Validation of stochastic difference equation models for empirical time series

Let the given covariance stationary process y(.) obey a linear stochastic difference equation made up of y(k) and its n1 lagged values, the observable input u(k) and Its n3 lagged values, the unobservable input w(k) derived from a zero mean white noise sequence, and n2 deterministic sinusoidal trend terms ??j(k), J=1,...,n2, the coefficients of all the terms being unknown. Our intention is to identify those terms in the difference equation whose coefficients are identically equal to zero on the basis of the observations y(k), u(k), k=1,...,N. This problem is equivalent to the following multiple hypothesis testing problem, i.e., determine the correct model among a finite number, say n4, of linear difference equation models, all the coefficients in each model being non-zero, but otherwise unknown. With the help of the given observations, we will construct a statistic Di for the ith model which denotes the predictive capabilities of the model, having the following property for all i = 1,..., n4: E(Di) ?? E(Dj) if ith model is true. Consequently, we prefer the ith model if the statistic Di has the smallest value among {D1,..,Dn 4}.