Time-Varying Periodicity in Intraday Volatility Torben G. Andersen, Martin Thyrsgaard and Viktor Todorov CREATES Research Paper 2018-5 Department of Economics and Business Economics Email: [email protected] Aarhus University Tel: +45 8716 5515 Fuglesangs Allé 4 DK-8210 Aarhus V Denmark Time-Varying Periodicity in Intraday Volatility∗ Torben G. Andersen† Martin Thyrsgaard‡ Viktor Todorov§ January 9, 2018 Abstract We develop a nonparametric test for deciding whether return volatility exhibits time- varying intraday periodicity using a long time-series of high-frequency data. Our null hypothesis, commonly adopted in work on volatility modeling, is that volatility fol- lows a stationary process combined with a constant time-of-day periodic component. We first construct time-of-day volatility estimates and studentize the high-frequency returns with these periodic components. If the intraday volatility periodicity is in- variant over time, then the distribution of the studentized returns should be identical across the trading day. Consequently, the test is based on comparing the empiri- cal characteristic function of the studentized returns across the trading day. The limit distribution of the test depends on the error in recovering volatility from dis- cretereturndataandtheempiricalprocesserrorassociatedwithestimatingvolatility moments through their sample counterparts. Critical values are computed via easy- to-implement simulation. In an empirical application to S&P 500 index returns, we find strong evidence for variation in the intraday volatility pattern driven in part by the current level of volatility. When market volatility is elevated, the period pre- ceding the market close constitutes a significantly higher fraction of the total daily integrated volatility than is the case during low market volatility regimes. JEL classification: C51, C52, G12. Keywords: high-frequencydata,periodicity,semimartingale,specificationtest,stochas- tic volatility. ∗Andersen’s and Todorov’s research is partially supported by NSF grant SES-1530748. †Department of Finance, Kellogg School, Northwestern University, NBER and CREATES. ‡CREATES, Department of Economics and Business Economics, Aarhus University. §Department of Finance, Kellogg School, Northwestern University. 1 1 Introduction Stock returns have time-varying volatility and this has important theoretical as well as practical ramifications. Most existing work on volatility has modeled it as a stationary process. However, there is both theoretical (see, e.g., [1, 25]) and empirical evidence (see, e.g., [2, 3]) for the presence of intraday periodicity in volatility. To illustrate this phe- nomenon, on Figure 1, we plot the average level of the S&P 500 index return volatility as a function of time-of-day. As seen from the figure, the intraday periodic component of volatility is nontrivial. Indeed, the average volatility at the market close is about three times the average volatility around lunch. 2.5 2 1.5 1 0.5 0 09:00 10:00 11:00 12:00 13:00 14:00 15:00 Time of Day (CST) Figure 1: Intraday Volatility Periodicity for the S&P 500 Index. The plot presents smoothed estimates of the average time-of-day volatility, normalized by the trading day volatility. Details regarding the construction of the series are provided in Section 5. High-frequency data is increasingly used, as it offers very significant efficiency gains for measuring and forecasting volatility, see, e.g., [5]. The pronounced periodic pattern 2 exhibited in Figure 1 has strong implications regarding the appropriate methodology for studying volatility using the intraday return data. The usual approach in the literature assumes that the time-of-day component of volatility is constant across days and then standardizes the high-frequency returns by the corresponding estimates, see, e.g., [2, 3], [9] and [35] among many others. However, this only annihilates the intraday volatility component from the returns, if the latter remains time-invariant. The goal of the current paper is to test this (null) hypothesis within a general nonparametric setting. Moreover, if the null hypothesis is rejected, we provide techniques that can help identify the sources of variation in the intraday periodic component. The statistical analysis is conducted using a long span of high-frequency return data. The major challenges in designing the test stem from the fact that volatility is not directly observed and both the stationary and periodic component of volatility can change over the course of the day. We take advantage of the long time span as well as the short distance between the intraday observations to circumvent these latency problems. We first estimate the average periodic component of volatility from the high-frequency returns. Then we standardize the returns with these estimated time-of-day volatility components. Under the null hypothesis, this studentization of the returns is sufficient to annihilate the periodic volatility component. Therefore, the studentized high-frequency returns should have the exact same distribution regardless of time-of-day, if the null is true. In contrast, this is violated under the alternative hypothesis, where the studentized return distribution isgivenbyaconvolutionofthedistributionsforthestationaryvolatilitycomponentandthe standardized (time-varying) periodic volatility component. As a result, the distribution of the studentized returns depends on the time-of-day, when the periodic volatility component varies over time. Given the discrepancy in the distributional properties of the studentized returns under 3 the null and alternative hypotheses, our test statistic is designed to measure the distance between the studentized return distribution for different parts of the trading day. In par- ticular, we rely on a weighted L2 norm of the difference in the real parts of the empirical characteristic functions of the studentized returns. We use only the real parts of the empirical characteristic functions, because the high-frequency returns are approximately conditionally normally distributed, when conditioning on the information at the beginning of the return interval. Under the null hypothesis, the limiting distribution of our test statistic depends on the error in recovering volatility from the returns as well as the empirical process error asso- ciated with estimating population moments of volatility using their sample counterparts. The contribution of the first of these errors to the limiting distribution is a distinctive feature of our test, setting it apart from other estimation and testing problems involving joint in-fill and long-span asymptotics for high-frequency return data, where this error is negligible asymptotically. The reason for the added complication is that, due to the nature of the testing problem at hand, we only use a limited number of high-frequency returns per day in forming the statistic. Hence, we cannot derive the limiting distribution assuming that volatility, effectively, is observed, which is a convenient simplification in determining the asymptotic properties of existing joint in-fill and long-span inference procedures. As a consequence, the limit distribution of our test statistic is non-standard, but its quantiles are readily evaluated through simulation. We implement our new testing procedure on high-frequency return data for the S&P 500 index. Even after excluding trading days comprising scheduled macroeconomic an- nouncements, our test rejects the null hypothesis of a time-invariant intraday periodicity in volatility. Additional analysis shows that a significant driver of the variation in the periodic volatility component is the concurrent level of volatility, as proxied by the VIX volatility 4 index at the market open. Upon separating the trading days into regimes of low, medium, and high volatility, according to the level of the VIX, we find that our test rejects signifi- cantly less on these three subsamples. Specifically, when volatility is elevated, the period before the market close contributes a substantially higher fraction of the total integrated daily volatility compared with regimes featuring lower volatility. Our paper is related to several strands of earlier work. First, there is a large literature on detecting and modeling periodicity in discrete time series. Examples include [14], [20], [22], [27], [32], [34] and [36]. Second, there is a sizable literature that estimates (assumed) constant intraday volatility patterns. This includes empirical work by [2, 3], [21], [23] and [38]. The (constant) intraday periodicity is further explicitly modeled or accounted for in papers estimating volatility models and detecting jumps, such as [9], [13], [17] and [28]. [24] study the commonality in the intraday periodicity across many assets. Finally, [15] assume that the stationary volatility component is constant within the day and test whether the periodic volatility component can explain the full dynamic evolution across each day. We reiterate that a common feature of the above literature is the assumed invariance for the periodicity of volatility, while the goal of the current paper is to test this underlying assumption of existing work, and to explore potential sources of deviation from this hypothesis. Third, [4] consider testing for changes in the periodic component of volatility at a specific (known) point in time and in a parametric volatility setting (both for the stationary and periodic components). Unlike that paper, the analysis here is fully nonparametric and we can test for changes in the periodic component, which can happen at unknown times and be stochastic. Fourth, our paper is related to a voluminous statistical literature that tests for the equality of two distributions by using a weighted L2 distance between the associated empirical characteristic functions. Applications of this approach for testing dependence between two variables can be found in, e.g., [7, 8], [16], [19] and 5 [33]. The empirical characteristic function has also been used to study serial dependence in time series by [26] and to test for Gaussianity in stationary time series by [18]. The major difference between this strand of the literature and the current paper is that the variables, whose distributions are compared, are not directly observable and need to be “filtered” from the data, and this filtering procedure affects the limiting distribution of the test statistic. The rest of the paper is organized as follows. Section 2 presents the setting and in- troduces the statistics for assessing stochastic time-variation in the periodicity of intraday volatility. In Section 3 we derive the asymptotic limit theory for these statistics, and we then apply it for developing a feasible testing procedure for whether the intraday volatility pattern is time-invariant. Section 4 summarizes the results obtained from a large-scale sim- ulation study, and Section 5 presents an empirical implementation of the testing procedure. Section 6 concludes. All formal assumptions and proofs are deferred to a Supplementary Appendix. 2 Setup and Estimation of Periodic Volatility The (log) price process X is defined on some filtered probability space (cid:0)Ω,F,(F ) ,P(cid:1). t t≥0 Consistent with the absence of arbitrage, it follows an Itˆo semimartingale of the form, (cid:90) dX = a dt + σ˜ dW + xµ(dt,dx), (1) t t t t R where a is the drift, W a Brownian motion, σ˜ the (diffusive) stochastic volatility, µ the t t t counting measure for jumps in X with compensator b dt ⊗ F(dx), where b is a ca`dla`g t t process and F : R → R . Our main focus is the stochastic volatility component. Beyond + the customary stationary part, we assume it contains a periodic component with a cycle 6 spanning one unit of time. Specifically, σ˜2 = σ2f for some stationary process σ and t t (cid:98)t(cid:99),t−(cid:98)t(cid:99) t time-of-day function f : N × [0,1] → R with f = f , where the time unit is one + + (cid:98)t(cid:99),0 (cid:98)t(cid:99),1 day. In the standard setting, adopted in most current work, f is deterministic and depends only on the time-of-day, t − (cid:98)t(cid:99). In fact, high-frequency data is increasingly used as it offers very significant efficiency gains for measuring and forecasting volatility, see e.g., [5].. However, it is plausible that the periodic component might vary with the concurrent level of (the stationary component of) volatility, as well as the occurrence of events such as prescheduled macroeconomic announcements and, more generally, any shifts in the orga- nization and operation of the financial markets. The goal of the current paper is to test whether the time-of-day periodic component of volatility changes over time. TheinferencewillbebasedondiscreteobservationsoftheprocessX atequidistanttimes 0, 1, 2,...,T, where the integer T represents the time span, and the integer n indicates the n n number of times we sample within a unit interval. We denote the length of the sampling intervalby∆ = 1/nandthehigh-frequencyincrementsofX by∆n X = X − n t,κ ((t−1)n+(cid:98)κn(cid:99))/n X , for t ∈ N and κ ∈ (0,1]. The asymptotic setting involves n → ∞ and ((t−1)n+(cid:98)κn(cid:99)−1)/n + T → ∞, where, intuitively, the increasing sampling frequency assists in the nonparametric identification of the level of stochastic volatility from discrete observations of X, and the long time span allows us to separate the stationary and periodic components of volatility. Our estimate for the time-of-day component of volatility is given by, T n π (cid:88) f(cid:98)κ = T 2 |∆nt,κX||∆nt,κ−∆X|1{Ant,κ}, Ant,κ = {|∆nt,κX| ≤ vn ∩ |∆nt,κ−∆nX| ≤ vn}, (2) t=1 for v = α∆(cid:36) with (cid:36) ∈ (0,1/2) and α > 0. Under appropriate conditions, f(cid:98) converges n n κ in probability to E(f σ2 ), for t ∈ N . Therefore, up to the constant E(σ2), f(cid:98) provides t,κ t+κ + t κ an estimate for the periodic component of volatility, when the latter is time-invariant. 7 We test for invariance of the intraday component of volatility by comparing the distri- bution of estimates for volatility deseasonalized by f(cid:98) over different parts of the trading κ day. Under the null hypothesis, these distributions are identical, while they differ under the alternative. The inference for the distribution of volatility at different parts of the day will be based on the result in [35] that (the real part of) the empirical characteristic function of the high-frequency increments in X is an estimate for the Laplace transform of stochastic volatility. Therefore, we introduce, 1 (cid:88)T (cid:18)√ (cid:113) (cid:19) L(cid:98)n(u) = cos 2un∆n X/ f(cid:98) , u ∈ R , (3) κ T t,κ κ + t=1 and, as shown in the next section, L(cid:98)n converges in probability (in a functional sense) to, κ (cid:104) (cid:105) L (u) = E e−uft,κσt2+κ/E[ft,κσt2+κ] , for t ∈ N and κ ∈ (0,1]. (4) κ + 3 Testing for Time-Invariant Periodicity of Volatility We proceed with the formal asymptotic results for L(cid:98)n, which in turn will allow us to κ construct a feasible test for detecting time-varying intraday volatility periodicity. 3.1 Infeasible Limit Theory Our results will be based on the function L(cid:98)n(u) in u, and the functional convergence results κ below take place in the Hilbert space L2(w), (cid:26) (cid:12)(cid:90) (cid:27) L2(w) = f : R → R(cid:12)(cid:12) |f(u)|2w(u)du < ∞ , (5) + (cid:12) R + for some positive-valued continuous weight function w with exponential tail decay. As usual, we denote the inner product and the norm on L2(w) by (cid:104)·,·(cid:105) and ||·||, respectively. Convergence in probability for L(cid:98)n is established in the following theorem. κ 8 Theorem 1. Suppose Assumptions 1-3 in the Supplementary Appendix hold with K = {κ}, for some κ ∈ (0,1], and (cid:36) ∈ [1, 1). Then, as n → ∞ and T → ∞, we have, 8 2 P L(cid:98)n →− L . (6) κ κ The intuition behind the above result is the following. First, f(cid:98) is an estimate of κ E(f σ2 ). Second, over small time intervals, we have ∆n X ≈ σ ∆n W. From t,κ t+κ t,κ (cid:101)t−1+(cid:98)κn(cid:99) t,κ n here, the result in Theorem 1 follows by a Law of Large Numbers. Theorem 1 requires both n → ∞ and T → ∞, but imposes no restriction on their relative rate of growth. We emphasize that the above result is functional, i.e., we recover the Laplace transform L as a function of u. As is well known, the Laplace transform κ of a positive-valued random variable uniquely identifies its distribution. Therefore, any differences in L for different times-of-day (different values of κ) must stem from time vari- κ ation in the periodic component of volatility. In this case, studentizing the high-frequency (cid:113) increments by the time-of-day estimate f(cid:98) will not be enough to eliminate the intraday κ periodic component. WenextderiveaCentralLimitTheorem(CLT)forthedifferenceinL(cid:98)n, fortwodifferent κ values of κ, under the null hypothesis. Theorem 2. Suppose Assumptions 1-3 in the Supplementary Appendix hold with K = {κ,κ(cid:48)} and f ≡ f (constant time-of-day periodicity) for t ∈ N . Let (cid:36) ∈ [1, 2]. Then, t,κ κ + 8 5 for any κ, κ(cid:48) ∈ (0,1], as n → ∞ and T → ∞ with T ∆ → 0, we have, n √ (cid:16) (cid:17) T L(cid:98)n −L(cid:98)n −→L N(0,K), (7) κ κ(cid:48) where K is a covariance integral operator characterized by, (cid:90) Kh(z) = k(z,u)h(u)w(u)du, ∀h ∈ L2(w), (8) R+ 9