On the use of the bispectrum to detect and model non-linearity
Adrian Gerard Barnett · Bulletin of the Australian Mathematical Society · 2003
Informally a discrete time series is a set of repeated and, normally, equally spaced observations from the same process over time.The statistical analysis of time series has two functions: to understand better the generating process underlying the time series, and to forecast future values.The first analytical methods developed were based upon linear series.A linear series can be represented as a linear function of its own past and current values and the past and current values of some noise process, which can be interpreted as the innovations to the system.A non-linear series has a generally more complex structure that depends upon non-linear interactions between its past and current values and the sequence of innovations.Existing linear statistical methods can only approximate nonlinear series.As there is evidence to show that non-linear series are common in real life, two important problems are to detect and then to classify non-linearity.In moving from a linear to a non-linear structure the choice of possible models has moved from a countably infinite to an uncountably infinite set.Hence the need for methods that not only detect non-linearity, but classify the non-linear relationship between the past and current values and innovations.The third order moment is the expectation of the product of three series values lagged in time.The bispectrum is the double Fourier transform of the third order moment.Both statistics are useful tools for eliciting information on non-linear time series.There are concerns with the assumption of asymptotic independence between the values of the bispectrum estimate used by an existing test of non-linearity.We develop a method with a greater power than this existing method to detect non-linear series by using a model-based bootstrap.Further we show how patterns in the bispectrum are useful for classifying the locations of the non-linear interactions [1].To understand better tests of non-linearity and related inference, we investigate the variance of two estimates of the bispectrum.The two estimates are shown to have different inferential properties.One estimate is generally better able than the other to detect non-linearity and give information on the location of the non-linear interactions.