the market and other factors add more price variability around the trend. Therefore, this
stochastic trend model is also fitted to price data.
The model consists of one measurement equation and two transition equations:
(6)
yt = μt + εt
μt = μt-1+ βt—1+ ηt
βt = βt-1 +ςt
where yt is the independent variable indexed by time t ;
is the state vector; εt is the random
error describing the short run randomness with mean zero and variance σε2 ;3
2
ση
I 0
σ2
is the error vector describing the long run randomness in the
transition equation that governs the evolution of the state vector. Both of the errors in the
measurement equation follow normal distributions and are independent of each other.
In the basic specification, μt, the mean component of the dependent variable, is shown
as a random walk with a drift. Therefore the final generalization shows that the mean of the
dependent variable grows at a random rate.
3 The model also allows for a non-normal errors when εt is assumed to be generated by an inverse
hyperbolic sine transformation from normality: et= (τt-δ)~N(0,1), and
τt =θ-1 ln θεt+
where δ is the non-centrality parameter; δ > 0(< 0) denotes the
distribution is skewed to the right (left) and if δ = 0 the distribution is symmetric. θ is associated with
the degree of kurtosis with θ ≠ 0 denoting a kurtotic distribution. Thus, the error term can be expressed
as εt
eθτt
e-θτt
2θ
12
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