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Sample text

Then in the representation (12) the data vector is expressible as Yt = P ~t + [I O]Q TIt + ut ' where the matrix Q is chosen as the matrix of row eigenvectors of the matrix A as described above. 4. Unit Roots A unit root is also introduced if (first) differences of data are used to construct state space models, as mentioned, for example, in connection with the two alternative choices in time series modeling discussed by Nelson and Plosser [1982]. To illustrate this. consider Yt =a +bt + ut where a and b are constant parameters and the exogenous disturbance, ut ' is modeled as q,(L ) ut =eeL ) Et (14) where Et is the basic exogenous stationary serially-uncorrelated stochastic process.

By interpreting = 0 as the long-run equilibrium relation, this representation is an error-correction model in the sense of Sargan [1964] because deviation from the equilibrium affects short-run (adjustment) dynamics. We next present two simple examples which may be considered to be model-based decompositions in which some a priori considerations impose particular forms of trend or seasonal variational components of time series. Example. 4 we apply these aggregation schemes to vector-valued data-generating processes.

Suppose now that the Kalman filter is activated at some point in the remote past. First, replace the subscript t by t - toto indicate that the filter is switched on at time to. If the noise sequences are wide-sense stationary and A is asymptotically stable, then letting to recede into the infinite past has the same effect on the recursions for {n t } as letting t approach infinity, and similarly for {L,}, and {31 }. Denoting the steady state (limiting) value of n t by n. , (19) shows that it satisfies an algebraic matrix equation n.

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