Likelihood of ARMA models. X12 implementation

We suppose that $(y_t, \quad 1 \le t \le n)$ follows an ARMA model.

The X12 implementation computes the exact likelihood in two main steps

Overview

We consider the transformation

\[z_t = \begin{pmatrix} z_{1t} \\ z_{2t} \end{pmatrix} = \begin{cases} y_t, & 1 \le t \le p \\ \Phi\left(B\right) y_t, & p \lt t \le n\end{cases}\]

It is obvious that $p\left(y_t\right) = p\left(z_t\right)$

We will estimate $p\left(z_t\right)=p\left(z_{2t}\right)p\left(z_{1t}\vert z_{2t}\right)$

Step 1: likelihood of a pure moving average process
Step 2: conditional distribution of the initial observations

$p\left(z_{1t}\right \vert z_{2t} )$ is obtained by considering the join distribution of

\[\left( z_{1t}, z_{2t} \right) \sim N \left( 0, \begin{pmatrix} \Sigma_{11} & \Sigma{12} \\ \Sigma_{21} & \Sigma{22} \end{pmatrix}\right)\]

It is distributed as

\[N\left( \Sigma_{12} \Sigma_{22}^{-1}v, \Sigma_{11}-\Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21} \right)\] \[= N\left( \Sigma_{12} T'Tv, \Sigma_{11}-\Sigma_{12} T'T\Sigma_{21} \right)\] \[= N\left( U'Tv, \Sigma_{11}-U'U \right)\]

where

  • $v$ is obtained by applying the auto-regressive polynomial on the observations
  • $T$ is the linear transformation defined in step 1
  • $U = T \Sigma_{21}$
Bibliography

[1] Otto M. C., Bell W.R., Burman J.P. (1987), “An Iterative GLS Approach to Maximum Likelihood Estimation of Regression Models with Arima Errors”, Bureau of The Census, SRD Research Report CENSUS/SRD/RR_87/34.