11.1 Local Trend Model
Consider the univariate time series yt satisfying
where {et} and {ηt} are two independent Gaussian white noise series and t = 1, … , T. The initial value μ1 is either given or follows a known distribution, and it is independent of {et} and {ηt} for t > 0. Here μt is a pure random walk of Chapter 2 with initial value μ1, and yt is an observed version of μt with added noise et. In the literature, μt is referred to as the trend of the series, which is not directly observable, and yt is the observed data with observational noise et. The dynamic dependence of yt is governed by that of μt because {et} is not serially correlated.
The model in Eqs. (11.1) and (11.2) can readily be used to analyze realized volatility of an asset price; see Example 11.1. Here μt represents the underlying log volatility of the asset price and yt is the logarithm of realized volatility. The true log volatility is not directly observed but evolves over time according to a random-walk model. On the other hand, yt is constructed from high-frequency transactions data and subjected to the influence of market microstructure noises. The standard deviation of et denotes the scale used to measure the impact of market microstructure noises.
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