How Important is Modeling Long Memory in Realized Volatility? Richard T Baillie Michigan State University, USA King’s College London, UK Rimini Center for Economic Analysis, Italy Fabio Calonaci Queen Mary University of London, UK Dooyeon Cho Sungkyunkwan University, Republic of Korea Seunghwa Rho Amazon Incorporated, Berlin, Germany Presentation at University of Kent: Fifty Years of Econometrics at Keynes College 7 September, 2018 Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 1 / 57 MSU
Summary of Paper Construction of observable RV series from high frequency financial market data is standard practice in finance. RV is direct measurement of volatility. RV time series are characterized by strong persistence in their autocorrelations for a wide range of financial assets. RV series appear to have long memory components. Develops an ARFIMA − EHAR model to includes long memory, jumps and good and bad volatility. ARFIMA − EHAR is found to be superior to time varying HAR models. Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 2 / 57
Contributions of this paper Develops the ARFIMA − EHAR model which includes long memory parameterization plus jumps and good and bad volatility. ARFIMA − EHAR appears necessary to capture the strong persistence as well as jumps and good and bad volatiltiy. Contrast with a time varying parameter ( TVP ), kernel weighted regression approach to estimate EHAR models. However, the TVP model is heavily parameterized and BIC indicates preference for ARFIMA − EHAR model. Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 3 / 57
Basics of Realized Volatility RV is model free measurement of financial market volatility; Andersen and Bollerslev (1998), Andersen, Bollerslev, Diebold and Labys (2001, 2003) and Barndorff-Nielsen and Shephard (2002). Continuous time diffusion process for the log of price ( p t ) , as dp ( t ) = µ ( t ) dt + σ ( t ) dW ( t ) , t ≥ 0 , where dp ( t ) is the change in the logarithmic price, µ ( t ) denotes the drift term which has continuous and locally bounded variations, σ ( t ) is a strictly positive volatility process and W ( t ) is standard Brownian motion. Daily returns are � t � t r t = p ( t ) − p ( t − 1 ) = t − 1 µ ( s ) ds + t − 1 σ ( s ) dW ( s ) . Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 4 / 57
Basics of Realized Volatility Distribution of returns depends on both the drift and spot volatility components. � � t � � t t − 1 σ 2 ( s ) dW ( s ) r t ∼ N t − 1 µ ( s ) ds , RV at day t is RV t and is defined as the sum of high frequency, intra day squared returns. Hence m ∑ r 2 RV t = t , τ τ = 1 where r t , τ = p t , τ − p t , τ − 1 is the intra day return given the m number of intra-day log-prices of the asset { p t , τ } m τ = 1 within day t observed at m fixed time intervals of τ = 1 , . . . , m . Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 5 / 57
Basics of Realized Volatility Under suitable conditions, including the absence of serial correlation in the intra-day returns, then RV t is a consistent estimator of integrated volatility, IV t . Hence � t m r 2 t − 1 σ 2 ∑ RV t = t , τ → s ds p τ = 1 Microstructure noise suggests basic RV model should have effects of jump components. If log-price process are a Brownian Semi-Martingale with Jumps, then dp ( t ) = µ ( t ) dt + σ ( t ) dW ( t ) + κ ( t ) dq ( t ) t ≥ 0 . where the jump component is κ ( t ) dq ( t ) , with κ ( t ) as the size of the jump. Also dq ( t ) is a continuous process with dq ( t ) = 1 if there is a jump at time t and is 0 otherwise. Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 6 / 57
Discrete Time daily returns � t � t N ( t ) t − 1 σ 2 ( t + τ − 1 ) dW ( τ ) + ∑ r t = t − 1 µ ( t + τ − 1 ) d τ + κ j ( t ) , j = 1 where N ( t ) counts the number of jumps occurring with possibly time varying intensity. In the presence of jumps, RV t converges uniformly in probability to � t t − 1 σ 2 κ 2 ( s ) . ∑ RV t → s ds + p t − 1 ≤ s ≤ t Hence, RV t is a consistent estimator of IV t only in absence of jumps, while otherwise it converges to a quantity that also accounts for the jump process, ∑ t − 1 ≤ s ≤ t κ 2 ( s ) . Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 7 / 57
Basics of Realized Volatility Decompose volatility into components associated with depreciations and appreciations. Realized Semivariance of Barndorff-Nielsen and Shephard (2007); where the positive (negative) realized semivariance is � t m 1 ( ∆ p s ) 2 I { ∆ p s > 0 } RS + ∑ r 2 t − 1 d σ 2 ∑ t = t , τ I { r t , τ > 0 } → s ds + 2 p i = 1 t − 1 d < s ≤ t and � t m 1 ( ∆ p s ) 2 I { ∆ p s < 0 } RS − r 2 t − 1 d σ 2 ∑ ∑ t = t , τ I { r t , τ < 0 } → s ds + 2 p i = 1 t − 1 d < s ≤ t where I ( · ) is an indicator function. Note that RV t = RS + t + RS − t . Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 8 / 57
Basics of Realized Volatility Signed jump variation is RS + − RS − ∆ J 2 = t ( ∆ p s ) 2 I { ∆ p s > 0 } − ( ∆ p s ) 2 I { ∆ p s < 0 } . ∑ ∑ → t − 1 d < s ≤ t t − 1 d < s ≤ t Signed jump variation is decomposed as ∆ J 2 t = ∆ J 2 + + ∆ J 2 − t t Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 9 / 57
RV FX Data Spot exchange rates of the Australian dollar ( AUD ), the Canadian dollar ( CAD ), the Euro ( EUR ), the UK British pound ( GBP ), and the Japanese Yen ( JPY ) all against the numeraire US dollar ( USD ). For January 2, 2004 through December 29, 2017. trading patterns from Friday 21:00 GMT through Sunday 22:00 GMT FX rates are the midpoint of the logarithms of the bid and ask rates. This provides a sample size of T = 4 , 382 daily observations. Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 10 / 57
RV Equity Data S & P 500 index consists of five minute tick interpolated prices from January 2, 2001 through December 31, 2016. Daily RV s are computed with the two-scales estimator proposed by Zhang et al. (2005). The trading hours span from 9:30 through 16:00 with a total of 78 intraday observations and the total number of the trading days after adjustments is T = 4 , 174 observations. Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 11 / 57
Figure 1. Realized volatility for each financial series Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 12 / 57
Figure 2. Autocorrelation function for each financial series Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 13 / 57
Table 1. Estimates of Long Memory Parameter d AUD CAD EUR GBP JPY S&P500 ARFIMA ( p , d , 0 ) p 7 4 3 4 0 6 d 0.786 0.625 0.553 0.598 0.396 0.728 ln ( L ) —2919.626 14.173 84.690 —714.267 —2106.755 —6086.334 BIC 5921.213 29.028 —120.202 1485.907 4238.098 12247.694 LW ( m = � T b � ) b = 0 . 3 d 0.316 0.606 0.385 0.381 0.541 0.312 0 . 5 d 0.495 0.731 0.756 0.779 0.501 0.473 0 . 7 d 0.678 0.598 0.558 0.643 0.380 0.734 FELW ( m = � T b � ) b = 0 . 3 d 0.274 0.495 0.294 0.344 0.390 0.245 0 . 5 d 0.501 0.708 0.743 0.794 0.510 0.474 0 . 7 d 0.606 0.564 0.518 0.556 0.336 0.681 CUSUM- ∆ d 2.020 ∗∗ 1.912 ∗∗ b = 0 . 5 0.671 0.546 0.674 1.067 Key: ARFIMA ( p , d , 0 ) models are estimated for p = 0 , 1 , . . . , 10 and the model with the smallest BIC is chosen. The LW and FELW estimators are estimated with band- widths ( m ) = � T b � with b ∈ { 0 . 3 , 0 . 5 , 0 . 7 } . CUSUM- ∆ d statistic is the usual CUSUM statistic applied to the d (estimated with FELW, m = T 0 . 5 ) fractionally filtered series. ln ( L ) represents the maximized log-likelihood and BIC represents Bayesian information criterion. ∗∗ denotes significance at 5% level. Estimates of d are significant for all the cases and ∗ notation is suppressed. Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 14 / 57
HAR Model for RV Corsi (2009) suggested the Heterogeneous Autoregressive ( HAR ) model of RV . HAR is an attempt to provide purely economic explanation for long memory. Basic HAR model is an additive cascade of partial volatilities from high frequencies to low frequencies; with each additive cascade having close to an AR ( 1 ) structure. Multiple components in the volatility process is justified in terms of the differences of agents risk profiles, institutional structures, temporal horizons, etc. Extended HAR or EHAR model include jumps and good and bad volatility. Richard T Baillie (MSU) Modeling Long Memory in Realized Volatility August 31, 2018 15 / 57
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