# ARIMA The ARIMA model is an extension of the ARMA model that applies to non-stationary time series (the kind of time series with one or more integrated unit-roots).By definition, the auto-regressive integrated moving average (ARIMA) process is an ARMA process for the differenced time series:.

$(1-\phi_1 L - \phi_2 L^2 -\cdots - \phi_p L^p)(1-L)^d x_t - \phi_o= (1+\theta_1 L+\theta_2 L^2 + \cdots + \theta_q L^q)a_t$ $y_t = (1-L)^d x_t$

Where:

• $$x_t$$ is the original non-stationary output at time t.
• $$y_t$$ is the observed differenced (stationary) output at time t.
• $$d$$ is the integration order of the time series.
• $$a_t$$ is the innovation, shock or error term at time t.
• $$p$$ is the order of the last lagged variables.
• $$q$$ is the order of the last lagged innovation or shock.
• $$\{a_t\}$$ time series observations are independent and identically distributed (i.e. i.i.d) and follow a Gaussian distribution (i.e. $$\Phi(0,\sigma^2)$$
Remarks
1. The variance of the shocks is constant or time-invariant.
2. Assuming $$y_t$$ (i.e. $$(1-L)^d x_t$$ ) is a stationary process with a long-run mean of \mu, then taking the expectation from both sides, we can express $$\phi_o$$ as follows: $\phi_o = (1-\phi_1-\phi_2-\cdots -\phi_p)\mu$
3. Thus, the ARIMA(p,d,q) process can now be expressed as: $(1-\phi_1 L - \phi_2 L^2 - \cdots - \phi_p L^p) (y_t-\mu) = (1+\theta_1 L + \theta_2 L^2 + \cdots + \theta_q L^q ) a_t$ $z_t=y_t-\mu$ $(1-\phi_1 L - \phi_2 L^2 - \cdots - \phi_p L^p) z_t = (1+\theta_1 L + \theta_2 L^2 + \cdots + \theta_q L^q ) a_t$
4. In sum, $$z_t$$ is the differenced signal after we subtract its long-run average.
5. The order of an ARIMA process is solely determined by the order of the last lagged variable with a non-zero coefficient. In principle, you can have fewer number of parameters than the order of the model.
6. Example: Consider the following ARIMA(12,2) process:

$(1-\phi_1 L -\phi_{12} L^{12} ) (y_t-\mu) = (1+\theta_2 L^2 ) a_t$
Requirements