ReMeDIAsymptoticVariance: Asymptotic variance of ReMeDI estimator

In highfrequency: Tools for Highfrequency Data Analysis

Asymptotic variance of ReMeDI estimator

Description

Estimates the asymptotic variance of the ReMeDI estimator.

Usage

ReMeDIAsymptoticVariance(pData, kn, lags, phi, i)

Arguments

pData	xts or data.table containing the log-prices of the asset
kn	numerical value determining the tuning parameter kn this controls the lengths of the non-overlapping interval in the ReMeDI estimation
lags	numeric containing integer values indicating the lags for which to estimate the (co)variance
phi	tuning parameter phi
i	tuning parameter i

Details

Some notation is needed for the estimator of the asymptotic covariance of the ReMeDI estimator. Let

\delta\left(n, i\right) = t_{i}^{n}-t_{t-1}^{n}, i\geq 1,

\hat{\delta}_{t}^{n}=\left(\frac{k_{n}\delta\left(n,i+1+k_{n}\right)-t_{i+2+2k_{n}}^{n}+t_{i+2+k_{n}}^{n}}{\left(t_{i+k_{n}}^{n}-t_{i}^{n}\right)\vee\phi_{n}}\right)^{2},

U\left(1\right)_{t}^{n}=\sum_{i=0}^{n_{t}-\omega\left(1\right)_{n}}\hat{\delta}_{i}^{n},

U\left(2,\boldsymbol{j}\right)_{t}^{n}=\sum_{i=0}^{n_{t}-\omega\left(2\right)_{n}}\hat{\delta}_{i}^{n}\Delta_{\boldsymbol{j}}\left(Y\right)_{i+\omega\left(2\right)_{2}^{n}}^{n},

U\left(3,\boldsymbol{j},\boldsymbol{j}'\right)_{t}^{n}=\sum_{i=0}^{n_{t}-\omega\left(3\right)_{n}}\hat{\delta}_{i}^{n}\Delta_{\boldsymbol{j}}\left(Y\right)_{i+\omega\left(3\right)_{2}^{n}}^{n}\Delta_{\boldsymbol{j}'}\left(Y\right)_{i+\omega\left(3\right)_{3}^{n}}^{n},

U\left(4;\boldsymbol{j},\boldsymbol{j}'\right)_{t}^{n}=-\sum_{i=2^{q-1}k_{n}}^{n_{t}-\omega\left(4\right)_{n}}\Delta_{\boldsymbol{j}}\left(Y\right)\Delta_{\boldsymbol{j}^{\prime}}\left(Y\right)_{i+\omega\left(3\right)_{3}^{n}}^{n},

U\left(5,k;\boldsymbol{j},\boldsymbol{j}'\right)_{t}^{n}=\sum_{Q_{q}\in\mathcal{Q}_{q}}\sum_{i=2^{e\left(Q_{q}\right)}k_{n}}^{n_{t}-\omega\left(5\right)_{n}}\Delta_{\boldsymbol{j}_{Q_{q}\oplus\left(\boldsymbol{j}\prime_{Q_{q'}}\left(+k\right)\right)}}\left(Y\right)_{i}^{n}\prod_{\ell:l_{\ell}\in Q_{q}^{c}}\Delta_{\left(j_{l_{\ell}},j\prime_{l_{\ell}}+k\right)\left(Y\right)_{i+\omega\left(5\right)_{\ell+1}^{n}\prime}},

U\left(6,k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)=\sum_{j_{l}\in\boldsymbol{j},j_{l^{\prime}}^{\prime}\in\boldsymbol{j}^{\prime}}\sum_{i=2k_{n}}^{n_{t}-\omega\left(6\right)n}\Delta_{\left(j_{l},j_{l^{\prime}}^{\prime}+k\right)}\left(Y\right)_{i}^{n}\Delta_{\boldsymbol{j}_{-l}}\left(Y\right)_{i+\omega\left(6\right)_{2}^{n}}^{n}\Delta_{\boldsymbol{j}_{-l^{\prime}}^{\prime}}\left(Y\right)_{i+\omega\left(6\right)_{3}^{n}}^{n} \\ -\sum_{j_{l}\in\boldsymbol{j}}\sum_{i=2^{q}k_{n}}^{n_{t}-\omega^{\prime}\left(6\right)_{n}}\Delta_{\left\{ j_{l}\right\} \oplus\boldsymbol{j}^{\prime}\left(+k\right)}\left(Y\right)_{i}^{n}\Delta_{\boldsymbol{j}-l}\left(Y\right)_{i+\omega^{\prime}\left(6\right)_{2}^{n}}^{n} \\ -\sum_{j_{l^{\prime}\in\boldsymbol{j}^{\prime}}^{\prime}}\sum_{i=2^{q}k_{n}}^{n_{t}-\omega^{\prime\prime}\left(6\right)n}\Delta_{\left\{ j_{l^{\prime}}^{\prime}+k\right\} \oplus\boldsymbol{j}}\left(Y\right)_{i}^{n}\Delta_{\boldsymbol{j}_{-l^{\prime}}^{\prime}}\left(Y\right)_{i+\omega^{\prime\prime}\left(6\right)_{2}^{n}\prime}^{n},

U\left(7,k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=ReMeDI\left(\boldsymbol{j}\oplus\boldsymbol{j}^{\prime}\left(+k\right)\right)_{t}^{n},

U\left(k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=\sum_{\ell=5}^{7}U\left(\ell,k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n},

U\left(k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=\sum_{\ell=5}^{7}U\left(\ell,k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n},

Where the indices are given by:

\omega\left(1\right)_{n}=2+2k_{n},\ \omega\left(2\right)_{2}^{n}=2+\left(3+2^{q-1}\right)k_{n},\ \omega\left(2\right)_{n}=\omega\left(2\right)_{2}^{n}+j_{1}+k_{n},

\omega\left(3\right)_{2}^{n}=2+\left(3+2^{q-1}\right)k_{n},\ \omega\left(3\right)_{3}^{n}=2+\left(5+2^{q-1}+2^{q^{\prime}-1}\right)k_{n}+j_{1},

\omega\left(3\right)_{n}=\omega\left(3\right)_{3}^{n}+j_{1}^{\prime}+k_{n},\ \omega\left(4\right)_{2}^{n}=2k_{n}+q_{n}^{\prime}+j_{1},\ \omega\left(4\right)_{n}=\omega\left(4\right)_{2}^{n}+j_{1}^{\prime}+k_{n},

e\left(Q_{q}\right)=\left(2\left|Q_{q}\right|+q^{\prime}-q-1\right)\vee1,\ \omega\left(5\right)_{\ell+1}^{n}=4\ell k_{n}+\sum_{\ell^{\prime}=1}^{\ell}j_{l_{\ell^{\prime}}}\vee\left(j_{l_{\ell}}^{\prime}+k\right)\textrm{for}\ell\geq 1,

\omega\left(5\right)_{n}=\omega\left(5\right)_{\left|Q_{q}^{c}\right|+1}^{n}+j_{l_{\left|Q_{q}^{c}\right|}}\vee\left(j_{l_{\left|Q_{q}^{c}\right|}}+k\right)+k_{n},

\omega\left(6\right)_{2}^{n}=\left(2^{q-2}+2\right)k_{n}+j_{\ell}\vee\left(j_{\ell^{\prime}}^{\prime}+k\right),\ \omega\left(6\right)_{3}^{n}=\left(2^{q-2}+2^{q^{\prime}-2}+2\right)k_{n}+j_{1}+j_{\ell}\vee\left(j_{\ell}^{\prime}+k\right),

\omega^{\prime}\left(6\right)_{2}^{n}=\left(2^{q-2}+2\right)k_{n}+j_{\ell}\vee\left(j_{1}^{\prime}+k\right),\ \omega^{\prime\prime}\left(6\right)_{2}^{n}=\left(2^{q^{\prime}-2}+1\right)k_{n}+\left(j_{\ell^{\prime}}^{\prime}+k\right)\vee j_{1},

\omega\left(6\right)_{n}=\omega\left(6\right)_{3}^{n}+j^{\prime}+k_{n},\ \omega^{\prime}\left(6\right)_{n}=\omega^{\prime}\left(6\right)_{2}^{n}+j_{1}+k_{n},\ \omega^{\prime\prime}\left(6\right)_{n}=\omega^{\prime\prime}\left(6\right)_{2}^{n}j_{1}^{\prime}+k_{n},

The asymptotic variance estimator is then given by

\hat{\sigma}\left(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=\frac{1}{n_{t}}\sum_{\ell=1}^{3}\hat{\sigma}_{\ell}\left(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n},

where

\hat{\sigma}_{1}\left(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=U\left(0;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)+\sum_{k=1}^{i_{n}}\left(U\left(k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}\right)+\left(2i_{n}+1\right)U\left(4;\boldsymbol{j},\boldsymbol{j}\right)_{t}^{n},

\hat{\sigma}_{2}\left(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=U\left(3;\boldsymbol{j},\boldsymbol{j}^{\prime}\right),

\hat{\sigma}_{3}\left(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=\frac{1}{n_{t}^{2}}\textrm{ReMeDI}\left(Y,\boldsymbol{j}\right)_{t}^{n}\textrm{ReMeDI}\left(Y,\boldsymbol{j}^{\prime}\right)_{t}^{n}U\left(1\right)_{t}^{n}\\,

-\frac{1}{n_{t}}\left(\textrm{ReMeDI}\left(Y,\boldsymbol{j}\right)_{t}^{n}U\left(2,\boldsymbol{j}^{\prime}\right)_{t}^{n}+\textrm{ReMeDI}\left(Y,\boldsymbol{j}^{\prime}\right)_{t}^{n}U\left(2,\boldsymbol{j}\right)_{t}^{n}\right),

Value

a list with components ReMeDI and asympVar containing the ReMeDI estimation and it's asymptotic variance respectively

Note

We Thank Merrick Li for contributing his Matlab code for this estimator.

Examples

kn <- knChooseReMeDI(sampleTDataEurope[, list(DT, PRICE)])

remedi <- ReMeDI(sampleTDataEurope[, list(DT, PRICE)], kn = kn, lags = 0:15)

asympVar <- ReMeDIAsymptoticVariance(sampleTDataEurope[, list(DT, PRICE)], 
                                     kn = kn, lags = 0:15, phi = 0.9, i = 2)

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