This function computes the empirical cross-covariance of two stationary multivariate time series. If only one time series is provided it determines the empirical autocovariance function.
cov.structure(X, Y = X, lags = 0)An object of class timedom. The list contains
operators \(\quad\) an array. Element [,,k] contains the covariance matrix related to lag \(\ell_k\).
lags \(\quad\) returns the lags vector from the arguments.
vector or matrix. If matrix, then each row corresponds to a timepoint of a vector time series.
vector or matrix. If matrix, then each row corresponds to a timepoint of a vector time series.
an integer-valued vector \((\ell_1,\ldots, \ell_K)\) containing the lags for which covariances are calculated.
Let \([X_1,\ldots, X_T]^\prime\) be a \(T\times d_1\) matrix and \([Y_1,\ldots, Y_T]^\prime\) be a \(T\times d_2\) matrix. We stack the vectors and assume that \((X_t^\prime,Y_t^\prime)^\prime\) is a stationary multivariate time series of dimension \(d_1+d_2\). This function determines empirical lagged covariances between the series \((X_t)\) and \((Y_t)\). More precisely it determines \(\widehat{C}^{XY}(h)\) for \(h\in\) lags, where \(\widehat{C}^{XY}(h)\) is the empirical version of \(\mathrm{Cov}(X_h,Y_0)\). For a sample of size \(T\) we set \(\hat\mu^X=\frac{1}{T}\sum_{t=1}^T X_t\) and \(\hat\mu^Y=\frac{1}{T}\sum_{t=1}^T Y_t\) and $$\hat C^{XY}(h) = \frac{1}{T}\sum_{t=1}^{T-h} (X_{t+h} - \hat\mu^X)(Y_{t} - \hat\mu^Y)'$$ and for \(h < 0\) $$\hat C^{XY}(h) = \frac{1}{T}\sum_{t=|h|+1}^{T} (X_{t+h} - \hat\mu^X)(Y_{t} - \hat\mu^Y)'.$$