2020-09-29t14-40-11z.md

date	title	id
2020-09-29 07:40:11 -0700	ARIMA models	2020-09-29t14-40-11z

In time series forecasting, when dealing with non-stationary data, the autoregressive-integrated-moving-average model can come in handy.

As the name suggests, ARIMA is simply an extension of ARMA that manages to handle non-stationary data through an integration.

More specifically, similarly to what we do in this note, we re-write the series in terms of the difference of consecutive values, apply ARMA on this new series, and then work our way backwards to the original series.

Mathematically, given a non-stationary series $y_t$, for a first order differenced (d = 1) ARIMA model we write the new series as

$$ z_t = a_{t+1} - a_t $$

We can then apply the ARMA model to this new series to get

$$ {z_{t}=c+\varepsilon _{t}+\sum _{i=1}^{p}\varphi {i}z{t-i}+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}.,} $$

Now obviously, we wish to recover $a_t$. To do this, we set shift the original equation, writing it as

$$ \begin{aligned} z_{t-1} &= a_t - a_{t-1} a_t &= z_{t-1} - a_{t-1} \end{aligned} $$

Noticing the structure of the latter equation, we can then apply a recurrence relation to replace all instances of $a$ on the RHS, obtaining

$$ a_t = a_{t-1} + \sum_{i=1}^{k-t} z_{k-i}. $$

We can then substitute the ARMA model of $z_t$ we defined earlier to have our full ARIMA model depending on on 3 parameters, $p. q. d$. $p$ and $q$ are the same parameters found in ARMA and $d$ parametrizes the amount of integrations we perform.

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Note, throughout this derivation, we utilized a degree of differencing $d = 1$. While this is the most commonly used value for the parameter, it doesn't allow us to generalize the model. A degree of differencing of 1 was used only to help with intuition.