Solved – Why ARIMA is prefered over any other time series analysis method

arimaintuitiontime series

I am new to time series analysis, and I am self learner. I am using R language to learn how to do time series analysis. I started by studying the concepts and the theory behind such analysis, however I see a great concentration on the ARIMA method, whereas there is a very small attention for other methods.

Could somebody one tell me why ARIMA is preferred over the other methods.

Best Answer

ARIMA models are not generally preferred over any other time series analysis method. There are certainly not preferred when the series demonstrate non-stationaries unable to be modelled using the ARIMA framework.

However, there is an important reason why the ARIMA might be preferred when the series are stationary. And this reason is the Wold's decomposition theorem - any covariance stationary process has a linear representation: a linear deterministic component ($V_t$) and a linear indeterministic components ($\varepsilon_t$)

Suppose that ${X_t}$ is a covariance stationary process with $\mathbb{E}[X_t] = 0$ and covariance function, $\gamma(j) = \mathbb{E}[X_t X_{t−j}]$ , $ \forall j$. Then

$$X_t = \sum_{j=0}^{\infty} \psi_j \varepsilon_{t−j} + V_t$$

where

  • $\psi_0=1$, $\sum_{j=0}^{\infty} \psi_j^2<\infty$
  • $\varepsilon_{t−j} \sim WN(0, \sigma_{\varepsilon}^2)$
  • $\mathbb{E}[\varepsilon_t V_s] = 0, \forall s,t>0$
  • $\varepsilon_t = X_t - \mathbb{E}[X_t|X_{t-1},X_{t-2},...]$

As you may see, the first part of the representation looks like an $MA(\infty)$ process with square summable moving average terms. The second part is the deterministic part of $X_t$ because $V_t$ is perfectly predictable based on past observations on $X_t$. And we know that models of $MA(\infty)$ representations are in their most general form $ARMA(p,q)$ representations: as long as the roots of the autoregressive part of an ARMA process are less than unity in absolute value, the process has a $MA(\infty)$ representation.

However, note, while an ARMA process generates an $MA(\infty)$ with square summable weights, it is not the only form that does this. A process that is square summable is not necessarily absolutely summable. $ARMA(p,q)$ models have ‘short memory’ relative to the entire class representations envisioned by the Wold representation. But Wold representation - despite covering more general cases- provides us with a strong argument of why modelling with ARMA is justifiable on stationary, short memory series.

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