Realized variance and market microstructure noise pdf

It is worthwhile to note [a, b] interval. In fact, the quadratic variation of a white noise m Next, we formulate results for RV AC1 that are similar to those process is unbounded as is the r-tic variation for any other in- teger. Given Assumptions 1 and 3 a , we have that m, is increased.

RV m in Lemma 2. The first result relies on only Assump- d tion 3 a ; c is not needed for the variance expression. A somewhat remarkable result of Lemma 3 is that the noise leads to a bias that diverges to infinity.

This result was first m derived in an unpublished thesis by Fang Usually, bias correction is accompanied by a moments of the return noise, ei,m.

Thus, in the absence of 2 noise-to-signal is as small as we find it to be in practice. In noise, we see an increase in the asymptotic variance as a result other words, RV m AC1 permits more frequent sampling than does of the bias correction. The estimates are based on high-frequency stock returns Next, we compare and RV m in terms of their MSEs RV AC1 of Alcoa left panels and Microsoft right panels in year the and their respective optimal sampling frequencies for a special The MSEs are m t0,m ,.

The lower panels show the percentage increase in the RMSE for different sampling frequencies caused by market microstructure 0 noise.

MSFT plot suggests that the time dependence lasts for This result is not surprising, because recent developments in 30 ticks, perhaps longer.

The noise is autocorrelated. We provide additional evidence of this fact, based on other A second, very interesting aspect that can be analyzed based empirical quantities, in the following sections. Following earlier versions of the present article, oretical accuracy of such confidence intervals, including those issues related to time dependence and noise—price correlation of Barndorff-Nielsen and Shephard The upper panels of Fig- b , Frijns and Lehnert , and Zhang Thus 4.

The ver- Assumption 4. At minute sampling, the dis- noise assumption used in Section 3. Suppose that Assumptions 1, 2, and 4 hold and was reduced to 1 cent. The main explanation for this A drawback of RV ACqis that it may produce a negative m is that CTS will sample the same price multiple times when m is estimate of volatility, because the covariances are not scaled large, which induces artificial autocorrelation in intraday re- downward in a way that would guarantee positivity.

This is par- turns. The plots in rows 3 and 4 are signature plots been observed in high-frequency intraday returns constructed when intraday returns are sampled in tick time.

These also re- from transaction prices. To rule out the possibility of a negative x tick veal a bias in RV AC1 at the highest frequencies, which shows estimate, one could use a different kernel, such as the Bartlett that the noise is time dependent in tick time. For example, the kernel. The two top rows are based on calendar time sampling; the bot- tom rows are based on tick time sampling.

Inter- as 15 seconds MSFT, year We comment on the lower estingly, Barndorff-Nielsen et al. In the time series literature, the lag length, qm , is typically 4. But if the noise were dependent ural to sample in tick time, such that the same observation is not in calendar time, then this would be inappropriate, because it sampled multiple times.

So the former q would cover 15 minutes, sumed away. Under As- crostructure noise that is time-dependent in tick time and corre- sumptions 2 and 4, the autocorrelation in intraday returns is lated with efficient returns. This also makes RV AC traday returns at the highest possible frequency in tick time. We more comparable across different frequencies, m. Thus we set suppress the subscript m to simplify the notation.

It m RV ACq. The upper four panels of Figure 5 represent a new type of 1 sec In this simple example, ui is only contemporaneously corre- signature plots for RV ACq. Thus these signature plots provide informa- complicated time dependence in tick time. In this situation we tion on time dependence in the noise process. The fact that the 1 tick could use RV ACq , with a q sufficiently large to capture the 1 sec RV ACq of the four price series differ and have not leveled off time dependence.

Thus in the upper four panels, Assumption 4 and Theorem 2 are formulated for the case where we sample in calendar time, it appears that the depen- with CTS, but a similar estimator can be defined under depen- dence lasts for as long as 2 minutes AA, year or as short dence in tick time.

The x -axis is the number of autocovariance terms, 1 sec q, included in RVAC. The results for are the panels in rows 1 and 3, and those q 1 tick for are in rows 2 and 4. In- were responsible for this result. However, the upward-sloping terestingly, Zhou showed that the subsample version of volatility signature until q is about 30 is actually found in most his estimator [see 6 ] has a variance that is at most of order 1 tick daily plots of RV ACq against q for MSFT in the year It appears that Zhou may turns, which we now generalize by including higher-order auto- have considered k as fixed in his asymptotic analysis, because 1 tick he referred to this estimator as being inconsistent see Zhou covariances.

We did this to make the estimator, RV ACq , robust , p. Therefore, the great virtues of subsample-based to both time dependence in the noise and correlation between estimators in this context were first recognized by Zhang et al. Interestingly, Zhou also pro- As is the case for RV 1 ACq , this estimator is robust to time dependence that is finite in tick 5. Here vidually, but give some of the more detailed results only for the m need not be equal to N unlike the situation in the previous ex- years and to conserve space.

We use the following notation for such skip-k to avoid mixing mix days with different tick sizes, we drop most intraday returns: of the days during January from our sample. This leads to the identity We filtered the raw data for outliers, and discarded transac- k tick tions outside the period AM— PM and removed days RV AC1 with less than 5 hours of trading from the sample.

The subsample version RV AC1 , proposed by Details of the filtering procedure are described in a techni- Zhou , can be expressed as cal appendix available at our website. We use all price observations in our analy- sis. Rearranging the terms, we see that this sum the RV, but has an impact on the autocorrelation of intraday is approximately given by returns. The percentage of observations for which the transaction price or one of the quoted prices was different from the previous one is given in parentheses.

The last column is the total number of data in our sample. Clearly, this is problematic for all ities. The corresponding figures for the other 28 DJIA equities three estimators. Here we use RV 1 AC1 as our 5. Be- [a, b] interval. The noise is smaller than one might think. This is particularly so in the more recent years see, e. In fact, the one-scale estimator proposed by Zhang et al.

From signal ratio. All numbers have been multiplied by The plotted series are the annual averages over the days in the year By plugging not just about half of them are found to be negative for the these numbers into the formulas of Corollary 2, we find the re- quotation data. Figures will also differ across days. For- correspond to ask, mid, and bid quotes. This is obviously in even longer for some days. Thus when the the dynamic impact on quotes and transaction prices as a re- time dependence in tick time is converted into calendar time, sponse to a change in the efficient price.

This can be done with the time dependence in and is about the same. The standard impulse response analysis. For alternative ways of de- results for MSFT are in some respects very different from those composing the observed price series, see, e.

For the quoted price series, the time dependence is slightly more involved, particularly for McInish, Shoesmith, and Wood , Hasbrouck , and mid-quotes.

Harris, McInish, and Wood , among others. Our analysis Comparing the results in Figures 6 and 7 shows that the differs from this literature because we apply cointegration tech- noise properties have changed after decimalization of the tick niques to quotes and transactions in conjunction, not to transac- size.

For transaction data, the change in the noise properties is tion prices from different exchanges. Because it is possible to most evident for AA, whereas in quoted prices the change is obtain the prevailing bid and ask prices at the time at which a most pronounced for MSFT.

For example, the first-order au- transaction occurs, we avoid issues related to nonsynchronous tocovariances for bid and ask quotes have opposite signs in trading. Our impulse response analysis, which we believe to be and Thus, based on the results in Table 2 and Fig- novel, shows how bid, ask, and transaction prices dynamically ures 6 and 7, we are led to the following fact.

The properties of the noise have changed over time. Suppose that the dynamics of the vector of prices, pti , can 6. It is reasonable to assume that each of the three observed all series when estimating the volatility of the latent efficient price series shares the same stochastic trend such that any pair price.

We choose included in the analysis. This shows which price series are most informative about the efficient price. The estimation is described in Appendix C. The Granger representation theorem is due to Johansen and Hansen , who obtained a recur- 6.

Some additional details about the estimation are given in Appendix C. There are a number of ways to define the common stochastic Assuming that the ultra-high—frequency price observations trend from the Granger representation see Johansen The are well described by 7 is problematic for several reasons. The reader This definition has the desired martingale property, because the should be aware that we have ignored all such issues, and thus corresponding intraday returns, the results that we draw from this analysis should be viewed as a rough approximation.

In our empirical analysis the elements are uncorrelated. This relation was dis- efficient price is larger for NYSE stocks, and, consequently, the cussed by Johansen see also Hansen and Johansen quoted prices are more closely related to the efficient price for , pp.

We find this and Baillie, Booth, Tse, and Zabotina see also to be a very interesting empirical observation that is likely tied Harris et al. Table 5 reports daily averages for the different there is no immediate evidence that a profit can be made from price series.

It is comforting to see that our estimate of the efficient price is in line with the quoted Define the two matrices, bid and ask prices.

Table 5. The efficient price and the noise processes were deduced from daily estimates of a cointegration vector autoregression model. The shaded area shows the best bid and ask quotes, and the star represent actual transaction prices.

An interesting question is how transaction prices and equation quotes are affected by a change in the efficient price. At low sam- see Hansen Thus the conclusions of these viewed as only indicative of the dynamic effect on bid, ask, articles may hold as long as intraday returns are not sampled and transaction prices as a response to a change in the efficient more frequently than, say, every 30 ticks.

On the other hand, price. We established these results in Section 4, Figure 9 displays the estimated IRFs for transaction prices, where we used a general specification for the noise process bid and ask quotes as a response to a change in the efficient that can accommodate both types of dependency.

Our cointe- price. These results are based on the daily estimates for Although most gration analysis showed that the negative correlation is found of the price change occurs instantaneously, the estimated IRFs in all price series, including transaction prices. This is likely explained by the differences standard measure of realized variance revealed a substantial im- between specialist quotes on the NYSE and the competitive provement in the precision, because the theoretical reduction of quotes on NASDAQ.

These gains were achieved with a simple bias correction that incorporates the first-order autoco- 7. For example, We have analyzed the properties of market microstructure the kernel estimators of Barndorff-Nielsen et al.

Our use sample estimators of Zhang et al. Among the estimators ties about market microstructure noise, and we have shown that tick kernel-based estimators are very useful in this context. However, there are characteristics of market microstructure noise, the most notable potential gains from allowing for some bias in exchange for a of which are that the noise process is time-dependent and that reduction of the variance.

This may particularly be the case in the noise process is correlated with the latent efficient returns. Thus, correcting for a smaller number of auto- 1 tick quotation data and were found to hold for intraday returns based covariances may be better in terms of the RMSE, so RV ACNW10 1 tick on both calendar time and tick time sampling.

Exploiting the discrete- Proof of Lemma 2 ness of the data as in Large ; Oomen is another possible avenue for improving existing estimators. All errors assumption. For the second sum, we find that remain the responsibility of the authors. Financial support from the Danish Research Agency grant no. Further, by propriate quantile from the standard normal distribution. An approximate confidence nomics, 61, 43— Andrews, D. Although Bai, X. On the other hand, we do not want to Baillie, R. Bandi, F. Although the model is simple to estimate by least squares Bandi, F.

Barndorff-Nielsen, O. AA stock over a sample period that spans the five year from January 2, to December 31, The raw data were filtered for outliers and we discarded transactions outside period from am to pm, and days with less than five hours of trading were removed from the sample, which reduced the sample by 13 days.

The RVs are calculated for the hours that the market is open, approximately minutes per day 6. The left panel also shows that the RVAC 1 is less sensitive to the choice of m. In our application the implications seem to fail once intraday returns are sampled more frequently than every 30 seconds. This can be un- derstood from the fact that the iid noise assumption causes the first-order autocorrelation of ei,m m and hence yi,m to be non-zero, whereas higher-order covariances are all zero.

If higher-order autocorrelations of yi,m are non-zero, which could be the case if the noise component, u t , was dependent across time different from iid noise , then the m RVAC 1 would be biased for large ms. This problem is evident from the signature plots in Figure m 2 that show that the RVAC 1 is biased for sampling frequency above 30 seconds.

For example, with m 1-second sampling the bias is quite severe and close to that of the standard RV, however the RVAC 1 generally has a smaller bias. The volatility signature plot of RVAC1 indicate that the time-dependence in u persists for less than 30 seconds, because the signature plot is quite constant for the frequencies that are below a second sampling.

Concluding Remarks m We have derived the bias and variance properties of RVAC 1 , which equals the standard realized vari- ance plus a bias correction that is given from the first-order autocorrelation of intraday returns.

Most of the existing theoretical studies of the RV in the presence of market mi- crostructure effects are based on this assumption, however our empirical analysis revealed that this assumption does not hold in practice.

While it may be true or approximately true for sampling at low frequencies, it does not hold when returns are sampled more frequently than every 30 seconds m in our empirical analysis. While the RVAC 1 is biased when sampling at high frequencies, its bias was less severe than that of m the standard RV, and RVAC 1 was found to dominate the standard RV m when the former is based on a less aggressive sampling, such as second sampling.

However, our analysis has revealed a need to study the properties of RV-measures under a more general specification for the noise process. Acknowledgements We thank Neil Shephard for valuable comments. Financial support from the Danish Research Agency, grant no.

All errors remain our responsibility. Next, we derive derive the expressions of each of these five terms. Thus Pm m 2. Proof of Corollary 4. Andersen, T. I, Elsevier-North Holland, Amsterdam.

Andreou, E. Bandi, F. Barndorff-Nielsen, O. Billingsley, P. Bollen, B. Corsi, F. Dacorogna, M. French, K. Montreal, November Meddahi, N. Newey, W. Oomen, R. Wasserfallen, W. Zhang, L. Zhou, B. An unbiased measure of realized variance By Peter Reinhard Hansen. An optimal and unbiased measure of realized variance based on intermittent high-frequency data By Peter Reinhard Hansen. Download PDF.