Graduate Macro Theory II: Notes on Time Series

A time series is a realization of a sequence of a variable indexed by time. The notation we will use to denote this is xt, t = 1, 2, . . . , T . A variable is said to be “random” if its realizations are stochastic. Unlike cross-sectional data, time series data can typically not be modeled as independent across observations (i.e. independent across time). Rather, time series are persistent in the sense that observations of random variables are typically positively correlated across time. Most of what we do in macro involves variables with such dependence across time. Before discussing the unique issues presented by time series data, we quickly review expectations and first and second moments.