Cross-covariance & Cross-correlation¶
- Random variable: \(X_{i}\)
- Random vector: \(( X_{1},X_{2},\dots,X_{p} )\)
- Random vector: \((Y_{t},Y_{t+1},Y_{t+2},\dots,Y_{t+n})\)
\[
E(X) = [E(X_{1}), E(X_{2}),\dots E(X_{p})]' = (\mu_{1},\mu_{2},\dots,\mu_{p})'
\]
The variance-covariance matrix:
\[
E(X-\mu)(X-\mu)' =
\]
"Dispersion matrix"
\[
\begin{bmatrix}
\sigma_{11} & \dots & \sigma_{1p} \\
\vdots & \ddots & \vdots \\
\sigma_{p1} & \dots & \sigma_{pp} \\
\end{bmatrix}
\]
Cross Covariance Matrix¶
\[
\Sigma_{xy} = E(X-\mu_{x})(Y-\mu_{y})'
\]
- \(ij\)-th element of \(\Sigma_{xy}\) is the covariance between \(X_{i}\) and \(Y_{j}\)
Stationary Multivariate Time Series¶
- \(Y_{t} = (Y_{1t},\dots,Y_{kt})'\) where \(t \in T \equiv (\pm 1, \pm 2,\dots)\)
- \(Y\) is a collection of different time series \(Y_{1}, Y_{2},\dots Y_{k}\). We are looking at the time \(t\) value of this collection.
- \(Y_{t}\) is stationary if
- joint probability distribution of \((Y_{t_{1}},Y_{t_{2}},\dots Y_{t_{n}})\)
- is the same as that of \((Y_{t_{1}+l},Y_{t_{2}+l},\dots Y_{t_{n}+l})\)
- for all time stamps \(\{ t_{1}\dots t_{n},n \}\) and all leads and lags.
Moments of the Process¶
- Mean vector \(E(Y_{t}) =\mu = (\mu_{1},\mu_{2},\dots, \mu_{k})'\)
- where \(\mu_{i}\) is the mean of an individual time series part of \(Y_{t}\)
- Covariance matrix: \(E(Y_{t}-\mu)(Y_{t}-\mu)' \equiv \Sigma\)
Cross Covariance & Cross Correlation¶
- Autocovariance function for \(y_{it}\) (one of the individual processes from \(Y_{t}\))
\[
\gamma_{ii}(l) = E(y_{it}-\mu_{i})(y_{i,t+l}- \mu_{i})
\]
- Cross covariance between \(y_{it}, y_{jt}\) at lag \(l\):
\[
\gamma_{ij}(l) = E(y_{it}-\mu_{i})(y_{j,t+l}-\mu_{j})
\]
- Cross correlation
\[
\rho_{ij}(l) = \dfrac{\gamma_{ij}(l)}{\{\gamma_{ii}(0)\gamma_{jj}(0)\}^{1/2}}
\]
- Cross covariance matrix
\[
\Gamma(l) = E(Y_{t}-\mu)(Y_{t}-\mu)'
\]
- \(\rho(l)=V^{1/2}\rho(l)V^{-1/2}\)
- where \(V = diag(\gamma_{11}(0),\gamma_{22}(0),\dots,\gamma_{kk}(0))\)
Vector White Noise Process¶
\[
\{ e_{t} \} \sim WN(0,\Sigma)
\]
if and only if,
- \(e_{t}\) is stationary with mean \(\vec{0}\) and variance covariance matrix
\[
\Gamma(k) = \begin{cases}
\Sigma, & k=0 \\
0, & \text{otherwise}
\end{cases}
\]