We will begin our study of time series by looking at two famous but very simple examples. These will not only give us a feel for the field, but as we will see later on, they will also become integral building blocks to describe more complex time series.
A basic but very important type of time series is known as discrete white noise, or simply white noise. In a white noise time series, the random variables that are generated all have a mean of 0, finite and identical variance σw2, and the random variables at different time steps are uncorrelated with each other. Although some texts do not enforce this requirement, most texts also specify that the variables are also independent and identically distributed (iid) random variables.
The iid property essentially requires that each random variable come from the exact same distribution, such as a normal distribution with a particular mean and standard deviation. The property also requires that two variables from...