4.2 Summary of nonparametric spectral estimation

In summary, the different steps to undertake are, in order,

1. Differencing (or detrending) should be performed prior to any analysis. The series should be centered around zero by subtracting the mean from the data to remove the frequencies near \(\omega=0\), which would otherwise dominate.
2. For tapering, multiply centered observations by a data window. The cosine bell taper, also known as Hann taper, is the default choice in the R function spectrum.
3. Zero-pad to increase the resolution and so that the length of your time series is a highly composite number, typically \(2^l>n\). If you want to convert your spectrum to an autocovariance, zero-padding helps avoid “circular autocorrelation”. It also reduces the computation burden for the FFT calculation.
4. Smooth the periodogram. It amounts to convoluting the periodogram with a kernel. The amount of averaging should increase as \(n \to \infty\). This reduces the variance, but introduces leakage. The amount of smoothing is controlled by the bandwidth parameter.