Estimates the asymptotic variance of the ReMeDI estimator.
ReMeDIAsymptoticVariance(pData, kn, lags, phi, i)xts or data.table containing the log-prices of the asset
numerical value determining the tuning parameter kn this controls the lengths of the non-overlapping interval in the ReMeDI estimation
numeric containing integer values indicating the lags for which to estimate the (co)variance
tuning parameter phi
tuning parameter i
a list with components ReMeDI and asympVar containing the ReMeDI estimation and it's asymptotic variance respectively
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 $$ \hat{\sigma}_{1}\left(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=U\left(0;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)+\sum_{k=1}^{i_{n}}\left(U\left(k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}\right)+\left(2i_{n}+1\right)U\left(4;\boldsymbol{j},\boldsymbol{j}\right)_{t}^{n}, $$
$$ \hat{\sigma}_{2}\left(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=U\left(3;\boldsymbol{j},\boldsymbol{j}^{\prime}\right), $$ $$ \hat{\sigma}_{3}\left(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=\frac{1}{n_{t}^{2}}\textrm{ReMeDI}\left(Y,\boldsymbol{j}\right)_{t}^{n}\textrm{ReMeDI}\left(Y,\boldsymbol{j}^{\prime}\right)_{t}^{n}U\left(1\right)_{t}^{n}\\, $$ $$ -\frac{1}{n_{t}}\left(\textrm{ReMeDI}\left(Y,\boldsymbol{j}\right)_{t}^{n}U\left(2,\boldsymbol{j}^{\prime}\right)_{t}^{n}+\textrm{ReMeDI}\left(Y,\boldsymbol{j}^{\prime}\right)_{t}^{n}U\left(2,\boldsymbol{j}\right)_{t}^{n}\right), $$