Time Series 6

with a noise term ut . ut is another noise term, with arbitrary distribution but serially independent; we take ut independent of wt . The filtering problem is to deduce information about xt , given the series of observations to time t: yt, yt−1, . . . , which we denote collectively by y≤t . The prediction problem would extrapolate information about xt+1, xt+2, . . . given information only through time t, and the smoothing problem would try to retroactively improve the estimates of xt−1, xt−2, . . . using information through t. Since xt is a random variable, information about it means constructing some model for its entire distribution