11.1 Fitting an AR Model

-what is the order of the process
-how can we estimate the parameters of the process

Suppose we have an AR(p) with mean \(\mu \) given by \[X_t-\mu =\alpha _1 (X_{t-1}-\mu )+\alpha _2 (X_{t-2}-\mu )+\cdots \cdots +\alpha _p (X_{t-p}-\mu )+Z_t\]

If \(X_1,X_2,\cdots \cdots ,X_n\) are the observations of a time series. The parameter i.e \(\alpha _1 ,\alpha _2 ,\cdots \cdots ,\alpha _p\) can be estimated by least square, by minimizing
\[S=\sum ^{n}_{t=p+1}\Big [X_t-\mu -\alpha _1 (X_{t-1}-\mu )-\alpha _2 (X_{t-2}-\mu )-\,\cdots \cdots \,-\alpha _p (X_{t-p}-\mu )\Big ]^2\] with respect to \(\mu ,\alpha _1 ,\alpha _2 ,..........., \alpha _p\).

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