The Impact of Microstructure Noise on the Distributional Properties of Daily Stock Returns Standardized by Realized Volatility

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1 The Impact of Microstructure Noise on the Distributional Properties of Daily Stock Returns Standardized by Realized Volatility Jeff Fleming, Bradley S. Paye Jones Graduate School of Management, Rice University Abstract Previous studies find that daily stock returns standardized by realized volatility are approximately standard normal. This evidence suggests that jumps are not an empirically relevant feature of stock prices, which is inconsistent with a growing body of research that directly tests for and finds evidence of jumps. This paper resolves the apparent contradiction. We show that upward bias in realized volatility estimates due to microstructure noise can artificially reduce the variance and increase the kurtosis of standardized returns and lead to the false appearance that returns are approximately standard normal. Using a bias-corrected realized volatility estimator, we find that standardized returns exhibit substantial departures from the standard normal and, in fact, are platykurtotic, consistent with a process in which jumps are empirical relevant. Keywords: Asset prices, Integrated volatility, Highfrequency returns, Jumps, Stock returns. 1 Introduction Assessing the empirical relevance of jumps in stock prices is an important area of research. It is relatively common in the literature to model stock prices as a continuous diffusion process because this class of models is both flexible and tractable. This class of models, however, is also potentially restrictive. It explicitly rules out the possibility that stock prices can exhibit occasional jumps, which intuitively could arise from the release of firm-specific, market-wide, and/or macroeconomic information. On the other hand, modeling stock prices as a jump-diffusion process entails substantially greater costs because these models are more difficult to estimate and may include additional sources of risk that are difficult to price empirically using cross-sectional data. According to some studies in the realized volatility literature, the empirical relevance of modeling jumps may be small. These studies (see, for example, Andersen, Bollerslev, Diebold and Ebens (21)) find that daily stock returns standardized by realized volatility are approximately standard normal. As Andersen, Bollerslev, Diebold and Labys (23) demonstrate, this finding is consistent with a continuous diffusion model for stock prices, which indirectly suggests that jumps are not empirically relevant. This conclusion, however, is starkly at odds with the evidence from another, more recent strand of the literature which directly tests for jumps. Andersen, Benzoni and Lund (22) and Chernov, Gallant, Ghysels and Tauchen (23), for example, find evidence of jumps based on parametric estimation of jump diffusion models, and Andersen, Bollerslev and Diebold (25), Huang and Tauchen (25), Jiang and Oomen (25), Tauchen and Zhou (25) and Barndorff-Nielsen and Shephard (25, 26) find evidence of jumps using nonparametric techniques that exploit the information in high-frequency intraday returns. This paper resolves the apparent empirical contradiction. The essence of our argument is that the distribution of standardized returns constructed using the standard realized volatility estimator can be distorted due to the impact of microstructure noise. We begin by illustrating that the distribution is highly sensitive to the sampling frequency of the returns used to construct realized volatility. As it turns out, using the five-minute sampling frequency, which is popular in practice, standardized returns are indeed approximately normal. However, the realized volatility estimates obtained using this sampling frequency exhibit clear evidence of bias, and using finer sampling reduces the variance of the standardized returns and increases the kurtosis. We demonstrate that each of these effects is consistent with the distortions caused by microstructure noise. Finally, using a bias-corrected realized volatility estimator to control for microstructure noise, we find that standardized returns are not normally distributed and in fact are markedly platykurtotic. As argued by Andersen, Bollerslev and Dobrev (26), this is precisely what one would expect if jumps are empirically relevant. These findings provide empirical guidance regarding the specification of continuous time models for stock prices, namely, that jumps are important and should not be ignored. The results also demonstrate the value of recently developed high-frequency variance estimators that remain conditionally unbiased and/or consistent in the presence of market microstructure noise. 1 While such estimators are theoretically appealing, evidence continues to accumulate regarding their improvements in practice relative to the standard realized variance estimator based on an intermediate sampling frequency selected using ad hoc judgment or signature plots (Andersen, Bollerslev, Diebold and Labys (2)). Our analysis provides a strik- 1 See, for example, Hansen and Lunde (23, 26), Zhang (24), Ait-Sahalia, Mykland and Zhang (25), Zhang, Mykland and Ait-Sahalia (25), and Barndorff-Nielsen, Hansen, Lunde and Shephard (26).

2 ing example of a situation where using a kernel-based estimator debunks a stylized fact, i.e., that daily stock returns standardized by integrated variance are standard normal. While the realized variance estimator computed at a sampling frequency based on a signature plot is unconditionally unbiased, it is not conditionally unbiased, and this is the key behind the ability of the kernel-based estimator to deliver very different results. The remainder of the paper is organized as follows. Section 2 reviews the theory underlying the mixture of normals hypothesis and discusses the use of realized volatility to test the hypothesis. Section 3 describes the data used for our analysis. Section 4 presents the results using the standard realized volatility estimator. Section 5 presents the results using the bias-corrected realized volatility estimator. Section 6 concludes. 2 The Mixture of Normals Hypothesis In a frictionless market with no arbitrage opportunities the log stock price p t must obey a semimartingale process on some filtered probability space (Ω, F, (F t ) t,p), as detailed by Back (1991). A common specification is to assume that prices follow a jump diffusion process: p t = μ(s)ds + N(t) σ (s) dw (s)+ κ j (1) j=1 where u(s) is a local finite variation process, σ is a cadlag volatility process, dw (s) represents increments to a Brownian motion W (t), N is a counting process, and κ j are the jump increments. Note that this process is a quite general. It allows a time-varying drift, stochastic volatility (including multi-factor volatility processes as studied by Chernov, Gallant, Ghysels and Tauchen (23)), and correlation between the price and volatility innovations to accommodate the so-called leverage effect. A leading special case of (1) is the pure diffusion model: p t = μ(s)ds + σ (s) dw (s) (2) which exhibits continuous sample paths so that no jumps in stock prices are permitted. Under the additional assumption that the mean and volatility processes μ(s) and σ (s) are independent of the innovation process W (s) over the period [t Δ,t], Andersen, Bollerslev, Diebold and Labys (23) show that ( r t,δ F t Δ,t N μ(s)ds, t Δ t Δ ) σ 2 (s)ds (3) where r t,δ = p t p t Δ denotes the Δ-period return ending at time t and F t Δ,t is shorthand for the σ-field generatedby {μ(s),σ(s)} overtheinterval [t Δ,t]. As Andersen, Bollerslev, Diebold and Labys (23) point out, the distributional result in (3) conditions on ex post sample path realizations of μ(s) and σ (s). At short horizons, it isoftenassumedthatμ(s) = because conditional mean variation is empirically an order of magnitude smaller than return variation. Thus, under equation (3), discretely sampled returns follow a normal mixture, where the mixture is governed by the daily integrated variance of returns. The class of pure diffusion models without leverage effects is often used as a convenient representation of stock prices, however, it rules out features (e.g., jumps and leverage effects) that may be important empirically. If imposing these restrictions is valid, then intraday returns standardized by the square root of the daily integrated variance will follow a standard normal distribution. 2.1 Realized Volatility Since the integrated variance series is latent, the empirical analysis of equation (3) is typically conducted using the realized volatility constructed from high-frequency intraday prices. The realized volatility is closely related to the quadratic variation (QV) of the price process, which is defined as t i t [p] t p lim (p m ti p ti 1 ) 2 (4) i=1 for any sequence of partitions = t <t 1 <... < t m = t with sup i {t i+1 t i } form. For the jumpdiffusion process in (1), QV may be decomposed as [p] t = N(t) σ 2 (s)ds + κ 2 j. (5) j=1 where σ2 (s)ds is the integrated variance (IV) of the price process and N(t) j=1 κ2 j is the sum of squared jumps. The increment in the QV process on trading day n is given by [p] n C [p] n O = n C n O σ 2 (s)ds + N(n C ) j=n(n O )+1 κ 2 j, (6) where n O and n C denote the open and close of trading on day n. Intuitively, this increment captures the total intraday variation in the price process, which may be decomposed into the sum of the integrated variance on day n plus the sum of the day s squared jumps. Realized volatility is an econometric measure of the increments in quadratic variation. In its simplest form, we construct the realized variance (RV) on day n by dividing the trading day into m subperiods of length Δ = 1/mand then summing the squared returns for each subperiod, RV m n m 1 = i= r 2 n C iδ,δ. (7) The realized volatility is the square root of RV. Andersen, Bollerslev, Diebold and Labys (21) and Barndorff- Nielsen and Shephard (21, 22) show that under weak

3 regularity conditions, the theory of quadratic variation implies RV m n IV n almostsurelyasδ. Constructing realized volatility allows a feasible examination of the standard normal hypothesis for standardized daily returns. If stock prices follow a pure diffusion process (i.e., no jumps), the decomposition in (5) implies that the quadratic variation simply equals the integrated variance. We can theoretically obtain an effectively errorfree measure of quadratic variation by constructing realized volatility using returns sampled at sufficiently high frequencies. 2.2 Impact of Microstructure Noise In practice, realized volatility can be biased because the true price process is unobservable and the price increments must be estimated from observed prices. Serial correlation in returns can be induced by bid-ask bounce, price discreteness, and price reporting errors. These effects tend to be more pronounced with more frequent sampling (i.e., increasing m), which suggests that the bias can potentially offset the efficiency gains associated with more frequent sampling. A number of researchers have analyzed the impact of microstructure noise on realized volatility (see, e.g., Zhou (1996), Hansen and Lunde (23, 26), Bandi and Russell (25, 26), and Zhang, Mykland and Ait-Sahalia (25)). Our objective, however, is slightly different because we are interested in the impact of these effects on the distribution of daily returns standardized by realized volatility. More specifically, our objective is twofold: (1) to assess whether microstructure noise can distort the distribution of standardized returns in such a way that leads to the false appearance of normality, and (2) to assess whether standardized returns, constructed using a bias-corrected realized volatility measure which is robust to microstructure noise, are in fact nonnormal and consistent with the empirical relevance of jumps. 3 Data The data consist of intradaily prices for the 2 stocks in the Major Market Index (MMI). 2 We obtain these data from the Trade and Quote (TAQ) database distributed by the New York Stock Exchange (NYSE). The sample period is January 4, 1993 to December 31, 23. We construct a database of intradaily transaction prices using all trades executed on any exchange but excluding records that have an out-of-sequence time, a price of zero, a TAQ correction code greater than two (errors and corrections), or a TAQ condition code (nonstandard settlement). In addition, we apply two filters designed to 2 These stocks are American Express, AT&T, ChevronTexaco, Coca-Cola, Disney, Dow Chemical, DuPont, Eastman Kodak, Exxon-Mobil, General Electric, General Motors, International Business Machines, International Paper, Johnson & Johnson, McDonald s, Merck, 3M, Philip Morris, Procter and Gamble, and Sears. eliminate obvious price reporting errors. First, we eliminate records that imply a price change greater than 2 percent in magnitude; and, second, we eliminate records that imply a price change greater than two percent in magnitude, followed by a reversal greater than two percent, if either (a) the price change is more than two times greater than the next largest price change on the observation date, or (b) the price exceeds the daily high or low (excluding that price) by more than the next largest price change on the observation date. 3.1 Constructing the Intraday Returns We use our database of transaction prices to construct the intraday returns. On most days, the trading day begins at 9:3am (EST) and ends at 4:pm, a period of 39 minutes. We divide this period into m intervals to construct returns with a sampling frequency of 39/m minutes. We consider a range of values for m, from m = 78 (i.e., 3-second intervals) to m = 6.5 (i.e., 6-minute intervals). We estimate the prices at the endpoints of each interval using the algorithm described by Fleming and Paye (26) and then take the first differences of the log prices to estimate the 39/m-minute intraday returns. 3.2 Characteristics of the Intraday Returns An underlying premise of the paper is that microstructure noise can distort observed returns and lead to biased realized volatility estimates. The empirical relevance of microstructure noise is typically gauged by examining the serial correlation of returns. Factors such as bid-ask bounce, the discreteness of prices, and/or price reporting errors induce negative serial correlation in observed returns and the serial correlation becomes more pronounced at finer sampling frequencies. Table 1 reports the serial correlation of the intraday returns for various sampling frequencies, with the results averaged across the 2 MMI stocks. Consistent with our premise regarding the relevance of microstructure effects, the average first-order serial correlation is negative Table 1: Serial correlations of intraday returns constructed using different sampling frequencies, coefficients averaged across all MMI stocks. Frequency Serial Correlation Coefficients (in minutes) ρ 1 ρ 2 ρ

4 (.7 for five-minute returns) and the magnitude increases with finer sampling frequencies (.14 for oneminute returns). with those reported by Andersen, Bollerslev, Diebold and Ebens (21), who conclude that daily stock returns standardized by realized volatility are approximately standard normal. 4 Returns Standardized by Realized Volatility We begin our empirical analysis by considering the properties of standardized returns constructed using the standard realized volatility estimator (7). In theory, realized volatility should be constructed using returns sampled arbitrarily finely. Of course, this is not practical because microstructure noise leads to upward bias in realized volatility at finer sampling frequencies. As a result, constructing realized volatility in practice involves a tradeoff between bias and efficiency. 4.1 Five-minute Sampling Frequency In order to balance this tradeoff, it has become common practice to construct realized volatility using returns sampled at a five-minute frequency. We consider this case first to provide a benchmark for our analysis. We construct the series of standardized returns for each stock by dividing the daily return by the realized volatility obtained from equation (7) implemented using the fiveminute returns over the course of each trading day. Table 2 reports the first four sample moments of the standardized returns for four of the MMI stocks: American Express (AXP), General Electric (GE), McDonald s (MCD), and Exxon-Mobil (XOM). These stocks were selected based on the alphabetical order of their ticker symbols and the results are intended to be representative. The skewness and kurtosis statistics for most of the stocks (including those not shown in the table) are roughly in line with the corresponding values under the standard normal distribution. The skewness statistics, although small in magnitude, tend to be positive, which may be indicative of the impact of leverage effects. More significantly, the variance of the scaled returns tends to be less than one, and for some of the stocks the departures are substantial. Table 2: Sample moments of daily returns standardized by realized volatility constructed with five-minute returns. Ticker Mean Std Skew Kurt Figure 1: QQ plot of daily returns standardized by realized volatility constructed with five-minute returns 4.2 Robustness to the Sampling Frequency Our initial choice of a five-minute sampling frequency conforms with the common choice in practice. However, as shown by the serial correlations of returns reported in Table 2, microstructure noise appears to be empirically relevant at this sampling frequency, and this may affect statistical inference. To assess the potential impact, we next investigate the robustness of the moments of the standardized returns to the choice of sampling frequency. We construct the daily realized volatilities for each stock, for a range of sampling frequencies, from m = 78 to m =6.5. Figure 2 plots the average realized variance, averaged across days and across stocks, for each sampling frequency. The plot labeled RV shows the results for realized volatilities constructed according to equation (7); we defer discussion of the plot labeled NW until Section 5. The figure is similar to the volatility signature plot proposed by Andersen, Bollerslev, Diebold and AXP GE MCD XOM Figure 1 provides QQ plots of the standardized returns for these four stocks against the standard normal distribution. Not surprisingly, given the sample moments reported in Table 2, the plots are visually consistent Figure 2: Volatility signature plots

5 Labys (2) to identify the optimal sampling frequency. ABDL recommend sampling as finely as possible until the point at which microstructure noise begins to distort the estimates. As the figure shows, the realized volatilities constructed using the five-minute sampling frequency exhibit clear evidence of upward bias. These estimates, on average, are more than 13 percent greater than the average daily squared return, which (although noisy) should be an unbiased estimate of the daily integrated variance. At sampling frequencies finer than five minutes the bias increases sharply, while sampling less finely reduces bias entails greater sampling error. Figures 3 and 4 illustrate how the moments of the daily returns standardized by realized volatility vary with the sampling frequency. Figure 3 shows the signature plot for the standard deviation of the standardized returns. The Figure 3: Standard deviation of standardized returns standard deviation is close to 1. for realized volatilities constructed using 6-minute returns but the standard deviation decreases consistently as the sampling frequency increases. At the finest sampling frequencies, the standard deviation is close to.8. Figure 4 shows the signature plot for the kurtosis of the standardized returns. 3 The kurtosis systematically Figure 4: Kurtosis of standardized returns increases as the sampling frequency increases. The kurtosis is close to 2.2 for realized volatilities constructed us- 3 The signature plot for the skewness is not shown because the skewness is relatively insensitive to the sampling frequency. ing 6-minute returns and increases to about 3.8 using 3- second returns. As it turns out, the kurtosis is close to 3. using a five-minute sampling frequency but, as the volatility signature plot indicates, the effects of microstructure bias on realized volatility have not been purged at this sampling frequency. Using any other choice of sampling frequency would lead to strong rejections of the normality of standardized returns. 4.3 Sources of the Dependence on Sampling Frequency The extent to which the patterns in Figures 3 and 4 can be linked to the pattern in Figure 2 suggests that microstructure noise can lead to distortions in the properties of standardized returns. Another possible source of distortion is discretization error: since realized volatility cannot be constructed using returns sampled arbitrarily finely, especially in the presence of microstructure noise, the integrated variance is measured with error, and the magnitude of the error increases as the sampling frequency decreases. Peters and de Vilder (24) investigate the impact of discretization error for the special case of a pure diffusion model in which the instantaneous volatility remains constant. They show that the density function for standardized returns converges to the standard normal density as m but that severe distortions can occur for small m. In the limiting case of m = 1 (i.e., realized variance equals the daily squared return), the kurtosis equals one. This dependence on the sampling frequency is similar to that depicted in Figure 4 for sampling frequencies longer than five minutes. More generally, Fleming and Paye (26) show that for any finite sampling frequency there exist (potentially complex) volatility dynamics such that similar severe distortions occur. These findings suggest that interpreting thin-tailed standardized returns as consistent with jumps involves implicitly ruling out such highly erratic (but cadlag) volatility processes. However, discretization effects alone cannot explain the standard deviations substantially less than 1. (Figure 3) or the kurtosis values greater than 3. (Figure 4) at sampling frequencies finer than five minutes. Microstructure noise, on the other hand, can generate both of these effects. Hansen and Lunde (26), Bandi and Russell (25, 26), and Zhang, Mykland and Ait- Sahalia (25), for example, demonstrate cases in which microstructure noise can lead to increasing upward bias in RV as m, causing daily returns to be standardized by too large a value, and driving the standard deviation of the standardized returns below 1.. The observed patterns in the kurtosis can be explained by dependence between the relative impact of the microstructure noise and the level of the daily integrated variance. Suppose, for example, that the distortions caused by microstructure noise are relatively less on high-volatility days. If this were the case, then the larger absolute returns typically observed on these days would tend to be scaled by

6 relatively less upward-biased realized volatilities, which, in turn, would lead to spurious fat-tailed behavior. Table 3 provides some empirical support for this explanation. The table reports the serial correlation of intraday returns on days in which the daily absolute return is greater than 1.5 percent. These days, on average, represent about 2 percent of the sample. The results indicate that the serial correlation is substantially lower on these days (compare to Table 1), especially at the finest sampling frequencies, which suggests that prices are relatively less distorted by microstructure noise. These findings are consistent with our microstructure-based explanation of the pattern in the kurtosis signature plot. Table 3: Serial correlations of intraday returns on largereturn days, coefficients averaged across all MMI stocks. Frequency Serial Correlation Coefficients (in minutes) ρ 1 ρ 2 ρ Returns Standardized by a Bias-Corrected Measure of Realized Volatility In this section, we consider an alternative realized volatility estimator which remains (approximately) unbiased in the presence of microstructure noise. Specifically, we use this estimator to evaluate the extent to which the deficiencies observed using the standard realized volatility estimator can be remedied, and whether the estimator delivers more reliable inference regarding the mixture of normals hypothesis. 5.1 Newey-West Realized Volatility A number of high-frequency integrated variance estimators that are robust to microstructure noise have been proposed in the literature. These include, for example, the kernel-based estimators of Hansen and Lunde (23, 26) and Barndorff-Nielsen, Hansen, Lunde and Shephard (26) and the subsampling estimators of Zhang, Mykland and Ait-Sahalia (25) and Ait-Sahalia, Mykland and Zhang (25). We adopt a kernel-based approach proposed by Hansen and Lunde (23) which directly accounts for serial correlation in observed returns. Specifically, we construct a Newey-West (1987) estimator of realized variance, NW m n 2 q m j=1 m 1 = i= r 2 n C iδ,δ + ( ) m j 1 j 1 r q m +1 n C iδ,δr n C (i+j)δ,δ (8) i= where q m denotes the lag length captured by the covariance terms. The advantage of this approach is that the estimator removes the effects of serial correlation on the variance, offering the potential of sampling returns at very high frequencies in order to incorporate the most information. Hansen and Lunde (26) propose a different weighting scheme in (8) which delivers conditionally unbiased estimates, however, their estimator can (and frequently does using our sample) generate negative variance estimates. The estimator in (8), on the other hand, is guaranteed to be positive but at the cost of being neither consistent nor conditionally unbiased. Nonetheless, the NW estimator affords the potential of a nearly conditionally unbiased estimator which provides an alternative to the standard realized volatility estimator for conducting feasible tests for the normality of standardized returns. We construct the realized volatilities using (8) for the MMI stocks, for a range of sampling frequencies, from m = 78 to m =6.5, and a range of window lengths (w) for the autocovariance terms. To implement a given w window length, we set q m = ceil( 39/m ), where ceil(x)denotes the smallest integer greater than or equal to x. Note that this specification implies that q m increases at finer sampling frequencies, consistent with Hansen and Lunde (26). The logic is that microstructure noise takes a certain period of time to die out (e.g., 15 minutes), and q m should cover this period, independent of the sampling frequency. Examining the properties of the resulting realized volatility estimates, we conclude that a 3-minute window width is sufficient to account for the structure of the microstructure noise. Figure 2 (shown earlier) provides the signature plot for the NW realized volatility estimates, averaged across the MMI stocks. In general, the estimates are not very sensitive to the sampling frequency, and certainly much less so than the standard RV estimates. The average NW estimates, for any choice of sampling frequency, are within two percent of the average daily squared return. This suggests that returns can be sampled at very fine sampling frequencies without incurring bias due to microstructure noise. 5.2 Properties of Standardized Returns We now consider the daily returns standardized by the NW realized volatility estimates. First, we construct the standardized returns for each stock using the realized volatilities obtained from (8) using 3-second returns.

7 Using this sampling frequency delivers the most precise realized volatility estimates. Table 4 reports the first four sample moments of the standardized returns for the four representative MMI stocks. The results differ markedly from those reported in Table 2. Specifically, using the bias-corrected realized volatility estimator pushes the sample standard deviations closer to one and the skewness statistics closer to zero. In addition, the kurtosis statistics are sharply lower, and typically on the order of 2.3 to 2.4 for most stocks (including those not shown in the table). As a result, a formal test of the hypothesis that the standardized returns follow the standard normal distribution would produce a sound rejection for every stock. Table 4: Sample moments of daily returns standardized by Newey-West realized volatility. Ticker Mean Std Skew Kurt AXP GE MCD XOM The inadequacy of the standard normal distribution is visually apparent in the QQ plots shown in Figure 5. The plots display a marked S-shaped pattern indicative of thin-tailed behavior relative to the standard normal. Thus, it is clear from both inspection of the QQ plots and from a classical hypothesis testing viewpoint that daily returns standardized using a bias-corrected realized volatility estimator are not standard normal. Figure 5: QQ plot of returns standardized by Newey- West realized variance constructed using a five-minute sampling frequency Finally, we consider the robustness of these results to the choice of sampling frequency. Figures 3 and 4 (shown earlier) provide the signature plots for the standard deviation and kurtosis of daily returns standardized using the NW realized volatility estimates. The patterns in these plots contrast sharply with those for the RV estimates. In particular, the standard deviation is flat with respect to the sampling frequency, hovering around a value of about.96. The kurtosis plot is also relatively flat, with kurtosis estimates of roughly 2.4 for sampling frequencies ranging from 3 seconds out to 5 minutes. At lower sampling frequencies, the discretization effects documented by Peters and de Vilder (24) and Andersen, Bollerslev and Dobrev (26) begin to kick-in, which produces lower kurtosis estimates. Based on this analysis, we conclude that daily returns standardized by NW realized volatility are decidedly thin-tailed. This evidence leads us to reject the pure diffusion model for stock returns; but the evidence is consistent with the reasoning of Andersen, Bollerslev and Dobrev (26) who argue that returns standardized using realized volatility will be thin-tailed in the presence of jumps. 6 Conclusion Prior research has found that daily returns standardized by realized volatility are approximately standard normally distributed, a finding which implies that jumps are not empirically relevant. Our paper, however, provides strong evidence to the contrary. We find that standardized returns are decided thin-tailed, which is consistent with the findings of other recent empirical research that directly tests for and finds evidence of jumps. The key to our results is accounting for the impact of microstructure noise on realized volatility. We demonstrate that, using the standard realized volatility estimator, the variance and kurtosis of standardized returns are highly sensitive to the sampling frequency of the returns used to construct realized volatility. As it turns out, using the popular five-minute sampling frequency, the kurtosis tends to be close to three, the theoretical value in the absence of jumps, for most of the stocks. However, as the sampling gets finer, the kurtosis becomes substantially greater than three and the variance of standardized returns decreases consistently. Since the latter effect cannot be attributed to the small sample effects in standardized returns documented by Peters and de Vilder (24) and Andersen, Bollerslev and Dobrev (26), we conclude that the results are driven by biases induced by microstructure noise. We then construct standardized returns using a kernelbased estimator of realized volatility that is (nearly) unbiased. Now, having removed the biases induced by microstructure noise, we find that the standardized returns are clearly platykurtotic, and we can soundly reject the null of normality. Moreover, these results are insensitive to the choice of sampling frequency. The mixture of normals hypothesis places important restrictions on the stock price process. The price path must be continuous and the volatility process must evolve

8 independently of the Brownian motion governing returns. Our results suggest that, consistent with findings based on parametric approaches, the first restriction is violated in the data. Acknowledgements We thank Peter Hansen, Nour Meddahi, Barbara Ostdiek, and James Weston for many useful discussions regarding our research in this area. We also thank Scott Baggett for research assistance in constructing the price dataset from TAQ data. References Ait-Sahalia, Y., Mykland, P., and Zhang, L. (25), Ultra-high frequency volatility estimation with dependent microstructure noise, Working paper, National Bureau of Economic Research (w1138). Andersen, T.G., Benzoni, L., and Lund, J. (22), An empirical investigation of continuous-time equity return models, Journal of Finance, 57, Andersen, T.G., Bollerslev, T., and Diebold, F.X. (25), Roughing it up: Including jump components in the measurement, modeling and forecasting of return volatility, Review of Economics and Statistics, Forthcoming. Andersen, T.G., Bollerslev, T., Diebold, F.X., and Ebens, H. (21), The distribution of realized stock return volatility, Journal of Financial Economics, 61, Andersen, T.G., Bollerslev, T., Diebold, F.X., and Labys, P. (2), Great realizations, Risk Magazine, 13, Andersen, T.G., Bollerslev, T., Diebold, F.X., and Labys, P. (21), The distribution of realized exchange rate volatility, Journal of the American Statistical Association, 96, Andersen, T.G., Bollerslev, T., Diebold, F.X., and Labys, P. (23), Modeling and forecasting realized volatility, Econometrica, 71, Andersen, T.G., Bollerslev, T., and Dobrev, D. (26), Noarbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects and jumps: Theory and testable distributional implications, Journal of Econometrics, Forthcoming. Back, K. (1991), Asset prices for general processes, Journal of Mathematical Economics, 2, Bandi, F.M., and Russell, J.R. (25), Microstructure noise, realized variance, and optimal sampling, Working paper, University of Chicago. Bandi, F.M., and Russell, J.R. (26), Separating market microstructure noise from volatility, Journal of Financial Economics, 79, Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., and Shephard, N. (26), Regular and modified kernel-based estimators of integrated variance: The case with independent noise, Working paper, Oxford University. Barndorff-Nielsen, O.E., and Shephard, N. (21), Non-Gaussian Ornstein Uhlenbeck-based models and some of their uses in financial economics (with discussion), Journal of the Royal Statistical Society, Series B, 63, Barndorff-Nielsen, O.E., and Shephard, N. (22), Econometric analysis of realized volatility and its use in estimating stochastic volatility models, Journal of the Royal Statistical Society, Series B, 64, Barndorff-Nielsen, O.E., and Shephard, N. (25), Variation, jumps, market frictions and high-frequency data in financial econometrics, Working paper, Oxford University. Barndorff-Nielsen, O.E., and Shephard, N. (26), Econometrics of testing for jumps in financial economics using bipower variation, Journal of Financial Econometrics, 4, 1 3. Chernov, M., Gallant, A.R., Ghysels, E., and Tauchen, G. (23), Alternative models for stock price dynamics, Journal of Econometrics, 116, Fleming, J., and Paye, B.S. (26), High-frequency returns, jumps and the mixture of normals hypothesis, Working paper, Rice University. Hansen, P.R., and Lunde, A. (23), An optimal and unbiased measure of realized variance based on intermittent highfrequency data, Mimeo prepared for the CIREQ-CIRANO Conference: Realized Volatility. Hansen, P.R., and Lunde, A. (26), Realized variance and market microstructure noise, Journal of Business and Economic Statistics, 24, Huang, X., and Tauchen, G. (25), The relative contribution of jumps to total price variance, Journal of Financial Econometrics, 3, Jiang, G.J., and Oomen, R.C.A. (25), A new test for jumps in asset prices, Working paper, University of Arizona and the University of Warwick. Newey, W.K., and West, K.D. (1987), A simple, positive semidefinite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica, 55, Peters, R.T., and de Vilder, R.G. (24), Testing the diffusion model for the S&P 5, Working paper, University of Amsterdam. Tauchen, G., and Zhou, H. (25), Identifying realized jumps on financial markets, Working paper, Duke University and Board of Governors of the Federal Reserve. Zhang, L. (24), Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach, Working paper, Carnegie Mellon University. Zhang, L., Mykland, P.A. and Ait-Sahalia, Y. (25), A tale of two time scales: Determining integrated volatility with noisy high-frequency data, Journal of the American Statistical Association, 1, Zhou, B. (1996), High-frequency data and volatility in foreignexchange rates, Journal of Business and Economic Statistics, 14,