Temporal aggregation and long memory for asset price volatility

Perron, Pierre; Shi, Wendong Article Temporal aggregation and long memory for asset price volatility Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Perron, Pierre; Shi, Wendong (2020) : Temporal aggregation and long memory for asset price volatility, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 13, Iss. 8, pp. 1-18, https://doi.org/10.3390/jrfm13080182 This Version is available at: https://hdl.handle.net/10419/239249 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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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. https://creativecommons.org/licenses/by/4.0/ Journal of Risk and Financial Management Article Temporal Aggregation and Long Memory for Asset Price Volatility Pierre Perron 1,* and Wendong Shi 2 1Department of Economics, Boston University, 270 Bay State Rd., Boston, MA 02215, USA 2School of Economics, Renmin University of China, 59 Zhongguancun Street, Beijing 100872, China; [email protected] *Correspondence: perr[email protected] Received: 3 August 2020; Accepted: 12 August 2020; Published: 15 August 2020   Abstract: The effects of temporal aggregation and choice of sampling frequency are of great interest in modeling the dynamics of asset price volatility. We show how the squared low-frequency returns can be expressed in terms of the temporal aggregation of a high-frequency series. Based on the theory of temporal aggregation, we provide the link between the spectral density function of the squared low-frequency returns and that of the squared high-frequency returns. Furthermore, we analyze the properties of the spectral density function of realized volatility series, constructed from squared returns with different frequencies under temporal aggregation. Our theoretical results allow us to explain some findings reported recently and uncover new features of volatility in financial market indices. The theoretical findings are illustrated via the analysis of both low-frequency daily Standard and Poor’s 500 (S&P 500) returns from 1928 to 2011 and high-frequency 1-min S&P 500 returns from 1986 to 2007. Keywords: long memory; stochastic volatility; temporal aggregation; semiparametric estimators; random level shifts 1. Introduction Long-memory processes, especially the possibility of confusing them with structural changes, are of great interest in the field of time series. Applications of long-memory models are numerous, in particular in relation to stock return volatility in financial markets. Ding et al. (1993) argue that stock return volatility series can be well described by long-memory processes. However, it has also been shown that the estimate of the long-memory parameter, d , is biased away from 0 and the autocovariance function exhibits a slow rate of decay when a stationary short-memory process is contaminated by structural changes in level. In other words, a spurious long-memory process can arise when there are structural changes in a short-memory process. This idea extends that advanced by Perron (1989,1990) , who shows that structural changes and unit roots (d= 1 ) are easily confused in the sense that the estimate of the sum of the autoregressive coefficients is biased towards 1 and that tests of the null hypothesis of a unit root are biased towards non-rejection with a stationary process contaminated by structural changes. Relevant literature on this issue include Diebold and Inoue (2001), Engle and Smith (1999), Gourieroux and Jasiak (2001), Granger and Hyung (2004). Perron and Qu (2007,2010) analyze the properties of the autocorrelation function, the periodogram, and the log-periodogram (LP) estimate of the long-memory parameter for short-memory processes with random level shifts (RLS). They show that the autocorrelation function, the periodogram, and the LP estimates for the log squared daily returns for the Standard and Poor’s 500 (S&P 500) during 1928–2002 can be explained by a level shift model with a short-memory component, instead of a long-memory process without level shifts. Lu and Perron (2010) present a method to directly estimate J. Risk Financial Manag. 2020,13, 182; doi:10.3390/jrfm13080182 www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2020,13, 182 2 of 18 the level shift model using an extension of the Kalman filter and apply it to the log absolute returns for the S&P 500, AMEX, DJIA and NASDAQ stock market return indices. Their point estimates imply few level shifts for all series but once these are taken into account there is no evidence of long-memory in the sense that little serial correlation is found in the remaining noise. McCloskey and Perron (2013) provide simple trimmed versions of the LP estimator, which are consistent and asymptotically normal with the same limiting variance as the standard LP estimator regardless of whether the underlying long/short-memory process is contaminated by level shifts or deterministic trends. Using these robust LP estimators, they study the log-squared daily return series of the S&P 500, the Dow Jones Industrial Average (DJIA), the NASDAQ and the AMEX stock market indices, which are also examined by Lu and Perron (2010). Their robust LP estimator is 0.007, indicating the (near) absence of long-memory for the S&P 500 volatility series, which contradicts the estimate of 0.51, given by the standard LP estimator. Very similar results emerge for the DJIA and the AMEX. An interesting finding is that the robust LP estimator is 0.51, indicating a highly persistent long-memory process, when using the log of daily realized volatility series constructed from 5-min returns of the S&P 500 futures index from 21 April 1982 to 2 March 2007. The contrasting results raise an interesting puzzle. More recently, Varneskov and Perron (2018) extend the work of Lu and Perron (2010) and adopt a full parametric model including both random level shifts and ARFIMA components. They estimate the long-memory parameter, d , for the realized volatility series of the S&P 500 data generated from 5-min returns, which is the same data used by McCloskey and Perron (2013) . They also estimate the long-memory parameter for the squared daily returns of the S&P 500 data for the time period 1929–2004, using a RLS+ARFIMA ( 0, d , 0 ) model. The estimate for the former is 0.36, indicating the existence of long-memory processes, while it is 0.053 for the latter, indicating the (near) absence of long-memory processes. They also consider other RLS+ARFIMA models and the 30-year T-bonds data and the Dollar-AUS exchange rate, all of which present similar results. The realized volatility series are essentially the aggregation of the squared high-frequency returns, and the daily return series are the aggregated intraday returns. In this regard, it may be possible to explain the above puzzle by means of temporal aggregation effects. There is a vast amount of papers on temporal aggregation, e.g., Wei (1978) and Chambers (1998), but little attention has been paid to its effects in the frequency domain. In a series of papers, Souza (2005,2007,2008) discuss the effect of temporal aggregation and bandwidth selection in estimating the degree of long memory. Ohanissian et al. (2008) propose a test to distinguish between true and spurious long memory by exploiting the invariance of the long-memory parameter to temporal aggregation. Hassler (2011) investigates whether typical frequency domain assumptions made for semiparametric estimation and inference are closed with respect to aggregation. It is believed by many that various measures of volatility are perturbed fractionally integrated processes, related to stochastic volatility model (Hassler 2011). However, the aggregation of a stochastic volatility model has been little discussed. Bollerslev and Wright (2000) study the effect of the temporal aggregation of a long-memory stochastic volatility model in the time domain. Studying the effect of the temporal aggregation of a stochastic volatility model in the frequency domain is also important. First, the log periodogram (LP) estimator has become one of the most popular memory parameter estimator among empiricists due to simplicity, intuitiveness and ease of use, as discussed by McCloskey and Perron (2013). Therefore, it is necessary to study the aggregation of the stochastic volatility in the frequency domain. Second, by using a stochastic volatility model, we can better understand the difference between the various measures of volatilities, e.g., the realized volatility and squared daily returns. The aim of the paper is first to show how the squared low-frequency returns can be expressed in terms of the temporal aggregation of a high-frequency series. We build a bridge between the spectral density function of squared low-frequency returns and that of the squared high-frequency returns. Furthermore, we analyze the properties of the spectral density function of realized volatility, constructed from the squared returns with different frequencies under temporal aggregation. J. Risk Financial Manag. 2020,13, 182 3 of 18 These results will allow us to tackle a second aim and help us explain the following puzzles. First, the trimmed LP estimates for the log squared daily return series are very closed to zero, while they are relatively large (around 0.4) for the log daily realized volatility constructed from high-frequency returns. Second, the trimmed LP estimates for the log squared daily return series are near zero, while they are very large for the log aggregated squared daily returns. Third, the trimmed LP estimates for the log realized return series are large using high-frequency return data, while they are close to zero for the disaggregated original high-frequency returns when a large bandwidth is used. The theoretical findings are illustrated through the analysis of both low-frequency daily S&P 500 returns from 1928 to 2011 and high-frequency 1-min S&P 500 returns from 1986 to 2007. We consider the estimation of the long-memory parameter using the standard and the trimmed LP estimate. Overall, the results indicate that random level shifts are needed to explain the empirical features documented. For the low frequency data on S&P 500 returns, one cannot infer whether the noise is stationary long memory. On the other hand, a long-memory process appears needed to explain the features related to high frequency S&P 500 futures. This is in line with the findings of Varneskov and Perron (2018). The remainder of this paper is structured as follows. Section 2introduces the stochastic volatility model and the different aggregation processes for the realized volatility and the squared daily returns. Section 3analyzes the properties of the long-memory parameter estimates with different aggregation levels and with random level shifts. Section 4focuses on empirical applications of the theoretical findings to S&P500 data with different sampling frequencies. Section 5contains brief concluding remarks and comments about potential avenues for future research. A mathematical appendix contains the technical derivations. 2. Alternative Volatility Measures We first review the stochastic volatility models and the aggregation mechanisms. We then present theoretical results about temporal aggregation in the frequency domain. 2.1. Stochastic Volatility Model It is evident that the number of observations for stock market returns during a fixed period of time is inversely related to the length of each return interval. That is, with prices fully available, a shorter interval means that the returns are observed more frequently and therefore more observations can be obtained. High-frequency returns are now available since many financial time series are available on a tick-by-tick basis, which is virtually continuous. In this paper, high-frequency returns are classified according to the numbers of time units included in their intervals. To simplify, the returns obtained every one unit of time period are defined as 1-period returns, and similarly k -period returns denote the returns observed every kunits of time periods. Here, kis an integer larger than 1. Let rt,n be the n th intraday log-return (i.e., rt,n=log (Pt,n)−log (Pt,n−1) , where Pt,n is the price index) at day tsuch that rt,n= (1/√s)ht,nεt,n with n= 1, . . . , s is the highest sampling frequency and t= 1, . . . , T is the number of days. The component ht,nintends to capture the volatility level. Assumption 1. We assume that εt,n is i.i.d. standard normal, and ht,n and εt,n are mutually independent. It is further assumed that the demeaned squared volatility level yt,n=h2 t,n−Eh2 t,n is covariance stationary long-memory with integrable spectral density fy(λ). Remark 1. This assumption considers the case of yt,n being long memory, with spectral density function fy(y)≈λ−2das λ→0. J. Risk Financial Manag. 2020,13, 182 4 of 18 2.2. Temporal Aggregation According to the classification of high-frequency returns in Section 2.1, a sample of s 1-period returns contains s/k k -period returns. We assume that k is chosen such that s=kS for some integer S . Let r(k) t,p=∑pk n=k(p−1)+1rt,n=∑k−1 j=0Ljrt,kp denote the continuously compounded k -period return, so that r(k) t,p= (1/√s)∑k−1 j=0Ljht,kpεt,kp for p= 1, . . . , S and t= 1, . . . , T . Here, L is the backshift operator. Note that the backshift operator, L , is applied to the second subscript, i.e., Lht,kpet,kp =ht,kp−1et,kp−1. Therefore, the k -period return r(k) t,p can be written as r(k) t,p= (1/√s)zt,pr∑k−1 j=0Ljh2 t,kp where zt,pis i.i.d. standard normal. Then the squared k-period return r(k)2 t,pis given by r(k)2 t,p= (1/s)z2 t,p∑k−1 j=0Ljh2 t,kp which can be expressed in terms of temporal aggregation, as r(k)2 t,p=h(1/s)h2 t,nz2 t,[n/k]+1i(k) where [n/k] denotes the integer part of n/k and [·](k) denotes the k -period non-overlapping temporal aggregation of the series xt, i.e., x(k) p=∑k−1 j=0xkp−j,p=1, . . . , P. Note that in the case of aggregated variables, we use a square bracelet operator, i.e., [·](k) . This should not confused with [·] without a subscript (k) which denotes the integer part. Therefore, the squared daily return r2 t=h(1/s)h2 t,nz2 ti(s) is the temporal aggregation of ( 1 /s)h2 t,nz2 t , over a day. Here, zt is i.i.d. standard normal for t= 1, . . . , T . The realized volatility constructed from the k-period returns r(k) t,p, RVt=∑S p=1r(k)2 t,p is the temporal aggregation of the squared k-period returns r(k)2 t,p, such that RVt=hr(k)2 t,pi(S) 2.3. Temporal Aggregation in the Frequency Domain Assumption 2. We assume that the demeaned squared volatility level yt,n=h2 t,n−Eh2 t,n is covariance stationary with integrable spectral density fy(λ). J. Risk Financial Manag. 2020,13, 182 5 of 18 Proposition 1. Under Assumptions 1-2, the spectral density of the squared r(k) t,pis fr(k)2 t,p (λ)=1 s2k sin λ 2 2 ∑k−1 j=0 sin λ+2jπ 2k −2 fyλ+2jπ 2k(1) +1 s2πk2Eh2 t,n2+Var h(k) t,n Proof. See the Appendix A. The first term on the right hand side of (1) corresponds to the spectral density of the temporal aggregation of the k -period demeaned squared volatility level, [yt,n](k) . The remaining two terms are constant, induced by the noise component ht , which does not carry any information about the long-memory. Remark 2. When k =s, we have the spectral density of the squared daily returns fr2 t(λ)=1 s3 sin λ 2 2 ∑s−1 j=0 sin λ+2jπ 2s −2 fyλ+2jπ 2s(2) +1 s2πs2Eh2 t,n2+var h(s) t,n The first term of (2) is such that 1 s3 sin λ 2 2 ∑s−1 j=0 sin λ+2jπ 2s −2 fyλ+2jπ 2s→1 sfyλ 2s as λ→ 0. Therefore, the spectral density of the demeaned squared daily volatility decreases as s increases. When s is large enough, the spectral density of the squared daily returns fr2 t(λ) will be dominated by the second and third terms of (2), which implies that the squared daily return series is dominated by noise. Proposition 2. The spectral density of the realized volatility obtained from k-period returns, r(k) t,p,is fhr(k)2 t,pi(S)(λ)=1 s3 sin λ 2 2 ∑s−1 j=0 sin λ+2jπ 2s −2 fyλ+2jπ 2s(3) +1 s2πsk Eh2 t,n2+Svar h(k) t,n Proof. See the Appendix A. Remark 3. When S=s , i.e., the realized volatility is obtained from 1-period returns, rt,n , we have the following form of the spectral density f[r2 t,n](s)(λ)=1 s3 sin λ 2 2 ∑s−1 j=0 sin λ+2jπ 2s −2 fyλ+2jπ 2s(4) +1 sπEh2 t,n2+var h2 t,n For the spectral density of realized volatility, we have from (4) f[r2 t,n](s)(λ)→1 sππfyλ 2s+Eh2 t,n2+var (ht,n) J. Risk Financial Manag. 2020,13, 182 6 of 18 as λ→ 0. This means that the three terms of the spectral density of the realized volatility in equation (4) decrease at the same rate when s increases. Comparing the spectral density of the squared daily returns (2) with that of the realized volatility (4), note that their first terms are identical, i.e., 1 s3 sin λ 2 2 ∑s−1 j=0 sin λ+2jπ 2s −2 fyλ+2jπ 2s corresponding to the spectral density of the temporal aggregation of the demeaned squared volatility level, yt,n=h2 t,n−Eh2 t,n , over a day. This is the only part that contains information about the long memory and the remaining terms are simply noise. Therefore, both realized volatility and squared daily returns contain the same information about long memory. A difference between the spectral density of the squared daily returns and that for the realized volatility series occurs only in the second and the third terms, fr2 t(λ)−fr(k)2 t,p(S)(λ)=(s−k) sπEh2 t,n2+1 s2πhvar y(s) t,n−Svar h(k) t,ni (5) which is independent of the value of λ . Furthermore, Var([yt,n](s))−Svar([ht,n](k)) is positive in general because most financial series have positive autocorrelation even at large lag. Therefore, the difference between the spectral density of the squared daily returns and the realized volatility will be positive and larger than [(s−k)/sπ] (Eh2 t,n)2. 3. Long-Memory Parameter Estimates across Aggregation Levels We first describe in Section 3.1 the standard log-periodogram regression, both regular and trimmed as suggested by McCloskey and Perron (2013). Then, in Section 3.2, we show the equivalence of the estimates across aggregation levels. 3.1. Log-Periodogram Regressions A long-memory process typically has a spectral density function which is proportional to λ−2d as λ goes to zero, where d is the memory parameter. The fractionally integrated model, proposed by Granger and Joyeux (1980) and Hosking (1981), is a long-memory generalization of an ARMA model whose autocorrelations decay exponentially. When d∈(0, 0.5) , the autocorrelations decay slowly, a characteristic of long-memory processes. Various estimators of d have been proposed, among which semiparametric estimators have become widely used as they do not require a distributional assumption on the process generating the difference of order d of the series. A popular semiparametric estimator is the LP regression estimator proposed by Geweke and Porter-Hudak (1983), which uses only frequencies near zero to avoid possible misspecification caused by high frequency movements. The LP regression estimator was analyzed by, among others, Robinson (1995). The LP regression estimator is based on the following spectral characterization of a long-memory process: log f(λ)≈c−2dlog λ as λ→ 0 + , where f is the spectral density function of the process. The periodogram of the time series at λjis defined as Ixλj≡wxλj 2=wx(λj)wxλj∗, J. Risk Financial Manag. 2020,13, 182 7 of 18 where wxλj is the discrete Fourier transform of {xt}T t=1 evaluated at the Fourier frequency λj= 2 πj/T , and c∗ denotes the complex conjugate of any complex number c . Ixλj can be viewed as a noisy approximation to f. Therefore, the LP regression is: log Ixλj=c+dXj+ej,j=l, . . . , m, where Xj=−log 2−2 cos λjfor j=l, . . . , m. The LP regression estimator is ˆ d=−0.5 ∑m j=lYj−Ylog Ij ∑m j=lYj−Y2, where Yj= ( 1 / 2 )log 2−2 cos λj and Y=(1/ (m−l+1)) ∑m k=lYk . When l= 1, this is the standard LP regression estimator. We can trim some of the lower frequencies, as in McCloskey and Perron (2013), to obtain consistency and asymptotic normality with the same limiting variance as the standard LP regression estimator regardless of whether the underlying long/short-memory process is contaminated by level shifts or deterministic trends. 3.2. Equivalence of Estimates across Aggregation Levels Lemma 1. Under Assumption 1, for the spectral densities of the aggregated series and the original squared return series, we have fhr(k)2 t,pi(S)(λ)−S fr(k)2 t,p (λ/S)→0as λ→0. Proof. See the Appendix A. Lemma 1 implies that the spectral densities of the aggregated series and the original squared return series have the same slope near frequency zero. Hence, aggregation does not change the value of the long-memory parameter, consistent with the results of Chambers (1998), Souza (2005) and Hassler (2011). Corollary 1. The periodogram is a finite sample version of the spectral density; hence, a similar relation holds approximately, i.e., Ihr(k)2 t,pi(S),j−SIr(k)2 t,p,j→p0 as T →∞for λ→0. A similar result was obtained for stationary long memory series by Ohanissian et al. (2008). Remark 4. Lemma 1 and Corollary 1 do not depend on the stochastic volatility specification. The validity of the results only require that the original process be stationary. Remark 5. As discussed by Souza (2008), a better estimator is not necessarily generated by temporally aggregating a time series, because the same estimate can be obtained when the same bandwidths are used on the original time series, which offers a wider choice of bandwidths. That is, the original time series could provide potentially improved estimates in the sense that it allows for more flexible bandwidth selection. Remark 6. The microstructure noise is not taken into consideration here. However, adding a microstructure noise process will not change the result in Lemma 1, because Assumption 1 will still be satisfied even adding a stationary noise, and the above results hold as long as the time series is stationary. According to Perron and Qu (2010), the autocorrelation function, the periodogram, and the LP estimate for the log squared daily returns for the S&P 500 can be explained by a simple level J. Risk Financial Manag. 2020,13, 182 8 of 18 shift model with a short-memory component, instead of a long-memory process without level shifts. Lu and Perron (2010) estimate a random level shifts model and find that few level shifts are present, but once these are accounted for, there is no evidence for the existence of long-memory processes in the sense that little serial correlation is found in the remaining noise. Therefore, random level shifts, which have not been included in Assumption 1, should be considered here to generalize Lemma 1. We consider the following random level shift model proposed by Perron and Qu (2010), uT,t=∑t j=1δT,j,δT,t=πT,tηt(6) where ηt∼i . i . d . N0, σ2 η and πT,t∼i . i . d . Bernoulli (p/T, 1) . It is also assumed that the components πT,t and ηt are mutually independent. According to Proposition 3 of Perron and Qu (2010), the limit of the expectation of the periodogram has the following form lim T→∞T−1EIu,j=pσ2 η 4π3 1 j2as T→∞. Lemma 2. Under the data generating process (6), we have the following relation between the k -period aggregated series and the original random level shift series, lim T→∞EhIu(k),ji−klim T→∞EhIu(1),ji=0as λ→0. Proof. See Appendix A. Therefore, Remarks 4–6 still hold when random level shifts are taken into consideration. 4. S&P 500 Volatility We consider two series of returns related to the S&P 500 data, namely low-frequency and high-frequency returns. The low-frequency data consist of 22,000 daily return observations for the period from 13 August 1928 to 30 December 2011. Among these daily returns, the observations from 13 August 1928 to 30 October 2002 were kindly provided by William Schwert. The source of the data for the period 4 January 1928 through 2 July 1962 is Schwert (1990). From 3 July 1962 to 30 October 2002 it is from the CRSP daily returns file, and the returns for the time period after 30 October 2002 were obtained from the Yahoo Finance website. Because of the need to construct various aggregate measures, the effective initial date for estimate is 13 August 1928. The high-frequency data pertain to S&P 500 futures and includes 1-min returns from 7 October 1986 to 2 March 2007, amounting to 5000 trading days in total. These were purchased from http://www.grainmarketresearch.com/. These futures contracts expire within one year after their inception. Specifically, contracts incepted in January, April, July, and October expire in March, June, September, and December, respectively. The cleaned version was provided by Shin Ikeda as described in Appendix Aof Ikeda (2015). The span of the data was mostly dictated by the data availability, though it conveniently avoids the turbulent period of the great recession. In order to eliminate the effect of outliers in the data, we use a logarithmic transformation of the observations. Since there are some zeros in the original high-frequency and daily data, we demean our data first, as in Deo et al. (2006). Other methods were proposed in the literature, e.g., Perron and Qu (2010) and Lu and Perron (2010), who add a small value to the squared returns. 4.1. Low Frequency Data We first start our analysis with low frequency data, i.e., the daily data series. Table 1shows the LP estimates for the log realized S -day return series, which is simply calculated by cumulating S neighboring squared daily returns that do not overlap. More specifically, S= 1, 5, 10, and 20 stands for the squared daily returns, the realized weekly (every five business days), biweekly, and J. Risk Financial Manag. 2020,13, 182 15 of 18 Author Contributions: Conceptualization, P.P. and W.S.; formal analysis, P.P. and W.S.; investigation, P.P. and W.S.; methodology, P.P. and W.S.; writing—original draft, P.P. and W.S.; writing—review & editing, P.P. and W.S. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: We wish to thank Shinsuke Ikeda for kindly sharing high frequency cleaned data for the S&P 500 futures. We are also grateful to Zhongjun Qu for useful comments. Conflicts of Interest: The authors declare no conflict of interest. Appendix A We first state a lemma that will be used in subsequent proofs. Lemma A1 (Souza (2005), Hassler (2011)) . Let vt be a covariance stationary discrete-time process with spectral density function fv(λ), the spectral density of the k-period aggregation, v(k), is, fv(k)(λ)=1 k sin λ 2 2 ∑k−1 j=0 sin λ+2jπ 2k −2 fvλ+2jπ 2k for λ∈[0, 2π]. Proof of Proposition 1. As shown in Section 2, the squared k-period return r(k)2 t,pis given by r(k)2 t,p=1 sz2 t,p∑k−1 j=0Ljh2 kp =1 sh∑k−1 j=0Ljh2 kp +z2 t,p−1∑k−1 j=0Ljh2 kpi =1 sh∑k−1 j=0Ljh2 t,kp −Eh2 t,kp+∑k−1 j=0LjEh2 t,kpi +1 shz2 t,p−1∑k−1 j=0Ljh2 t,kp −Eh2 t,kp+z2 t,p−1∑k−1 j=0LjEh2 t,kpi =1 sh∑k−1 j=0Ljyt,kp +∑k−1 j=0LjEh2 t,kpi +1 shz2 t,p−1∑k−1 j=0Ljyt,kp +z2 t,p−1∑k−1 j=0LjEh2 t,kpi which can be expressed as r(k)2 t,p=1 sh[yt,n](k) p+kEh2 t,n+[yt,n](k) pz2 t,p−1+kEh2 t,nz2 t,p−1i From Lemma A1, the spectral density of the squared k-period return r(k)2 t,pis given by fr(k)2 t,p =1 s2k sin λ 2 2 ∑k−1 j=0 sin λ+2jπ 2k −2 fyλ+2jπ 2k +1 s2πvar y(k) t,n+k2Eh2 t,n2. J. Risk Financial Manag. 2020,13, 182 16 of 18 Proof of Proposition 2. The realized volatility constructed from the k-period returns r(k) t,pis RVt=∑S p=1r(k)2 t,p =1 s∑S p=1h∑k−1 j=0Ljyt,kp +∑k−1 j=0LjEh2 t,kpi +1 s∑S p=1hz2 t,p−1∑k−1 j=0Ljyt,kp +z2 t,p−1∑k−1 j=0LjEh2 t,kpi =1 sh∑S p=1∑k−1 j=0Ljyt,kp +∑S p=1∑k−1 j=0LjEh2 t,kpi +1 sn∑S p=1hz2 t,p−1∑k−1 j=0Ljyt,kpi+∑S p=1hz2 t,p−1∑k−1 j=0LjEh2 t,kpio =1 s∑s p=1Ljyt,n+sEh2 t,kp +∑S p=1z2 t,p−1y(k) t,np+kEh2 t,kp ∑S p=1z2 t,p−1 which can be expressed as RVt=1 s[yt,n](s) p+sEh2 t,kp +hz2 t,p−1y(k) t,ni(S)+kEh2 t,kp hz2 t,p−1i(S). From Lemma A1, the spectral density of the realized volatility constructed from the k -period returns r(k) t,p,RVt, is given by fRV =1 s3 sin λ 2 2 ∑s−1 j=0 sin λ+2jπ 2s −2 fyλ+2jπ 2s +1 s2πSvar y(k) t,n+Sk2Eh2 t,n2. Proof of Lemma 1. We have the following relation fhr(k)2 t,pi(s)(λ)−S fr(k)2 t,p (λ/S)= 1 s sin λ 2 2 ∑s−1 j=0 sin λ+2jπ 2s −2 fyλ+2jπ 2s −S k sin λ 2S 2 ∑k−1 j=0 sin λ+2jπ 2s −2 fyλ+2jπ 2s =1 s2[U(λ)+V(λ)] where V(λ)=1 s sin λ 2 2 ∑s−1 j=1 sin λ+2jπ 2s −2 fyλ+2jπ 2s −S k sin λ 2S 2 ∑k−1 j=1 sin λ+2jπ 2ks  −2 fyλ+2jπ 2ks  →0 since |sin (λ/2)|→0 for λ→0, and J. Risk Financial Manag. 2020,13, 182 17 of 18 U(λ)=1 s sin λ 2 2 sin λ 2s −2 fyλ 2s−S k sin λ 2S 2 sin λ 2s −2 fyλ 2ks  =1 sλ 22 sin λ 2s −2 fyλ 2s−S kλ 2S2 sin λ 2s −2 fyλ 2ks  →p0 since |sin (λ/2)|→pλ/2 and sin (λ/2S)→pλ/2Sfor λ→0. Proof of Lemma 2. After the k-period non-overlapping temporal aggregation, the random level shift component, ηq∼i . i . d . 0, k2σ2 η , and the total number of level shifts is unchanged as long as the sample size is large enough. 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Washington, DC: US Department of Commerce, Bureau of Census. c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).