The variance density spectral estimate of a time series may be defined as the Discrete Fourier Transform of the variance-covariance function of the sample. It follows that the periodogram of an unselfcorrelated, N-long, sample time series is a consistent estimator of the population spectrum biased by convolution with the sincĀ² function. Models fitted to time series can be tested by examining residuals for self-correlation using well established methods such a Ljung-Box. Autoregressive Moving Average (ARMA) models are well suited to this process and allow peaks due to sinusoidal components to be tested for significance using the *F*-distribution.

Time series having power law spectra with negative indices, i.e. with “red” spectra, are common in nature due to integration effects. These often appear to be deterministic functions of elapsed time or to be correlated with the time series of similar red spectra quantities. This is known as *spurious regression* or* spurious correlation* and was first noted by Yule (1926) as “Nonsense Correlations”. Such spurious effects may be avoided by fitting an ARMA model and testing its drift coefficient for significance.

**References:**

Reid, J. (2017) “There is no significant trend in global average temperature”. *Energy & Environmen*t, **28**, 3, pp 302-317 doi: 10.1177/0958305X16686447

Part II of *The Fluid Catastrophe* by John Reid