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Figure from:Structural Time Series Models

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ERR SSS He ‘ga SS ieee a, To illustrate these general points we will put the univariate structural model (1.1) of trends, seasonals and cycles discussed in Section 1 into time invariant state space form by writing a,=(,, B, 6, 6_,,...,8_..5, W&, w*)’, where

Figure 1 ERR SSS He ‘ga SS ieee a, To illustrate these general points we will put the univariate structural model (1.1) of trends, seasonals and cycles discussed in Section 1 into time invariant state space form by writing a,=(,, B, 6, 6_,,...,8_..5, W&, w*)’, where

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An approximation to (2.21) can be obtained for many models by running
An approximation to (2.21) can be obtained for many models by running
Probability that ML estimator of signal—noise ratio q is exactly equal to zero Table 2 SS a a a ee a en wwe ————ee rreaaeooerroeer eee v o U* “The term log |S,| can have a significant effect on small sample properties of maximum likelihood (ML) estimators in certain circumstances. This can be seen by looking at some results from the paper by Shephard and Harvey (1990) which analyses the sampling behaviour of the ML estimator of q, the ratio of the variances of y, and ¢,, in the local level model (1.12). When q is zero the reduced form of the local level model is strictly noninvertible. Evaluating the probability that q is estimated to be exactly zero for various true values of q
Probability that ML estimator of signal—noise ratio q is exactly equal to zero Table 2 SS a a a ee a en wwe ————ee rreaaeooerroeer eee v o U* “The term log |S,| can have a significant effect on small sample properties of maximum likelihood (ML) estimators in certain circumstances. This can be seen by looking at some results from the paper by Shephard and Harvey (1990) which analyses the sampling behaviour of the ML estimator of q, the ratio of the variances of y, and ¢,, in the local level model (1.12). When q is zero the reduced form of the local level model is strictly noninvertible. Evaluating the probability that q is estimated to be exactly zero for various true values of q
Principal components analysis of estimated covariance matrix of trend disturbances Table 3
Principal components analysis of estimated covariance matrix of trend disturbances Table 3
As yp, is unknown, this model cannot be analysed by using the Kalman filter. Instead, an estimate of a, can be made at time ¢ — 1, written a,,_,, and it can be used to replace y, in the variance. One of the problems with this approach is that this model does not constrain the observations to lie between zero and one, as e, is assumed Gaussian. Although this could be a problem if yu, were to be close to zero or one, this is unlikely to pose a difficulty for moderate sample sizes. This idea can be used to construct a model of opinion polls. Suppose there are just two parties. If the level of support for one party is modelled as a logistic transformation of a Gaussian random walk and the measurement error originates from using a simple random sample size n,, then
As yp, is unknown, this model cannot be analysed by using the Kalman filter. Instead, an estimate of a, can be made at time ¢ — 1, written a,,_,, and it can be used to replace y, in the variance. One of the problems with this approach is that this model does not constrain the observations to lie between zero and one, as e, is assumed Gaussian. Although this could be a problem if yu, were to be close to zero or one, this is unlikely to pose a difficulty for moderate sample sizes. This idea can be used to construct a model of opinion polls. Suppose there are just two parties. If the level of support for one party is modelled as a logistic transformation of a Gaussian random walk and the measurement error originates from using a simple random sample size n,, then
The Kalman filter can be applied in the standard way once the logist transformation has been Taylor expanded. The resulting model is An approach similar to this. but using a multivariate continuous time model to
The Kalman filter can be applied in the standard way once the logist transformation has been Taylor expanded. The resulting model is An approach similar to this. but using a multivariate continuous time model to
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