rMRCov: Modulated realized covariance
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rMRCov

R Documentation

Modulated realized covariance

Description

Calculate univariate or multivariate pre-averaged estimator, as defined in Hautsch and Podolskij (2013).

Usage

rMRCov(
  pData,
  pairwise = FALSE,
  makePsd = FALSE,
  theta = 0.8,
  crossAssetNoiseCorrection = FALSE,
  ...
)

Arguments

`pData`	a list. Each list-item contains an `xts` or `data.table` object with the intraday price data of a stock.
`pairwise`	boolean, should be `TRUE` when refresh times are based on pairs of assets. `FALSE` by default.
`makePsd`	boolean, in case it is `TRUE`, the positive definite version of rMRCov is returned. `FALSE` by default.
`theta`	a `numeric` controlling the preaveraging horizon. Detaults to `0.8` as recommended by Hautsch and Podolskij (2013)
`crossAssetNoiseCorrection`	a `logical` denoting whether to apply the bias correction term on the off-diagonals (covariance) terms. We set this to `FALSE` by default as noise is typically seen as independent across assets.
`...`	used internally, do not change.

Details

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 (rMRCov) 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

Value

A d \times d covariance matrix.

Author(s)

Giang Nguyen, Jonathan Cornelissen, Kris Boudt, and Emil Sjoerup.

References

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.

Examples

## 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)]))
rMRCov(a, pairwise = TRUE, makePsd = TRUE)


# We can also add use data.tables and use a named list to convey asset names
a <- list(foo = sampleOneMinuteData[as.Date(DT) == "2001-08-04", list(DT, MARKET)],
          bar = sampleOneMinuteData[as.Date(DT) == "2001-08-04", list(DT, STOCK)])
rMRCov(a, pairwise = TRUE, makePsd = TRUE)


## End(Not run)

highfrequency documentation built on Oct. 4, 2023, 5:08 p.m.