Modulated Realized Covariance (MRC): Return univariate or multivariate preaveraged estimator.

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Description

Function returns univariate or multivariate preaveraged estimator, as difined in Hautsch and Podolskij (2013).

Usage

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MRC(pdata, pairwise=FALSE,makePsd= FALSE,...)

Arguments

pdata

a list. Each list-item contains an xts 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 MRC is returned. FALSE by default.

...

additional arguments.

Details

In practice, market microstructure noise leads to a departure from the pure semimartingale model. We consider the process Y in period τ:

\mbox{Y}_{τ} = X_{τ} + ε_{τ}

where, the observed d dimensional log-prices are the sum of underlying Brownian semimartingale process X and a noise term ε_{τ}.

ε_{τ} 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 τ of a list (after refeshing data). Let r_{τ_i} be a return (with i=1, …,N) of an asset in period τ. Assume there is d assets.

In order to define the univariate pre-averaging estimator, we first define the pre-averaged returns as

\bar{r}_{τ_j}^{(k)}= ∑_{h=1}^{k_N-1}g≤ft(\frac{h}{k_N}\right)r_{τ_{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} = \lfloorθ N^{1/2}\rfloor. We use θ = 0.8 as recommendations in (Hautsch & 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}}{θ ψ_2}∑_{i=0}^{N-k_N+1} (\bar{r}_{τ_i})^2-\frac{ψ_1^{k_N}N^{-1}}{2θ^2ψ_2^{k_N}}∑_{i=0}^{N}r_{τ_i}^2

with

ψ_1^{k_N}= k_N ∑_{j=1}^{k_N}≤ft(g≤ft(\frac{j+1}{k_N}\right)-g≤ft(\frac{j}{k_N}\right)\right)^2,\quad

ψ_2^{k_N}= \frac{1}{k_N}∑_{j=1}^{k_N-1}g^2≤ft(\frac{j}{k_N}\right).

ψ_2= \frac{1}{12}

The multivariate counterpart is very similar. The estimator is called the Modulated Realized Covariance (MRC) and is defined as

\mbox{MRC}= \frac{N}{N-k_N+2}\frac{1}{ψ_2k_N}∑_{i=0}^{N-k_N+1}\bar{\boldsymbol{r}}_{τ_i}\cdot \bar{\boldsymbol{r}}'_{τ_i} -\frac{ψ_1^{k_N}}{θ^2ψ_2^{k_N}}\hat{Ψ}

where \hat{Ψ}_N = \frac{1}{2N}∑_{i=1}^N \boldsymbol{r}_{τ_i}(\boldsymbol{r}_{τ_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} = \lfloorθ N^{1/2+δ}\rfloor. δ = 0.1 results in a consistent estimate without the bias correction and a PSD estimate, in which case:

\mbox{MRC}^{δ}= \frac{N}{N-k_N+2}\frac{1}{ψ_2k_N}∑_{i=0}^{N-k_N+1}\bar{\boldsymbol{r}}_i\cdot \bar{\boldsymbol{r}}'_i

Value

an d x d matrix

Author(s)

Giang Nguyen, Jonathan Cornelissen and Kris Boudt

References

Hautsch, N., & 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(2), 165-183.

Examples

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