The covariance matrix of asset returns is the key input for many problems in finance and economics. This paper introduces a Bayesian nonparametric method to estimate the ex post covariance matrix from high-frequency data. The proposed estimator is robust to independent market microstructure noise and nonsynchronous trading and has several desirable features. First, pooling is employed to cluster high-frequency observations with similar covariance to improve estimation accuracy. Second, data augmentation is incorporated in synchronization to reduce the bias from nonsynchronous trading. Third, the proposed estimator is guaranteed to be positive definite. Monte Carlo simulation shows that the Bayesian nonparametric method provides more precise covariance estimates in both ideal and realistic settings. Empirical applications evaluate the proposed covariance estimator from an economic perspective and show that it offers improved out-of-sample performance compared with several classical estimators.
The covariance matrix of asset returns is the key input for many financial problems, such as portfolio allocation and risk management. Ever since the availability of high-frequency data, estimation of realized volatility and covariance using intraday returns has become a very active area of research. The main challenge in high-frequency covariance estimation is how to mitigate the influence of market microstructure noise and nonsynchronous trading. Several approaches have been adopted for the covariance estimation of noisy and nonsynchronously arrived prices. Model-free estimators include the multivariate realized kernel (MRK) (Barndorff-Nielsen et al 2011), two-scales covariance (TSRC) (Zhang 2011), pre-averaged realized covariance (PARC) (Christensen et al 2010), cumulative covariance (Hayashi and Yoshida 2005; Voev and Lunde 2007) and range-based covariance (Bannouh et al 2009). Another branch of covariance estimation methods relies on parametric models. Examples include the quasi-maximum likelihood estimator (Aït-Sahalia et al 2010), moving-average-based estimator (Hansen et al 2008), Bayesian covariance estimator (Peluso et al 2014) and Kalman smoother-based estimator (Corsi et al 2015). Other estimation approaches include those used by Malliavin and Mancino (2002), Renó (2003), Bandi and Russell (2005) and Large (2007).
Mykland and Zhang (2009) suggest that volatility estimation efficiency can be improved by blocking consecutive returns with constant variance. The Bayesian nonparametric variance estimation method introduced by Griffin et al (2019) provides a more flexible way of pooling data, which does not require returns to be consecutive in time and allows the number of clusters to be determined endogenously. This paper designs a Bayesian nonparametric covariance (BNC) estimator by integrating the pooling method suggested in Griffin et al (2019) with a vector-moving-average (VMA) adjustment and synchronization with data augmentation. The proposed covariance estimator is robust to microstructure noise and nonsynchronous trading.
The proposed high-frequency covariance estimation approach has three desirable features. First, the multivariate extension of the pooling method proposed in Griffin et al (2019) improves the accuracy of covariance matrix estimation. Unlike classical approaches that calculate covariance estimates using observations directly, the Bayesian nonparametric method applies a Dirichlet process mixture (DPM) model to cluster observations and constructs the estimator using posterior estimates that combine the information in each cluster and the prior. Such a mechanism improves the estimation accuracy by pooling observations and offering shrinkage for outliers. Second, inspired by Peluso et al (2014), this paper incorporates data augmentation in synchronizing observations. Under the previous-tick sampling scheme, the nonupdated prices on grid points are treated as missing observations and estimated conditional on observed data and the model structure. Such a synchronization method leads to returns with weaker cross-autocorrelations and makes it easier for the VMA filter to correct the bias from nonsynchronous trading. Third, by placing an inverse Wishart prior on intraperiod covariance, the proposed covariance matrix estimator is guaranteed to be positive definite.
Monte Carlo simulation assesses the estimation accuracy of the BNC estimator in both ideal and realistic settings. Compared with several benchmark estimators, the proposed estimator yields lower root-mean-square errors in estimating the covariance matrix in most cases. The optimal sampling frequency for the proposed estimator is also investigated based on simulated data. Empirical applications to equity data evaluate the out-of-sample performance of the proposed estimator via portfolio optimization. Under scenarios with different data dimensions and transaction costs, the BNC estimator consistently results in leading portfolio allocation outcomes. In addition, the proposed estimator captures similar time series dynamics of correlation and realized beta as the multivariate realized kernel (MRK).
This paper is organized as follows. Section 2 defines the estimation target. Section 3 discusses the BNC model, the proposed covariance estimator and the synchronization method with data augmentation. Section 4 conducts data simulation and compares the proposed estimator with competing alternatives. Empirical applications are presented in Section 5. Section 6 presents our conclusions. The online appendix collects additional results.
2 Ex post covariance
Suppose the log prices of assets are generated from
where is a vector of drift terms, is the instantaneous volatility matrix and is a vector of standard Brownian motions. As the true measure of the return covariance on day , the integrated covariance is the quantity of interest and is defined as
In practice, assets have different trading frequencies and their prices are not updated simultaneously. Let be the intraday log price of asset observed at . Synchronized using the previous-tick scheme with grid length , the th regularly spaced price of asset is defined as
Let denote the vector of synchronized prices. Due to the bid–ask bounce, discrete price changes and measurement error, price observations are contaminated with market microstructure noise. Based on noisy prices, the th intraday return vector is given as
where is the vector of microstructure errors. The presence of microstructure noise leads to autocorrelated return series and makes error-related terms part of the covariance of . Further, due to nonsynchronous trading, previous-tick returns have cross-autocorrelations, which results in underestimating the off-diagonal elements of the covariance matrix, especially when the sampling frequency is high. This phenomenon was documented as the Epps effect in Epps (1979).
In this section, we will first discuss the Bayesian nonparametric pooling framework and the synchronization approach with data augmentation, and then we will introduce the BNC estimator.
3.1 Bayesian nonparametric mixture model
In the Bayesian framework, the estimation target is
where is the true covariance of the th efficient return vector and . Intraday return vectors with similar covariation can be pooled to increase the estimation accuracy of . Clustering high-frequency returns using parametric models may suffer from misspecification, and the determination of the optimal number of clusters is challenging. The Bayesian nonparametric mixture framework suggested by Griffin et al (2019) provides a flexible way to exploit pooling. As a nonparametric version of the mixture model, the DPM allows the number of clusters to be nonfixed and inferred based on data. To correct the bias caused by microstructure noise and nonsynchronous trading, we incorporate the pooling framework with a VMA parameterization. The DPM-VMA model is given as
where is the vector of error terms, is the moving-average coefficient matrix and is the state-dependent intraperiod covariance matrix. Intraday return vectors within the same cluster share the common covariance matrix. If belongs to the th cluster, then , where is the unique covariance matrix that labels cluster .
The distribution of , denoted by , is a discrete distribution with a varying number of groups. Such a flexible structure is achieved by setting the Dirichlet process as the prior of . A draw from the DP is a discrete distribution that is centered around the base measure . The precision parameter influences the degree of pooling. As increases, the pooling process is more likely to have more clusters and the effect of pooling diminishes. The base measure is set to be an inverse Wishart distribution, , which ensures the positive definiteness of intraperiod covariance matrix estimates.
The base measure needs to be calibrated based on the volatility and covariation pattern within the target period. The degree of freedom is set to and the scale matrix is set to
where is an unbiased covariance estimator, such as the five-minute realized covariance (RC).11 1 For , the mean is . The center of is . Such a setting makes the prior informative by adopting useful information from another covariance estimator. The precision parameter is treated as unknown and a hierarchical prior is placed on it to add flexibility. A univariate Gaussian prior is placed on each element of and .
3.2 Synchronization with data augmentation
Synchronization is the process of placing observed prices on grid points. Although the previous-tick method synchronizes observations, two problems remain. First, the return periods are not exactly matched. As shown in Figure 1(a), the th asset 1 return and asset 2 return do not cover the same time interval, and there exists dependence between adjacent returns, such as and . Second, the regularly spaced returns contain zeros due to the absence of transactions in one or more interval(s). An example of zero return can be found in Figure 1(b). Zero returns not only influence the estimation of the model parameters but also lead to lag cross-autocorrelation, where . A model needs to be equipped with moving-average coefficient matrixes to filter all cross-autocorrelations.
In related work, Peluso et al (2014) apply data augmentation based on a dynamic linear model to synchronize observations. The method proposed in this paper also employs data augmentation but exploits pooling to increase estimation accuracy. In the example shown in Figure 2, no observation exists in the interval for both asset 1 and asset 2 or the interval for asset 3. , and can be considered missing. In the Bayesian framework, missing observations on common grid points can be estimated along with other model parameters. We refer the reader to the online appendix for detailed data augmentation steps. By filling in the missing observation gaps using data augmentation, zero returns can be eliminated and cross-autocorrelations in intraday returns are weakened.
The sampling frequency has a large impact on the quality of the proposed covariance estimator. Sampling data at low frequencies reduces the influence of nonsynchronous trading and microstructure noise but leads to more information loss. In contrast, increasing the sampling frequency diminishes the data augmentation value because more grid points contain missing observations, which requires inference. The optimal sampling frequency is investigated using simulated data in Section 4.
3.3 BNC estimator
Hansen et al (2008) show that the volatility estimator based on moving-average residuals needs to be scaled to estimate the integrated variance unbiasedly. Under the DPM-VMA model, the posterior mean of integrated covariance matrix is given as
where is the identity matrix. Equation (3.6) offers an unbiased estimator of the ex post covariance in the presence of independent microstructure noise and nonsynchronous trading, if only adjacent returns have cross-sectional dependence.22 2 The proof is available in the online appendix. can be estimated by integrating out the parameters and distributional uncertainties. Based on Markov chain Monte Carlo (MCMC) outputs, the BNC estimator of is defined as
The BNC estimator takes advantage of pooling and shrinkage to increase estimation accuracy. Conditional on return vectors in the th group and the base function , the posterior distribution of the intraperiod covariance matrix is
The posterior mean of is expressed as
which combines information from both data and the prior. Instead of assigning equal weights to all observations, the proposed approach assigns weights to observed information in a different manner and provides shrinkage for outliers. As shown in (3.9), the information from an outlier cluster is downweighted, which could reduce the estimation noise. Further, by weakening cross-autocorrelation with the help of data augmentation, the BNC estimator requires fewer moving-average terms compared with the moving-average-based estimator proposed in Hansen et al (2008). Estimating multiple MA coefficient matrixes is challenging, and imprecise parameter estimates may jeopardize the quality of the covariance estimator.
Another desirable feature is that the proposed covariance matrix estimator is guaranteed to be positive definite. Using an inverse Wishart prior ensures the positive definiteness of the intraperiod covariance matrixes. In addition, data augmentation removes the zero returns caused by nonupdated observations, which further eliminates the chance of obtaining singular estimates.
3.4 Model estimation
MCMC is employed to estimate the DPM-VMA model. Expressing the DP prior as in the stick-breaking representation in Sethuraman (1994), the following model is written in the form of a mixture model with infinite states:
where is the weight associated with the th component.
The infinite state space challenges the estimation of model parameters. The slice sampler from Kalli et al (2011) is applied to simplify the parameter estimation. Conditional on a set of auxiliary variables , the infinite state space can be truncated to a finite space, which makes the estimation of model parameters feasible. A set of latent state variables , where , is also introduced to label each observation’s cluster. The number of clusters is adjusted over the MCMC iterations. A new cluster with a covariance matrix can be created, and redundant clusters can be merged.
Combining the priors and data information, the joint posterior of the model parameters and is
The posterior sampling contains the following steps.
Sample for .
Sample for .
Sample for .
Update based on and .
Sample for .
Sampling high-dimensional parameters such as and using the Metropolis–Hastings algorithm results in low mixing and satisfactory proposals are difficult to find. The Hamiltonian Monte Carlo method introduced in Neal (2011) is applied to sample and . Compared with the random walk Metropolis algorithm, the Hamiltonian Monte Carlo produces distant proposals and explores the high-dimensional posterior distribution more efficiently. The Gibbs sampler handles the estimation of covariance matrixes . The concentration parameter is sampled using the method in Escobar and West (1994). After model parameter estimates are updated, the missing observations are estimated based on the method discussed in Section 3.2. The model parameter estimation details are provided in the online appendix.
4 Simulation results
This section assesses the estimation accuracy of the proposed estimator via Monte Carlo simulation. First, in an ideal scenario, we investigate whether pooling improves covariance estimation. We then compare the proposed estimator with five alternative high-frequency covariance estimators in realistic settings with different noise-to-signal ratios and nonsynchronous levels.
4.1 Synchronous data without noise
Following Barndorff-Nielsen et al (2011), fundamental log prices are generated from a multivariate factor stochastic volatility model:
where for , and are standard Brownian motions and . To produce data similar to equity prices with 6.5 trading hours, values in 23 400 sub-intervals are generated in each period. Based on simulated prices, returns at five-minute, one-minute and ten-second frequencies are formed and cases of three assets and ten assets are considered. The true covariance matrix is calculated as , where and
Since the simulated data are synchronous and free of error, we consider the BNC estimator without moving-average adjustment and compare it with the RC. For the Bayesian nonparametric approach, the center of the base function is set as the average of five-minute RCs on days and , the prior of ’s element is and the hierarchical prior on is . Parameter estimation is based on 5000 MCMC runs after 10 000 burn-in draws. Table 1 reports the RC and BNC estimators’ root mean squared errors (RMSEs) in estimating the integrated covariance matrix. Even though the RC provides the gold standard as it is the consistent estimator of the integrated covariance in the absence of microstructure noise and nonsynchronous trading, the Bayesian nonparametric approach offers additional improvement. In five of six cases, the BNC estimator has lower RMSE norms than the RC.33 3 The RMSE norm is defined as , where . For example, in the five-minute returns of the 10-asset case, switching from RC to BNC reduces the norm of the RMSE matrix from 1.133 to 1.111. The improvement is mainly attributed to the better estimation of the off-diagonal elements of the matrix. The comparison confirms that pooling and shrinkage improve the estimation accuracy of integrated covariance in finite samples.
4.2 Nonsynchronous data with microstructure noise
where represents the noise-to-signal ratio that governs the size of microstructure noise. The error terms are assumed to be uncorrelated, and the error variance increases with price volatility. Poisson processes are applied to generate arrival times of nonsynchronously spaced data. The parameter in the Poisson process governs the trading frequency of the simulated data. For example, means that asset prices arrive every five seconds on average.
The BNC estimator is compared with the five-minute RC (), MRK, PARC, TSRC and moving-average-based covariance estimator (RC-MA). The details of calculating those benchmark estimators are provided in the online appendix. The sampling frequencies in the calculation of the RC-MA and BNC estimators are 25, 45 and 75 seconds for cases , and , respectively.44 4 In the DGP of three assets, represents , represents and represents . In the DGP of ten assets, represents , represents and represents . MRK and PARC are calculated based on refresh-time sampled data. We set and for the TSRC estimator.55 5 The data frequencies for TSRC are 30 seconds for the case, 20 seconds for the case and 10 seconds for the case.
Table 2 reports the RMSEs of six covariance estimators in estimating the ex post covariance matrix in various cases with different data frequencies, microstructure noise levels and dimensions. provides less precise estimates because only 78 observations are used. The estimation accuracies of MRK, PARC, TSRC, RC-MA and BNC are improving as the data frequency increases. The top performer is the BNC estimator, which yields the lowest RMSE norms in 9 of 12 cases. A comparison of the results in the two microstructure noise cases ( and 0.003) suggests that the BNC estimator performs better when the market microstructure noise is low. The BNC estimator outperforms the RC-MA estimator in all scenarios, which confirms the benefits of incorporating pooling and data augmentation in covariance estimation.
|(a) Three assets|
|freq. (s)||(%)||BNC with DA||BNC without DA|
|(b) Ten assets|
|freq. (s)||(%)||BNC with DA||BNC without DA|
The sampling frequency plays an important role in ex post covariance estimation. Increasing the sampling frequency provides more information but aggravates the nonsynchronous problem. For the BNC estimator, the quality of data augmentation could be influenced by the ratio between updated and nonupdated prices. To investigate the optimal sampling frequency for the Bayesian nonparametric approach, we evaluate the BNC estimator at different sampling frequencies conditional on data simulated from DGPs with noise-signal ratio and . As shown in Table 3, the RMSE norm of the BNC estimator exhibits a U-shape as the proportion of zero returns increases. In both the three- and ten-asset cases, the estimation errors are reduced to a low level when the proportion of zero returns is approximately 10–15%. Thus, we recommend adjusting the sampling frequency to make synchronized returns containing around 10% to 15% zeros. Table 3 also shows that data augmentation provides significant improvement to covariance estimation. Comparing the BNC estimator with and without data augmentation, the latter has higher estimation errors in almost all cases, especially when the sampling frequency is high. Figure 3 plots 30-second BNC estimates with and without data augmentation, along with the true covariance values. The BNC estimator without data augmentation clearly underestimates the covariance values.
5 Empirical applications
This section reports the results obtained by applying the BNC estimation approach to high-frequency equity data. The data consist of 20 Standard & Poor’s 100 index companies’ intraday transactions from January 3, 2011 to December 31, 2018. The stock symbols are AXP, BAC, C, CAT, CVX, DIS, GS, HD, HON, IBM, JNJ, JPM, KO, MCD, NKE, PFE, PG, VZ, WMT and XOM.66 6 The company names are American Express, Bank of America, Citigroup, Caterpillar, Chevron, Walt Disney, Goldman Sachs, Home Depot, Honeywell, International Business Machine, Johnson and Johnson, JPMorgan Chase, Coca-Cola, McDonald’s, Nike, Pfizer, Procter & Gamble, Verizon Communications, Walmart and ExxonMobil. The data up to December 31, 2014 are obtained from the New York Stock Exchange Trade and Quote database, and Tick Data provides the data after January 2, 2015. We follow the procedure in Barndorff-Nielsen et al (2009) to clean the data.
We consider both 10- and 20-asset applications. BAC, CAT, DIS, GS, IBM, JNJ, KO, PG, WMT and XOM compose the 10-asset group. The intraday simple returns are converted into continuously compounded returns and are scaled by 100. Table 4 reports the summary statistics of 5-minute and 30-second previous-tick returns of the 10 assets. , refresh-time-based MRK, 10-second TSRC, 30-second PARC and BNC estimators are applied to estimate daily covariance matrixes. The synchronization methods and estimation details are the same as those discussed in Section 4. Table 5 provides summary statistics of the diagonal and off-diagonal estimates of the 10-asset covariance matrix. Compared with competing estimators, the has higher variance in estimating both realized variance and covariance, suggesting that the covariance estimator constructed using low-frequency data has larger estimation noise.
|(a) Five-minute returns|
|(b) 30-second returns|
5.1 Portfolio allocation evaluation
We assess the out-of-sample performance of the proposed covariance estimator from a portfolio optimization perspective. Based on the predicted covariance matrix, both the global minimum-variance portfolio and a target return portfolio are formed following the approach applied by de Pooter et al (2008). The optimization problem of the global minimum-variance portfolio is
where represents portfolio weights, is the predicted covariance matrix based on information up to day , and is a vector of 1s. The solution to (5.1) is
For an investor holding a target return portfolio, the portfolio weights can be obtained by solving the following minimization problem given the desired portfolio return :
where is the return mean vector. The optimal weight for the target return portfolio is
where stands for the weights of the maximum Sharpe ratio portfolio and is given by
The realized portfolio return on day is .
Ex post covariance estimates based on intraday returns over trading hours measure the covariance of open-to-close returns. Overnight information plays a nontrivial role in predicting future volatility. Following Fleming et al (2003), we incorporate overnight information to obtain estimates of close-to-close return covariance matrixes , with defined as
The next-period covariance is predicted using an exponential smoother with the decay rate :
|(a) 10 assets|
|Min. var.||Target return portfolio|
|(b) 20 assets|
|Min. var.||Target return portfolio|
The out-of-sample period spans from July 1, 2011 to the end of 2018 and the portfolios are rebalanced daily. The return mean vector is set to be the sample average of close-to-close returns. Table 6 summarizes the performance of both the global minimum-variance portfolio and the target return portfolio in 10- and 20-asset cases, respectively. Based on and a 12.5% annual target return, the portfolio based on the BNC estimator has the highest Sharpe ratio in the 10-asset case and is ranked second-best in the 20-asset case. For example, the Sharpe ratios of 10-asset portfolios based on BNC and MRK are 1.117 and 1.115, respectively, whereas , PARC and TSRC do not lead to portfolios with Sharpe measures higher than 1.1. Moreover, the global minimum-variance portfolio based on BNC has the lowest return standard deviation in the 20-asset case.
Following Fleming et al (2003), we also evaluate the target return portfolios using a utility-based approach. Assume that the investor has the following quadratic utility function:
where stands for the risk aversion coefficient. The performance fee is the cost that an investor would pay to switch from benchmark portfolio to alternative portfolio . is the value that equalizes the following equation:
Using the portfolio as the benchmark, the performance fees that an investor with quadratic utility would like to pay to switch to alternative portfolios are reported in Table 6. Both a less risk-averse investor () and a more conservative investor () would be willing to pay higher performance fees for BNC or MRK estimates than for PARC and TSRC.
|(a) 10 assets|
|Min. var.||Target return portfolio|
|(b) 20 assets|
|Min. var.||Target return portfolio|
We further evaluate portfolio performance in the presence of transaction costs. Following de Pooter et al (2008), we calculate the portfolio return under transaction cost as
Table 7 shows the out-of-sample performance of the target return portfolio and global minimum-variance portfolio in the presence of a 2% annual commission fee. The ranking of five covariance estimators is consistent with the no-transaction-cost case. BNC and MRK remain the top two estimators in terms of the Sharpe ratio and utility levels. Comparing the results in Tables 6 and 7 shows that transaction costs influence portfolios based on RC, PARC and TSRC more than BNC or MRK-based portfolios. Investors favor MRK or BNC more and are willing to pay higher performance fees when transaction costs are present. Transaction costs have a more significant impact on portfolios with a more substantial daily turnover. As shown in Table 5, , PARC and TSRC have higher average variances of estimating covariances than BNC and MRK, which could lead to more dramatic weight adjustments and higher transaction costs.
5.2 Correlation and realized beta
The ex post covariance matrix measures facilitate the analysis of the dynamics of correlation and realized beta. Figure 4 provides the correlation series between BAC and CAT based on 10-asset , MRK and BNC estimates. The correlation estimates based on MRK and BNC share similar dynamics, while the correlation implied from the five-minute RC is very volatile. Conditional on estimates of the covariance between the market index and an individual stock, the realized beta is defined as , where the second asset is the SPDR S&P 500 ETF (SPY). Figure 5 plots the realized beta of BAC based on the covariance measures of BAC as well as SPY estimated by , MRK and BNC. and have very similar paths, while the realized beta based on RC fluctuates widely. Table 8 shows the estimation results of the ARMA(1,1) model for the five versions of the realized beta, which also confirms that BNC captures similar time series dynamics of the realized beta to MRK.
This paper proposes a Bayesian nonparametric method of estimating the covariance matrix for nonsynchronous prices contaminated with independent microstructure noise. The proposed estimator provides three benefits in covariance estimation. First, pooling observations with similar covariance increases the precision of ex post covariance estimation. Second, the estimated covariance estimator is guaranteed to be positive definite. Third, a new synchronization method with data augmentation is introduced to reduce nonsynchronous bias in previous-tick returns.
Monte Carlo simulation confirms that the BNC estimator is very competitive with existing estimators in both ideal and more realistic settings. Empirical applications to equity returns show that the correlation and realized beta implied on the BNC estimator have similar time series dynamics to the MRK. Portfolio allocation experiments show that the proposed BNC estimator offers improved out-of-sample performance compared with several classical estimators.
Declaration of interest
The author reports no conflicts of interest. The author alone is responsible for the content and writing of the paper.
The author is grateful to the editor and two anonymous referees for their inspiring input and useful comments.
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