R: CholCov estimator

rCholCov {highfrequency}

R Documentation

CholCov estimator

Description

Positive semi-definite covariance estimation using the CholCov algorithm. The algorithm estimates the integrated covariance matrix by sequentially adding series and using 'refreshTime' to synchronize the observations. This is done in order of liquidity, which means that the algorithm uses more data points than most other estimation techniques.

Usage

rCholCov(
  pData,
  IVest = "rMRCov",
  COVest = "rMRCov",
  criterion = "squared duration",
  ...
)

Arguments

`pData`	a list. Each list-item i contains an `xts` object with the intraday price data (in levels) of stock `i` for day `t`. The order of the data does not matter as it will be sorted according to the criterion specified in the `criterion` argument
`IVest`	integrated variance estimator, default is `"rMRCov"`. For a list of implemented estimators, use `listCholCovEstimators()`.
`COVest`	covariance estimator, default is `"rMRCov"`. For a list of implemented estimators, use `listCholCovEstimators()`.
`criterion`	criterion to use for sorting the data according to liquidity. Possible values are `"squared duration"`, `"duration"`, `"count"`, defaults to `"squared duration"`.
`...`	additional arguments to pass to `IVest` and `COVest`. See details.

Details

Additional arguments for IVest and COVest should be passed in the ... argument. For the rMRCov estimator, which is the default, the theta and delta parameters can be set. These default to 1 and 0.1 respectively.

The CholCov estimation algorithm is useful for estimating covariances of d series that are sampled asynchronously and with different liquidities. The CholCov estimation algorithm is as follows:

First sort the series in terms of decreasing liquidity according to a liquidity criterion, such that series 1 is the most liquid, and series d the least.
Step 1:

Apply refresh-time on {a} = \{1\} to obtain the grid \tau^{a}.

Estimate \hat{g}_{11} using an IV estimator on f_{\tau^{a}_j}^{(1)}= \hat{u}_{\tau^{a}_j}^{(1)}.
Step 2:

Apply refresh-time on {b} = \{1,2\} to obtain the grid \tau^{b}.

Estimate \hat{h}^{b}_{21} as the realized beta between f_{\tau^{b}_j}^{(1)} and \hat{u}_{\tau^{b}_j}^{(2)}. Set \hat{h}_{21}=\hat{h}^{b}_{21}.

Estimate \hat{g}_{22} using an IV estimator on f_{\tau^{b}_j}^{(2)}= \hat{u}_{\tau^{b}_j}^{(2)}-\hat{h}_{21}f_{\tau^{b}_j}^{(1)}.
Step 3:

Apply refresh-time on {c} = \{1,3\} to obtain the grid \tau^{c}.

Estimate \hat{h}^{c}_{31} as the realized beta between f_{\tau^{c}_j}^{(1)} and \hat{u}_{\tau^{c}_j}^{(3)}. Set \hat{h}_{31}= \hat{h}^{c}_{31}.

Apply refresh-time on {d} = \{1,2,3\} to obtain the grid \tau^{d}.

Re-estimate \hat{h}_{21}^{d} at the new grid, such that the projections f_{\tau^{d}_j}^{(1)} and f_{\tau^{d}_j}^{(2)} are orthogonal.

Estimate \hat{h}^{d}_{32} as the realized beta between f_{\tau^{d}_j}^{(2)} and \hat{u}_{\tau^{d}_j}^{(3)}. Set \hat{h}_{32} = \hat{h}^{d}_{32}.

Estimate \hat{g}_{33} using an IV estimator on f_{\tau^{d}_j}^{(3)}= \hat{u}_{\tau^{d}_j}^{(3)}-\hat{h}_{32}f_{\tau^{d}_j}^{(2)} -\hat{h}_{31}f_{\tau^{d}_j}^{(1)}.
Step 4 to d:

Continue in the same fashion by sampling over {1,...,k,l} to estimate h_{lk} using the smallest possible set.

Re-estimate the h_{nm} with m<n\leq k at every new grid to obtain orthogonal projections.

Estimate the g_{kk} as the IV of projections based on the final estimates, \hat{h}.

Value

a list containing the covariance matrix "CholCov", and the Cholesky decomposition "L" and "G" such that \code{L} \times \code{G} \times \code{L}' = \code{CholCov}.

Author(s)

Emil Sjoerup

References

Boudt, K., Laurent, S., Lunde, A., Quaedvlieg, R., and Sauri, O. (2017). Positive semidefinite integrated covariance estimation, factorizations and asynchronicity. Journal of Econometrics, 196, 347-367.