3.4 What is a time series model?

We use a time series model to analyze time series data. A time series model for \(\{x_t\}\) is a specification of the joint distributions of a sequence of random variables \(\{X_t\}\), of which \(\{x_t\}\) is thought to be a realization.

Here is a plot of many realizations from a time series model.

Distribution of realizations

Figure 3.3: Distribution of realizations

These lines represent the distribution of possible realizations. However, we have only one realization. The time series model allows us to use the one realization we have to make inferences about the underlying joint distribution from whence our realization came.

Blue line is our one realization.

Figure 3.4: Blue line is our one realization.