View source: R/realizedMeasures.R
ReMeDIAsymptoticVariance | R Documentation |
Estimates the asymptotic variance of the ReMeDI estimator.
ReMeDIAsymptoticVariance(pData, kn, lags, phi, i)
pData |
|
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 |
Some notation is needed for the estimator of the asymptotic covariance of the ReMeDI estimator. Let
δ≤ft(n, i\right) = t_{i}^{n}-t_{t-1}^{n}, i≥q 1,
\hat{δ}_{t}^{n}=≤ft(\frac{k_{n}δ≤ft(n,i+1+k_{n}\right)-t_{i+2+2k_{n}}^{n}+t_{i+2+k_{n}}^{n}}{≤ft(t_{i+k_{n}}^{n}-t_{i}^{n}\right)\veeφ_{n}}\right)^{2},
U≤ft(1\right)_{t}^{n}=∑_{i=0}^{n_{t}-ω≤ft(1\right)_{n}}\hat{δ}_{i}^{n},
U≤ft(2,\boldsymbol{j}\right)_{t}^{n}=∑_{i=0}^{n_{t}-ω≤ft(2\right)_{n}}\hat{δ}_{i}^{n}Δ_{\boldsymbol{j}}≤ft(Y\right)_{i+ω≤ft(2\right)_{2}^{n}}^{n},
U≤ft(3,\boldsymbol{j},\boldsymbol{j}'\right)_{t}^{n}=∑_{i=0}^{n_{t}-ω≤ft(3\right)_{n}}\hat{δ}_{i}^{n}Δ_{\boldsymbol{j}}≤ft(Y\right)_{i+ω≤ft(3\right)_{2}^{n}}^{n}Δ_{\boldsymbol{j}'}≤ft(Y\right)_{i+ω≤ft(3\right)_{3}^{n}}^{n},
U≤ft(4;\boldsymbol{j},\boldsymbol{j}'\right)_{t}^{n}=-∑_{i=2^{q-1}k_{n}}^{n_{t}-ω≤ft(4\right)_{n}}Δ_{\boldsymbol{j}}≤ft(Y\right)Δ_{\boldsymbol{j}^{\prime}}≤ft(Y\right)_{i+ω≤ft(3\right)_{3}^{n}}^{n},
U≤ft(5,k;\boldsymbol{j},\boldsymbol{j}'\right)_{t}^{n}=∑_{Q_{q}\in\mathcal{Q}_{q}}∑_{i=2^{e≤ft(Q_{q}\right)}k_{n}}^{n_{t}-ω≤ft(5\right)_{n}}Δ_{\boldsymbol{j}_{Q_{q}\oplus≤ft(\boldsymbol{j}\prime_{Q_{q'}}≤ft(+k\right)\right)}}≤ft(Y\right)_{i}^{n}∏_{\ell:l_{\ell}\in Q_{q}^{c}}Δ_{≤ft(j_{l_{\ell}},j\prime_{l_{\ell}}+k\right)≤ft(Y\right)_{i+ω≤ft(5\right)_{\ell+1}^{n}\prime}},
U≤ft(6,k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)=∑_{j_{l}\in\boldsymbol{j},j_{l^{\prime}}^{\prime}\in\boldsymbol{j}^{\prime}}∑_{i=2k_{n}}^{n_{t}-ω≤ft(6\right)n}Δ_{≤ft(j_{l},j_{l^{\prime}}^{\prime}+k\right)}≤ft(Y\right)_{i}^{n}Δ_{\boldsymbol{j}_{-l}}≤ft(Y\right)_{i+ω≤ft(6\right)_{2}^{n}}^{n}Δ_{\boldsymbol{j}_{-l^{\prime}}^{\prime}}≤ft(Y\right)_{i+ω≤ft(6\right)_{3}^{n}}^{n} \\ -∑_{j_{l}\in\boldsymbol{j}}∑_{i=2^{q}k_{n}}^{n_{t}-ω^{\prime}≤ft(6\right)_{n}}Δ_{≤ft\{ j_{l}\right\} \oplus\boldsymbol{j}^{\prime}≤ft(+k\right)}≤ft(Y\right)_{i}^{n}Δ_{\boldsymbol{j}-l}≤ft(Y\right)_{i+ω^{\prime}≤ft(6\right)_{2}^{n}}^{n} \\ -∑_{j_{l^{\prime}\in\boldsymbol{j}^{\prime}}^{\prime}}∑_{i=2^{q}k_{n}}^{n_{t}-ω^{\prime\prime}≤ft(6\right)n}Δ_{≤ft\{ j_{l^{\prime}}^{\prime}+k\right\} \oplus\boldsymbol{j}}≤ft(Y\right)_{i}^{n}Δ_{\boldsymbol{j}_{-l^{\prime}}^{\prime}}≤ft(Y\right)_{i+ω^{\prime\prime}≤ft(6\right)_{2}^{n}\prime}^{n},
U≤ft(7,k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=ReMeDI≤ft(\boldsymbol{j}\oplus\boldsymbol{j}^{\prime}≤ft(+k\right)\right)_{t}^{n},
U≤ft(k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=∑_{\ell=5}^{7}U≤ft(\ell,k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n},
U≤ft(k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=∑_{\ell=5}^{7}U≤ft(\ell,k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n},
Where the indices are given by:
ω≤ft(1\right)_{n}=2+2k_{n},\ ω≤ft(2\right)_{2}^{n}=2+≤ft(3+2^{q-1}\right)k_{n},\ ω≤ft(2\right)_{n}=ω≤ft(2\right)_{2}^{n}+j_{1}+k_{n},
ω≤ft(3\right)_{2}^{n}=2+≤ft(3+2^{q-1}\right)k_{n},\ ω≤ft(3\right)_{3}^{n}=2+≤ft(5+2^{q-1}+2^{q^{\prime}-1}\right)k_{n}+j_{1},
ω≤ft(3\right)_{n}=ω≤ft(3\right)_{3}^{n}+j_{1}^{\prime}+k_{n},\ ω≤ft(4\right)_{2}^{n}=2k_{n}+q_{n}^{\prime}+j_{1},\ ω≤ft(4\right)_{n}=ω≤ft(4\right)_{2}^{n}+j_{1}^{\prime}+k_{n},
e≤ft(Q_{q}\right)=≤ft(2≤ft|Q_{q}\right|+q^{\prime}-q-1\right)\vee1,\ ω≤ft(5\right)_{\ell+1}^{n}=4\ell k_{n}+∑_{\ell^{\prime}=1}^{\ell}j_{l_{\ell^{\prime}}}\vee≤ft(j_{l_{\ell}}^{\prime}+k\right)\textrm{for}\ell≥q 1,
ω≤ft(5\right)_{n}=ω≤ft(5\right)_{≤ft|Q_{q}^{c}\right|+1}^{n}+j_{l_{≤ft|Q_{q}^{c}\right|}}\vee≤ft(j_{l_{≤ft|Q_{q}^{c}\right|}}+k\right)+k_{n},
ω≤ft(6\right)_{2}^{n}=≤ft(2^{q-2}+2\right)k_{n}+j_{\ell}\vee≤ft(j_{\ell^{\prime}}^{\prime}+k\right),\ ω≤ft(6\right)_{3}^{n}=≤ft(2^{q-2}+2^{q^{\prime}-2}+2\right)k_{n}+j_{1}+j_{\ell}\vee≤ft(j_{\ell}^{\prime}+k\right),
ω^{\prime}≤ft(6\right)_{2}^{n}=≤ft(2^{q-2}+2\right)k_{n}+j_{\ell}\vee≤ft(j_{1}^{\prime}+k\right),\ ω^{\prime\prime}≤ft(6\right)_{2}^{n}=≤ft(2^{q^{\prime}-2}+1\right)k_{n}+≤ft(j_{\ell^{\prime}}^{\prime}+k\right)\vee j_{1},
ω≤ft(6\right)_{n}=ω≤ft(6\right)_{3}^{n}+j^{\prime}+k_{n},\ ω^{\prime}≤ft(6\right)_{n}=ω^{\prime}≤ft(6\right)_{2}^{n}+j_{1}+k_{n},\ ω^{\prime\prime}≤ft(6\right)_{n}=ω^{\prime\prime}≤ft(6\right)_{2}^{n}j_{1}^{\prime}+k_{n},
The asymptotic variance estimator is then given by
\hat{σ}≤ft(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=\frac{1}{n_{t}}∑_{\ell=1}^{3}\hat{σ}_{\ell}≤ft(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n},
where
\hat{σ}_{1}≤ft(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=U≤ft(0;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)+∑_{k=1}^{i_{n}}≤ft(U≤ft(k;\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}\right)+≤ft(2i_{n}+1\right)U≤ft(4;\boldsymbol{j},\boldsymbol{j}\right)_{t}^{n},
\hat{σ}_{2}≤ft(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=U≤ft(3;\boldsymbol{j},\boldsymbol{j}^{\prime}\right),
\hat{σ}_{3}≤ft(\boldsymbol{j},\boldsymbol{j}^{\prime}\right)_{t}^{n}=\frac{1}{n_{t}^{2}}\textrm{ReMeDI}≤ft(Y,\boldsymbol{j}\right)_{t}^{n}\textrm{ReMeDI}≤ft(Y,\boldsymbol{j}^{\prime}\right)_{t}^{n}U≤ft(1\right)_{t}^{n}\\,
-\frac{1}{n_{t}}≤ft(\textrm{ReMeDI}≤ft(Y,\boldsymbol{j}\right)_{t}^{n}U≤ft(2,\boldsymbol{j}^{\prime}\right)_{t}^{n}+\textrm{ReMeDI}≤ft(Y,\boldsymbol{j}^{\prime}\right)_{t}^{n}U≤ft(2,\boldsymbol{j}\right)_{t}^{n}\right),
a list with components ReMeDI
and asympVar
containing the ReMeDI estimation and it's asymptotic variance respectively
We Thank Merrick Li for contributing his Matlab code for this estimator.
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)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.