Model selection for time series

The following problem is used to illustrate a number of important model selection techniques. Let X^T=(X_t ,t=1,. . .,T) be a sequence of observations generated by an unknown model M*. A family of plausible models is denoted by M^T=(M_k(θ), k=1,. . .,K), where M_k(θ) describes an autoregressive model of order k. For simplicity, emphasis is placed on the problem of choosing the `best´ model from the family M^T