Modeling of time series arrays by multistep prediction or likelihood methods

摘要

An estimation theory is provided for the fitting of possibly incorrect, invcrtible, short-memory models to (short- or long-memory) time series or time scries arrays by multistep predietion error minimization or Gaussian likelihood maximization. By array, we mean data v_t(T), 1<=t <=T, that depend on the number of observations T, such as regression or other cstimatcd-model residuals, or the outputs of time varying filters, for example seasonal adjustments. Our theory only requires the modeled arrayto have basic properties: for a.s. [i.p.] convergence of parameter estimates, the array's sample lagged second moments must converge a.s. [i.p.]. and its end values y_(l-j) (T) and y_(T-j) (T) must be of order less than T~(12). Or an appropriately differenced version of the observed array must have these properties. In Findley et at. (Ann. Statist. 29 (2001) 815), broad classes of arrays were shown to have these properties. Even for the special case of auloregressive moving average models tit to stationary Gaussian time scries data, our result on the convergence of parameter estimates minimizing p-step-ahead prediction error sums of squares is new.