Lecture 2 : ARMA Models ∗ 1 ARMA Process

As we have remarked, dependence is very common in time series observations. To model this time series dependence, we start with univariate ARMA models. To motivate the model, basically we can track two lines of thinking. First, for a series xt, we can model that the level of its current observations depends on the level of its lagged observations. For example, if we observe a high GDP realization this quarter, we would expect that the GDP in the next few quarters are good as well. This way of thinking can be represented by an AR model. The AR(1) (autoregressive of order one) can be written as: xt = φxt−1 + t where t ∼ WN(0, σ2 ) and we keep this assumption through this lecture. Similarly, AR(p) (autoregressive of order p) can be written as: