A general smoother for nonGaussian parameterdriven models

We might also want to obtain the estimation of ht given all the information available, that is conditional on YT. Such a procedure is called smoothing and as before it is possible to derive a formal backward algorithm which delivers the smoothed densities f(ht\YT).

Let f(ht+1\YT) be the input of the iteration,14 t = T - 1,T - 2,..., 2,1. We can decompose the joint density of ht+1,ht, conditional on the information set Yt, in the product of the transition density by the filtered density (available from the filter):

By conditioning with the prediction density obtained from the filter, we obtain the following conditional density:

The joint density of ht+1, ht, conditional on the information set YT, is given by the product of the conditional density f(ht\ht+1, YT) by the input of the algorithm f(ht+1\YT). The information set ht+1, YT is included in the information set ht+1, Yt, ef+1, nT+2, where ef+1 = (et+1,..., eT) and qf+2 = (qt+2,..., nT). Given that ef+1, nf+2 is independent of ht, ht+1, Yt, we can conclude that f(ht\ht+1, YT) = f(ht\ht+1, Yt) (computed at step 2) and then:

step 3 f(ht+1,ht\YT) = f(ht\ht+1, YT)f(ht+1\YT) = f(ht\ht+1, Y)f(ht+1 \YT) Finally, by marginalisation we obtain the smoothed density of ht (output):

step 4 f(ht\Yt) =J f(ht+1 ,ht\YT)dht+1 = j f(ht\ht+1,Yt)f(ht+1\YT)dht+1

Again, only in the linear and Gaussian case, and in the Markovian and discrete case is it possible to obtain an analytic backward recursion: the Kalman smoother and the Kim smoother (Kim, 1994).

8.4.2.1 The Kalman smoother for continuous SV models

Let h t+1/t = E(ht+1\LYT) = E(ht+1\YT) and Qt+VT = MSE (ht+1\LYT) = MSE (ht+1\Yt). The Kalman smoother15 computes these quantities recursively for t = T - 1, T - 2,..., 2,1:

where ht/t, Qt/t, ht+1/t, Qt+1/t are stored from the Kalman filter.

14 For the first iteration (t = T - 1), the input is simply the final output of the filter f(hT\YT).

15 See also de Jong (1989).

For the log transformation of the continuous SV model (8.6), the Kalman smoother is useful in estimating the unobserved log-volatility, in fact it provides the best linear unbiased estimator of ht given (y1, y2,..., yT).

8.4.2.2 The Kim smoother for discrete SV models

The input is the smoothed probability P(ht+1\YT) and the recursion is simply:

st+1

where P(ht\Yt) and P(ht+1\Yt) are stored from the Hamilton filter.

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