arch.covariance.kernel.CovarianceEstimator.opt_bandwidth¶
- property CovarianceEstimator.opt_bandwidth : float[source]¶
Estimate optimal bandwidth.
Notes
Computed as
\[\hat{b}_T = c_k \left[\hat{\alpha}\left(q\right) T \right]^{\frac{1}{2q+1}}\]where \(c_k\) is a kernel-dependent constant, T is the sample size, q determines the optimal bandwidth rate for the kernel.
The constants and the values of q are from [1]. For example, \(c_k\) is 1.1447 and q is 1 for the Bartlett kernel, 2.6614 and 2 for the Parzen kernel, and 1.3221 and 2 for the quadratic-spectral kernel. \(\hat{\alpha}\left(q\right)\) is not the parametric (AR(1)) estimate that is suggested in [1]. It is the nonparametric estimate of [2],
\[\hat{\alpha}\left(q\right) = \left( \frac{\sum_{j=-n}^{n} \left|j\right|^q \hat{\sigma}_j} {\sum_{j=-n}^{n} \hat{\sigma}_j} \right)^2\]where \(\hat{\sigma}_j\) is the lag-j autocovariance of the data combined using
weights, which is the real part of the autocovariance \(T^{-1}\sum_t v_t\bar{v}_{t-j}\) of the combined series \(v_t\) if x is complex, and \(n = \lceil 4 \left(T / 100\right)^{r} \rceil\). The rate r is 2/9 for the Bartlett kernel, 4/25 for the Parzen kernel and 2/25 for the quadratic-spectral kernel, as in [2].References