How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise

Transcription

1 How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise Yacine Aït-Sahalia Princeton University and NBER Per A. Mykland The University of Chicago Lan Zhang Carnegie Mellon University In theory, the sum of squares of log returns sampled at high frequency estimates their variance. When market microstructure noise is present but unaccounted for, however, we show that the optimal sampling frequency is finite and derives its closed-form expression. But even with optimal sampling, using say 5-min returns when transactions are recorded every second, a vast amount of data is discarded, in contradiction to basic statistical principles. We demonstrate that modeling the noise and using all the data is a better solution, even if one misspecifies the noise distribution. So the answer is: sample as often as possible. Over the past few years, price data sampled at very high frequency have become increasingly available in the form of the Olsen dataset of currency exchange rates or the TAQ database of NYSE stocks. If such data were not affected by market microstructure noise, the realized volatility of the process (i.e., the average sum of squares of log-returns sampled at high frequency) would estimate the returns variance, as is well known. In fact, sampling as often as possible would theoretically produce in the limit a perfect estimate of that variance. We start by asking whether it remains optimal to sample the price process at very high frequency in the presence of market microstructure noise, consistently with the basic statistical principle that, ceteris paribus, more data are preferred to less. We first show that, if noise is present but unaccounted for, then the optimal sampling frequency is finite, and we We are grateful for comments and suggestions from the editor, Maureen O Hara, and two anonymous referees, as well as seminar participants at Berkeley, Harvard, NYU, MIT, Stanford, the Econometric Society and the Joint Statistical Meetings. Financial support from the NSF under grants SBR-4 (Aït-Sahalia), DMS-4639 (Mykland and Zhang), and the NIH under grant RO AG34- (Zhang) is also gratefully acknowledged. Addresscorrespondenceto: Yacine Aït-Sahalia, BendheimCenter for Finance, Princeton University, Princeton, NJ 854, (69) or [email protected]. The Review of Financial Studies Vol. 8, No. ª 5 The Society for Financial Studies; all rights reserved. doi:.93/rfs/hhi6 Advance Access publication February, 5

2 The Review of Financial Studies / v 8 n 5 derive a closed-form formula for it. The intuition for this result is as follows. The volatility of the underlying efficient price process and the market microstructure noise tend to behave differently at different frequencies. Thinking in terms of signal-to-noise ratio, a log-return observed from transaction prices over a tiny time interval is mostly composed of market microstructure noise and brings little information regarding the volatility of the price process since the latter is (at least in the Brownian case) proportional to the time interval separating successive observations. As the time interval separating the two prices in the log-return increases, the amount of market microstructure noise remains constant, since each price is measured with error, while the informational content of volatility increases. Hence very high frequency data are mostly composed of market microstructure noise, while the volatility of the price process is more apparent in longer horizon returns. Running counter to this effect is the basic statistical principle mentioned above: in an idealized setting where the data are observed without error, sampling more frequently can be useful. What is the right balance to strike? What we show is that these two effects compensate each other and result in a finite optimal sampling frequency (in the root mean squared error sense) so that some time aggregation of the returns data is advisable. By providing a quantitative answer to the question of how often one should sample, we hope to reduce the arbitrariness of the choices that have been made in the empirical literature using high frequency data: for example, using essentially the same Olsen exchange rate series, these somewhat ad hoc choices range from 5-min intervals [e.g., Andersen et al. (), Barndorff-Nielsen and Shephard (), Gençay et al. () to as long as 3 min [e.g., Andersen et al. (3)]. When calibrating our analysis to the amount of microstructure noise that has been reported in the literature, we demonstrate how the optimal sampling interval should be determined: for instance, depending upon the amount of microstructure noise relative to the variance of the underlying returns, the optimal sampling frequency varies from 4 min to 3 h, if day s worth of data are used at a time. If a longer time period is used in the analysis, then the optimal sampling frequency can be considerably longer than these values. But even if one determines the sampling frequency optimally, the fact remains that the empirical researcher is not making full use of the data at his disposal. For instance, suppose that we have available transaction records on a liquid stock, traded once every second. Over a typical 6.5 h day, we therefore start with 3,4 observations. If one decides to sample once every 5 minutes, then whether or not this is the optimal sampling frequency it amounts to retaining only 78 observations. Stated differently, one is throwing away 99 out of every 3 transactions. From a statistical perspective, this is unlikely to be the optimal solution, even 35

3 Sampling and Market Microstructure Noise though it is undoubtedly better than computing a volatility estimate using noisy squared log-returns sampled every second. Somehow, an optimal solution should make use of all the data, and this is where our analysis is headed next. So, if one decides to account for the presence of the noise, how should one proceed? We show that modeling the noise term explicitly restores the first order statistical effect that sampling as often as possible is optimal. This will involve an estimator different from the simple sum of squared log-returns. Since we work within a fully parametric framework, likelihood is the key word. Hence we construct the likelihood function for the observed log-returns, which include microstructure noise. To do so, we must postulate a model for the noise term. We assume that the noise is Gaussian. In light of what we know from the sophisticated theoretical microstructure literature, this is likely to be overly simplistic and one may well be concerned about the effect(s) of this assumption. Could it be more harmful than useful? Surprisingly, we demonstrate that our likelihood correction, based on the Gaussianity of the noise, works even if one misspecifies the assumed distribution of the noise term. Specifically, if the econometrician assumes that the noise terms are normally distributed when in fact they are not, not only is it still optimal to sample as often as possible (unlike the result when no allowance is made for the presence of noise), but the estimator has the same variance as if the noise distribution had been correctly specified. This robustness result is, we think, a major argument in favor of incorporating the presence of the noise when estimating continuous time models with high frequency financial data, even if one is unsure about the true distribution of the noise term. In other words, the answer to the question we pose in our title is as often as possible, provided one accounts for the presence of the noise when designing the estimator (and we suggest maximum likelihood as a means of doing so). If one is unwilling to account for the noise, then one has to rely on the finite optimal sampling frequency we start our analysis with. However, we stress that while it is optimal if one insists upon using sums of squares of log-returns, this is not the best possible approach to estimate volatility given the complete high frequency dataset at hand. In a companion paper [Zhang, Mykland, and Aït-Sahalia (3)], we study the corresponding nonparametric problem, where the volatility of the underlying price is a stochastic process, and nothing else is known about it, in particular no parametric structure. In that case, the object of interest is the integrated volatility of the process over a fixed time interval, such as a day, and we show how to estimate it using again all the data available (instead of sparse sampling at an arbitrarily lower frequency of, say, 5 min). Since the model is nonparametric, we no longer use a likelihood approach but instead propose a solution based on subsampling and averaging, which involves estimators constructed on two different time 353

4 The Review of Financial Studies / v 8 n 5 scales, and demonstrate that this again dominates sampling at a lower frequency, whether arbitrarily or optimally determined. This article is organized as follows. We start by describing in Section our reduced form setup and the underlying structural models that support it. We then review in Section the base case where no noise is present, before analyzing in Section 3 the situation where the noise is ignored. In Section 4, we examine the concrete implications of this result for empirical work with high frequency data. Next, we show in Section 5 that accounting for the presence of the noise through the likelihood restores the optimality of high frequency sampling. Our robustness results are presented in Section 6 and interpreted in Section 7. In Section 8, we study the same questions when the observations are sampled at random time intervals, which are an essential feature of transaction-level data. We then turn to various extensions and relaxation of our assumptions in Section 9; we added a drift term, then serially correlated and cross-correlated noise respectively. In Section concludes. All proofs are in the appendix.. Setup Our basic setup is as follows. We assume that the underlying process of interest, typically the log-price of a security, is a time-homogeneous diffusion on the real line dx t ¼ mðx t ; uþdt þ sdw t, ðþ where X ¼, W t is a Brownian motion, m(.,.) is the drift function, s the diffusion coefficient, and u the drift parameters, where u Q and s >. The parameter space is an open and bounded set. As usual, the restriction that s is constant is without loss of generality since in the univariate case a one-to-one transformation can always reduce a known specification s(x t ) to that case. Also, as discussed in Aït-Sahalia and Mykland (3), the properties of parametric estimators in this model are quite different depending upon whether we estimate u alone, s alone, or both parameters together. When the data are noisy, the main effects that we describe are already present in the simpler of these three cases, where s alone is estimated, and so we focus on that case. Moreover, in the high frequency context we have in mind, the diffusive component of () is of order (dt) / while the drift component is of order dt only, so the drift component is mathematically negligible at high frequencies. This is validated empirically: including a drift which actually deteriorates the performance of variance estimates from high frequency data since the drift is estimated with a large standard error. Not centering the log returns for the purpose of variance estimation produces more accurate results [see Merton (98)]. So we simplify the analysis one step further by setting m ¼, which we do 354

5 Sampling and Market Microstructure Noise until Section 9., where we then show that adding a drift term does not alter our results. In Section 9.4, we discuss the situation where the instantaneous volatility s is stochastic. But for now, X t ¼ sw t : ðþ Until Section 8, we treat the case where our observations occur at equidistant time intervals D, in which case the parameter s is estimated at time T on the basis of N þ discrete observations recorded at times t ¼, t ¼ D,...,t N ¼ N D ¼ T. In Section 8, we let the sampling intervals themselves be random variables, since this feature is an essential characteristic of high frequency transaction data. The notion that the observed transaction price in high frequency financial data is the unobservable efficient price plus some noise component due to the imperfections of the trading process is a well established concept in the market microstructure literature [see, for instance Black (986)]. So, we depart from the inference setup previously studied [Aït-Sahalia and Mykland (3)] and we now assume that, instead of observing the process X at dates t i, we observe X with error: ~X ti ¼ X ti þ U ti, ð3þ where the U ti s are i.i.d. noise with mean zero and variance a and are independent of the W process. In this context, we view X as the efficient log-price, while the observed ~X is the transaction log-price. In an efficient market, X t is the log of the expectation of the final value of the security conditional on all publicly available information at time t. It corresponds to the log-price that would be, in effect, in a perfect market with no trading imperfections, frictions, or informational effects. The Brownian motion W is the process representing the arrival of new information, which in this idealized setting is immediately impounded in X. By contrast, U t summarizes the noise generated by the mechanics of the trading process. We view the source of noise as a diverse array of market microstructure effects, either information or non-information related, such as the presence of a bid-ask spread and the corresponding bounces, the differences in trade sizes and the corresponding differences in representativeness of the prices, the different informational content of price changes owing to informational asymmetries of traders, the gradual response of prices to a block trade, the strategic component of the order flow, inventory control effects, the discreteness of price changes in markets that are not decimalized, etc., all summarized into the term U. That these phenomena are real and important and this is an accepted fact in the market microstructure literature, both theoretical and empirical. One can in fact argue that these phenomena justify this literature. 355

6 The Review of Financial Studies / v 8 n 5 We view Equation (3) as the simplest possible reduced form of structural market microstructure models. The efficient price process X is typically modeled as a random walk, that is, the discrete time equivalent of Equation (). Our specification coincides with that of Hasbrouck (993), who discusses the theoretical market microstructure underpinnings of such a model and argues that the parameter a is a summary measure of market quality. Structural market microstructure models do generate Equation (3). For instance, Roll (984) proposes a model where U is due entirely to the bid-ask spread. Harris (99b) notes that in practice there are sources of noise other than just the bid-ask spread, and studies their effect on the Roll model and its estimators. Indeed, a disturbance U can also be generated by adverse selection effects as in Glosten (987) and Glosten and Harris (988), where the spread has two components: one that is owing to monopoly power, clearing costs, inventory carrying costs, etc., as previously, and a second one that arises because of adverse selection whereby the specialist is concerned that the investor on the other side of the transaction has superior information. When asymmetric information is involved, the disturbance U would typically no longer be uncorrelated with the W process and would exhibit autocorrelation at the first order, which would complicate our analysis without fundamentally altering it: see Sections 9. and 9.3 where we relax the assumptions that the Us are serially uncorrelated and independent of the W process. The situation where the measurement error is primarily due to the fact that transaction prices are multiples of a tick size (i.e., ~X ti ¼ m i k where k is the tick size and m i is the integer closest to X ti /k) can be modeled as a rounding off problem [see Gottlieb and Kalay (985), Jacod (996), Delattre and Jacod (997)]. The specification of the model in Harris (99a) combines both the rounding and bid-ask effects as the dual sources of the noise term U. Finally, structural models, such as that of Madhavan, Richardson, and Roomans (997), also give rise to reduced forms where the observed transaction price ~X takes the form of an unobserved fundamental value plus error. With Equation (3) as our basic data generating process, we now turn to the questions we address in this article: how often should one sample a continuous-time process when the data are subject to market microstructure noise, what are the implications of the noise for the estimation of the parameters of the X process, and how should one correct for the presence of the noise, allowing for the possibility that the econometrician misspecifies the assumed distribution of the noise term, and finally allowing for the sampling to occur at random points in time? We proceed from the simplest to the most complex situation by adding one extra layer of complexity at a time: Figure shows the three sampling schemes we consider, starting with fixed sampling without market 356

7 Sampling and Market Microstructure Noise Figure Various discrete sampling modes no noise (Section ), with noise (Sections 3 7) and randomly spaced with noise (Section 8) microstructure noise, then moving to fixed sampling with noise and concluding with an analysis of the situation where transaction prices are not only subject to microstructure noise but are also recorded at random time intervals. 357

8 The Review of Financial Studies / v 8 n 5. The Baseline Case: No Microstructure Noise We start by briefly reviewing what would happen in the absence of market microstructure noise, that is when a ¼. With X denoting the log-price, the first differences of the observations are the log-returns Y i ¼ ~X ti ~X ti, i ¼,..., N. The observations Y i ¼ s(w tiþ W ti ) are then i.i.d. N(, s D) so the likelihood function is Y l s ¼N ln ps D = s D Y, ð4þ where Y ¼ (Y,...,Y N ). The maximum-likelihood estimator of s coincides with the discrete approximation to the quadratic variation of the process ^s ¼ Yi T, ð5þ i¼ which has the following exact small sample moments: E ^s X N ¼ EYi N s D ¼ ¼ s, T T i¼ " # var ^s XN ¼ var Y T i ¼ X N T var Y! i ¼ N T s4 D s 4 D ¼ T i¼ i¼ and the following asymptotic distribution T = ^s s! Nð, vþ, ð6þ T! where v ¼ avar ^s h ¼ DE l s i ¼ s 4 D: ð7þ Thus selecting D as small as possible is optimal for the purpose of estimating s. X N 3. When the Observations are Noisy but the Noise is Ignored Suppose now that market microstructure noise is present but the presence of the Us is ignored when estimating s. In other words, we use the log-likelihood function (4) even though the true structure of the observed log-returns Y i s is given by an MA() process since Y i ¼ ~X ti ~X ti ¼ X ti X ti þ U ti U ti ¼ sðw ti W ti ÞþU ti U ti «i þ h«i, ð8þ 358

9 Sampling and Market Microstructure Noise where the «i s are uncorrelated with mean zero and variance g (if the Us are normally distributed, then the «i s are i.i.d.). The relationship to the original parametrization (s, a ) is given by g þ h ¼ var ½ Yi Š ¼ s D þ a, ð9þ g h ¼ covðy i, Y i Þ ¼ a : ðþ Equivalently, the inverse change of variable is given by g ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ s D þ s Dð4a þ s DÞ, ðþ h ¼ a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a s D þ s Dð4a þ s DÞ : ðþ Two important properties of the log-returns Y i s emerge from Equations (9) and (). First, it is clear from Equation (9) that microstructure noise leads to spurious variance in observed log-returns, s D þ a versus s D. This is consistent with the predictions of theoretical microstructure models. For instance, Easley and O Hara (99) develop a model linking the arrival of information, the timing of trades, and the resulting price process. In their model, the transaction price will be a biased representation of the efficient price process, with a variance that is both overstated and heteroskedastic due to the fact that transactions (hence the recording of an observation on the process ~X) occur at intervals that are timevarying. While our specification is too simple to capture the rich joint dynamics of price and sampling times predicted by their model, heteroskedasticity of the observed variance will also appear in our case once we allow for time variation of the sampling intervals (see Section 8). In our model, the proportion of the total return variance that is market microstructure-induced is a p ¼ s D þ a ð3þ at observation interval D. AsD gets smaller, p gets closer to, so that a larger proportion of the variance in the observed log-return is driven by market microstructure frictions, and correspondingly a lesser fraction reflects the volatility of the underlying price process X. Second, Equation () implies that < h <, so that logreturns are (negatively) autocorrelated with first order autocorrelation a /(s D þ a ) ¼p/. It has been noted that market microstructure noise has the potential to explain the empirical autocorrelation of returns. For instance, in the simple Roll model, U t ¼ (s/)q t where s is the bid/ask 359

10 The Review of Financial Studies / v 8 n 5 spread and Q t, the order flow indicator, is a binomial variable that takes the values þ and with equal probability. Therefore, var[u t ] ¼ a ¼ s /4. Since cov(y i,y i ) ¼a, the bid/ask spread can be recovered in this p model as s ¼ ffiffiffiffiffiffi r where r ¼ g h is the first-order autocorrelation of returns. French and Roll (986) proposed to adjust variance estimates to control for such autocorrelation and Harris (99b) studied the resulting estimators. In Sias and Starks (997), U arises because of the strategic trading of institutional investors which is then put forward as an explanation for the observed serial correlation of returns. Lo and MacKinlay (99) show that infrequent trading has implications for the variance and autocorrelations of returns. Other empirical patterns in high frequency financial data have been documented: leptokurtosis, deterministic patterns, and volatility clustering. Our first result shows that the optimal sampling frequency is finite when noise is present but unaccounted for. The estimator ^s obtained from maximizing the misspecified log-likelihood function (4) is quadratic in the Y i s [see Equation (5)]. In order to obtain its exact (i.e., small sample) variance, we need to calculate the fourth order cumulants of the Y i s since cov Yi, Y j ¼ cov Y i, Y j þ cum Yi, Y i, Y j, Y j ð4þ (see, e.g., Section.3 of McCullagh (987) for definitions and properties of the cumulants). We have the following lemma. Lemma. The fourth cumulants of the log-returns are given by cum Y i,y j,y k,y l 8 >< cum 4 ½UŠ, if i ¼ j ¼ k ¼ l, ¼ ðþ si;j;k;l ð Þ cum 4 ½UŠ, if maxði, j,k, lþ¼minði, j, k, lþþ, ð5þ >:, otherwise, where s(i, j, k, l) denotes the number of indices among (i, j, k, l) that are equal to min(i, j, k, l) and U denotes a generic random variable with the common distribution of the U ti s. Its fourth cumulant is denoted cum 4 [U]. Now U has mean zero, so in terms of its moments cum 4 ½UŠ ¼ EU 4 3 EU : ð6þ In the special case where U is normally distributed, cum 4 [U] ¼ and as a result of Equation (4) the fourth cumulants of the log-returns are all (since W is normal, the log-returns are also normal in that case). If the distribution of U is binomial as in the simple bid/ask model described above, then cum 4 [U ] ¼s 4 /8; since in general s will be a tiny percentage of the asset price, say s ¼.5%, the resulting cum 4 [U ] will be very small. 36

11 Sampling and Market Microstructure Noise We can now characterize the root mean squared error RMSE ^s ¼ E ^s s = þ var ^s of the estimator by the following theorem. Theorem. In small samples (finite T), the bias and variance of the estimator ^s are given by E ^s s ¼ a D, ð7þ var ^s s 4 D þ 4s Da þ 6a 4 þ cum 4 ½UŠ ¼ a4 þ cum 4 ½UŠ TD T : ð8þ Its RMSE has a unique minimum in D which is reached at the optimal sampling interval sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi =3 =3 3 D ¼ a4 T B s 4 3a4 þ cum 4 A 7s 4 a 8 T sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A=3 3 þ þ 3a4 þ cum 4 C A: ð9þ 7s 4 a 8 T As T grows, we have D ¼ =3 a 4=3 T =3 þ O : ðþ s 4=3 T =3 The trade-off between bias and variance made explicit in Equations (7) and (8) is not unlike the situation in nonparametric estimation with D playing the role of the bandwidth h. A lower h reduces the bias but increases the variance, and the optimal choice of h balances the two effects. Note that these are exact small sample expressions, valid for all T. Asymptotically in T, var½^s Š!, and hence the RMSE of the estimator is dominated by the bias term which is independent of T. And given the form of the bias (7), one would in fact want to select the largest D possible to minimize the bias (as opposed to the smallest one as in the no-noise case of Section ). The rate at which D should increase with T is given by Equation (). Also, in the limit where the noise disappears (a! and cum 4 [U ]! ), the optimal sampling interval D tends to. 36

12 The Review of Financial Studies / v 8 n 5 How does a small departure from a normal distribution of the microstructure noise affect the optimal sampling frequency? The answer is that a small positive (resp. negative) departure of cum 4 [U] starting from the normal value of leads to an increase (resp. decrease) in D, since r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! =3 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! þ a4 T s 4 a4 A T s 4 D ¼ D normal þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cum 4 ½UŠ 3 =3 a 4=3 T =3 a4 T s 4 s 8=3 þ O cum 4 ½UŠ, ðþ where D normal is the value of D corresponding to Cum 4 [U] ¼. And, of course, the full formula (9) can be used to get the exact answer for any departure from normality instead of the comparative static one. Another interesting asymptotic situation occurs if one attempts to use higher and higher frequency data (D!, say sampled every minute) over a fixed time period (T fixed, say a day). Since the expressions in Theorem are exact small sample ones, they can in particular be specialized to analyze this situation. With n ¼ T/D, it follows from Equations (7) and (8) that E ^s na ¼ T var ^s ð Þ ¼ ne U þ on ð Þ, T ½ Š þ on ð Þ ¼ 4nE U 4 þ on ¼ n 6a 4 þ cum 4 U T T þ on ð Þ ðþ ð3þ so ðt=nþ^s becomes an estimator of E [U ] ¼ a whose asymptotic variance is E [U 4 ]. Note in particular that ^s estimates the variance of the noise, which is essentially unrelated to the object of interest s. This type of asymptotics is relevant in the stochastic volatility case we analyze in our companion paper [Zhang, Mykland, and Aït-Sahalia (3)]. Our results also have implications for the two parallel tracks that have developed in the recent financial econometrics literature dealing with discretely observed continuous-time processes. One strand of the literature has argued that estimation methods should be robust to the potential issues arising in the presence of high frequency data and, consequently, be asymptotically valid without requiring that the sampling interval D separating successive observations tend to zero [see, e.g., Hansen and Scheinkman (995), Aït-Sahalia (996), Aït-Sahalia ()]. Another strand of the literature has dispensed with that constraint, and the asymptotic validity of these methods requires that D tend to zero instead of or in addition to, an increasing length of time T over which these observations 36

13 Sampling and Market Microstructure Noise are recorded [see, e.g., Andersen et al. (3), Bandi and Phillips (3), Barndorff-Nielsen and Shephard ()]. The first strand of literature has been informally warning about the potential dangers of using high frequency financial data without accounting for their inherent noise [see, e.g., Aït-Sahalia (996, p. 59)], and we propose a formal modelization of that phenomenon. The implications of our analysis are most important for the second strand of the literature, which is predicated on the use of high frequency data but does not account for the presence of market microstructure noise. Our results show that the properties of estimators based on the local sample path properties of the process (such as the quadratic variation to estimate s ) change dramatically in the presence of noise. Complementary to this are the results of Gloter and Jacod () which show that the presence of even increasingly negligible noise is sufficient to adversely affect the identification of s. 4. Concrete Implications for Empirical Work with High Frequency Data The clear message of Theorem for empirical researchers working with high frequency financial data is that it may be optimal to sample less frequently. As discussed in the Introduction, authors have reduced their sampling frequency below that of the actual record of observations in a somewhat ad hoc fashion, with typical choices 5 min and up. Our analysis provides not only a theoretical rationale for sampling less frequently, but also gives a precise answer to the question of how often one should sample? For that purpose, we need to calibrate the parameters appearing in Theorem, namely s, a, cum 4 [U], D and T. We assume in this calibration exercise that the noise is Gaussian, in which case cum 4 [U ] ¼. 4. Stocks We use existing studies in empirical market microstructure to calibrate the parameters. One such study is Madhavan, Richardson, and Roomans (997), who estimated on the basis of a sample of 74 NYSE stocks that approximately 6% of the total variance of price changes is attributable to market microstructure effects (they report a range of values for p from 54% in the first half hour of trading to 65% in the last half hour, see their Table 4; they also decompose this total variance into components due to discreteness, asymmetric information, transaction costs and the interaction between these effects). Given that their sample contains an average of 5 transactions per hour (their Table ), we have in our framework p ¼ 6%, D ¼ = ð5 7 5Þ: ð4þ These values imply from Equation (3) that a ¼.6% if we assume a realistic value of s ¼ 3% per year. (We do not use their reported 363

14 The Review of Financial Studies / v 8 n 5 Table Optimal sampling frequency T Value of a day year 5 years (a) s ¼ 3% Stocks.% min 4 min 6 min.5% 5 min 3 min 53 min.% min.3 h. h.5% min. h 3.8 h.% 3 min 3.3 h 5.6 h.3% 57 min 5.6 h.5 day.4%.4 h.3 day. days.5% h.7 day.9 days.6%.6 h. days 3.7 days.7% 3.3 h.7 days 4.6 days.8% 4. h 3. days. week.9% 4.9 h 3.8 days.3 week.% 5.9 h 4.3 days.5 week (b) s ¼ % Currencies.5% 4 min 3 min 39 min.% 9 min 58 min.6 h.% 3 min.4 h 4. h.5%.3 h 8. h 4. h.% 3.5 h.7 h.5 day This table reports the optimal sampling frequency D given in Equation 9 for different values of the standard deviation of the noise term a and the length of the sample T. Throughout the table, the noise is assumed to be normally distributed (hence cum 4 [U] ¼ in formula 9). In panel (a), the standard deviation of the efficient price process is s ¼ 3% per year, and at s ¼ % per year in panel (b). In both panels, year ¼ 5 days, but in panel (a), day ¼ 6.5 hours (both the NYSE and NASDAQ are open for 6.5 h from 9:3 to 6: EST), while in panel (b), day ¼ 4 hours as is the case for major currencies. A value of a ¼.5% means that each transaction is subject to Gaussian noise with mean and standard deviation equal to.5% of the efficient price. If the sole source of the noise were a bid/ask spread of size s, then a should be set to s/. For example, a bid/ask spread of cents on a $ stock would correspond to a ¼.5%. For the dollar/euro exchange rate, a bid/ask spread of s ¼.4% translates into a ¼.%. For the bid/ask model, which is based on binomial instead of Gaussian noise, cum 4 [U] ¼s 4 /8, but this quantity is negligible given the tiny size of s. volatility number since they apparently averaged the variance of price changes over the 74 stocks instead of the variance of the returns. Since different stocks have different price levels, the price variances across stocks are not directly comparable. This does not affect the estimated fraction p, however, since the price level scaling factor cancels out between the numerator and the denominator.) The magnitude of the effect is bound to vary by type of security, market and time period. Hasbrouck (993) estimates the value of a to be.33%. Some authors have reported even larger effects. Using a sample of NAS- DAQ stocks, Kaul and Nimalendran (99) estimate that about 5% of the daily variance of returns is due to the bid-ask effect. With s ¼ 4% (NASDAQ stocks have higher volatility), the values p ¼ 5%, D ¼ =5 364

15 Sampling and Market Microstructure Noise yield the value a ¼.8%. Also on NASDAQ, Conrad, Kaul and Nimalendran (99) estimate that % of the variance of weekly returns (see their Table 4, middle portfolio) is due to bid-ask effects. The values p ¼ %, D ¼ =5 imply that a ¼.4%. In Table, we compute the value of the optimal sampling interval D implied by different combinations of sample length (T ) and noise magnitude (a). The volatility of the efficient price process is held fixed at s ¼ 3% in Panel (A), which is a realistic value for stocks. The numbers in the table show that the optimal sampling frequency can be substantially affected by even relatively small quantities of microstructure noise. For instance, using the value a ¼.5% calibrated from Madhavan, Richardson, and Roomans (997), we find an optimal sampling interval of minutes if the sampling length is day; longer sample lengths lead to higher optimal sampling intervals. With the higher value of a ¼.3%, approximating the estimate from Hasbrouck (993), the optimal sampling interval is 57 min. A lower value of the magnitude of the noise translates into a higher frequency: for instance, D ¼ 5 min if a ¼.5% and T ¼ day. Figure displays the RMSE of the estimator as a function of D and T, using parameter values s ¼ 3% and a ¼.5%. The figure illustrates the fact that deviations from the optimal choice of D lead to a substantial increase in the RMSE: for example, with T ¼ month, the RMSE more than doubles if, instead of the optimal D ¼ h, one uses D ¼ 5 min. Figure RMSE of the estimator ^s when the presence of the noise is ignored 365

16 The Review of Financial Studies / v 8 n 5 Table Monte Carlo simulations: bias and variance when market microstructure noise is ignored Sampling interval Theoretical mean Sample mean Theoretical stand. dev. Sample stand. dev. 5 min min min h h day week This table reports the results of M ¼, Monte Carlo simulations of the estimator ^s, with market microstructure noise present but ignored. The column theoretical mean reports the expected value of the estimator, as given in Equation (7) and similarly for the column theoretical standard deviation (the variance is given in Equation (8)). The sample columns report the corresponding moments computed over the M simulated paths. The parameter values used to generate the simulated data are s ¼.3 ¼.9 and a ¼ (.5%) and the length of each sample is T ¼ year. 4. Currencies Looking now at foreign exchange markets, empirical market microstructure studies have quantified the magnitude of the bid-ask spread. For example, Bessembinder (994) computes the average bid/ask spread s in the wholesale market for different currencies and reports values of s ¼.5% for the German mark, and.6% for the Japanese yen (see Panel B of his Table ). We calculated the corresponding numbers for the 996 period to be.4% for the mark (followed by the euro) and.6% for the yen. Emerging market currencies have higher spreads: for instance, s ¼.% for Korea and.% for Brazil. During the same period, the volatility of the exchange rate was s ¼ % for the German mark, % for the Japanese yen, 7% for Brazil and 8% for Korea. In Panel B of Table, we compute D with s ¼ %, a realistic value for the euro and yen. As we noted above, if the sole source of the noise were a bid/ ask spread of size s, then a should be set to s/. Therefore, Panel B reports the values of D for values of a ranging from.% to.%. For example, the dollar/euro or dollar/yen exchange rates (calibrated to s ¼ %, a ¼.%) should be sampled every D ¼ 3 min if the overall sample length is T ¼ day, and every. h if T ¼ year. Furthermore, using the bid/ask spread alone as a proxy for all microstructure frictions will lead, except in unusual circumstances, to an understatement of the parameter a, since variances are additive. Thus, since D is increasing in a, one should interpret the value of D read off on the row corresponding to a ¼ s/ as a lower bound for the optimal sampling interval. 4.3 Monte Carlo Evidence To validate empirically these results, we perform Monte Carlo simulations. We simulate M ¼, samples of length T ¼ year of the process X, add microstructure noise U to generate the observations ~X, and then 366

17 Sampling and Market Microstructure Noise the log returns Y. We sample the log-returns at various intervals D ranging from 5 min to week, and calculate the bias and variance of the estimator ^s over the M simulated paths. We then compare the results to the theoretical values given in Equations (7) and (8) of Theorem. The noise distribution is Gaussian, s ¼ 3% and a ¼.5% the values we calibrated to stock returns data above. Table shows that the theoretical values are in close agreement with the results of the Monte Carlo simulations. Table also illustrates the magnitude of the bias inherent in sampling at too high a frequency. While the value of s used to generate the data is.9, the expected value of the estimator when sampling every 5 min is.8, so on average the estimated quadratic variation is twice as big as it should be in this case. 5. Incorporating Market Microstructure Noise Explicitly So far we have stuck to the sum of squares of log-returns as our estimator of volatility. We then showed that, for this estimator, the optimal sampling frequency is finite. However, this implies that one is discarding a large proportion of the high frequency sample (99 out of every 3 observations in the example described in the introduction), in order to mitigate the bias induced by market microstructure noise. Next, we show that if we explicitly incorporate the Us into the likelihood function, then we are back in a situation where the optimal sampling scheme consists in sampling as often as possible that is, using all the data available. Specifying the likelihood function of the log-returns, while recognizing that they incorporate noise, requires that we take a stand on the distribution of the noise term. Suppose for now that the microstructure noise is normally distributed, an assumption whose effect we will investigate below in Section 6. Under this assumption, the likelihood function for the Ys is given by l h, g ¼ln detðvþ= N ln pg = g Y V Y, ð5þ where the covariance matrix for the vector Y ¼ (Y,...,Y N ) is given by g V, where þ h h h þ h h».. V ¼ v ij i; j¼;... ;N ¼ h þ h» ð6þ B. h A h þ h 367

18 The Review of Financial Studies / v 8 n 5 Further, detðvþ ¼ hnþ h ð7þ and, neglecting the end effects, an approximate inverse of V is the matrix V ¼ [v ij ] i, j¼,..., N where v ij ¼ h ðhþ jijj [see Durbin (959)]. The product VV differs from the identity matrix only on the first and last rows. The exact inverse is V ¼ [v ij ] i, j ¼,...,N where v ij ¼ h h Nþ n ðhþ jijj ðhþ iþj ðhþ Nijþ h ð Þ Nþjijjþ þðhþ Nþijþ þðhþ Niþjþo ð8þ [see Shaman (969), Haddad (995)]. From the perspective of practical implementation, this estimator is nothing else than the MLE estimator of an MA() process with Gaussian errors: any existing computer routines for the MA() situation can, therefore, be applied [see e.g., Hamilton (995, Section 5.4)]. In particular, the likelihood function can be expressed in a computationally efficient form by triangularizing the matrix V, yielding the equivalent expression: l h, g ¼ X N i¼ lnðpd i Þ X N i¼ ~Y i d i, where d i ¼ g þ h þþh i þ h þ þh ði Þ and the ~Y i s are obtained recursively as ~Y ¼ Y and for i ¼,..., N: ~Y i ¼ Y i h þ h þ þh ði Þ ~Y þ h þ þh ði Þ i : ð9þ This latter form of the log-likelihood function involves only single sums as opposed to double sums if one were to compute Y V Y by brute force using the expression of V given above. We now compute the distribution of the MLE estimators of s and a, which follows by the delta method from the classical result for the MA() estimators of g and h by the following proposition. 368

19 Sampling and Market Microstructure Noise Proposition. When U is normally distributed, the MLE ð^s, ^a Þ is consistent and its asymptotic variance is given by avar normal ^s, ^a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 s 6 Dð4a þ s DÞþ s 4 D s Dh D, s, a D A, a þ s D h D, s, a with h D, s, a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ s Dð4a þ s DÞþ s D: ð3þ Since avar normal ð^s Þ is increasing in D, it is optimal to sample as often as possible. Further, since avar normal ^s ¼ 8s 3 ad = þ s 4 D þ oðdþ, ð3þ the loss of efficiency relative to the case where no market microstructure noise is present (and, if a ¼ is not estimated, avarð^s Þ¼s 4 D as given in Equation (7), or if a ¼ is estimated, avar(s)¼6^s 4 d) is at order D /. Figure 3 plots the asymptotic variances of ^s as functions of D with and without noise (the parameter values are again s ¼ 3% and a ¼.5%). Figure 4 reports histograms of the distributions of ^s and ^a from, Monte Carlo simulations with the solid curve plotting the asymptotic distribution of the estimator from Proposition. The sample path is of length T ¼ year, the parameter values are the same as above, and the Figure 3 Comparison of the asymptotic variances of the MLE ^s without and with noise taken into account 369

20 The Review of Financial Studies / v 8 n 5 Figure 4 Asymptotic and Monte Carlo distributions of the MLE (^s, ^a ) with Gaussian microstructure noise process is sampled every 5 min since we are now accounting explicitly for the presence of noise, there is no longer a reason to sample at lower frequencies. Indeed, the figure documents the absence of bias and the good agreement of the asymptotic distribution with the small sample one. 6. The Effect of Misspecifying the Distribution of the Microstructure Noise We now study a situation where one attempts to incorporate the presence of the Us into the analysis, as in Section 5, but mistakenly assumes a 37

21 Sampling and Market Microstructure Noise misspecified model for them. Specifically, we consider the case where the Us are assumed to be normally distributed when in reality they have a different distribution. We still suppose that the Us are i.i.d. with mean zero and variance a. Since the econometrician assumes the Us to have a normal distribution, inference is still done with the log-likelihood l(s, a ), or equivalently l(h, g ) given in Equation (5), using Equations (9) and (). This means that the scores l _ s and l _ a, or equivalently Equations (C.) and (C.) are used as moment functions (or estimating equations ). Since the first order moments of the moment functions only depend on the second order moment structure of the log-returns (Y,...,Y N ), which is unchanged by the absence of normality, the moment functions are unbiased under the true distribution of the Us: h i h i lh _ ¼ E true lg _ ¼ ð3þ E true and similarly for l _ s and l _ a. Hence the estimator ð^s, ^a Þ based on these moment functions is consistent and asymptotically unbiased (even though the likelihood function is misspecified). The effect of misspecification, therefore, lies in the asymptotic variance matrix. By using the cumulants of the distribution of U, we express the asymptotic variance of these estimators in terms of deviations from normality. But as far as computing the actual estimator, nothing has changed relative to Section 5: we are still calculating the MLE for an MA() process with Gaussian errors and can apply exactly the same computational routine. However, since the error distribution is potentially misspecified, one could expect the asymptotic distribution of the estimator to be altered. This does not happen, as far as ^s is concerned: see the following theorem. Theorem. The estimators ð^s, ^a Þ obtained by maximizing the possibly misspecified log-likelihood function (5) are consistent and their asymptotic variance is given by avar true ^s, ^a ¼ avarnormal ^s, ^a þ cum4 ½UŠ, ð33þ D where avar normal ð^s, ^a Þ is the asymptotic variance in the case where the distribution of U is normal, that is, the expression given in Proposition. In other words, the asymptotic variance of ^s is identical to its expression if the Us had been normal. Therefore, the correction we proposed for the presence of market microstructure noise relying on the assumption that the noise is Gaussian is robust to misspecification of the error distribution. 37

22 The Review of Financial Studies / v 8 n 5 Documenting the presence of the correction term through simulations presents a challenge. At the parameter values calibrated to be realistic, the order of magnitude of a is a few basis points, say a ¼.% ¼ 3. But if U if of order 3,cum 4 [U] which is of the same order as U 4, is of order. In other words, with a typical noise distribution, the correction term in Equation (33) will not be visible. Nevertheless, to make it discernible, we use a distribution for U with the same calibrated standard deviation a as before, but a disproportionately large fourth cumulant. Such a distribution can be constructed by letting U ¼ vt n where v > is constant and T n is a Student t distribution with v degrees of freedom. T n has mean zero, finite variance as long as v > and finite fourth moment (hence finite fourth cumulant) as long as v > 4. But as v approaches 4 from above, E½Tn 4 Š tends to infinity. This allows us to produce an arbitrarily high value of cum 4 [U] while controlling for the magnitude of the variance. The specific expressions of a and cum 4 [U] for this choice of U are given by a ¼ var½uš ¼ v n n, ð34þ 6v 4 n cum 4 ½UŠ ¼ ðn 4Þðn Þ : ð35þ Thus, we can select the two parameters (v, n) to produce desired values of (a,cum 4 [U ]). As before, we set a ¼.5%. Then, given the form of the asymptotic variance matrix Equation (33), we set cum 4 [U ] so that cum 4 ½UŠD ¼ avar normal ð^a Þ=. This makes avar true ð^a Þ by construction 5% larger than avar normal ð^a Þ. The resulting values of (v, n) from solving Equations (34) and (35) are v ¼.5 and v ¼ As above, we set the other parameters to s ¼ 3%, T ¼ year, and D ¼ 5 minutes. Figure 5 reports histograms of the distributions of ^s and ^a from, Monte Carlo simulations. The solid curve plots the asymptotic distribution of the estimator, given now by Equation (33). There is again good adequacy between the asymptotic and small sample distributions. In particular, we note that as predicted by Theorem, the asymptotic variance of ^s is unchanged relative to Figure 4 while that of ^a is 5% larger. The small sample distribution of ^s appears unaffected by the non-gaussianity of the noise; with a skewness of.7 and a kurtosis of.95, it is closely approximated by its asymptotic Gaussian limit. The small sample distribution of ^a does exhibit some kurtosis (4.83), although not large relative to that of the underlying noise distribution (the values of v and n imply a kurtosis for U of 3 þ 6/(n 4) ¼ ). Similar simulations but with a longer time span of T ¼ 5 years are even closer to the Gaussian asymptotic limit: the kurtosis of the small sample distribution of ^a goes down to

23 Sampling and Market Microstructure Noise Figure 5 Asymptotic and Monte Carlo distributions of the QMLE (^s, ^a ) with misspecified microstructure noise 7. Robustness to Misspecification of the Noise Distribution Going back to the theoretical aspects, the above Theorem has implications for the use of the Gaussian likelihood l that go beyond consistency, namely that this likelihood can also be used to estimate the distribution of ^s under misspecification. With l denoting the log-likelihood assuming that the Us are Gaussian, given in Equation (5), lð^s, ^a Þ denote the observed information matrix in the original parameters s and a. Then ^V ¼ davar normal ¼ T l ^s, ^a 373

24 The Review of Financial Studies / v 8 n 5 is the usual estimate of asymptotic variance when the distribution is correctly specified as Gaussian. Also note, however, that otherwise, so long as ð^s, ^a Þ is consistent, ^V is also a consistent estimate of the matrix avar normal ð^s, ^a Þ. Since this matrix coincides with avar true ð^s, ^a Þ for all but the (a, a ) term (see Equation (33)), the asymptotic variance of T = ð^s s Þ is consistently estimated by ^V s s. The similar statement is true for the covariances, but not, obviously, for the asymptotic variance of T = ð^a a Þ. In the likelihood context, the possibility of estimating the asymptotic variance by the observed information is due to the second Bartlett identity. For a general log likelihood l,ifs E true ½ l _ l _ Š=N and D E true ½ lš=n (differentiation refers to the original parameters (s, a ), not the transformed parameters (g, h)) this identity says that S D ¼ : ð36þ It implies that the asymptotic variance takes the form avar ¼ D DS ¼ D DD : ð37þ It is clear that Equation (37) remains valid if the second Bartlett identity holds only to first order, that is, S D ¼ oðþ ð38þ as N!, for a general criterion function l which satisfies E true ½ _ lš¼oðnþ. However, in view of Theorem, Equation (38) cannot be satisfied. In fact, we show in Appendix E that where g ¼ g s ¼ g a S D ¼ cum 4 ½UŠgg þ oðþ, a 4 D = sð4a þ s DÞ 3= D ð = s 6a þ s D ð4a þ s DÞ 3= C Þ A : ð39þ ð4þ From Equation (4), we see that g 6¼ whenever s >. This is consistent with the result in Theorem that the true asymptotic variance matrix, avar true ð^s, ^a Þ; does not coincide with the one for Gaussian noise, avar normal ð^s, ^a Þ. On the other hand, the matrix gg is of rank, signaling that there exist linear combinations that will cancel out the first column of S D. From what we already know of the form of the correction matrix, D gives such a combination that the asymptotic variance of the original parameters (s, a ) will have the property that its first column is not subject to correction in the absence of normality. 374

25 Sampling and Market Microstructure Noise A curious consequence of Equation (39) is that while the observed information can be used to estimate the asymptotic variance of ^s when a is not known, this is not the case when a is known. This is because the second Bartlett identity also fails to first order when considering a to be known, that is, when differentiating with respect to s only. Indeed, in that case we have from the upper left component in the matrix Equation (39): h S s s D s s ¼ N E true i s s s, a i h þ N E true ls s s, a i ¼ cum 4 ½UÞðg s Þ þ oðþ, which is not o() unless cum 4 [U ] ¼. To make the connection between Theorem and the second Bartlett identity, one needs to go to the log profile likelihood ls sup l s, a : ð4þ a Obviously, maximizing the likelihood l(s, a ) is the same as maximizing l(s ). Thus one can think of s as being estimated (when a is unknown) by maximizing the criterion function l(s ), or by solving _lð^s Þ¼. Also, the observed profile information is related to the original observed information by l ^s ¼ h l ^s, ^a i s s, ð4þ that is, the first (upper left hand corner) component of the inverse observed information in the original problem. We explain this in Appendix E, where we also show that E true ½ _lš ¼oðNÞ. In view of Theorem, lð^s Þ can be used to estimate the asymptotic variance of ^s under the true (possibly non-gaussian) distribution of the Us, and so it must be that the criterion function l satisfies Equation (38), that is h N E true _ls i þ N E true ls ¼ oðþ: ð43þ This is indeed the case, as shown in Appendix E. This phenomenon is related, although not identical, to what occurs in the context of quasi-likelihood [for comprehensive treatments of quasilikelihood theory, see the books by McCullagh and Nelder (989) and Heyde (997), and the references therein, and for early econometrics examples, see Macurdy (98) and White (98)]. In quasi-likelihood situations, one uses a possibly incorrectly specified score vector which is 375

26 The Review of Financial Studies / v 8 n 5 nevertheless required to satisfy the second Bartlett identity. What makes our situation unusual relative to quasi-likelihood is that the interest parameter s and the nuisance parameter a are entangled in the same estimating equations ( _ l s and _ l a from the Gaussian likelihood) in such a way that the estimate of s depends, to first order, on whether a is known or not. This is unlike the typical development of quasi-likelihood, where the nuisance parameter separates out [see, e.g., McCullagh and Nelder (989, Table 9., p. 36)]. Thus only by going to the profile likelihood l can one make the usual comparison to quasi-likelihood. 8. Randomly Spaced Sampling Intervals One essential feature of transaction data in finance is that the time that separates successive observations is random, or at least time-varying. So, as in Aït-Sahalia and Mykland (3), we are led to consider the case where D i ¼ t i t i are either deterministic and time-varying, or random in which case we assume for simplicity that they are i.i.d., independent of the W process. This assumption, while not completely realistic [see Engle and Russell (998) for a discrete time analysis of the autoregressive dependence of the times between trades] allows us to make explicit calculations at the interface between the continuous and discrete time scales. We denote by N T the number of observations recorded by time T. N T is random if the Ds are. We also suppose that U ti can be written U i, where the U i are i.i.d. and independent of the W process and the D i s. Thus, the observation noise is the same at all observation times, whether random or nonrandom. If we define the Y i s as before, in the first two lines of Equation (8), though the MA() representation is not valid in the same form. We can do inference conditionally on the observed sampling times, in light of the fact that the likelihood function using all the available information is LY ð N, D N,..., Y, D ; b, c Þ ¼ LY ð N,..., Y jd N,..., D ; bþ LðD N,..., D ; cþ, where b are the parameters of the state process, that is (s, a ), and c are the parameters of the sampling process, if any (the density of the sampling intervals density L(D NT,..., D ; c) may have its own nuisance parameters c, such as an unknown arrival rate, but we assume that it does not depend on the parameters b of the state process). The corresponding loglikelihood function is X N n¼ ln LY ð N,..., Y jd N,..., D ; b Þþ XN n¼ ln LðD N,..., D ; cþ ð44þ 376

27 Sampling and Market Microstructure Noise and since we only care about b, we only need to maximize the first term in that sum. We operate on the covariance matrix of the log-returns Ys, now given by s D þ a a a s D þ a a». ¼ a s D 3 þ a» a a s D n þ a : ð45þ C A Note that in the equally spaced case, ¼ g V. But now Y no longer follows an MA() process in general. Furthermore, the time variation in D i s gives rise to heteroskedasticity as is clear from the diagonal elements of. This is consistent with the predictions of the model of Easley and O Hara (99) where the variance of the transaction price process ~X is heteroskedastic as a result of the influence of the sampling times. In their model, the sampling times are autocorrelated and correlated with the evolution of the price process, factors we have assumed away here. However, Aït-Sahalia and Mykland (3) show how to conduct likelihood inference in such a situation. The log-likelihood function is given by ln LY ð N,..., Y jd N,..., D ; bþ l s, a ¼ln detðþ= NlnðpÞ= Y Y=: ð46þ In order to calculate this log-likelihood function in a computationally efficient manner, it is desirable to avoid the brute force inversion of the N N matrix. We extend the method used in the MA() case (see Equation (9)) as follows. By Theorem 5.3. in Dahlquist and Bj orck (974), and the development in the proof of their Theorem 5.4.3, we can decompose in the form ¼ LDL T, where L is a lower triangular matrix whose diagonals are all and D is diagonal. To compute the relevant quantities, their Example shows that if one writes D ¼ diag(g,..., g n ) and k». L ¼ k 3», ð47þ B. A k n 377

28 The Review of Financial Studies / v 8 n 5 then the g k s and k k s follow the recursion equation g ¼ s D þ a and for i ¼,..., N: k i ¼a =g i and g i ¼ s D i þ a þ k i a : ð48þ Then, define ~Y ¼ L Y so that Y Y ¼ ~Y D ~Y. From Y ¼ L ~Y, it follows that ~Y ¼ Y and, for i ¼,..., N: ~Y i ¼ Y i k i ~Y i : And det() ¼ det(d) since det(l) ¼. Thus we have obtained a computationally simple form for Equation (46) that generalizes the MA() form in Equation (9) to the case of non-identical sampling intervals: l s, a X N ¼ lnðpg i Þ X N ~Y i : ð49þ g i¼ i¼ i We can now turn to statistical inference using this likelihood function. As usual, the asymptotic variance of T = ð^s s, ^a a Þ is of the form h i h i avar ^s, ^a E l s s E l ¼ lim T s a B T h ia : ð5þ T! E l a a T To compute this quantity, suppose in the following that b and b can represent either s or a. We start with: Lemma. Fisher s Conditional Information is given by h E i l b b D ¼ q ln det : ð5þ qb b To compute the asymptotic distribution of the MLE of (b, b ), one would then need to compute the inverse of E½ l b b Š¼E D ½E½ l b b jdšš where E D denotes expectation taken over the law of the sampling intervals. From Equation (5), and since the order of E D and q /qb b can be interchanged, this requires the computation of E D ½ln det Š ¼ E D ½ln det DŠ ¼ XN E D ½lnðg i ÞŠ, where from Equation (48) the g i s are given by the continuous fraction g ¼ s D þ a, i¼ g ¼ s D þ a a 4 a 4 s D þ a, g 3 ¼ s D 3 þ a s D þ a a 4 s D þ a 378

29 Sampling and Market Microstructure Noise and in general g i ¼ s D i þ a a 4 s D i þ a a4» It, therefore, appears that computing the expected value of ln(g i ) over the law of (D, D,..., D i ) will be impractical. 8. Expansion around a fixed value of D To continue further with the calculations, we propose to expand around a fixed value of D, namely D ¼ E [D]. Specifically, suppose now that D i ¼ D ð þ ej i Þ, ð5þ where e and D are nonrandom, the j i s are i.i.d. random variables with mean zero and finite distribution. We will Taylor-expand the expressions above around e ¼, that is, around the non-random sampling case we have just finished dealing with. Our expansion is one that is valid when the randomness of the sampling intervals remains small, that is, when var[d i ] is small, or o(). Then we have D ¼ E [D] ¼ O() and var½d i Š¼D e var½j i Š. The natural scaling is to make the distribution of j i finite, that is, var[j i ] ¼ O(), so that e ¼ O(var[D i ]) ¼ o(). But any other choice would have no impact on the result since var[d i ] ¼ o() implies that the product e var[j i ]iso() and whenever we write remainder terms below they can be expressed as O p (e 3 j 3 ) instead of just O(e 3 ). We keep the latter notation for clarity given that we set j i ¼ O p (). Furthermore, for simplicity, we take the j i s to be bounded. We emphasize that the time increments or durations D i do not tend to zero length as e!. It is only the variability of the D i s that goes to zero. Denote by the value of when D is replaced by D, and let X denote the matrix whose diagonal elements are the terms D j i, and whose off-diagonal elements are zero. We obtain the following theorem. Theorem 3. The MLE ð^s, ^a Þ is again consistent, this time with asymptotic variance avar ^s, ^a ¼ A ð Þ þ e A ðþ þ O e 3, ð53þ where A ðþ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 s 6 D ð4a þ s D þ s 4 D s D h D, s, a D A a þ s D h D, s, a 379

30 The Review of Financial Studies / v 8 n 5 and A ðþ ¼ var½š j ð4a þ D s Þ A ðþ A ðþ s s s a A ð Þ a a! with pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ðþ ¼4ðD s s s6 þ D 3= s 5 4a þ D s Þ, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ðþ ¼ D 3= s a s 3 4a þ D s ða þ 3D s ÞþD s4 ð8a þ 3D s Þ, p A ðþ ¼D a a s ða þ s ffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 4a þ D s þ D s Þ : In connection with the preceding result, we underline that the quantity avarð^s, ^a Þ is a limit as T!, as in Equation (5). Equation (53), therefore, is an expansion in e after T!. Note that A () is the asymptotic variance matrix already present in Proposition, except that it is evaluated at D = E[D]. Note also that the second order correction term is proportional to var[j], and is therefore zero in the absence of sampling randomness. When that happens, D ¼ D with probability one and the asymptotic variance of the estimator reduces to the leading term A (), that is, to the result in the fixed sampling case given in Proposition. 8. Randomly spaced sampling intervals and misspecified microstructure noise Suppose now, as in Section 6, that the Us are i.i.d., have mean zero and variance a, but are otherwise not necessarily Gaussian. We adopt the same approach as in Section 6, namely to express the estimator s properties in terms of deviations from the deterministic and Gaussian case. The additional correction terms in the asymptotic variance are given in the following result. Theorem 4. The asymptotic variance is given by avar true ^s, ^a ¼ A ð Þ þ cum 4 ½UŠB ðþ þ e A ðþ þ cum 4 ½UŠB ðþ þ O e 3 ð54þ where A () and A () are given in the statement of Theorem 3 and ð Þ ¼ D B 38

31 Sampling and Market Microstructure Noise while B ðþ ¼ s s D 3= s 5 4a þ D s Þ ð B ðþ ¼ var½š j B ðþ B ðþ s s s a B ð Þ a a 4D þ s6 6a 4 þ a D s þ D s4 5=! ða þ D s Þ 3 ð4a þ D s Þ D s4 B ðþ ¼ s a ða þ D s Þ 3 ð4a þ D s Þ 5= pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4a þ D s 3a 6 þ 64a 4 D s þ 35a D s4 þ 6D 3 s6 þ D = s 6a 6 þ 6a 4 D s þ 47a D s4 þ 6D 3 s6 B ðþ 6a 8 D 5= s 3 3a 4 þ a D s þ D ¼ s4 a a ða þ D s Þ 3 ð4a þ D s Þ 5= p a þ s D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : s Dð4a þ s DÞ The term A () is the base asymptotic variance of the estimator, already present with fixed sampling and Gaussian noise. The term cum 4 [U ]B () is the correction due to the misspecification of the error distribution. These two terms are identical to those present in Theorem. The terms proportional to e are the further correction terms introduced by the randomness of the sampling. A () is the base correction term present even with Gaussian noise in Theorem 3, and cum 4 [U ]B () is the further correction due to the sampling randomness. Both A () and B () are proportionalto var[j] and hence vanish in the absence of sampling randomness. 9. Extensions In this section, we briefly sketch four extensions of our basic model. First, we show that the introduction of a drift term does not alter our conclusions. Then we examine the situation where market microstructure noise is serially correlated; there, we show that the insight of Theorem remains valid, namely that the optimal sampling frequency is finite. Third, we turn to the case where the noise is correlated with the efficient price signal. Fourth, we discuss what happens if volatility is stochastic. In a nutshell, each one of these assumptions can be relaxed without affecting our main conclusion, namely that the presence of the noise gives rise to a finite optimal sampling frequency. The second part of our analysis, dealing with likelihood corrections for microstructure noise, will not 38

32 The Review of Financial Studies / v 8 n 5 necessarily carry through unchanged if the assumptions are relaxed (for instance, there is not even a known likelihood function if volatility is stochastic, and the likelihood must be modified if the assumed variancecovariance structure of the noise is modified). 9. Presence of a drift coefficient What happens to our conclusions when the underlying X process has a drift? We shall see in this case that the presence of the drift does not alter our earlier conclusions. As a simple example, consider linear drift, that is, replace Equation () with X t ¼ mt þ sw t : ð55þ The contamination by market microstructure noise is as before: the observed process is given by Equation (3). As before, we first-difference to get the log-returns Y i ¼ ~X ti ~X ti þ U ti U ti. The likelihood function is now lnlðy,...,y N jd N,...,D ; bþ l s,a,m ¼lndetðÞ=N lnðpþ=ðy mdþ ðy mdþ=, where the covariance matrix is given in Equation (45), and where D ¼ (D,..., D N ).Ifb denotes either s or a, one obtains lmb ¼ D q qb ðy mdþ, so that E½ l mb jdš ¼ no matter whether the Us are normally distributed or have another distribution with mean and variance a. In particular, h E i l mb ¼ : ð56þ Now let E½ lš be the 3 3 matrix of expected second likelihood derivatives. Let E½ lš¼te½dšd þ oðtþ. Similarly define covð l, _ lþ¼te½dšs _ þ oðtþ. As before, when the Us have a normal distribution, S ¼ D, and otherwise that is not the case. The asymptotic variance matrix of the estimators is of the form avar ¼ E [D]D SD. Let D s ;a be the corresponding matrix when estimation is carried out on s and a for known m, and D m is the asymptotic information on m for known s and a. Similarly define S s ;a and avar s ;a. Since D is block diagonal by Equation (56), D ¼ D s ; a, D m 38

33 Sampling and Market Microstructure Noise it follows that D ¼ D s ;a D m Hence avar ^s, ^a ¼ E½DŠD s ;a S s ;a D s ;a : ð57þ The asymptotic variance ofð^s, ^a Þ is thus the same as if m were known, in other words, as if m ¼, which is the case that we focused on in all the previous sections. 9. Serially correlated noise We now examine what happens if we relax the assumption that the market microstructure noise is serially independent. Suppose that, instead of being i.i.d. with mean and variance a, the market microstructure noise follows! : du t ¼bU t dt þ cdz t, ð58þ where b >, c > and Z is a Brownian motion independent of W. U D ju has a Gaussian distribution with mean e bd U and variance c /b( e bd ). The unconditional mean and variance of U are and a = c /b. The main consequence of this model is that the variance contributed by the noise to a log-return observed over an interval of time D is now of order O(D), that is of the same order as the variance of the efficient price process s D, instead of being of order O() as previously. In other words, log-prices observed close together have very highly correlated noise terms. Because of this feature, this model for the microstructure noise would be less appropriate if the primary source of the noise consists of bid-ask bounces. In such a situation, the fact that a transaction is on the bid or ask side has little predictive power for the next transaction, or at least not enough to predict that two successive transactions are on the same side with very high probability [although Choi, Salandro, and Shastri (988) have argued that serial correlation in the transaction type can be a component of the bid-ask spread, and extended the model of Roll (984) to allow for it]. On the other hand, the model (58) can better capture effects such as the gradual adjustment of prices in response to a shock such as a large trade. In practice, the noise term probably encompasses both of these examples, resulting in a situation where the variance contributed by the noise has both types of components, some of order O(), some of lower orders in D. The observed log-returns take the form Y i ¼ ~X ti ~X ti þ U ti U ti ¼ sðw ti W ti ÞþU ti U ti w i þ u i, 383

34 The Review of Financial Studies / v 8 n 5 where the w i s are i.i.d. N(, s D), the u i s are independent of the w i s, so we have var½y i Š¼s D þ E½u i Š, and they are Gaussian with mean zero and variance Eu h i i ¼ E ð Uti U tiþ ¼ c e bd ¼ c D þ oðdþ ð59þ b instead of a. In addition, the u i s are now serially correlated at all lags since EU ½ ti U tk Š ¼ c e b bd ð ik for i k. The first-order correlation of the log-returns is now covðy i, Y i Þ ¼ c e bd b Þ ¼ c b D þ o D instead of h. The result analogous to Theorem is as follows. If one ignores the presence of this type of serially correlated noise when estimating s, then follows the theorem. Theorem 5. In small samples (finite T), the RMSE of the estimator ^s is given by RMSE ^s c 4 e bd c 4 e bd T D ebd þ e Tb ¼ b D þ T b ð þ e bd Þ : þ TD s D þ c e bd! = b D ¼ c bc D þ s þ c c þ O D þ O T T ð6þ so that for large T, starting from a value of c in the limit where D!, increasing D first reduces RMSE [^s ]. Hence the optimal sampling frequency is finite. One would expect this type of noise to be not nearly as bad as i.i.d. noise for the purpose of inferring s from high frequency data. Indeed, the variance of the noise is of the same order O(D) as the variance of the efficient price process. Thus log returns computed from transaction prices sampled close together are not subject to as much noise as previously 384

35 Sampling and Market Microstructure Noise (O(D) versus O()) and the squared bias b of the estimator ^s no longer diverges to infinity as D! : it has the finite limit c 4. Nevertheless, b first decreases as D increases from, since b ¼ E ^s s ¼ c 4 e bd b D e bd and qb /qd!bc 4 < asd!. For large enough T, this is sufficient to generate a finite optimal sampling frequency. To calibrate the parameter values b and c, we refer to the same empirical microstructure studies we mentioned in Section 4. We now have p ¼ E½u i Š=ðs D þ E½u i ŠÞ as the proportion of total variance that is microstructure-induced; we match it to the numbers in Equation (4) from Madhavan, Richardson, and Roomans (997). In their Table 5, they report the first-order correlation of price changes (hence returns) to be approximately r ¼. at their frequency of observation. Here r ¼ cov(y i, Y i )/var[y i ]. If we match p ¼.6 and r ¼., with s ¼ 3% as before, we obtain (after rounding) c ¼.5 and b ¼ 3 4. Figure 6 displays the resulting RMSE of the estimator as a function of D and T. The overall picture is comparable to Figure. As for the rest of the analysis of the article, dealing with likelihood corrections for microstructure noise, the covariance matrix of the logreturns, g V in Equation (6) should be replaced by the matrix whose diagonal elements are var Yi ¼ Ew i þ Eu i ¼ s D þ c e bd b Figure 6 RMSE of the estimator ^s when the presence of serially correlated noise is ignored 385

36 The Review of Financial Studies / v 8 n 5 and off-diagonal elements i > j are: covðy i, Y j Þ¼EY i Y j ¼ E ð wi þ u i Þðw j þ u j Þ ¼ Eu i u j ¼ E ð Uti U tiþðu tj U tj Þ ¼ EU ti U tj EUti U tj U EUti t j þ U EUti t j e ¼ c e bd bdði j Þ : b Having modified the matrix g V, the artificial normal distribution that assumes i.i.d. Us that are N(, a ) would no longer use the correct second moment structure of the data. Thus we cannot relate a priori the asymptotic variance of the estimator of the estimator ^s to that of the i.i.d. normal case, as we did in Theorem. 9.3 Noise correlated with the price process We have assumed so far that the U process was uncorrelated with the W process. Microstructure noise attributable to informational effects is likely to be correlated with the efficient price process, since it is generated by the response of market participants to information signals (i.e., to the efficient price process). This would be the case for instance in the bid-ask model with adverse selection of Glosten (987). When the U process is no longer uncorrelated from the W process, the form of the variance matrix of the observed log-returns Y must be altered, replacing g v ij in Equation (6) with cov Y i, Y j ¼ cov s ð Wti W tiþþu ti U ti, s W tj W tj þ Utj U tj ¼ s Dd ij þ cov sðw ti W ti Þ, U tj U tj þ cov s W tj W tj, Uti U ti þ cov U ti U ti, U tj U tj, where d ij is the Kronecker symbol. The small sample properties of the misspecified MLE for s analogous to those computed in the independent case, including its RMSE, can be obtained from E ^s X N ¼ EY i T i¼ var ^s X N ¼ T i¼ var Yi X N þ T i¼ X i j¼ cov Yi j : Specific expressions for all these quantities depend upon the assumptions of the particular structural model under consideration: for instance, in the 386

37 Sampling and Market Microstructure Noise Glosten (987) model (see his Proposition 6), the Us remain stationary, the transaction noise U ti is uncorrelated with the return noise during the previous observation period, that is, U ti U ti, and the efficient return s(w ti W ti ) is also uncorrelated with the transaction noises U ti+ and U ti. With these in hand, the analysis of the RMSE and its minimum can then proceed as above. As for the likelihood corrections for microstructure noise, the same caveat as in serially correlated U case applies: having modified the matrix g V, the artificial normal distribution would no longer use the correct second moment structure of the data and the likelihood must be modified accordingly. 9.4 Stochastic volatility One important departure from our basic model is the case where volatility is stochastic. The observed log-returns are still generated by Equation (3). Now, however, the constant volatility assumption () is replaced by dx t ¼ s t dw t : ð6þ The object of interest in much of the literature on high frequency volatility estimation [see, e.g., Barndorff-Nielsen and Shephard (), Andersen et al. (3)] is then the integral Z T s t dt ð6þ over a fixed time period [,T ], or possibly several such time periods. The estimation is based on observations = t < t < < t n ¼ T, and asymptotic results are obtained when maxdt i!. The usual estimator for Equation (6) is the realized variance X n i¼ : ~X tiþ ~X ti ð63þ In the context of stochastic volatility, ignoring market microstructure noise leads to an even more dangerous situation than when s is constant and T!. We show in the companion paper Zhang, Mykland, and Aït-Sahalia (3) that, after suitable scaling, the realized variance is a consistent and asymptotically normal estimator but of the quantity a. This quantity has, in general, nothing to do with the object of interest Equation (6). Stated differently, market microstructure noise totally swamps the variance of the price signal at the level of the realized variance. To obtain a finite optimal sampling interval, one needs that a! as n!, that is, the amount of noise must disappear asymptotically. For further developments on this topic, we refer to Zhang, Mykland, and Aït-Sahalia (3). 387

38 The Review of Financial Studies / v 8 n 5. Conclusions We showed that the presence of market microstructure noise makes it optimal to sample less often than would otherwise be the case in the absence of noise, and we determined accordingly the optimal sampling frequency in closed-form. We then addressed the issue of what to do about it, and showed that modeling the noise term explicitly restores the first order statistical effect that sampling as often as possible is optimal. We also demonstrated that this remains the case if one misspecifies the assumed distribution of the noise term. If the econometrician assumes that the noise terms are normally distributed when in fact they are not, not only is it still optimal to sample as often as possible, but the estimator has the same asymptotic variance as if the noise distribution had been correctly specified. This robustness result is, we think, a major argument in favor of incorporating the presence of the noise when estimating continuous time models with high frequency financial data, even if one is unsure about what is the true distribution of the noise term. Hence, the answer to the question we pose in our title is as often as possible, provided one accounts for the presence of the noise when designing the estimator. Appendix A. Proof of Lemma To calculate the fourth cumulant cum(y i, Y j, Y k, Y l ), recall from Equation (8) that the observed log-returns are Y i ¼ sðw ti W ti ÞþU ti U ti : First, note that the t i are nonrandom, and W is independent of the Us, and has Gaussian increments. Second, the cumulants are multilinear, so cum Y i, Y j, Y k, Y l ¼ cum s ð Wti W ti ÞþU ti U ti, s W tj W tj þ Utj U, tj sðw tk W tk ÞþU tk U tk, sðw tl W tl ÞþU tl U tl Þ ¼ s 4 cum W ti W ti, W tj W tj, W tk W tk, W tl W tl þ s 3 cum W ti W ti, W tj W tj, W tk W tk, U tl U tl ½Š 4 þ s cum W ti W ti, W tj W tj, U tk U tk, U tl U tl ½Š 6 þ scum W ti W ti, U tj U tj, U tk U tk, U tl U tl ½Š 4 þ cum U ti U ti, U tj U tj, U tk U tk, U tl U tl : Out of these terms, only the last is nonzero because W has Gaussian increments (so all cumulants of its increments of order greater than two are zero), and is independent of the Us (so all cumulants involving increments of both W and U are also zero). Therefore, cum Y i, Y j, Y k, Y l ¼ cum Uti U, U ti U, U U, U tj tj U tk tk tl tl : If i ¼ j ¼ k ¼ l, we have: cumðu ti U ti, U ti U ti, U ti U ti, U ti U ti Þ ¼ cum 4 ðu ti U ti Þ ¼ cum 4 ðu ti ¼ cum 4 ½UŠ Þþcum 4 ðu ti Þ 388

39 Sampling and Market Microstructure Noise with the second equality following from the independence of U ti and U ti, and the third from the fact that the cumulant is of even order. If max(i, j, k, l ) ¼ min(i, j, k, l ) þ, two situations arise. Set m ¼ min(i, j, k, l ) and M ¼ max(i, j, k, l ). Also set s ¼ s(i, j, k, l ) ¼ #{i, j, k, l ¼ m}. If s is odd, say s ¼ with i ¼ m, and j, k, l ¼ M ¼ m þ, we get a term of the form cum U tm U tm, U tmþ U tm, U tmþ U tm, U tmþ U tm ¼cum4 ðu tm Þ: By permutation, the same situation arises if s ¼ 3. If instead s is even, that is, s ¼, then we have terms of the form cum U tm U tm, U tm U tm, U tmþ U tm, U tmþ U tm ¼ cum4 ðu tm Þ: Finally, if at least one pair of indices in the quadruple (i, j, k, l ) is more than one integer apart, then cum U ti U ti, U tj U tj, U tk U tk, U tl U tl ¼ by independence of the Us. Appendix B. Proof of Theorem Given the estimator (5) has the following expected value E ^s ¼ T X N i¼ The estimator s variance is var ^s ¼ N s D þ a ¼ EY i " # XN var T i¼ Y i T ¼ T X N i;j¼ ¼ s þ a D : cov Yi j : Applying Lemma in the special case where the first two indices and the last two respectively are identical yields 8 >< cum 4 ½UŠ, if j ¼ i, cum Y i, Y i, Y j, Y j ¼ cum 4 ½UŠ, if j ¼ i þ orj ¼ i, ðb:þ >:, otherwise: In the middle case, that is, whenever j ¼ i þ orj ¼ i, the number s of indices that are equal to the minimum index is always. Combining Equation (B.) with Equation (4), we have var ^s X N ¼ T i¼ ¼ T X N i¼ cov Yi, Y i þ n covðy i, Y i i¼ N X T cov Yi, Y iþ X N þ T i¼ Þ þ cum 4 ½U o Š þ N Xn o T covðy i, Y iþ Þ þ cum 4 ½UŠ þ T X N ¼ N T i¼ n var½y i n o covðy i, Y i Þ þ cum 4 ½UŠ Š þ cum 4 ½U i¼ cov Yi, Y i o Š þ ðn Þ n o covðy i, Y i Þ þ cum 4 ½UŠ T 389

40 The Review of Financial Studies / v 8 n 5 with var[y i ] and cov(y i, Y i ) = cov(y i, Y iþ ) given in Equations (9) and (), so that var ^s N n ¼ T s D þ a o þ cum4 ½UŠ þ ðn Þ T a 4 þ cum 4 ½UŠ ¼ s4 D þ 4s Da þ 6a 4 þ cum 4 ½UŠ a4 þ cum 4 ½UŠ TD T, since N ¼ T/D. The expression for the RMSE follows from those for the expected value and variance given in Equations (7) and (8): RMSE ^s 4a 4 ¼ D þ s4 D! = þ 4s Da þ 6a 4 þ cum 4 ½UŠ a4 þ cum 4 ½UŠ TD T : ðb:þ The optimal value D of the sampling interval given in Equation (9) is obtained by minimizing RMSE½^s Š over D. The first order condition that arises from setting qrmse½^s Š=qD to is the cubic equation in D: D 3 3a4 þ cum 4 ½UŠ s 4 D 4a4 T s 4 ¼ : ðb:3þ We now show that Equation (B.3) has a unique positive root, and that it corresponds to a minimum of RMSE½^s Š. We are, therefore, looking for a real positive root in D ¼ z to the cubic equation z 3 þ pz q ¼, ðb:4þ where q > and p < since from Equation (6): 3a 4 þ cum 4 ½UŠ ¼ 3a 4 þ EU 4 3E U ¼ EU 4 > : Using Vièta s change of variable from z to w given by z ¼ w p/(3w) reduces, after multiplication by w 3, the cubic to the quadratic equation y qy p3 7 ¼ ðb:5þ in the variable y w 3. Define the discriminant D ¼ p 3 q : þ 3 The two roots of Equation (B.5) are y ¼ q þ D=, y ¼ q D= are real if D (and distinct if D > ) and complex conjugates if D <. Then the three roots of Equation (B.4) are z ¼ y =3 þ y =3, þ i 3= i 3= z ¼ y=3 þ y =3 z 3 ¼ y=3 þ y =3 y =3 y =3 y =3 y =3, ðb:6þ [see, e.g., Abramowitz and Stegun (97, Section 3.8.)]. If D >, the two roots in y are both real and positive because p < and q > imply y > y > and hence of the three roots given in Equation (B.6), z is real and positive and z and z 3 are complex conjugates. If D ¼, then y ¼ y ¼ q/ > and the three roots are real (two of which 39

41 Sampling and Market Microstructure Noise are identical) and given by z ¼ y =3 þ y =3 ¼ =3 q =3, z ¼ z 3 ¼ y=3 þ y =3 ¼ z : Of these, z > and z ¼ z 3 <. If D <, the three roots are distinct and real because so and therefore so that y ¼ q þ iðdþ= re iu, y ¼ q ið DÞ= re iu y =3 ¼ r =3 e iu=3, y =3 ¼ r =3 e iu=3 y =3 þ y =3 ¼ r =3 cosðu=3þ, y =3 y =3 ¼ ir =3 sinðu=3þ z ¼ r =3 cosðu=3þ z ¼r =3 cosðu=3þþ3 = r =3 sinðu=3þ z 3 ¼r =3 cosðu=3þ 3 = r =3 sinðu=3þ: Only z is positive because q > and (D) / > imply that < u < p/. Therefore cos(u/3) >, so z > ; sin(u/3) >, so z 3 < ; and cosðu=3þ> cosðp=6þ ¼ 3= ¼ 3= sinðp=6þ> 3 = sinðu=3þ, so z <. Thus Equation (B.4) has exactly one root that is positive, and it is given by z in Equation (B.6). Since RMSE½^s Š is of the form RMSE ^s TD 3 s 4 D a 4 4a Ts þcum 4 ½UŠ þd 6a 4 T þtcum 4 ½UŠ þ4a 4 T ¼ ¼ a 3D 3 þa D þa Dþa T D! = T D with a 3 >, it tends to þ when D tends to þ. Therefore, that single positive root corresponds to a minimum of RMSE½^s Š which is reached at D ¼ y =3 þ y =3 ¼ q =3 þ q =3: D= þ D= Replacing q and p by their values in the expression above yields Equation (9). As shown above, if the expression inside the square root in formula (9) is negative, the resulting D is still a positive real number.! = Appendix C. Proof of Proposition The result follows from an application of the delta method to the known properties of the MLE estimator of an MA() process [Hamilton (995, Section 5.4)], as follows. Because we 39

42 The Review of Financial Studies / v 8 n 5 re-use these calculations below in the proof of Theorem (whose result cannot be inferred from known MA() properties), we recall some of the expressions of the score vector of the MA() likelihood. The partial derivatives of the log-likelihood function (5) have the form i h ¼ q ln detðvþ qh qv Y g qh Y ðc:þ and i g ¼ N g þ g 4 Y V Y, ðc:þ so that the MLE for g is ^g ¼ N Y V Y: ðc:3þ At the true parameters, the expected value of the score vector is zero: E [i h ] ¼ E [i g ] ¼. Hence it follows from Equation (C.) that qv E Y qh Y q ln det V ¼g ð Þ ¼g h ð þ N ÞhN þ Nh þn ð Þ, qh ð h Þð h þn ð Þ Þ thus as N! qv E Y qh Y ¼ hg ð h Þ þ oðþ: Similarly, it follows from Equation (C.) that EY V Y ¼ Ng : Turning now to Fisher s information, we have N E i g g ¼ g 4 þ g 6 EY V N Y ¼ g 4, whence the asymptotic variance of T = ð^g g Þ is g 4 D. We also have that h E i l g h ¼ qv E Y g4 qh Y h ¼ g ð h Þ þ oðþ, whence the asymptotic covariance of T = ð^g g Þ and T = ð^h hþ is zero. To evaluate E½ l hh Š, we compute " # h E i l hh ¼ q ln detðvþ qh þ g E Y q V qh Y ðc:4þ ðc:5þ ðc:6þ and evaluate both terms. For the first term in Equation (C.6), we have from Equation (7): ( q ln detðvþ þ h þ h þn 3h h N qh ¼ ð h þn Þ ð h Þ ) Nh N 3 þ h þn 4N h N ¼ þ h ð h Þ þ oðþ: ðc:7þ 39

43 Sampling and Market Microstructure Noise For the second term, we have for any non-random N N matrix Q: EY ½ QY Š ¼ E½Tr½Y QYŠŠ ¼ E½Tr½QYY ŠŠ ¼ Tr½E½QYY ŠŠ ¼ Tr QE½YY ½ ŠŠ ¼ Tr Qg V ¼ g Tr½QVŠ, where Tr denotes the matrix trace, which satisfies Tr[AB] ¼ Tr[BA]. Therefore " # E Y q V qh Y " # ¼ g Tr q V qh V ¼ g X N X N i¼ j¼! q v ij qh v ij! X N ¼ g q v ii qh þ h N X q v i;iþ þ qh h þ XN q v i;i qh h i¼ i¼ i¼ ( g ¼ ð h þn Þ 4þ h þ h þn 4h h N ð h Þ ) N þ hn 6 6h þ h þn 3h 4þN þ ð h þ 8N h N Þ ¼ g N ð h Þ þ on ð Þ: ðc:8þ Combining Equations (C.7) and (C.8) with Equation (C.6), it follows that " # h E i l hh ¼ q ln detðv N Þ qh þ g E Y q VN qh Y N! N ð h Þ þ on ð Þ: ðc:9þ In light of that and Equation (C.5), the asymptotic variance of T = ð^h hþ is the same as in the g known case, that is, ( h )D (which of course confirms the result of Durbin (959) for this parameter). We can now retrieve the asymptotic covariance matrix for the original parameters (s, a ) from that of the parameters (g, h). This follows from the delta method applied to the change of variable [Equations (9) and ()]: s a!! ¼ f g D g ð þ hþ, h ¼ : ðc:þ g h Hence where! T = ^s ^a s a!!! N, avar ^s, ^a, T! avar ^s, ^a ¼rf g, h avar ^g, ^h rf g, h ð þ hþ g! ð þ hþ ð þ hþ D D A g4 D B D h g h g ð þ hþ D p 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 6 Dð4a þ s DÞþ s 4 D s Dh D, s, a D a þ s D h D, s, a A: h C A g 393

44 The Review of Financial Studies / v 8 n 5 Appendix D. Proof of Theorem We have that E true i h i g ¼ covtrue i h, i g! ¼ cov true X N qv ij g Y i Y j qh, X N g 4 Y k Y l v kl i;j¼ k;l¼ ¼ X N qv ij 4g 6 qh vkl cov true Y i Y j, Y k Y l i;j;k;l¼ ¼ X N qv ij 4g 6 qh vkl cum true Y i, Y j, Y k, Y l þ covtrue Y i, Y j covtrue ðy k, Y l Þ, i;j;k;l¼ ðd:þ where true denotes the true distribution of the Ys, not the incorrectly specified normal distribution, and Cum denotes the cumulants given in Lemma. The last transition is because cov true Y i Y j, Y k Y l ¼ Etrue Y i Y j Y k Y l Etrue Y i Y j Etrue ½Y k Y l Š ¼ k ijkl k ij k kl ¼ k i;j;k;l þ k i;j k k;l ½Šk 3 i;j k k;l ¼ k i;j;k;l þ k i;k k j;l þ k i;l k j;k ¼ cum true Y i, Y j, Y k, Y l þ covtrue ðy i, Y k Þcov true Y j, Y k þ cov true ðy i, Y l Þcov true Y j, Y k, since Y has mean zero [see, e.g., McCullagh (987, Section.3)]. The need for permutation goes away due to the summing over all indices (i, j, k, l ), and since V ¼ [v ij ] is symmetric. When looking at Equation (D.), note that cum normal (Y i, Y j, Y k, Y l ) ¼, where normal denotes a Normal distribution with the same first and second order moments as the true distribution. That is, if the Ys were normal we would have h i E normal lh _ lg _ ¼ X N qv ij 4g 6 qh vkl cov normal Y i, Y j covnormal ðy k, Y l Þ : i;j;k;l¼ Also, since the covariance structure does not depend on Gaussianity, cov true (Y i, Y j ) ¼ cov normal (Y i, Y j ). Next, we have h i h i h i E normal lh _ lg _ ¼E normal lhg ¼E true lhg ðd:þ with the last equality following from the fact that l hg depends only on the second moments of the Ys. (Note that in general E true ½ _ l h _ lg Š 6¼ E true ½ l hg Š because the likelihood may be misspecified.) Thus, it follows from Equation (D.) that h i E true lh _ lg _ It follows similarly that E true h ¼ E normal lh _ lg _ h ¼E true lhg lh _ ¼ var true lh _ i 4g 6 i 4g 6 h i ¼E true lhh þ 4g 4 X N i;j;k;l¼ X N i;j;k;l¼ X N i;j;k;l¼ qv ij qh vkl cum true Y i, Y j, Y k, Y l qv ij qh vkl cum true Y i, Y j, Y k, Y l : ðd:3þ qv ij qh qv kl qh cum true Y i, Y j, Y k, Y l ðd:4þ 394

45 Sampling and Market Microstructure Noise and E true lg _ ¼ var true lg _ h ¼E true lg g i þ 4g 8 X N i;j;k;l¼ v ij v kl cum true Y i, Y j, Y k, Y l : ðd:5þ We now need to evaluate the sums that appear on the right-hand sides of Equations (D.3) (D.5). Consider two generic symmetric N N matrices [n i, j ] and [v i, j ]. We are interested in expressions of the form X ðþ s n i;j v k;l ¼ XN X ðþ s n i;j v k;l i;j;k;l:m¼mþ h¼ i;j;k;l:m¼h;m¼hþ ¼ XN X 3 h¼ r¼ i;j;k;l:m¼h;m¼hþ;s¼r ¼ XN h¼ X ðþ r n i;j v k;l n n h;hþ v hþ;hþ n hþ;hþ v h;hþ þ n h;h v hþ;hþ þ n hþ;hþ v h;h þ 4n h;hþ v h;hþ n hþ;h v h;h n h;h v hþ;ho : ðd:6þ It follows that if we set then Y(n, v) ¼ cum 4 [U] C (n, v) where Yðn, vþ ¼ XN n i;j v k;l cum true Y i, Y j, Y k, Y l, ðd:7þ i;j;k;l¼ cn, ð vþ ¼ XN h¼ n h;h v h;h þ XN h¼ n n h;hþ v hþ;hþ n hþ;hþ v h;hþ þ n h;h v hþ;hþ þ n hþ;hþ v h;h þ 4n h;hþ v h;hþ n hþ;h v h;h n h;h v hþ;ho : ðd:8þ If the two matrices [n i,j ] and [v i,j ] satisfy the following reversibility property: n Nþi,Nþj ¼ n i,j and v Nþi,Nþj ¼ v i,j (so long as one is within the index set), then Equation (D.8) simplifies to: cn, ð vþ ¼ XN h¼ n h;h v h;h þ XN h¼ n 4n h;hþ v hþ;hþ 4n hþ;hþ v h;hþ þ n h;h v hþ;hþ þ 4n h;hþ v h;hþo : This is the case for V and its derivative qv /qh, as can be seen from the expression for v i,j given in Equation (8), and consequently for qv i,j /qh. Therefore, if we wish to compute the sums in Equations (D.3) (D.5) we need to find the three quantities c(qv/qh,v),c(qv/qh,qv/qh), and c(v,v), respectively. All are of order O(N), and only the first term is needed. Replacing the terms v i,j and qv i,j /qh by their expressions from 395

46 The Review of Financial Studies / v 8 n 5 Equation (8), we obtain: n cðv, vþ ¼ ð þ h Þð hþ 3 ð h þn ð Þ Þ ð þ hþ h N þ h þ h þn ð Þ þ h þn ð Þ þ Nð hþ þ h o þ h N þ h þn ð Þ þ 6h þn þ h þ4n ¼ 4N ð hþ þ on ð Þ, ðd:9þ c qv qh,v ¼ OðÞþN ð hþ þ h h þ h þ O h N þ N O h N hð hþ 4 ð þ h Þ ð h þn ð Þ Þ 3 ¼ 4N ð hþ 3 þ on ð Þ, ðd:þ c qv qh, qv 4 OðÞþ3N h 4 h þ h þ O h N þ N O h N þ N 3 O h N ¼ qh 3h ð þ hþð þ h Þ 3 ð hþ 5 ð h þn ð Þ Þ 4 ¼ 4N ð hþ 4 þ on ð Þ: ðd:þ The asymptotic variance of the estimator ð^g, ^hþ obtained by maximizing the (incorrectlyspecified) log-likelihood function (5) that assumes Gaussianity of the Us is given by ^g, ^h ¼ D D S D avar true where, from Equations (C.4), (C.5), and (C.9) we have D ¼ D ¼ hi N E true l ¼ hi N E normal l ¼ h N E normal _l l _ i h g ¼ 4 Ng ð h Þ þ o N B ð h Þ þ oðþ A ðd:þ and, in light of Equations (D.3) (D.5), S ¼ h N E _ true l l _ i ¼ hi N E true l þ cum 4 ½UŠ C ¼ D þ cum 4 ½UŠC, ðd:3þ where C ¼ g 8 cðv, vþ g 6 c qv qh, v B g 4 c qv qh, qv C A qh g ¼ 8 ð hþ þ oðþ g 6 ð hþ 3 þ oðþ B g 4 ð hþ 4 þ oðþ A ðd:4þ 396

47 Sampling and Market Microstructure Noise from the expressions just computed. It follows that avar true ^g, ^h ¼ D DDþ ð cum4 ½UŠCÞ D ¼ D DIdþ cum 4 ½UŠD C ¼ D Id þ cum 4 ½UŠD C D ¼ D Id þ cum 4 ½UŠD C D ¼ avar normal ^g, ^h þ Dcum4 ½UŠD CD, where Id denotes the identity matrix and 4 þ ð hþ avar normal ^g g 4 D, ^h ¼!, h D CD ð hþ g ð hþ ¼ B D ð þ hþ A g 4 ð hþ so that avar true 4 þ ð hþ ^g, ^h g4! ð hþ g ð hþ ¼ D h þ Dcum 4 ½UŠB ð þ hþ A : g 4 ð hþ By applying the delta method to change the parametrization, we now recover the asymptotic variance of the estimates of the original parameters: avar true ^s, ^a ¼rf g, h avartrue ^g, ^h rf g, h p 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 6 Dð4a þ s DÞþ s 4 D s Dh D, s, a ¼ B D a þ s D h D, s, a A : þ Dcum4 ½UŠ Appendix E. Derivations for Section 7 To see Equation (39), let orig (E.7) denote parametrization in (and differentiation with respect to) the original parameters s and a, while transf denotes parametrization and differentiation in g and h, and f inv denotes the inverse of the change of variable function defined in (C.), namely qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! g ¼ f inv s, a a þ s D þ s Dð4a þ s DÞ ¼ B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic ðe:þ a a s A D þ s Dð4a þ s DÞ and rf inv its Jacobian matrix. Then, from l _ orig ¼rf inv ðs ; a Þ l _ transf, we have lorig ¼rf inv s, a l transf rf inv s, a h i þ H ltransf _, where H½ _ l transf Š is a matrix whose terms are linear in _ l transf and the second partial derivatives of f inv. Now E true ½ _ l orig Š¼E true ½ _ l transf Š¼, and so E true ½H½ _ l transf ŠŠ ¼ from which it 397

48 The Review of Financial Studies / v 8 n 5 follows that h D orig ¼ N E true i l orig ¼rf inv s, a Dtransf rf inv s, a D = a þ s = D D s 3 ð4a þ s DÞ 3= sð4a þ s DÞ 3= ¼! B a 4 D= s 6a þ s D A þ oðþ ð4a þ s DÞ 3= ðe:þ with D transf ¼ N E true ½ l transf Š given in Equation (D.). Similarly, lorig _ l _ orig ¼ rf inv ðs, a Þ l _ transf l _ transf rf invðs, a Þ and so S orig ¼rf inv s, a Stransf rf inv s, a ¼rf inv s, a ð Dtransf þ cum 4 ½UŠCÞrf inv s, a ¼ D orig þ cum 4 ½UŠrf inv s, a C rfinv s, a ðe:3þ with the second equality following from the expression for S transf given in Equation (D.3). To complete the calculation, note from Equation (D.4) that C ¼ g transf gtransf þ oðþ, where Thus g transf ¼ g4 ð hþ g ð hþ! : rf inv s, a C rfinv s, a ¼ gorig gorig þ oðþ, ðe:4þ where g ¼ g orig ¼rf inv s, a gtransf ¼ a 4 D = sð4a þ s DÞ 3=! D= s 6a þ s C D A ð4a þ s DÞ 3= ðe:5þ which is the result (4). Inserting Equation (E.4) into Equation (E.3) yields the result (39). For the profile likelihood l, let ^a s denote the maximizer of l(s, a ) for given s. Thus by definition lðs Þ¼lðs, ^a s Þ. From now on, all differentiation takes place with respect to the original parameters, and we will omit the subscript orig in what follows. Since ¼ l _ a ðs, ^a s Þ, it follows that ¼ q qs _ l a s, ^a s so that ¼ l s a s, ^a s þ la a s, ^a q^a s s qs, q^a s qs ¼ s, ^a ls a s : ðe:6þ la a s, ^a s 398

49 Sampling and Market Microstructure Noise The profile score then follows _l s ¼ ls _ s, ^a s þ _ la s, ^a q^a s s qs so that at the true value of (s, a ), h i _l s ¼ ls _ s, a E true ls a h i_ la s, a þ Op ðþ, E true la a ðe:7þ ðe:8þ since ^a ¼ a þ O p ðn = Þ and D l s a N ls a s, ^a h i s N E true ls a D l a a N la a s, ^a s N E true la a h i ¼ O p N = ¼ O p N = as sums of random variables with expected value zero, so that q^a s qs ¼ N ls s, ^a a s N l a a s, ^a s h i N E true ls a þ D l s a ¼ h i N E true la a þ D l a a h i E true ls a ¼ h i þ D l a D l a þ o p N = E true la a s a h i ¼ E true ls a h i þ O p N =, E true la a while _l a s, a ¼ Op N = also as a sum of random variables with expected value zero. Therefore h i E true _ls h ¼ Etrue ls _ s, a i E true ls a h h ie true la _ s, a i þ OðÞ E true la a ¼ OðÞ, since E true ½ l _ s ðs, a ÞŠ ¼ E true ½ l _ a ðs, a ÞŠ ¼. In particular, E true ½ _lðs ÞŠ ¼ oðnþ as claimed. Further differentiating Equation (E.7), one obtains l s ¼ ls s s, ^a s þ la a s, ^a q^a s s qs þ l s a s, ^a q^a s s qs þ l _ a s, ^a q ^a s s q s ¼ l s s s, ^a ls a s s, ^a s þ _ l la a s, ^a a s, ^a q ^a s s q s s 399

50 The Review of Financial Studies / v 8 n 5 from Equation (E.6). Evaluated at s ¼ ^s, one gets ^a s ¼ ^a and l _ a ð^s, ^a Þ¼, and so l ^s ¼ ls s ^s, ^a ls a ^s, ^a la a ð ^s, ^a Þ ¼ h l ð^s, ^a Þ iss, ðe:9þ where ½ lð^s, ^a Þ Š s s is the upper left element of the matrix lð^s, ^a Þ. Thus Equation (4) is valid. Alternatively, we can see that the profile likelihood l satisfies the Bartlett identity to first order, that is, Equation (43). Note that by Equation (E.8), h i 3 h N E true _ls i ¼ N E true ls _ s, a E true 6 ls a h i_ la s, þ Op ðþ A7 4 5 E true la a h i 3 ¼ N E true ls _ s, a E true 6 ls a h i_ la s, A7 4 5 þ oðþ E true la a h i ¼ N E true ls _ s, a E true 6 ls a þ h i_ la s, a A E true la a h i 3 E true ls a h i_ la s, a _ls s, a 5 þ oðþ E true la a so that h N E true _ls i ¼ S s s þ D s a S D s a S D aa a a D a s þ o pðþ a a ¼ D s s þ D! s a D D s a D D aa a a D a s a a þ cum 4 ½UŠ g s þ D! s a g a D D s a g a a D s g a þ o p ðþ a a by invoking Equation (39). Continuing the calculation, h N E true _ls i ¼ D s s D s a D a a ¼ = D þ oðþ, s s þ cum 4 ½UŠ g s D s a g D a þ oðþ a a since from the expressions for D orig and g orig in Equations (E.) and (E.5) we have ðe:þ g s D s a D a a g a ¼ : ðe:þ Then by Equation (E.9) and the law of large numbers, we have N E true ls ¼= D þ oðþ, s s and Equation (43) follows from combining Equation (E.) with Equation (E.). ðe:þ 4

51 Sampling and Market Microstructure Noise Appendix F. Proof of Lemma Id implies that q qb q ¼ qb ðf:þ and, since is linear in the parameters s and a (see Equation (45)) we have so that q qb qb ¼ q ¼ q q qb qb qb qb q q ¼ q q þ q qb qb qb qb qb qb q q ¼ q q þ : qb qb qb qb ðf:þ ðf:3þ In the rest of this lemma, let expectations be conditional on the Ds. We use the notation E [ jd] as a shortcut for E [ jd N,..., D ]. At the true value of the parameter vector, we have, h ¼ E l _ i b jd ¼ q ln det q E Y YjD : ðf:4þ qb qb with the second equality following from Equation (46). Then, for any nonrandom Q, we have EY ½ QYŠ ¼ Tr½QE½YY ŠŠ ¼ Tr½QŠ: ðf:5þ This can be applied to Q that depends on the Ds, even when they are random, because the expected value is conditional on the Ds. Therefore, it follows from Equation (F.4) that q ln det ¼E Y q YjD ¼Tr q q ¼ Tr, qb qb qb qb ðf:6þ with the last equality following from Equation (F.) and so q ln det ¼ q q Tr qb qb qb qb q q ¼ Tr qb qb " # q q ¼Tr þ q qb qb qb qb q q ¼Tr qb qb, ðf:7þ again because of Equation (F.). In light of Equation (46), the expected information (conditional on the Ds) is given by " # h E i l b b jd ¼ q ln det þ qb b E Y q YjD : qb b 4

52 The Review of Financial Studies / v 8 n 5 Then, " # " # E Y q YjD ¼ Tr q qb b qb b q ¼ Tr qb q ¼ Tr qb q þ qb q qb q qb q qb with the first equality following from Equation (F.5) applied to Q ¼ q /qb b, the second from Equation (F.3), and the third from the fact that Tr[AB] ¼ Tr[BA]. It follows that h E i l b b jd ¼ q q q q Tr þ Tr qb qb qb qb ¼ q q Tr qb qb ¼ q ln det : qb b Appendix G. Proof of Theorem 3 In light of Equations (45) and (5), ¼ þ es X ðg:þ from which it follows that ¼ Id þ es ¼ Id þ es since X X ¼ es X þ e s 4 X ðid þ eaþ ¼ Id ea þ e A þ O e 3 : þ O e 3, ðg:þ Also, q ¼ q þ e qs X: qb qb qb Therefore, recalling Equation (F.6), we have q ln det q ¼ Tr qb qb ¼ Tr es X þ e s 4 X þ O e 3 q þ e qs X qb qb ¼ Tr q þ etr s q qb X þ qs qb qb X þ e Tr s 4 X q s qs qb qb X X þ O p e 3 : ðg:3þ 4

53 Sampling and Market Microstructure Noise We now consider the behavior as N!of the terms up to order e. The remainder term is handled similarly. Two things can be determined from the above expansion. Since the j i s are i.i.d. with mean, E [X] ¼, and so, taking unconditional expectations with respect to the law of the D i s, we obtain that the coefficient of order e is E Tr s q X þ qs qb qb X ¼ Tr E s q X þ qs qb qb X ¼ Tr s E½X q Š þ qs qb qb E½XŠ ¼ : Similarly, the coefficient of order e is E Tr s 4 X q s qs qb qb X h ¼ Tr s 4 E X i q h s qs E qb qb X i h ¼ Tr s E X i s q qs Id qb qb ¼ Tr s E X X s q qs Id : qb qb The matrix E½X XŠ has the following terms X X ¼ i;j XN X N k¼ l¼ X ik kl X lj ¼ D j ij j and since E [j i j j ] ¼ d ij var[j] (where d ij denotes the Kronecker symbol), it follows that E X X ¼ D var ½Šdiag j, ðg:4þ where diag½ Š is the diagonal matrix formed with the diagonal elements of. From this, we obtain that q ln det E ¼ Tr q þ e Tr s qb qb E X X s q qs Id þ O e 3 qb qb ¼ Tr q qb þ e D var½štr j s diag s q qs Id þ O e 3 : ðg:5þ qb qb To calculate E½ l b b Š, in light of Equation (5), we need to differentiate E [q ln det /qb ] with respect to b. Indeed " # h E i h h l b b ¼ E E ii l b b jd ¼ E q ln det ¼ q q ln det E, qb qb qb qb where we can interchange the unconditional expectation and the differentiation with respect to b because the unconditional expectation is taken with respect to the law of the D i s, which is independent of the b parameters (i.e., s and a ). Therefore, differentiating Equation (G.5) ij 43

54 The Review of Financial Studies / v 8 n 5 with respect to b will produce the result we need. (The reader may wonder why we take the expected value before differentiating, rather than the other way around. As just discussed, the results are identical. However, it turns out that taking expectations first reduces the computational burden quite substantially.) Combining with Equation (G.5), we therefore have h E i l b b ¼ q q ln det E qb qb ¼ q Tr q qb qb e D var½š j q Tr s diag s q qs Id þ O e 3 qb qb qb f ðþ þ e f ðþ þ O e 3 : ðg:6þ It is useful now to introduce the same transformed parameters (g, h) as in previous sections and write ¼ g V with the parameters and V defined as in Equations (9), (), and (6), except that D is replaced by D in these expressions. To compute f (), we start with Tr q ¼ Tr g V q g V qb qb qv ¼ Tr V þ Tr g V V qg ¼ Tr V qb qv qh qh qg þ Ng qb qb qb ðg:7þ with qg /qb and qh/qb to be computed from Equations () and (). If Id denotes the identity matrix and J the matrix with on the infra and supra-diagonal lines and everywhere else, we have V ¼ h Id þ hj, so that qv/qh ¼ hidþj. Therefore qv Tr V qh ¼ htr V þ Tr V J ¼ h XN v i;i þ XN v i;i þ v i;iþ þ v ; þ v N;N i¼ i¼ ¼ h hn N h þ ð h Þð h þn ð Þ Þ ¼ h ð h Þ þ oðþ: ðg:8þ Therefore, the first term in Equation (G.7) is O() while the second term is O(N) and hence q ¼ Ng qb Tr qg þ OðÞ: qb This holds also for the partial derivative of Equation (G.7) with respect to b. Indeed, given the form of Equation (G.8), we have that q qv Tr V qb qh ¼ q h qb ð h Þ þ oðþ¼o ðþ, since the remainder term in Equation (G.8) is of the form p(n)h q(n), where p and q are polynomials in N or order greater than or equal to and, respectively, whose differentiation 44

55 Sampling and Market Microstructure Noise with respect to h will produce terms that are of order o(n). Thus it follows that q Tr q ¼ N q qg g þ on ð Þ qb qb qb qb ( ) ¼ N qg qg þ g q g þ on ð Þ: ðg:9þ qb qb qb qb Writing the result in matrix form, where the (, ) element corresponds to (b, b )=(s, s ), the (, ) and (, ) elements to (b, b ) ¼ (s, a ) and the (, ) element to (b, b ) ¼ (a, a ), and computing the partial derivatives in Equation (G.9), we have f ðþ ¼ q Tr q qb qb D = a þ s = D D s ¼ N 3 ð4a þ s D Þ 3= sð4a þ s D Þ 3=! B a 4 D= s 6a þ s D A þ on ð Þ: ðg:þ ð4a þ s D Þ 3= As for the coefficient of order e, that is f () in Equation (G.6), define a Tr s diag s q qs Id qb qb so that ðg:þ f ðþ ¼ D var½šqa j : qb We have a ¼ s 4 Tr diag q s qs Tr diag qb qb ¼ s 4 g 6 Tr V diag V V q g V s qs g 4 Tr V diag V qb qb ¼ s 4 g 4 Tr V diag V V qv qh þ qh qb s 4 g 6 qg Tr V diag V qb s qs g 4 Tr V diag V qb ¼ s 4 4 qh g Tr V diag V V qv þ qb qh s 4 g 6 qg s qs g 4 Tr V diag V : qb qb Next, we compute separately Tr V diag V V qv ¼ Tr diag V V qv qh qh V ¼Tr diag V qv qh ¼ XN i;i qvi;i v qh i¼ ¼ OðÞNh þ h h 4 h 6 þ O h N þ N O h N ð þ h Þ ð h Þ 4 ð h þn ð Þ Þ 3 ¼ Nh ð h Þ 3 þ on ð Þ 45

56 The Review of Financial Studies / v 8 n 5 and Tr V diag V X N ¼ v i;i i¼ ¼ OðÞþN h4 þ O h N ð þ h Þð h Þ 3 ð h þn ð Þ Þ N ¼ ð h Þ þ on ð Þ: Therefore a ¼ s 4 4 qh g qb! Nh ð h Þ 3 þ s 4 6 qg g s qs g 4 qb qb N ð h Þ! þ on ð Þ, which can be differentiated with respect to b to produce qa/qb. As above, differentiation of the remainder term o(n) still produces a o(n) term because of the structure of the terms there (they are again of the form p(n)h q(n) ). Note that an alternative expression for a can be obtained as follows. Going back to the definition (G.), a ¼ s 4 Tr diag q s qs qb the first trace becomes Tr diag q qb Tr qb ¼ Tr diag ¼Tr diag ¼ XN diag q qb q qb q ii qb i¼ ii ¼ q X N qb ii i¼ ¼ q Tr diag, qb so that we have a ¼s 4 q Tr diag s qs Tr diag qb qb ¼s 4 q Tr qs 4 diag Tr diag qb qb ¼ q s 4 Tr diag qb ¼ q s 4 g 4 Tr V diag V qb!! ¼ q s 4 g 4 N qb ð h Þ þ on ð Þ! ¼ N q s 4 g 4 qb ð h Þ þ on ð Þ, ðg:þ 46

57 Sampling and Market Microstructure Noise where the calculation of Tr[V diag[v ]] is as before, and where the o(n) term is a sum of terms of the form p(n)h q(n) as discussed above. From this one can interchange differentiation and the o(n) term, yielding the final equality above. Therefore!! qa ¼ q s 4 g 4 N qb qb qb ð h Þ þ on ð Þ! ¼ N q s 4 g 4 qb qb ð h Þ þ on ð Þ: ðg:3þ Writing the result in matrix form and calculating the partial derivatives, we obtain f ðþ ¼ D var½šqa j ¼ ND var½š j a 8a s D D qb ð4a þ s D Þ 3 B 8s A þ on ð Þ: ðg:4þ D Putting it all together, we have obtained h i N E l b b ¼ N f ðþ þ e f ðþ þ O e 3 F ðþ þ e F ðþ þ O e 3 þ oðþ, where F ðþ ¼ ðg:5þ D = a þ s = D D s 3 ð4a þ s D Þ 3= sð4a þ s D Þ 3=! B a 4 D= s 6a þ s D C, A ð4a þ s D Þ 3= F ðþ D var½š j ð4a þ s D Þ 3 a 8a s D D B 8s A : D ðg:7þ The asymptotic variance of the maximum-likelihood estimators avarð^s, ^a Þ is, therefore, given by avar ^s, ^a ¼ E½DŠ F ðþ þ e F ðþ þ O e 3 h i F ¼ D F ðþ Id þ e F ðþ ðþ þ O e 3 h i F ¼ D Id þ e F ðþ ðþ þ O e 3 h F h i F ¼ D Id e F ðþ ðþ þ O e 3 h ðþ h i h i F ¼ D F ðþ e D F ðþ A ðþ þ e A ðþ þ O e 3, ðþ F ðþ i ðþ i F h i þ O e 3 where the final results for A () ¼ D [F () ] and A () ¼D [F () ] F () [F () ], obtained by replacing F () and F () by their expressions in Equation (G.5), are given in the statement of the theorem. 47

58 The Review of Financial Studies / v 8 n 5 Appendix H. Proof of Theorem 4 It follows as in Equations (D.3) (D.5) that h i E true lb _ lb _ jd ¼ cov true lb _, l _ b jd ¼ cov true X N q Y i Y j qb i;j¼ ij, ¼ X N q q 4 qb i;j;k;l¼ ij qb kl h i ¼E true lb b jd þ X N q 4 qb i;j;k;l¼ ij h i ¼E true lb b jd þ 4 cum 4½U X N k;l¼ q! Y k Y l qb cov true Y i Y j, Y k Y l jd q qb Šc q, q qb qb kl kl cum true Y i, Y j, Y k, Y l jd, ðh:þ since cum true (Y i, Y j, Y k, Y l jd) ¼,, or, cum true (U), as in Equation (5), and with c defined in Equation (D.8). Taking now unconditional expectations, we have h i E true lb _ lb _ ¼ cov true lb _, l _ b h ¼ E cov true lb _, l _ i h i h i b jd þ cov true E true lb _ jd, E true lb _ jd h ¼ E cov true lb _, l _ i b jd h i ¼E true lb b þ 4 cum 4½UŠE c q, q : ðh:þ qb qb with the first and third equalities following from the fact that E true ½ l _ bi jdš ¼. Since h i h i E true lb b jd ¼ E normal lb b jd and consequently h i h i E true lb b ¼ E normal lb b have been found in the previous subsection (see Equation (G.5)), what we need to do to obtain E true ½ l _ b lb _ Š is to calculate E c q, q : qb qb With given by Equation (G.), we have for i ¼, q qb i ¼ q qb i e q s qb X þ e q i qb i s 4 X þ O e 3 and, therefore, by bilinearity of c we have E c q, q ¼ c q, q ee c q q, s qb qb qb qb qb qb X ½Š þ e E c q q, s 4 qb qb X ½Š þ e q E c s q qb X, s qb X þ O e 3, ðh:3þ 48

59 Sampling and Market Microstructure Noise where the [] refers to the sum over the two terms where b and b are permuted. The first (and leading) term in Equation (H.3), ¼ c q g V qb qb qb ¼ c qg V þ g qb ¼ c qg V þ g qb c q, q, q g V qb qv qb qv qh, qg V qv þ g qb qb qh, qg V qv qh þ g qb qb qh qb corresponds to the equally spaced, misspecified noise distribution, situation studied in Section 6. The second term, linear in e, is zero since E c q q, s qb qb X ¼ c q, E q s qb qb X ¼ c q q, s qb qb E½X Š ¼ with the first equality following from the bilinearity of c, the second from the fact that the unconditional expectation over the D i s does not depend on the b parameters, so expectation and differentiation with respect to b can be interchanged, and the third equality from the fact that E [X] ¼. To calculate the third term in Equation (H.3), the first of two that are quadratic in e, note that a E c q q, s 4 qb qb X X ¼ c q q, s 4 qb qb E X X ¼ D var ½Šc j q q, s 4 diag qb qb ¼ D var½šc j q g V q, s 4 g 6 V diag V V, ðh:4þ qb qb with the second equality obtained by replacing E½X XŠ with its value given in Equation (G.4), and the third by recalling that ¼ g V. The elements (i, j) of the two arguments of c in Equation (H.4) are n i;j ¼ q g v i;j ¼ qg v i;j qvi;j qh þ g qb qb qh qb and! v k;l ¼ q s 4 g 6XN v k;m v m;m v m;l qb m¼ ¼ q s4 g 6 X N v k;m v m;m v m;l þ s 4 g 6 q qb qh m¼ X N m¼! v k;m v m;m v m;l qh qb from which c in Equation (H.4) can be evaluated through the sum given in Equation (D.8). 49

60 The Review of Financial Studies / v 8 n 5 Summing these terms, we obtain c q g V q, s 4 g 6 V diag V V qb qb ¼ 4NC ð V ð hþþc V C 3V Þ C W h þ CW C 3W ð þ 3hÞ ð hþ 7 ð þ hþ 3 þ on ð Þ, where C V ¼ qg, C V ¼ g, C 3V ¼ qh qb qb C W ¼ q s4 g 6, C W ¼ s 4 g 6, C 3W ¼ qh : qb qb The fourth and last term in Equation (H.3), also quadratic in e, q a E c s q qb X, s qb X is obtained by first expressing q c qb s q X, qb s X in its sum form and then taking expectations term by term. Letting now n i;j ¼ q s qb X, v k;l ¼ q s ij qb X we recall our definition of c(n, v) given in Equation (D.8) whose unconditional expected value (over the D i s, i.e., over X) we now need to evaluate in order to obtain a. We are thus led to consider four-index tensors l ijkl and to define ~c ðlþ XN h¼ l h;h;h;h þ XN n l h;hþ;hþ;hþ l hþ;hþ;h;hþ h¼ þ l h;h;hþ;hþ þ l hþ;hþ;h;h þ 4l h;hþ;h;hþ l hþ;h;h;h l h;h;hþ;ho, kl ðh:5þ where l ijkl is symmetric in the first two and the last two indices, respectively, that is, l ijkl ¼ l jikl and l ijkl ¼ l ijlk. In terms of our definition of c in Equation (D.8), it should be noted that cðn, vþ ¼~cðlÞ when one takes l ijkl = n i,j v k,l. The expression we seek is, therefore, q a ¼ E c s q qb X, s qb X ¼ ~c ðlþ, ðh:6þ where l ijkl is taken to be the following expected value l ijkl E n i;j v k;l " # q ¼ E s q qb X s ij qb X " kl ¼ E XN q s qb ir X rs q s sj r;s;t;u¼ qb kt X tu # ul ¼ XN q s q qb ir s sj r;s;t;u¼ qb kt E½X ul rs X tu Š ¼ D var½šxn j q s q qb ir s rj qb kr rl r¼ 4

61 Sampling and Market Microstructure Noise with the third equality following from the interchangeability of unconditional expectations and differentiation with respect to b, and the fourth from the fact that E[X rs X tu ] 6¼ only when r ¼ s ¼ t ¼ u, and E½X rr X rr Š ¼ D var½š: j and Thus we have l ijkl ¼ D var½šxn j r¼ ¼ D var½šxn j r¼ q s g 4 V qb ir V q s g 4 V rj qb kr V rl q qb s g 4 v i;r v r;j q qb s g 4 v k;r v r;l : ðh:7þ q s g 4 v i;r v r;j q s g 4 v k;r v r;l q s g 4 ¼ v i;r v r;j þ s g 4 q vi;r v r;j ð Þ qh qb qb qb qh qb q s g 4 v k;r v r;l þ s g 4 q vk;r v r;l qh : qb qh qb Summing these terms, we obtain D ~c var½jš N ðlþ ¼ ð hþ 7 ð þ hþ 3 ð þ h Þ 3 ðc lð h 4 ÞðC 5l C 6l ð þ h þ h þ h 3 ÞþC 4l ð h 4 ÞÞ þ C l C 3l ðc 5l C 6l ð þ h þ 4h þ 6h 3 þ 5h 4 þ 4h 5 þ 4h 6 Þ þ C 4l ð þ h þ h þ h 3 h 4 h 5 h 6 h 7 ÞÞÞ þ oðnþ, where C l ¼ q s g 4, C l ¼ s g 4, C 3l ¼ qh, qb qb C 4l ¼ q s g 4, C 5l ¼ s g 4, C 6l ¼ qh : qb qb Putting it all together, we have E c q, q ¼ c q, q þ e ða ½Šþa ÞþO e 3 : qb qb qb qb Finally, the asymptotic variance of the estimator ð^s, ^a Þ is given by avar true ^s, ^a ¼ E½DŠ D S, D ðh:8þ where D ¼ D ¼ hi N E true l ¼ hi N E normal l ¼ h N E normal _l l _ i F ðþ þ e F ðþ þ O e 3 4

62 The Review of Financial Studies / v 8 n 5 is given by the expression in the correctly specified case (G.5), with F () and F () given in Equations (G.6) and (G.7), respectively. Also, in light of Equation (H.), we have S ¼ h N E _ true l l _ i ¼ hi N E true l þ cum 4 ½UŠ C ¼ D þ cum 4 ½UŠC, where C ¼ E c q qs, q qs B C ðþ þ e C ðþ þ O e 3 : Since, from Equation (H.3), we have E c q, q ¼ c q qb qb qb qb it follows that C () is the matrix with entries C ðþ ¼ and E c q qs, q qa C E c q qa, q A qa, q þ e a ½Šþe a þ O e 3, 4N c q ; q, that is, qb qb = D D s ð4a þ D s Þ 3 a 4 sð4a þ D s Þ D =! 6a þ D s 3= ð4a þ D s Þ 3! B a 8 D= s 6a þ D s a4 6a þ 3D s A ð4a þ D s Þ 3= ð4a þ D s Þ 3 with 4N a ½Š¼var ½Š j 4N a þ oðþ, ð C ðþ ¼ ð 4N a ½Šþa Þ: D 3= 4a þ Ds Þs D 4a þ Ds ð4a þ Ds Þ 9= D 3= 4a 8 a 4 D s 4 þd 4 s 8 D sða þd s Þ 3 ð4a þd s Þ 9= ¼ var½š j ð Þð4a þ Ds Þ 3= D = ð 8Ds 4a þ Ds a 4 ð4a þ Ds Þ 9= ð Þ 3= D = ð a 4 ð4a þ Ds Þ 9= s 4a 4 þ a Ds þ D s4 s 6a þ Ds Þ þ oðþ, 3= s 44a 6 8a 4 D s þ7a D s 6 þ3d 3 s 6 ða þd s Þ 3 ð4a þd s Þ 9= D 3= s 3 5a þ4a D s þ9d s 4 ða þd s Þ 3 ð4a þd s Þ 9= þoðþ, It follows from Equation (H.8) that avar true ^s, ^a ¼ E½DŠ DDþ ð cum4 ½UŠCÞ D ¼ D DIdþ cum 4 ½UŠD C ¼ D Id þ cum 4 ½UŠD C D ¼ D Id þ cum 4 ½UŠD C D ¼ avar normal ^s, ^a þ D cum4 ½UŠD CD, Þ C A C A 4

63 Sampling and Market Microstructure Noise where avar normal ^s, ^a ¼ D D ¼ A ðþ þ e A ðþ þ O e 3 is the result given in Theorem 3, namely Equation (53). The correction term due to the misspecification of the error distribution is determined by cum 4 [U] times D D CD ¼ D F ðþ þe F ðþ þo e 3 C ðþ þe C ðþ þo e 3 F ðþ þe F ðþ þo e 3 h i F ¼ D Id e F ðþ ðþ þo e 3 h i F ðþ C ðþ þe C ðþ þo e 3 h i F Ide F ðþ ðþ þo e 3 h i ðþ h i C ¼ D F ðþ h i ðþ F ðþ F h i C h i h i þe D F ðþ ðþ F ðþ F ðþ F h i ðþ C h F ðþ h i C h i F ðþ ðþ F h i F ðþ ðþ F ðþ þo e 3 B ðþ þe B ðþ þo e 3 : i ðþ F ðþ The asymptotic variance is then given by avar true ^s, ^a ¼ A ðþ þ cum 4 ½UŠB ðþ þ e A ðþ þ cum 4 ½UŠB ðþ þ O e 3 with the terms A (), A (), B () and B () given in the statement of the theorem. Appendix I. Proof of Theorem 5 From EYi ¼ Ew i þ Eu i ¼ s D þ c e bd b it follows that the estimator (5) has the following expected value E ^s X N ¼ EY i T i¼ ¼ N T s D þ c e bd b ¼ s þ c e bd bd ¼ s þ c bc D þ O D : ði:þ The estimator s variance is " # var ^s XN ¼ var Y T i i¼ ¼ T X N i¼ var Yi X N þ T i¼ X i j¼ cov Yi j : 43

64 The Review of Financial Studies / v 8 n 5 Since the Y i s are normal with mean zero, var Yi ¼ var ½ Yi Š ¼ E Yi and for i > j cov Yi, Y j ¼ ¼ cov Y i, Y j E ui u j since cov Y i, Y j ¼ EYi Y j ¼ E ð wi þ u i Þ w j þ u j ¼ Eui u j : Now we have Eu i u j ¼ E ð Uti U ti Þ U tj U tj ¼ EU ti U tj EUti U tj U EUti tj þ U EUti tj e ¼ c e bd bdðijþ b so that e cov Yi, Y j ¼ c e bd bdðij b ¼ c4 bd ij e ð Þ e bd 4 b! Þ and consequently var ^s ( c 4 e bd Ne bd þ e NbD ¼ T b ð þ e bd Þ þ N s D þ c e bd ) b ði:þ with N ¼ T/D. The RMSE expression follows from Equations (I.) and (I.). As in Theorem, these are exact small sample expressions, valid for all (T, D). References Abramowitz, M., and I. A. Stegun, 97, Handbook of Mathematical Functions, Dover, New York. Aït-Sahalia, Y., 996, Nonparametric Pricing of Interest Rate Derivative Securities, Econometrica, 64, Aït-Sahalia, Y.,, Maximum-Likelihood Estimation of Discretely-Sampled Diffusions: A Closed- Form Approximation Approach, Econometrica, 7, 3 6. Aït-Sahalia, Y., and P. A. Mykland, 3, The Effects of Random and Discrete Sampling When Estimating Continuous-Time Diffusions, Econometrica, 7, Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys,, The Distribution of Exchange Rate Realized Volatility, Journal of the American Statistical Association, 96, Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys, 3, Modeling and Forecasting Realized Volatility, Econometrica, 7, Bandi, F. M., and P. C. B. Phillips, 3, Fully Nonparametric Estimation of Scalar Diffusion Models, Econometrica, 7, Barndorff-Nielsen, O. E., and N. Shephard,, Econometric Analysis of Realized Volatility and Its Use in Estimating Stochastic Volatility Models, Journal of the Royal Statistical Society B, 64,

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