Modeling of time series arrays by multistep prediction or likelihood methods

An estimation theory is provided for the fitting of possibly incorrect, invertible, short-memory models to (short- or long-memory) time series or time series arrays by multistep prediction error minimization or Gaussian likelihood maximization. By array, we mean data yt(T),1⩽t⩽T, that depend on the number of observations T, such as regression or other estimated-model residuals, or the outputs of time varying filters, for example seasonal adjustments. Our theory only requires the modeled array to have basic properties: for a.s. [i.p.] convergence of parameter estimates, the arrayʹs sample lagged second moments must converge a.s. [i.p.], and its end values y1+j(T) and yT−j(T) must be of order less than T1/2. Or an appropriately differenced version of the observed array must have these properties. In Findley et al. (Ann. Statist. 29 (2001) 815), broad classes of arrays were shown to have these properties. Even for the special case of autoregressive moving average models fit to stationary Gaussian time series data, our result on the convergence of parameter estimates minimizing p-step-ahead prediction error sums of squares is new.