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

### Description

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

### Usage

1 |

### 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

1 2 3 4 | ```
data(sample_5minprices_jumps)
a= list (sample_5minprices_jumps["2010-01-04",1],
sample_5minprices_jumps["2010-01-04",2] )
MRC(a, pairwise=TRUE,makePsd=TRUE)
``` |