Non-parametric estimation of intraday spot volatility: Disentangling Instantaneous Trend and Seasonality

Vatter, Thibault; Wu, Hau-Tieng; Chavez-Demoulin, Valérie; Yu, Bin Article Non-parametric estimation of intraday spot volatility: Disentangling Instantaneous Trend and Seasonality Econometrics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Vatter, Thibault; Wu, Hau-Tieng; Chavez-Demoulin, Valérie; Yu, Bin (2015) : Non-parametric estimation of intraday spot volatility: Disentangling Instantaneous Trend and Seasonality, Econometrics, ISSN 2225-1146, MDPI, Basel, Vol. 3, Iss. 4, pp. 864-887, https://doi.org/10.3390/econometrics3040864 This Version is available at: https://hdl.handle.net/10419/171849 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Econometrics 2015,3, 864-887; doi:10.3390/econometrics3040864 OPEN ACCESS econometrics ISSN 2225-1146 www.mdpi.com/journal/econometrics Article Non-Parametric Estimation of Intraday Spot Volatility: Disentangling Instantaneous Trend and Seasonality Thibault Vatter 1,*, Hau-Tieng Wu 2, Valérie Chavez-Demoulin 1and Bin Yu 3 1Faculty of Business and Economics (HEC), University of Lausanne, 1015 Lausanne, Switzerland; E-Mail: valerie.chav[email protected] 2Department of Mathematics, University of Toronto, Toronto M5S2E4, ON, Canada; E-Mail: [email protected] 3Department of Statistics, University of California, Berkeley 94720, CA, USA; E-Mail: bin[email protected]y.edu *Author to whom correspondence should be addressed; E-Mail: thibault.v[email protected]; Tel.: +41-21-693-61-04. Academic Editor: Nikolaus Hautsch Received: 19 August 2015 / Accepted: 26 November 2015 / Published: 18 December 2015 Abstract: We provide a new framework for modeling trends and periodic patterns in high-frequency financial data. Seeking adaptivity to ever-changing market conditions, we enlarge the Fourier flexible form into a richer functional class: both our smooth trend and the seasonality are non-parametrically time-varying and evolve in real time. We provide the associated estimators and use simulations to show that they behave adequately in the presence of jumps and heteroskedastic and heavy-tailed noise. A study of exchange rate returns sampled from 2010 to 2013 suggests that failing to factor in the seasonality’s dynamic properties may lead to misestimation of the intraday spot volatility. Keywords: intraday spot volatility; seasonality; foreign exchange returns; time-frequency analysis; synchrosqueezing JEL classifications: C14; C22; C51; C52; C58; G17 Econometrics 2015,3865 1. Introduction Over the last two decades, improved access to high-frequency data has offered a magnifying glass to study financial markets. Analysis of these data poses unprecedented challenges to econometric modeling and statistical analysis. As for traditional financial time series (e.g., daily closing prices), the most critical variable at higher frequencies is arguably the asset return: from the pricing of derivatives to portfolio allocation and risk management, it is a cornerstone for academic research and practical applications. Because its properties are of such importance, the literature has sought models consistent with the new observed features of the data. Among all characteristics of the asset return, empirical studies have found that its second moment structure is preponderant, partly because of its influence on assessments of market risk. Therefore, its time-varying nature has received a lot of attention in the literature (see, e.g., [1]). However, common heteroskedastic models have not addressed an empirical regularity of intraday data: the seasonality. It is now well-documented that patterns due to the cyclical nature of market activity are one of the main sources of misspecification for typical volatility models (see, e.g., [2–4]). Some periodicity in the volatility is inevitable due to market openings and closings around the world, but it is unaccounted for in the vast majority of econometrics models [5,6]. To avoid misspecification bias, one possibility is to explicitly incorporate the seasonality in traditional models, for instance with the periodic-GARCH of [7]. Alternatively, a pre-filtering step combined with a non-periodic model can be used as in [3,4,8–10]. Even though an explicit inclusion of seasonality is advocated as more efficient (see [7]), the resulting models are less flexible and computationally more expensive. Although they may potentially propagate errors, two-step procedures are more convenient and often consistent whenever each step is consistent (see [11]). As a result, seasonality pre-filtering has received a considerable amount of attention in the literature dedicated to high-frequency financial time series. In this context, an attractive approach is to generalize the removal of weekends and holidays from daily data and use the “business clock” (see [5]), a new time scale where time passes more quickly when the market is inactive and conversely during “power hours”, at the cost of synchronicity between assets. The other approach, which allows researchers to work in physical time (removing only closed market periods), is to model the periodic patterns directly, either non-parametrically with estimators of scale (e.g., in [8,10,12]) or using smoothing methods (e.g., splines in [13]) or parametrically with the Fourier flexible form (see [14]), introduced by [3,4] in the context of intraday volatility in financial markets. Until now, the standard assumption in the literature have been to consider a constant seasonality over the sample period. However, this assumption has seldom been verified empirically, and few studies have acknowledged this issue. In [15], the authors use frequency leakage as evidence in favor of a slowly varying seasonality. In [16], it is observed that smaller sample periods yield improved seasonality estimators. In this paper, we argue that the constant assumption can only hold for arbitrary small sample periods, because the entire shape of the market (and its periodic patterns) evolves over time. In contrast to seasonal adjustments considered in the literature, we relax the assumption of constant seasonality. In [17], the authors used a related approach to model the amplitude of the fundamental daily component as stochastic and the remainder cyclical components as deterministic. Econometrics 2015,3866 In this paper, all the cyclical components are assumed to vary smoothly with time. From a time-frequency decomposition of the data, we suggest a non-parametric framework to obtain instantaneous estimates of trend and seasonality, respectively from the lowest and highest frequencies. Our non-parametric framework is comparable to a dynamic combination of realized volatility or bipower variation and Fourier flexible form. The instantaneous trend and seasonality can be estimated using rolling moving averages (for the realized-volatility or bipower variation part) or rolling regressions (for the Fourier flexible form part). Our model differs in several aspects. First, in a single step it disentangles the trend from the seasonality. Second, it yields naturally smooth pointwise estimates. Third, it is data-adaptive in the sense that there are fewer parameters to fine tune. Fourth, the trend estimate is more robust to jumps in the log-price than traditional realized measures such as the realized volatility or bipower variation. There is a major reason why dynamic seasonality models are important when modeling intraday returns, which should be of interest to practitioners and academics alike: non-dynamic models may lead to severe underestimation or overestimation of intraday spot volatility. Hence, there are possible implications whenever seasonality pre-filtering is used as an intermediate step. From a risk management viewpoint, intraday measures such as Value-at-Risk and Expected-Shortfall under the assumption of constant seasonality may suffer from inappropriate high-quantile estimation. In other words, the assumption may lead to alternate periods of underestimated (respectively overestimated) risk when the seasonality is higher (respectively lower) than suggested by a constant model. In the context of jump detection, the assumption may induce an underestimation of the number and size of jumps when the seasonality is higher, and overestimation when the seasonality is lower. The rest of the paper is organized as follows. In Section 2, we describe our model for the intraday return, from a continuous-time perspective to the discretized process. To relax the constant seasonality assumption, we define a class of seasonality models which includes the Fourier flexible form as a special case. In Section 2.4, we provide a detailed exposition of a method to study the class of models defined in Section 2. We start by recalling the link between the Fourier flexible form and the Fourier transform. We then sketch the theoretical basics of time-frequency analysis, which was introduced to overcome this limitation (see [18,19]). Finally, we introduce the synchrosqueezing transform to study the class of models defined in Section 2, and we provide associated estimators of instantaneous trend and seasonality. In Section 3, we conduct simulations to study the properties of the estimators from Section 2.4. We show that they are robust to various heteroskedasticity and jumps specifications. In Section 4, we estimate the model using four years of high-frequency data on the CHF/USD, EUR/USD, GBP/USD, and JPY/USD exchange rates. To obtain confidence intervals for the estimated trend and seasonality, we develop tailor-made resampling procedures. We conclude in Section 5. 2. Intraday Seasonality Dynamics On a generic filtered probability space, suppose that the log-price of the asset, denoted as p(t), is determined by the continuous-time jump diffusion process dp(t) = µ(t)dt+σ(t)dW(t) + q(t)dI(t),(1) Econometrics 2015,3867 where µ(t)is a continuous and locally bounded variation process, σ(t)>0is the stochastic volatility with càdlàg sample paths, W(t)is a standard Brownian motion, and I(t)is a finite activity-counting process with jump size q(t) = p(t)−p(t−)independent from W(t). Although typical continuous-time models may display realistic features of asset price behavior at a daily frequency, seasonality is invariably missing. In contrast, the econometric literature on high-frequency data often decomposes the spot volatility into three separate components: the first being slowly time varying, the second periodic, and the third purely stochastic (see, e.g., [3,8,10]). As such, we may consider a continuous-time model where the volatility satisfies σ(t) = eT(t)+s(t)h(t),(2) with T(t)slowly varying, s(t)quickly time-varying and periodic and h(t)the intraday stochastic volatility (e.g., a square-root process [20,21]). In other words, we decompose the volatility into three components: T(t)is the trend, s(t)the seasonality, and h(t)is the intraday stochastic component. Although trading occurs in continuous time, the price process is only collected at discrete points in time. Ignoring the drift term, Equations (1) and (2) suggest a natural discrete-time model for the return process as rn=eTn+snhnwn+qnIn,(3) where n=t/τ with τas the sampling interval, and •Tnand snrepresenting the trend and seasonality in the volatility, •hnthe intraday volatility component, •wnthe white noise, •and qnInthe discretized finite activity counting process. Note that this discretized multiplicative construction for the volatility was introduced by [3] and frequently used afterward (see, e.g., [10,15,17]). The jumps part was later added by [8]. In Sections 2.1 and 2.2, we describe flexible continuous-time versions of the seasonality s(t)and trend T(t). Our aim is to make the model as flexible and adaptive as possible while keeping the mathematical tractability required to analyze and estimate it. For the seasonality, we use a class of functions that generalizes the “standard" model in the literature, namely the Fourier flexible form (see [3,14]). In this spirit, a sum of periodic components displaying amplitude and frequency modulations is arguably the most intuitive idea. For the trend, we suggest a class of functions that is slowly time-varying in order to describe the behavior of the volatility at larger timescales. Since our specification includes polynomials and harmonic functions of very low frequency, it is appropriate to describe long-term trends and cycles. In Section 2.3, we put all the pieces back together in an adaptive volatility model, in which seasonality and trend are discretely sampled from the continuous-time versions. In Section 2.4, we describe the tools aimed at studying and estimating this new class of models. Econometrics 2015,3868 2.1. The Adaptive Seasonality Model The seasonal behavior inside a time series can be described by repeating oscillations as time passes. Due to complicated underlying dynamics, this oscillatory behavior might change from time to time. Intuitively, to capture this effect, we consider the adaptive seasonality model s(t) = K X k=1 ak(t) cos {2πkφ(t) + ξk},(4) where K∈N,ak(t)>0,φ0(t)>0,φ(0) = 0 and ξk∈R. The function ak(t)is called the amplitude modulation, φ(t)the phase function, ξkthe phase shift, and φ0(t)the instantaneous frequency. When the phase function is linear (ωt with ω > 0) and amplitude modulations are constant (ak>0∀k), then the model reduces to the Fourier flexible form (see [14]). When the phase function is nonlinear, the instantaneous frequency generalizes the concept of frequency, capturing the number of oscillations that one observes during an infinitesimal time period. The amplitude modulation represents the instantaneous magnitude of the oscillation. Although those time-varying quantities allow us to capture momentary behavior, there is in general no unique representation for an arbitrary ssatisfying (4), even if K= 1. Indeed, there are an infinity of smooth pairs of functions α(t)and β(t)so that cos(t) = {1 + α(t)}cos {t+β(t)},1 + α(t)>0and 1 + β0(t)>0. This is known as the identifiability problem studied in [22]. To resolve this issue, it is necessary to restrict the functional class. Accordingly, we borrow the following definition from [22]: Definition 1 (Intrinsic mode function class Ac1,c2 ).For fixed choices of 0<  1and c1< c2< ∞, the space Ac1,c2 of intrinsic mode functions consists of functions f:R→R,f∈C1(R)∩L∞(R) having the form f(t) = a(t) cos {2πφ(t)},(5) where a:R→Rand φ:R→Rsatisfy the following conditions for all t∈R: a∈C1(R)∩L∞(R),inf t∈Ra(t)> c1,sup t∈R a(t)< c2,(6) φ∈C2(R),inf t∈Rφ0(t)> c1,sup t∈R φ0(t)< c2,(7) |a0(t)| ≤ |φ0(t)|,|φ00(t)| ≤ |φ0(t)|.(8) An intrinsic mode function mainly satisfying two requirements. First, the amplitude modulation and instantaneous frequency are continuously differentiable and bounded from above and below. Second, the rate of variation of both the amplitude modulation, and instantaneous frequency are small compared to the instantaneous frequency itself. With the extra conditions on the intrinsic mode function, the identifiability issue can be resolved in the following way ([22], Theorem 2.1). Suppose f(t) = a1(t) cos {2πφ1(t)}∈Ac1,c2 can be represented in a different form, for instance f(t) = a2(t) cos {2πφ2(t)}, which also satisfies the conditions of Ac1,c2 . Define α(t) = φ1(t)−φ2(t) and β(t) = a1(t)−a2(t), then α∈C2(R),β∈C1(R)and |α0(t)| ≤ C,|α(t)| ≤ C and |β(t)|< C for all t∈Rfor a constant Cdepending only on c1. If a member of Ac1,c2 can be represented in Econometrics 2015,3869 two different forms, then the differences in phase function, amplitude modulation and instantaneous frequency between the two forms are controllable by the small model constant . In this sense, we are able to define the amplitude modulation and instantaneous frequency rigorously. Because these conditions exclude jumps in amplitude modulation and instantaneous frequency, we make the working assumption that the seasonality component of the market evolves slowly over time. As such, we do not try to model abrupt changes. The usual tests for structural breaks are aimed at detecting if and where a break occurs, and they are not the modeling target of the class Ac1,c2 . To assess the validity of this class when modeling financial data, we assume that the seasonality belongs to the Ac1,c2 class and resort to confidence interval estimation. The model from Equation (4) comprises more than one oscillatory components so further conditions are necessary to resolve the identifiability problem. Definition 2 (Adaptive seasonality model Cc1,c2 ).The space Cc1,c2 of superpositions of intrinsic mode functions consists of functions fhaving the form f(t) = K X k=1 fk(t)and fk(t) = ak(t) cos {2πkφ(t) + ξk}(9) for some finite K > 0such that for each k= 1, . . . , K, ξk∈R, ak(t) cos {2πkφ(t) + ξk}∈Ac1,c2 and φ(0) = 0.(10) Functions in the class Cc1,c2 are composed of more than one intrinsic mode function satisfying the condition (10). An identifiability theorem similar to the one for Ac1,c2 can be proved similarly as that in ([22], Theorem 2.2): if a member of Cc1,c2 can be represented in two different forms, then the two forms have the same number of intrinsic mode functions, and the differences in their phase function, amplitude modulation, and frequency modulations for each intrinsic mode function are small. To summarize, with the Cc1,c2 model and its identifiability theorem, the amplitude modulation and instantaneous frequency are well-defined up to an uncertainty of order . A popular member of Cc1,c2 is the Fourier flexible form (see [14]), introduced by [3,4] in the context of intraday seasonality. Using a Fourier series to decompose any periodic function into simple oscillating building blocks, a simplified 1version of the Fourier flexible form reads s(t) = K X k=1 [akcos (2πkt) + bksin (2πkt)] ,(11) where the sum of sines and cosines with integer frequencies captures the intraday patterns in the volatility. Thus, we can view s(t)in (11) as a member of Cc1,c2 with φ(t) = t,ξk= arctan(ak/bk) and ak(t) = pa2 k+b2 kfor k= 1, . . . , K, that is with a fixed daily oscillation. 1[3,4] consider the addition of dummy variables to capture weekday effects or particular events such as holidays in particular markets, in addition to unemployment reports, retail sales figures, etc. While the periodic model captures most of the seasonal patterns, their dummy variables allow the quantification of the relative importance of calendar effects and announcement events. Econometrics 2015,3870 2.2. The Adaptive Trend Model For the remainder of this paper, we use Fy(ω) = R∞ −∞ y(t)e−i2πωtdtand Py(ω) = |Fy(ω)|2for the Fourier transform and power spectrum of a weakly stationary process y. We fix a Schwartz function ψso that suppFψ⊂[1 −∆,1 + ∆], where 0<∆<1and Fψdenotes its Fourier transform. Definition 3 (Adaptive trend model Tc1 ).For fixed choices of 0<  1and c1<∞, the space Tc1 consists of functions T:R→R,T∈C1(R)so that FTexists in the distribution sense, and ZT(t)1 √aψt−b adt≤CTand ZT0(t)1 √aψt−b adt≤CT(12) for all b∈Rand a∈0,1+∆ c1i, for some CT≥0. Because we ideally want a trend to be slowly time-varying, the intuition behind (12) act as a bound on how fast a function oscillates locally. A special case satisfying (12) is a continuous function Tfor which its Fourier transform FTexists and is compactly supported in −1−∆ 1+∆c1,1−∆ 1+∆c1. There are two well-known examples of such trend functions: the first is the polynomial function T(t) = PL l=1 αltl, where αl∈R, which is commonly applied to model trends. By a direct calculation, its Fourier transform is supported at zero. The second is a harmonic function T(t) = cos(2πωt)with very low-frequency (i.e., with |ω|<1−∆ 1+∆c1), as its Fourier transform is supported at ±ω. More generally, using the Plancheral theorem, one can verify that suppFT⊂−1−∆ 1+∆c1,1−∆ 1+∆c1 implies RT(t)1 √aψ(t−b a)(t)dt= 0 for all a∈(0,1+∆ c1]and all b∈R. In our case, however, the trend is non-parametric by assumption, and its Fourier transform might in all generality be supported everywhere in the Fourier domain. If this is so, then (12) describes a trend that essentially captures the slowly varying features of those models (i.e., its local behavior is similar to that of a polynomial or a very low frequency periodic function) but is more general. 2.3. The Adaptive Volatility Model Let the return process be as in (3). We define the log-volatility process as yn= log |rn|. We assume that the log-volatility process follows an adaptive volatility model, that is yn=Tn+sn+zn,(13) where •sn, discretely sampled from s(t)∈ Cc1,c2 , is the volatility seasonality, •Tn, discretely sampled from T(t)∈ Tc1 , is the volatility trend, •and zn= log hnwn+qnIne−(Tn+sn)is an additive noise process that satisfies E(zn)<∞ and var(zn)<∞. Econometrics 2015,3871 It should be noted that the boundedness of the mean and variance for the noise is a mild condition for reasonable econometric models (see the two remarks below). Remark 1. First assume that there are neither jumps (In= 0) nor intraday heteroskedasticity (hn= 1), and that wnfollows a standardized generalized error distribution, wn∼GED(ν), with density given by fν(w) = ν 21+1/νΓ(1/ν)λν e−|w/λν|ν 2, where λν=2−2/νΓ(1/ν)/Γ(3/ν)1/2and Γis the gamma function. As noted in [23], the generalized error distribution nests the normal distribution when ν= 2, has fatter (thinner) tails when ν < 2(ν > 2) and is log-concave for ν > 1. Because E(log w2 n) = {2ψ(1/ν)/ν + log Γ(1/ν)−log Γ(3ν)}and var (log w2 n) = (2/ν)2Ψ(1/ν)(see [23]) with ψand Ψas the digamma and trigamma functions (i.e., the first and second derivatives of log Γ), then E(zn)<E(log w2 n)<∞and var(zn)<var (log w2 n)<∞ when ν > 0. Remark 2. To introduce intraday heteroskedasticity (still without jumps), it is convenient to use the EGARCH(1,1) of [24]. Defining vn= log |hn|, the model is obtained by writing vn=γ+βvn−1+θwn−1+α(|wn−1|−E|wn−1|), where γ, β, θ, α ∈Rand |β|<1. For this model, it is straightforward that var log h2 nw2 n=θ2+α2var (|wn|) 1−β2+var log w2 n. Because var |wn|<∞for the generalized error distribution (see [23]), var(zn)<∞follows. For other heteroskedasticity models, note that E(log |hnwn|)2≤qE(log |wn|)2+qE(log |hn|)22 by Cauchy-Schwartz inequality. Hence, if both E(log |wn|)2and E(log |hn|)2are bounded, then var(zn)<∞follows. When jumps are added to the model, it is in general not possible to obtain a closed formula for E(zn)and var(zn). However, the boundedness of the mean and variance is strongly supported by our simulations for cases of practical interest. Within this framework, every specific choice of trend T, seasonality s, and noise zyields a different model. Due to their non-parametric nature, estimating T,s, or the amplitude modulation and instantaneous frequency inside sis non-trivial. In practice, the problem would be much easier in the Fourier flexible form case, that is, in the case of a constant seasonal pattern. However, this assumption is challenged by empirical evidence. Laakkonnen [16] observes that a Fourier flexible form that is estimated yearly, then quarterly, and finally weekly performs better and better at capturing periodicities in the data. Although this sub-sampling is interpretable as a varying seasonality, it is still piecewise constant. As a piecewise constant function implies that the system under study undergoes structural changes (i.e., shocks) at every break point, it is an unlikely candidate for the market’s seasonality. We take the perspective of [15], who use frequency leakage in the power spectrum of the absolute return to Econometrics 2015,3878 approximately normal. However, a bias correction proves to be necessary, which is achieved for the trend by substracting the bootstrapped bias PB b=1 b Tb(nτ)/B −b T(nτ)from the estimate b T(nτ). Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08 −4 −2 0 2 4 x 10−3 CHFUSD Return Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08 −4 −2 0 2 4 x 10−3 EURUSD Return Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08 −4 −2 0 2 4 x 10−3 GBPUSD Return Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08 −4 −2 0 2 4 x 10−3 JPYUSD Return (a) Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08 −30 −25 −20 −15 CHFUSD Log−volatility Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08 −30 −25 −20 −15 EURUSD Log−volatility Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08 −30 −25 −20 −15 GBPUSD Log−volatility Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08 −30 −25 −20 −15 JPYUSD Log−volatility (b) Figure 2. First week of returns and log-volatilities in January 2010. Top: CHF/USD (left) and EUR/USD (right); Bottom: GBP/USD (left) and JPY/USD (right). (a) Return rn; (b) Log-volatility yn= 2 log |rn−bµ|. Econometrics 2015,3879 12345 −0.05 0 0.05 0.1 0.15 CHFUSD Lag (day) Autocorrelation 12345 −0.05 0 0.05 0.1 0.15 EURUSD Lag (day) Autocorrelation 12345 −0.05 0 0.05 0.1 0.15 GBPUSD Lag (day) Autocorrelation 12345 −0.05 0 0.05 0.1 0.15 JPYUSD Lag (day) Autocorrelation (a) 1 2 3 4 5 10−4 10−2 100 102 CHFUSD Frequency (1/day) Power spectrum 1 2 3 4 5 10−4 10−2 100 102 EURUSD Frequency (1/day) Power spectrum 1 2 3 4 5 10−4 10−2 100 102 GBPUSD Frequency (1/day) Power spectrum 1 2 3 4 5 10−4 10−2 100 102 JPYUSD Frequency (1/day) Power spectrum (b) Figure 3. Autocorrelation γy(l)and power spectrum Py(ω)of the log-volatility. γy(l) = b Ehnyn−b E(yn)onyn−l−b E(yn)oiand Py(ω) = b Fy(ω) 2 . Top: CHF/USD (left) and EUR/USD (right); Bottom: GBP/USD (left) and JPY/USD (right). (a) Autocorrelation γy(l). The lag lis from 1to 1440 and the x axis ticks are divided by 288; (b) Power spectrum Py(ω). The frequency ωis from 0to 5. Econometrics 2015,3880 2010 2011 2012 2013 0.2 0.4 0.6 0.8 x 10−3 Trend CHFUSD exp(T) RV BV 2010 2011 2012 2013 0.2 0.4 0.6 0.8 x 10−3 Trend EURUSD exp(T) RV BV 2010 2011 2012 2013 0.2 0.4 0.6 0.8 x 10−3 Trend GBPUSD exp(T) RV BV 2010 2011 2012 2013 0.2 0.4 0.6 0.8 x 10−3 Trend JPYUSD exp(T) RV BV (a) Jul 2011 Aug 2011 Sep 2011 0.2 0.4 0.6 0.8 x 10−3 Trend CHFUSD exp(T) RV BV Jul 2011 Aug 2011 Sep 2011 0.2 0.4 0.6 0.8 x 10−3 Trend EURUSD exp(T) RV BV Jul 2011 Aug 2011 Sep 2011 0.2 0.4 0.6 0.8 x 10−3 Trend GBPUSD exp(T) RV BV Jul 2011 Aug 2011 Sep 2011 0.2 0.4 0.6 0.8 x 10−3 Trend JPYUSD exp(T) RV BV (b) Figure 4. Trend reconstruction results. In each panel: reconstructed trend (black line), realized volatility (red line) and bipower variation (blue line). In Panel 4b: 95% confidence intervals for the reconstructed trend (shaded area). Top: CHF/USD (left) and EUR/USD (right); Bottom: GBP/USD (left) and JPY/USD (right). (a) Trend reconstruction for 2010–2013; (b) Zoom during the summer of 2011. Econometrics 2015,3881 The results are separately presented for the trend in Figure 4and for the seasonality in Figures 5 and 6. In Panel 4a, our reconstructed trend is compared to the realized volatility and bipower variation for the 2010–2013 period. The estimates are only represented between 0and 10−3for the sake of visual clarity, and a few jumps in the CHF/USD and JPY/USD exchange rates generate peaks well above the upper limit. In Panel 4b, the three trend estimates are compared over a shorter time period in the summer of 2011, in the middle of United States debt-ceiling crisis and European sovereign debt crisis of 2011. Our reconstructed trend visibly constitutes a smoothed version of the usual volatility measures, which generally fall inside of the 95% confidence interval except for upward spikes due to jumps. Notice the steep volatility increase in all exchange rates as stock markets around the world started to fall, following the downgrading of United States’ credit rating on 6 August 2011 by Standard & Poor’s. This is especially true for the Swiss franc, which encountered increasing pressure from currency markets because it is considered a safe investment in times of economic uncertainty. In Figure 5, we show the amplitude modulations of the first periodic component. When a unique Fourier flexible form is estimated for the whole sample, the corresponding amplitude is very close to the mean amplitude obtained with the synchrosqueezing transform. We observe that the rolling Fourier flexible form closely tracks the synchrosqueezing transform, but is far wigglier. This was expected because the synchrosqueezing transform estimator is essentially an instantaneous (and smooth by construction) version of the Fourier flexible form. As expected from Figure 3, the amplitude modulations of the first component in the JPY/USD are much smaller than the three other exchange rates. 2010 2011 2012 2013 0.2 0.6 1 1.4 Amplitude 1 CHFUSD SST rFFF FFF 2010 2011 2012 2013 0.2 0.6 1 1.4 Amplitude 1 EURUSD SST rFFF FFF 2010 2011 2012 2013 0.2 0.6 1 1.4 Amplitude 1 GBPUSD SST rFFF FFF 2010 2011 2012 2013 0.2 0.6 1 1.4 Amplitude 1 JPYUSD SST rFFF FFF Figure 5. Amplitude modulations reconstruction results. In each panel: reconstructed amplitude modulations of the first component (black line) with 95% confidence intervals (shaded area), Fourier flexible form (dashed line) and rolling Fourier flexible form (red line). (Top) CHF/USD (left) and EUR/USD (right); (Bottom) GBP/USD (left) and JPY/USD (right). Econometrics 2015,3882 11−07−11 11−07−12 11−07−13 11−07−14 11−07−15 −1.5 −1 −0.5 0 0.5 1 1.5 Seasonality CHFUSD SST rFFF FFF 11−07−11 11−07−12 11−07−13 11−07−14 11−07−15 −1.5 −1 −0.5 0 0.5 1 1.5 Seasonality EURUSD SST rFFF FFF 11−07−11 11−07−12 11−07−13 11−07−14 11−07−15 −1.5 −1 −0.5 0 0.5 1 1.5 Seasonality GBPUSD SST rFFF FFF 11−07−11 11−07−12 11−07−13 11−07−14 11−07−15 −1.5 −1 −0.5 0 0.5 1 1.5 Seasonality JPYUSD SST rFFF FFF (a) 11−08−08 11−08−09 11−08−10 11−08−11 11−08−12 −1.5 −1 −0.5 0 0.5 1 1.5 Seasonality CHFUSD SST rFFF FFF 11−08−08 11−08−09 11−08−10 11−08−11 11−08−12 −1.5 −1 −0.5 0 0.5 1 1.5 Seasonality EURUSD SST rFFF FFF 11−08−08 11−08−09 11−08−10 11−08−11 11−08−12 −1.5 −1 −0.5 0 0.5 1 1.5 Seasonality GBPUSD SST rFFF FFF 11−08−08 11−08−09 11−08−10 11−08−11 11−08−12 −1.5 −1 −0.5 0 0.5 1 1.5 Seasonality JPYUSD SST rFFF FFF (b) Figure 6. Seasonality reconstruction results. In each panel: reconstructed seasonality (black line) with 95% confidence intervals (shaded area) and rolling Fourier flexible form (red line). Top: CHF/USD (left) and EUR/USD (right); Bottom: GBP/USD (left) and JPY/USD (right). (a) Second week of July 2011; (b) Second week of August 2011. In Panels 6a and 6b of Figure 6, we show the seasonality estimated as a sum of the four periodic components: the Fourier flexible form represents the average daily oscillation, the rolling Fourier flexible Econometrics 2015,3883 form is a wiggly estimate of the dynamics, and the synchrosqueezing transform extracts a smooth instantaneous seasonality. As in Figure 5, we observe that the overall magnitude of oscillation is much smaller in the JPY/USD exchange rate. In summary, a striking feature of our estimated trend is its robustness to jumps, which affect both the realized volatility and, albeit to a lesser extent, the bipower variation. Figures 5and 6strongly suggest that the seasonality evolves dynamically over time. Although similar estimates of trend and seasonality can be obtained with moving averages and rolling regressions respectively, the synchrosqueezing transform provides smooth estimates and does not require an arbitrary choice of the length of the window or the rolling overlap. One could argue that the choice of a mother wavelet is also arbitrary, the influence of this choice on the synchrosqueezing transform estimators is negligible (see [22,25]). In any case, the main message is that dynamic methods are necessary to properly understand the seasonality, as static models can lead to severe underestimation or overestimation of the intraday spot volatility. 5. Discussion Our disentangling of instantaneous trend and seasonality of the intraday spot volatility is an extension of classical econometric models that provides adaptivity to ever-changing markets. The proposed method suggests a realistic framework for the high-frequency time series behavior. First, the method allows the daily component of the volatility to be modeled as an instantaneous trend evolving in real time within the day. Second, the model allows the seasonality to be non-constant over the sample. In a simulation study using a realistic setting, numerical results confirm that the proposed estimators for the trend and seasonality components behave appropriately. We show that this result holds even in the presence of heteroskedastic and heavy-tailed noise, and it is robust to jumps. Using the CHF/USD, EUR/USD, GBP/USD, and USD/JPY exchange rates sampled every fite minutes between 2010 and 2013, we confirm empirically that the oscillation frequency is constant, as originally suggested in the Fourier flexible form from [3,4]. In the four exchange rates, we show that the amplitude modulations of the periodic components, and hence the overall magnitude of oscillation, evolve dynamically over time. As such, neglecting those modulations in the periodic part of the volatility would imply either an overestimation (when the periodic components are lower than their mean value), or an underestimation (when the periodic components are higher than their mean value). We show that the synchrosqueezing transform estimator produces results comparable to a smooth version of a rolling Fourier flexible form. However, the adaptivity of synchrosqueezing transform should still be emphasized, because it does not require ad-hoc choices for the rolling overlapping ratio or of the length of each window, as for a parametric regression. Although we illustrate our model by simultaneously disentangling the low- and high-frequency components of the intraday spot volatility in the foreign exchange market, it is possible to embed the new methodology into a forecasting exercise. This exercise may be useful in the context of an investor’s optimal portfolio choice or for risk-management purposes, which is beyond the scope of this paper’s aim of presenting the modeling framework. Further research directions include the extension of the framework to the multivariate data and the non-homogeneously sampled (“tick-by-tick”) data. We will return to these questions and related issues in future works. Econometrics 2015,3884 Acknowledgments Most of this research was conducted while the corresponding author was visiting the Berkeley Statistics Department. He is grateful for its support and helpful discussions with the members both in Bin Yu’s research group and the Coleman Fung Risk Management Research Center. This research is also supported in part by the Center for Science of Information (CSoI), a US NSF Science and Technology Center, under grant agreement CCF-0939370, and by NSF grants DMS-1160319 and CDS&E-MSS 1228246. Author Contributions Thibault Vatter and Hau-Tieng Wu conceived and developed the methodology. Hau-Tieng Wu contributed the theoretical aspects and the legacy code to perform a Synchrosqueezing Transform. Thibault Vatter adapted the code to the financial context and analyzed the data. Valérie Chavez-Demoulin and Bin Yu supervised the project and contributed to the writing. Conflicts of Interest The authors declare no conflict of interest. Appendix. Implementation Details In this section, we provide the numerical synchrosqueezing form implementation details. The MATLAB code is available from the authors upon request, and we refer the readers to [29] for more implementation details. We fix a discretely sampled time series y={yn}N n=1, where xn=x(nτ), with τas the sampling interval and N= 2Lfor L∈N+. Note that we use the bold notation to indicate the numerical implementation (or the discrete sampling) of an otherwise continuous quantity. Step 1: numerically implement Wy(t, a). We discretize the scale axis aby aj= 2j/nvτ,j= 1, . . . , Lnv, where nvis the voice number chosen by the user. In practice, we choose nv= 32. We denote the numerical continous wavelet transform as a N×namatrix Wy. This is a well-studied step, and our continous wavelet transform implementation is modified from that of wavelab5. Step 2: numerically implement ωy(a, t). The next step is to calculate the instantaneous frequency information function ωy(a, t)(14). The ∂tWy(a, t)term is implemented directly by finite difference at taxis, and we denote the result as an N×namatrix ∂tWy. The ωy(a, t)is implemented as an N×namatrix wyby the following entry-wise calculation: wx(i, j) = (−i∂tWy(i,j) 2πWy(i,j)when Wy(i, j)6= 0 NaN when Wy(i, j) = 0., 5http://statweb.stanford.edu/~wavelab/ Econometrics 2015,3885 where NaN is the IEEE arithmetic representation for Not-a-Number. Step 3: numerically implement Sy(t, ω). We now compute the synchrosqueezing transform Sy(15). We discretize the frequency domain [1 Nτ ,1 2τ]by equally spaced intervals of length ∆ω=1 Nτ . Here 1 Nτ and 1 2τare the minimal and maximal frequencies detectable by the Fourier transform theorem. Denote nω=b 1 2τ−1 Nτ ∆ωc, which is the number of the discretization of the frequency axis. Fix γ > 0,Syis discretized as an N×nωmatrix Syby the following evaluation Sy(i, j) = X k:|wy(i,k)−j∆ω|≤∆ω/2,|Wy(i,j)|≥γ log(2)√aj ∆ωnv Wy(i, k), where i= 1, . . . , N and j= 1, . . . , nω. Notice that the number γis a hard thresholding parameter, which is chosen to reduce the influence of noise and numerical error. In practice, we simply choose γ as the 0.1quantile of |Wy|. If the error is Gaussian white noise, the choice of γis suggested in [29]. In general, determining how to adaptively choose γis an open problem. Step 4: estimate the instantaneous frequency, amplitude modulation and trend from Sy. We fit a discretized curve c∗∈ZN nω, where Znω={1, . . . , nω}is the index set of the discretized frequency axis, to the dominant area of Sy, by maximizing the following functional over c∈ZN nω: hN X m=1 log |Sy(m, cm)| Pnω i=1 PN j=1 |Sy(j, i)|!−λ N X m=2 |cm−cm−1|2i, where λ > 0. The first term is used to capture the maximal value of Syat each time, and the second term is used to impose regularity on the extracted curve. In other words, the user-defined parameter λdetermines the smoothness of the resulting curve estimate. In practice, we simply choose λ= 10. Denote the maximizer of the functional as c∗∈RN. In that case, the estimator of the IF of the k-th component at time t=nτ is defined as φ0 k(n) := c∗(n)∆ω, where φ0 k∈RN. With c∗, the k-th component ak(t) cos(2πφk(t)) and its amplitude modulation, ak(t)at time t=nτ are estimated by: fk(n) := <2 Rψ∆ω c∗(n)+b∆/∆ωc X i=c∗(n)−b∆/∆ωc Sy(n, i), ak(n) :=  2 Rψ∆ω c∗(n)+b∆/∆ωc X i=c∗(n)−b∆/∆ωc Sy(n, i), where <is the real part, fk∈RN, and ak∈RN. Lastly, we estimate the trend T(t)at time t=nτ by T(n) := yn−< 2 Rψ∆ω nω X i=bωl/∆ωc Sy(n, i), where T∈RNand ωl>0is determined by the model. Econometrics 2015,3886 References 1. Andersen, T.G.; Bollerslev, T.; Diebold, F.X.; Labys, P. Modeling and Forecasting Realized Volatility. Econometrica 2003,71, 579–625. 2. Guillaume, D.M.; Pictet, O.V.; Dacorogna, M.M. On the intra-daily performance of GARCH processes, Working papers; Olsen and Associates: Zurich, Switzerland, 1994. 3. Andersen, T.G.; Bollerslev, T. Intraday periodicity and volatility persistence in financial markets. J. Empir. 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