Time Series 2

This week we will talk about linear time series models: AR, MA, ARMA, ARIMA, etc. First we will talk about theory and after we will talk about fitting the models to real data. Let {xt}t=−∞ be a scalar time series. That means that each xt is a real-valued random variable on some probability space, and the time series is completely defined by the distribution of each xt and of all the joint distributions. The index variable t has the interpretation of discrete time, and we consider the series to extend to infinity in both directions. This definition of a time series does not talk about any rule by which the values xt are generated; that is the subject of this section. The series {xt} is stationary if all joint distributions xt, . . . , xt−p are independent of t, that is, of where we are in the sequence. It is weakly stationary if only the first and second moments are independent of t. The series is Gaussian if all joint distributions are Gaussian. If the series is Gaussian then weak stationarity is equivalent to stationarity. For the linear theory we only worry about first and second moments, it will not really matter whether a series is stationary or only weakly so, or whether it is Gaussian. People with a background in stochastic analysis may wonder “what about filtrations?” Do we need a detailed construction to describe how information is revealed in time? Generally speaking, that concept stays in the background in time series analysis, because we are not doing optimal control. It is still present, and we would prefer that any real-time analysis methods include only data observed to date rather than future data. But unlike trading strategies, here the payoff for forward-looking is less dramatic since we are only doing statistical modeling. Sometimes we even consider analysis methods such as Fourier transforms that use the entire series of observations both future and past.