Integration

Motivation

Stationarity is a crucial subject for this course. Yet, it is also quite tricky at the same time. In practice, we won’t always be able to tell if a process is stationary. We can only hope that the realizations of the process that we see are representative of the data generation process. What’s more, even if that’s the case, sometimes, whether or not the process is stationary might not be obvious.

In the context of time series, our data series may not always exihibit stationarity. Several elements, such as seasonal or cyclical components, will make our time series depart for stationarity. We’ll have a look at these later in the chapter, but for now, let’s focus on the most important cause of non-stationarity, which is integration.

Trends

Generally speaking, a nonstationary series will show some form of trending. Here, by trending, we mean that the series increases or decreases over time. In other words, it doesn’t tend to have a constant value in the long run. Take the case of the price of Ethereum (ETH), a cryptocurrency in USD, which we can see below.

The chart shows a form of irregular trend, with sudden ups and downs that are very persistent over time. It almost looks like a random walk. As it happens, the random walk can be used to study a type of trend, the stochastic trend. Together with the deterministic trend, they constitute the two most basic types of nonstationary processes. Let’s see them in more detail.