8 Autoregressive processes
Definition 8.0.1. Let \(\{Z_t\}\) be a purely random process with mean \(0\) and variance \(\sigma _Z^2\). The process \(\{X_t\}\) is an autoregressive process of order \(p\), written AR(\(p\)), if \[ X_t = \alpha _1X_{t-1} + \alpha _2X_{t-2} + \dots + \alpha _pX_{t-p} + Z_t . \]
The name says what the model is: a multiple regression of \(X_t\) on its own past values rather than on separate regressors, with \(Z_t\) as the error. Each value is a fraction of the recent values plus a new shock, so a shock is remembered for ever, though with fading weight.
8.1 The first-order process
8.2 The general order
8.3 The second-order process
8.4 Practice problems
8.2 The general order
8.3 The second-order process
8.4 Practice problems
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