Calculate univariate or multivariate pre-averaged estimator, as defined in Hautsch and Podolskij (2013).
rMRC(pData, pairwise = FALSE, makePsd = FALSE)a list. Each list-item contains an xts object with the intraday price data of a stock.
boolean, should be TRUE when refresh times are based on pairs of assets. FALSE by default.
boolean, in case it is TRUE, the positive definite version of rMRC is returned. FALSE by default.
A \(d \times d\) covariance matrix.
In practice, market microstructure noise leads to a departure from the pure semimartingale model. We consider the process \(Y\) in period \(\tau\): $$ \mbox{Y}_{\tau} = X_{\tau} + \epsilon_{\tau}, $$ where the observed \(d\) dimensional log-prices are the sum of underlying Brownian semimartingale process \(X\) and a noise term \(\epsilon_{\tau}\).
\(\epsilon_{\tau}\) is an i.i.d. process with \(X\).
It is intuitive that under mean zero i.i.d. microstructure noise some form of smoothing of the observed log-price should tend to diminish the impact of the noise. Effectively, we are going to approximate a continuous function by an average of observations of \(Y\) in a neighborhood, the noise being averaged away.
Assume there is \(N\) equispaced returns in period \(\tau\) of a list (after refreshing data). Let \(r_{\tau_i}\) be a return (with \(i=1, \ldots,N\)) of an asset in period \(\tau\). Assume there is \(d\) assets.
In order to define the univariate pre-averaging estimator, we first define the pre-averaged returns as $$ \bar{r}_{\tau_j}^{(k)}= \sum_{h=1}^{k_N-1}g\left(\frac{h}{k_N}\right)r_{\tau_{j+h}}^{(k)} $$ where g is a non-zero real-valued function \(g:[0,1]\) \(\rightarrow\) \(R\) given by \(g(x)\) = \(\min(x,1-x)\). \(k_N\) is a sequence of integers satisfying \(\mbox{k}_{N} = \lfloor\theta N^{1/2}\rfloor\). We use \(\theta = 0.8\) as recommended in Hautsch and Podolskij (2013). The pre-averaged returns are simply a weighted average over the returns in a local window. This averaging diminishes the influence of the noise. The order of the window size \(k_n\) is chosen to lead to optimal convergence rates. The pre-averaging estimator is then simply the analogue of the realized variance but based on pre-averaged returns and an additional term to remove bias due to noise $$ \hat{C}= \frac{N^{-1/2}}{\theta \psi_2}\sum_{i=0}^{N-k_N+1} (\bar{r}_{\tau_i})^2-\frac{\psi_1^{k_N}N^{-1}}{2\theta^2\psi_2^{k_N}}\sum_{i=0}^{N}r_{\tau_i}^2 $$ with $$ \psi_1^{k_N}= k_N \sum_{j=1}^{k_N}\left(g\left(\frac{j+1}{k_N}\right)-g\left(\frac{j}{k_N}\right)\right)^2,\quad $$ $$ \psi_2^{k_N}= \frac{1}{k_N}\sum_{j=1}^{k_N-1}g^2\left(\frac{j}{k_N}\right). $$ $$ \psi_2= \frac{1}{12} $$ The multivariate counterpart is very similar. The estimator is called the Modulated Realized Covariance (rMRC) and is defined as $$ \mbox{MRC}= \frac{N}{N-k_N+2}\frac{1}{\psi_2k_N}\sum_{i=0}^{N-k_N+1}\bar{\boldsymbol{r}}_{\tau_i}\cdot \bar{\boldsymbol{r}}'_{\tau_i} -\frac{\psi_1^{k_N}}{\theta^2\psi_2^{k_N}}\hat{\Psi} $$ where \(\hat{\Psi}_N = \frac{1}{2N}\sum_{i=1}^N \boldsymbol{r}_{\tau_i}(\boldsymbol{r}_{\tau_i})'\). It is a bias correction to make it consistent. However, due to this correction, the estimator is not ensured PSD. An alternative is to slightly enlarge the bandwidth such that \(\mbox{k}_{N} = \lfloor\theta N^{1/2+\delta}\rfloor\). \(\delta = 0.1\) results in a consistent estimate without the bias correction and a PSD estimate, in which case: $$ \mbox{MRC}^{\delta}= \frac{N}{N-k_N+2}\frac{1}{\psi_2k_N}\sum_{i=0}^{N-k_N+1}\bar{\boldsymbol{r}}_i\cdot \bar{\boldsymbol{r}}'_i $$
Hautsch, N., and Podolskij, M. (2013). Preaveraging-based estimation of quadratic variation in the presence of noise and jumps: theory, implementation, and empirical Evidence. Journal of Business & Economic Statistics, 31, 165-183.
# NOT RUN {
library("xts")
# Note that this ought to be tick-by-tick data and this example is only to show the usage.
a <- list(as.xts(sampleOneMinuteData[as.Date(DT) == "2001-08-04", list(DT, MARKET)]),
as.xts(sampleOneMinuteData[as.Date(DT) == "2001-08-04", list(DT, STOCK)]))
rMRC(a, pairwise = TRUE, makePsd = TRUE)
# }
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