Comp: ut ⇠ t⌫ (0, 1) 0.018 (0.014) 0.025 (0.013) 0.402 (0.147) 0.543 (0.154) 0.911 (0.045) 0.508 (0.105) 0.141 (0.011) 0.528 (0.009) 0.089 (0.016) 5916.342
)
–
Using the predict intraday data is imp daily data as in our a flexible function fo important for the d marginally improves 500 but is essentially of returns with Nor volatility directed b function of log(RV) p of out-of-sample hor
Acknowledgements
We thank the edi Zhongfang He, Lars S conference on Realiz ment Conference, th and Risk Manageme the Federal Reserve B Guangyu Fu and Xia tance. We are also gr
John M. Maheu and Thomas H. McCurdy Do high-frequency measures of volatility References improve forecasts of retu The main features of our results are as follows. Bivariate
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Robustness/S&P Estimation
Mixed-Normal log (RV ) dynamics: improvement GARCH dynamics (not shown): no improvement In sample: OSV preferred Out of sample: no difference Parameter estimates very similar
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Conclusion
Joint models of return and RV Comparison by predictive likelihood Intraday info can be exploited RV improves forecasts
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
Introduction Data & RV estimation Return-RV models Density Forecasts Results Conclusion
Critique
Why 5 minutes? Could give more info on MSN; different solutions Jumps Many models, few reported Leverage not included in return-RV models More than 2 components? How is HAR specification chosen? Discussion of leverage not adjusted...
John M. Maheu and Thomas H. McCurdy
Do high-frequency measures of volatility improve forecasts of retu
