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In much of the literature I'm studying it's one of those terms that occurs frequently yet without a rigorous definition to be found. Specifically, I am told:

For time-indexed random variables (RVs) $\{X_t\}$, the additive decomposition model is given as

$$X_t = {ll}(X_{t-1}, X_{t-2}, \ldots) + {fc}(X_{t-1}, X_{t-2}, \ldots, \varepsilon_t, \varepsilon_{t-1}, \ldots)$$

where

  • $ll$ is the long-term level, which is a stochastic process and can be visualised as a smoothed version of $\{X_t\}$, not to be confused with trends which are deterministic patterns
  • $fc$ is the fluctuation component which represents changes in local level, assumed stationary and with zero mean level
  • $\{\varepsilon_t\}$ are innovations, and are IID mean-zero RVs

But what is the difference in meaning between trend vs. long-term level vs. local level vs. mean level?

Additionally, aren't the fluctutation component and innovations modelling the same thing, which is the noise associated with each observation? So why complicate things by including both?

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