12. ARIMA Time Series Models

In this chapter we will deal with the classical, linear time series analysis. At first we will define the general linear process.

Definition 12.1 (Linear Process)  
If the process $ X_t$ has the representation

$\displaystyle X_t = \mu + \sum_{i=-\infty}^{\infty} a_i \varepsilon_{t-i}
$

with white noise $ \varepsilon_t$ and absolute summability of the filter $ (a_i): \sum_{i=-\infty}^{\infty} \vert a_i\vert < \infty $, then it is a linear process.

The linear process $ X_t$ is covariance stationary, since $ \mathop{\text{\rm\sf E}}(X_t)=\mu$ and

$\displaystyle \mathop{\text{\rm Cov}}(X_t,X_{t+\tau}) = \sigma^2\sum_{i=-\infty...
...\boldsymbol{1}(\tau = i-j) = \sigma^2
\sum_{i=-\infty}^{\infty} a_i a_{i-\tau} $

with $ \mathop{\text{\rm Var}}(\varepsilon_t)=\sigma^2$.

In general in econometrics, especially in the area of financial markets, series are observed which indicate a non-stationary behavior. In the previous chapter we saw that econometric models, which are based on assumptions of rational expectations, frequently imply that the relevant levels of, for example, prices, follow a random walk. In order to handle these processes within the framework of the classical time series analysis, we must first form the differences in order to get a stationary process. We generalize the definition of a difference stationary process in the following definition.

Definition 12.2 (Integrated process)  
We say that the process $ X_t$ is integrated of order $ d$, $ I(d)$, when $ (1-L)^{d-1} X_t$ is non-stationary and $ (1-L)^d X_t$ is stationary.

White noise is, for example, $ I(0)$, a random walk $ I(1)$. In only a few cases processes are observed that are $ I(d)$ with $ d>1.$ This means that in most cases first differences are enough to form a stationary process. In the following we assume that the observed process $ Y_t$ is $ I(d)$ and we consider the transformed process $ X_t = (1-L)^d Y_t$, i.e., we will concentrate on stationary processes.