Identification of the coefficients in a non-linear time series of the quadratic type (Q1069633)

This paper presents a method for identifying the structure of the quadratic model \[ x(t)=\dot e(t)+\sum^{R}_{r=1}\sum^{S}_{s=0}a(r,s)e(t-s)e(t-r-s), \] where the e(t) are independent, identically distributed zero mean random variables. This finite parameter model is a simplified second-order Volterra expansion of a stable non-linear filter. The standard deviation of e(t) is not estimable from observations of the x(t). It is shown that the \((m,m+n)th\) third-order cumulant (called the bicovariance) of \(\{\) x(t)\(\}\) is equal to a(m,n) times the square of the variance of e(t). The large sample distribution of the sample bicovariance is used to determine which quadratic coefficients are significantly different from zero. The method is illustrated using artificial and real data; daily stock price series for several securities.