Empirical Evaluation of Competing High-Frequency Estimators of Quadratic Variation

Abstract

Those who wish to estimate the daily volatility of asset returns using intradaily data are spoiled for choice. The last three decades have seen many applicable estimators proposed, including the standard realized volatility (RV) estimator (Andersen et al. 2001; Barndorff-Nielsen and Shephard 2002; Bandi and Russell 2008), the first-order-autocorrelation adjusted RV estimator (French, Schwert, and Stambaugh 1987; Zhou 1996; Hansen and Lunde 2006), the two-scale (TSRV), and multi-scale RV (MSRV) estimators (Zhang, Mykland, and Ait-Sahalia 2005a; Zhang 2006), the Realized Kernel (RK) estimator (Barndorff-Nielsen et al. 2008), the quasi-maximum likelihood estimator (QMLE; Ait-Sahalia, Mykland, and Zhang 2005; Xiu 2010) and the preaveraged RV (PARV) estimator (Jacod et al. 2009), to name but a few of the most well-established methods. These estimators may be implemented using data on quote or trade prices, measured in calendar-time or tick-time, at a range of different frequencies. Many of these methods also require choices to be made about bandwidths, window widths, kernel functions, etc. This broad menu of estimators presents empirical analysts with a conundrum: which estimator should be used for a particular application of interest?

In a theoretical setting with no microstructure noise, there exists wide agreement that the best choice of estimator is generally the simple RV estimator computed using the highest frequency data available (Andersen et al. 2001). However, as is well known, in the presence of microstructure noise, the RV estimator becomes severely biased at high frequencies. Two broad approaches exist to circumvent this problem. The first is to implement the simple RV estimator using an intermediate data frequency—high enough for the variance to be reasonably small, but not so high as to create severe bias. Five-minute RV is often chosen as a trade-off between these two concerns, although this choice of frequency is often arbitrary. At best, the frequency might be chosen based on a visual inspection of a volatility signature plot (e.g. Awartani 2008; Degiannakis and Floros 2016; Shen, Urquhart, and Wang 2020; Bandi and Russell 2008). The second approach is to implement one of the many noise-robust estimators that have been proposed in the literature. In theory, these estimators are asymptotically unbiased, eliminating the need to choose a sampling frequency that trades off bias for variance. However, the extent to which unbiasedness is achieved by these estimators in empirical applications is unclear. Despite the volume of literature proposing and applying high-frequency volatility estimators, relatively little work has been done on the empirical evaluation of competing estimators.

Ait-Sahalia and Xiu (2019) introduce a Hausman test of the presence of microstructure noise in high-frequency data. This is constructed as a test of the difference between the RV estimator and a maximum likelihood estimator (MLE). It should be noted that the Ait-Sahalia and Xiu (2019) test detects bias only. If the aim is to trade bias for variance, then it provides only part of the necessary information. Patton (2011) proposes a method for testing the equality of values of a class of loss function for two high-frequency volatility estimators. The mean squared error (MSE) is a member of the class, so the method facilitates the empirical evaluation of competing estimators in terms of a particular bias-variance trade-off. However, it does not provide separate evaluations of the bias and variance. Liu, Patton, and Sheppard (2015) have used Patton’s method with a QLIKE loss function in a comprehensive analysis of over 400 different implementations of 8 different types of RV estimator applied to 31 different financial assets. They consider quote and transaction prices with tick- and calendar-time observations, with sampling frequencies ranging from 1 s to 15 min. They find little evidence that any of the estimators considered is superior to the 5-min RV estimator. This result has been widely cited in the literature,¹ usually as a justification of the use of the 5-min RV estimator instead of a noise-robust estimator in applied work. It should be noted that rankings of estimators based on the QLIKE loss function do not necessarily correspond to rankings based on the MSE. However, in Supplementary Appendix, Liu, Patton, and Sheppard (2015) report results computed using the MSE instead of QLIKE, and these also fail to find evidence of other estimators being superior to 5-min RV, a result that they attribute to a lack of power. Gatheral and Oomen (2010) adopt a different approach, comparing the bias and MSE of a range of RV estimators (that includes 5-min RV) using data generated by a simulated order book market. In contrast to the empirical work of Liu, Patton, and Sheppard (2015), they find that the simple RV estimator is consistently one of the worst-performing estimators, irrespective of the sampling frequency used. Their overall recommendation is that practitioners should use TSRV, MSRV, or RK computed with an ad-hoc choice of tuning parameter and the highest available frequency of data. Nonetheless, the fact that their data are simulated raises the question of whether their results would hold with data generated by real markets.

In this article, we propose tests of equality for the bias, error variance and MSE for pairs of estimators of quadratic variation estimated using intradaily returns data. We prove stationary bootstrap limit theorems that allow our tests to be implemented with multiple hypothesis testing methodologies including White’s (2000) reality check, Hansen’s (2005) superior predictive ability (SPA) test, the STEP-M and generalized STEP-M procedures of Romano and Wolf (2005) and Romano and Wolf (2007), and the Model Confidence Set of Hansen, Lunde, and Nason (2011).

Like the Ait-Sahalia and Xiu (2019) test, our test of bias may be applied to simple RV estimators of different frequencies to determine whether they are unduly impacted by microstructure noise. However, our test may also be applied to any other high-frequency volatility estimator under only very mild assumptions about the estimation errors. Also, while the Ait-Sahalia and Xiu (2019) test considers the null hypothesis of equal bias on a single trading day, our test considers the average bias over a large number of trading days. We are unaware of any previously published tests of the equality of variance of the errors of high-frequency volatility estimators. Our test of equal MSE differs from Patton’s in two main ways. First, Patton’s approach is based on the assumption that the latent interdaily volatility process is a simple random walk. In contrast, our approach assumes a standard diffusion process for the intradaily asset price, makes some mild assumptions about microstructure noise, and allows the first difference of the interdaily volatility to be a member of a fairly general class of near-epoch dependent (NED) processes. Second, the simulation experiment that we report in Section 2 suggests that our test of equal MSE has a considerable power advantage over the equivalent test proposed by Patton. We pay two prices for these advantages. First, since our method exploits particular properties of the MSE, it cannot be generalized to other Bregman-type loss functions such as QLIKE. Since the MSE loss function is easily interpreted and widely used, we don’t regard this as a significant drawback. Second, while our test of equal bias applies to all volatility estimators, our tests of equal variance and equal MSE exploit a property that is a feature of a particular class of volatility estimators. We show that this class includes the RV estimator at all frequencies, and the RK, TSRV,² MSRV, PARV,³ QMLE, and FOAC estimators. However, the applicability to estimators outside this class needs to be established on a case-by-case basis.

The remainder of this article is arranged as follows. In Section 1, we state and explain our assumptions, present our test statistics for bias, variance and MSE, and state some theoretical properties that justify their use. The proofs are presented in the Appendix. In Section 2, we present the results of simulation studies that examine one of our key assumptions and investigate the size and power of our three test statistics assuming a range of different models for intradaily asset prices and their volatilities. We also compare the performance of our test for equal MSE with the corresponding test proposed by Patton (2011). In Section 3, we present an empirical study of the comparative bias, error variance and MSE of the RV, TSRV, MSRV, RK, PARV, and QMLE estimators applied to fifty of the most liquid stocks traded on the NYSE using a range of bandwidths, window lengths and subsamples. We also consider optimal parameter selection methods. In contrast to Liu, Patton, and Sheppard (2015), we find considerable evidence that there exist estimators that beat 5-min RV for many stocks in terms of the MSE, and we are able to explain the relative performance in terms of the comparative biases and variances. In Section 4, we draw some conclusions.

1 Main Results

Let $t=1,…,T$ index a sequence of trading days and let $θt$ denote the quadratic variation of a variable on day t. Let x_kt, $k∈{i,j}$ denote a pair of estimators of $θt$⁠, such that $xkt=θt+ukt$⁠, with u_kt denoting the estimation error. Let $θ˜t$ denote a proxy for $θt$⁠, and $u˜t$ denote the corresponding proxy error $u˜t=θ˜t−θt$⁠.

While it provides an elegant solution to the identification problem and a simple statistic, the assumption that the latent daily volatility process $θt$ follows a simple random walk is strong, and may not be satisfied in practical applications.

In contrast, our approach is based on the following assumptions:

Assumption 1 may be satisfied by using, for example, a low-frequency RV estimator as the proxy⁴. Assumption 2 is motivated by our belief that the current values of the estimation errors u_kt do not provide useful information for predicting the future estimation errors or quadratic variation. The fact that all the popular high-frequency estimators of $θt$ are constructed using data from only day t suggests that our belief is widespread. Note that, while there exists evidence of intradaily autocorrelation of microstructure noise that spans several ticks (e.g. Li and Linton 2022; Li, Laeven, and Vellekoop 2020; Li et al. 2022), this implies at worst a negligible degree of dependence between the estimation errors on successive days and, since the estimation error is likely to be related to the sum of the squared microstructure noise terms, does not imply autocorrelation of the daily estimation error. Note also that Li et al. (2022) find evidence that the variance of the microstructure error has predictive power for the quadratic variation on future days—which may result in some correlation between the daily estimation error and the quadratic variation on the following day. However, they find that, while statistically significantly different from zero, the reduction in the out-of-sample root mean squared prediction error from including the microstructure noise in a heterogeneous autoregressive (HAR) model is only 0.054%, so any impact on the veracity of Assumption 2 is likely to be negligible (see Li et al. 2022, Table 6, Panel A, Row 2). In Supplementary Appendix to this article (Supplementary Appendix B), we estimate the values of $cov(θt+1,ut)/E(ut2), cov(θt,ut+1)/E(ut2)$ and $cov(ut+1,ut)/E(ut2)$ for a range of estimators using values simulated with the estimated HAR model of Li et al. (2022), including lagged values of the variance of microstructure noise, and find that they are inconsequentially small.

Assumption 3 limits the variability of the daily bias of the estimation errors. If the estimation errors are mean-stationary, then Assumption 3 holds trivially. Assumption 4 requires a more detailed justification. It follows from standard regression theory that if $cov(θt,ukt)≠0$ then there exists a recentered, rescaled version of x_kt that has a lower MSE. Consequently, $cov(θt,ukt)=0$ might be regarded as a property that a “good” estimator should possess. However, this observation does not guarantee that the estimators in which we are interested will have this property. Below, we argue that, under fairly general conditions, for comparisons of estimators from a particular class, the relevant covariances will cancel out so that Assumption 4 holds.

Barndorff-Nielsen and Shephard (2002) show that in the absence of the jump component, if $μt(τ)=0, pt(τ)$ is directly observable, no leverage effect exists, and x_kt is the simple RV estimator computed at any frequency, then $cov(θt,ukt)=0$⁠, so Assumption 4 is satisfied under these conditions for the RV estimator. Meddahi (2002) allows for the leverage effect to exist⁵ and for a non-zero drift, and finds that $cov(θt,ukt)≠0$ under these conditions. However, he also shows that, for a broad class of stochastic volatility models, $corr(θt,ukt)=O(Nt−32)$⁠. Consequently, while non-zero, the relevant covariance would be expected to converge to zero relatively quickly as the number of intraday observations grows. Meddahi (2002) also shows that, for the models estimated by Andersen, Benzoni, and Lund (2002), the empirical magnitude of $corr(θt,ukt)$ is negligible. For example, for 1-h RV, he finds values of the order of $10−5$ or smaller for this correlation (Table III in Meddahi 2002). For higher-frequency RVs, the magnitude is even smaller. In Supplementary Appendix A, we simulate some popular models of asset prices that exhibit the leverage effect and show that the impact of leverage on our results is inconsequential. For this reason, and to avoid unnecessarily complicating the analysis, we will assume an absence of leverage in our subsequent analysis. Nonetheless, the model of Barndorff-Nielsen and Shephard (2002) and Meddahi (2002) is still too restrictive for our purposes. Accordingly, we generalize it in three ways.

The rationale for Assumption 4 is now presented as Proposition 1.1, which is proved in the Appendix.

For estimators that cannot be written as a quadratic form as in Equation (4), Assumption 4 would need to be investigated on a case-by-case basis. In cases where this is mathematically challenging, an alternative approach is to use simulation to estimate $cov(θt,uit−ujt)$ for particular data-generating processes (DGPs) and estimators.⁶ In cases where $cov(θt,uit−ujt)$ is found to be negligibly small for a range of DGPs, our test might be considered useful, even in the absence of a formal proof of Assumption 4.

Note that $var(uit)−var(ujt)=var(xit)−var(xjt)−2cov(θt,uit−ujt)$⁠, so Assumption 4 is sufficient for the identification of the difference in the error variances of two estimators. More trivially, $E(uit)−E(ujt)=E(xit)−E(xjt)$ so the difference in the bias of the two estimators is identified. These results, combined with the unbiased proxy given by Assumption 1 are sufficient for the identification of the MSE. Thus, with the addition of some assumptions about weak dependence and the finiteness of moments, we are able to construct statistics for testing the equality of the bias, error variance, and MSE of two estimators x_it and x_jt. We now turn our attention to this task.

Firstly, we define the statistics that we will use to measure the differences in bias, variance, and MSE respectively:

• $1. γ¯B=1T∑t=1TγBt where γBt=xit−xjt$

  (5)
• $2. γ¯V=1T∑t=1TγVt where γVt=12(Δxit−Δx¯i)2−12(Δxjt−Δx¯j)2$

  (6)
• $3. γ¯M=1T∑t=1TγMt where γMt=12(Δxit−Δx¯i)2−12(Δxjt−Δx¯j)2+(xit−θ˜t)(x¯i−θ˜¯)−(xjt−θ˜t)(x¯j−θ˜¯)$

  (7)

The null hypotheses of interest are $HB:E(γ¯B)=0, HV:E(γ¯V)=0$⁠, and $HM:E(γ¯M)=0$ which are, respectively, the hypotheses that the difference in biases is zero, the difference in variances is zero, and the difference in MSEs is zero under Assumptions 1, 2, and 4. In subsequent theorems, we will make use of the following assumptions:

It should be noted that most published papers that introduce new high-frequency estimators of quadratic variation include limit theorems that usually prove that, when suitably centered and rescaled, the estimation error (u_kt) converges to a mixed normal distribution as the number of intradaily observations grows. This, and the fact that each daily estimator is computed using a different dataset, suggests that the assumptions made above about the properties of $u˜t$ and u_kt for $k∈{i,j}$ are quite mild.

Note also that we make mild assumptions about the dynamic behavior of the daily quadratic variation (⁠$θt$⁠). Specifically, our approach allows the first difference of the quadratic variation to be any member of a broad class of NED processes.

The following results state the relationship between the statistics and the objects of interest, and are proved in the Appendix:

Our objective is to be able to test multiple hypotheses of equality of bias, error variance, and MSE for large sets of estimators. For example, we might be interested in comparisons of RV estimators computed at many different frequencies, or RK estimators with different kernels and/or bandwidths, or a comparison of RV versus RK versus TSRV versus MSRV, etc. This requires a convergence result for a suitable bootstrap algorithm in order to justify the use of techniques such as White’s (2000) reality check, Hansen’s (2005) SPA test, the STEP-M and generalized STEP-M procedures of Romano and Wolf (2005) and Romano and Wolf (2007), and the model confidence set of Hansen, Lunde, and Nason (2011). For this purpose, we implement the stationary bootstrap of Politis and Romano (1994), and we refer the reader to that paper for details of the procedure. This requires another assumption:

In the Appendix, we prove the following results:

Notice that we are able to test hypotheses about equality of bias by assuming only mild moment and mixing conditions for the estimation errors. In particular, we require no assumptions about the intradaily or interdaily behavior of the efficient price or microstructure noise. Nor do we need to assume that $cov(θt,uit−1−ujt−1)=cov(θt−1,uit−ujt)=cov(ukt,ukt−1)=0, k∈{i,j}$ (Assumption 2), or $cov(θt,uit−ujt)=0$ (Assumption 4). Consequently, this test may be applied to any pair of high-frequency estimators of quadratic variation. Furthermore, when comparing the bias of two estimators, one of the estimators could be the unbiased proxy from Assumption 1, in which case the test becomes a test of absolute, rather than comparative, bias. Thus, for example, RV estimators computed using a range of frequencies could be tested to determine a set of frequencies at which there is no evidence of bias, providing an alternative approach to the day-specific Hausman test proposed by Ait-Sahalia and Xiu (2019) that requires only very mild assumptions. To test the equality of variances of two estimators, we also require moment and weak dependence assumptions for the daily change in the quadratic variation $Δθt$ (Assumptions 5(1)(a) and 5(2)(a)), we need to assume that $cov(θt,uit−1−ujt−1)=cov(θt−1,uit−ujt)=cov(ukt,ukt−1)=0, k∈{i,j}$ (Assumption 2), and we need to assume that $cov(θt,uit−ujt)=0$ (Assumption 4), which restricts the range of estimators to which the test may be applied (see the discussion preceding Proposition 1.1). Finally, in addition to the assumptions required for the bias and variance tests, the unbiased proxy provided by Assumption 1 is also required for our test of equal MSE.

2 Monte Carlo Simulations

In this section, we perform simulation experiments to investigate the finite-sample performance of our proposed tests for equality of bias, error variance, and MSE. In particular, we consider three matters of interest. First, we wish to confirm that the asymptotic results in Section 1 provide good approximations to the finite-sample size of each of the statistics proposed. Second, we want to compare the finite sample power of the statistics that we propose to that of the statistics proposed by Patton (2011). Third, since both our MSE statistic and the statistics proposed by Patton require the use of an unbiased proxy that may have a large variance, we wish to investigate the impact of changes in the variance of the proxy on the size and power of the statistics considered.

We model the latent daily quadratic variation, $θt$⁠, using the following DGPs:

1. Exponential martingale (EM): $θt=eσWt−σ2t2$ where $σ=0.05$ and W_t is a standard Brownian motion.

2. HAR-RV: This DGP was used by Corsi (2008). Let $RVD,t$ denote RV (the square root of realized variance) on Day t, and let $RVW,t=(1/5)∑s=t−4tRVD,s$⁠, and $RVM,t=(1/22)∑s=t−21tRVD,s$⁠. Then $θt=c+βDRVD,t−1+βWRVW,t−1+βMRVM,t−1+εt$⁠, where c = 0.781, $βD=0.372, βW=0.343, βM=0.224$⁠, and $εt∼TN(0,σ)$⁠, with $σ=0.5$⁠, where TN denotes a Truncated Normal distribution with a lower bound of $−(c+βDRVD,t−1+βWRVW,t−1+βMRVM,t−1)$ and an infinite upper bound. Corsi (2008) suggested the truncation of the left-tail of $εt$ to ensure the positivity of $θt$⁠. The parameter choices are those obtained by Corsi (2008) when estimating the model using S&P500 data. Let $n=1,…,N$ index intraday returns, with N = 78 corresponding to 5-min increments over a 6.5-h trading day. Intraday returns are simulated via $rt,n=θtZn$ where $Zn∼N(0,1)$⁠. For each t, these are the returns used to construct $RVD,t$⁠. We initialize the model with $θ0=c/(1−βD−βW−βM)$⁠, and use 100 days as a burn-in period.

3. Two-factor Diffusion (TF): This two-factor diffusion model was used in Andersen, Bollerslev, and Meddahi (2005) and Bollerslev and Zhou (2002). Let $θt=∫t−1tσt(τ)dτ$ where $σt(τ)2=σ1t(τ)2+σ2t(τ)2, dσ1t(τ)2=0.5708(0.3257−σ1t(τ)2)dt+0.2286σ1t(τ)dW1,t(τ)$⁠, and $dσ2t(τ)2=0.0757(0.1786−σ2t(τ)2)dt+0.1096σ2t(τ)dW2,t(τ)$⁠. $W1,t(τ)$ and $W2,t(τ)$ are independent standard Brownian motions. We set $σ1t(0)2=0.3$ and $σ2t(0)2=0.2$⁠.

4. Jump Diffusion (JD): This jump diffusion model was used by Eraker, Johannes, and Polson (2003). Let $θt=∫t−1tσt(τ)dτ$ with $dσt(τ)2=0.025(0.5585−σt(τ)2)dt+σt(τ)((−0.504)(0.0896)dW1,t(τ)+1−0.5042(0.0896)dW2,t(τ))+χvdJv,t$⁠, where $W1,t(τ)$ and $W2,t(τ)$ are independent standard Brownian motions, $Jv,t$ is a Poisson process with an intensity of 0.0055 and $χv∼ Exp(1.798)$⁠, where Exp denotes the Exponential distribution. We set $σt(0)=0.5$⁠.

5. Rough Fractional Stochastic Volatility (RFSV): This DGP was used by Gatheral, Jaisson, and Rosenbaum (2018). $θt=∫t−1tσt(τ)dτ$ where $dlogσt(τ)2=−0.0005(logσt(τ)2−(−5))dt+0.3dWH,t(τ)$⁠, and $WH,t(τ)$ is fractional Brownian motion with a Hurst index of 0.14. We set $σt(0)=exp(−5)$⁠.

This simulation setup is very similar in style to that of Patton (2011). In particular, the properties of the estimator and the proxy error are parameterized to provide a close agreement with the equivalent quantities in those simulations. It is worth noting that we also duplicated the simulation methodology of Patton (2011) and found the results to be qualitatively very similar to those reported here. In the interests of saving space, we do not report these results here, preferring instead the simulation design described above since it covers a much wider range of dependence structures.

The statistics examined are:

1. $γ¯B$ for difference in bias (see Equation 5).

2. $γ¯V$ for difference in error variance (see Equation 6).

3. $γ¯M$ for difference in MSE (see Equation 7).

4. $γ¯P:MSE=1T∑t=1T−1((xit−θ˜t+1)2−(xjt−θ˜t+1)2)$ which is the statistic for the difference in MSE method due to Patton (2011).

5. $γ¯P:QLIKE=1T∑t=1T−1(θ˜t+1xit−θ˜t+1xjt+ln(xitxjt))$ which is the statistic for the difference in QLIKE method due to Patton (2011).

6. $γ¯Inf=1T∑t=1T((xit−θt)2−(xjt−θt)2)$ for the infeasible difference in MSE.

In our simulations, we test null hypotheses that each of these statistics has an expected value of zero. The first three of these statistics are those that we propose in Section 1. The fourth and fifth are the statistics proposed by Patton (2011) for testing the difference in MSE and QLIKE for pairs of volatility estimators, assuming that the underlying latent daily volatility follows a simple random walk. The sixth statistic is the statistic that we would use if the daily volatility were actually observable. It represents an upper limit on the possible performance of the other statistics. We simulate 5000 samples of size T = 500 and, for each sample, we compute the daily value of the base case volatility estimator $x0t$ and each of the estimators $x1t,…,x5t$⁠. We then calculate each of the six statistics listed above using $x0t$ and, in turn, each of $x1t,…,x5t$⁷, and conduct t-tests using the stationary bootstrap of Politis and Romano (1994) with the corrected block length selection procedure of Politis and White (2004) and Patton, Politis, and White (2009), implemented with the bandwidth selection procedure of Politis (2003). Using each statistic, we record rejection rates computed using standard critical values for a 5% significance level, which we use to estimate the size of each statistic and its power to detect a range of departures from the null hypothesis.

In order to estimate the size of each statistic under the different DGPs for daily volatility, we compute each statistic using $x0t$ and $x1t$ using the parameter values $b1=0$ and $ζ1=1$ since this corresponds to the case where $x0t$ and $x1t$ have the same bias, error variance, and MSE. We estimate the power of each test statistic to detect two departures from this null hypothesis: difference in the biases, and difference in the variances. In order to estimate the power to detect difference in the biases, we compute the estimators $x2t,…,x5t$ using the parameter values $ζ2=ζ3=ζ4=ζ5=1$ and $b2=1.5,b3=3,b4=4.5$ and $b5=6$⁠. To estimate the power to detect difference in the variances, we compute the estimators $x2t,…,x5t$ using the parameter values $ζ2=1.125,ζ3=1.25,ζ4=1.375$ and $ζ5=1.5$ and $b2=b3=b4=b5=0$⁠. We report the rejection rates in Table 1.