= Common factor cancellation in an ARMA model
{title2=$\Phi(B)X_t=\Theta(B)W_t$}
An <ARMA> equation need not be a minimal-order representation when its <autoregressive polynomial> and <moving-average polynomial> have a common factor. For a stationary causal solution, a common stable factor can be cancelled using its convergent inverse filter. For example, $(1-0.5B)X=(1-0.5B)(1-0.9B)W$ reduces to $X=(1-0.9B)W$, an MA(1). Its <autocorrelation function> vanishes beyond lag one, despite the displayed ARMA(1,2) equation. Root conditions should be applied to a reduced representation when asserting necessary and sufficient <causality and invertibility root criteria for an ARMA model>.
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