I have been reading a paper and found this quote?

"Note that a linear conditional mean model with ARCH disturbances can be described by a nonlinear specification without ARCH, i.e. the bilinear model."

How would I actually go about writing the bilinear model?


A general bilinear model $BL(p, q, r, s)$ can be written as

$$ X_t - \sum_j^p \phi_j X_{t-j}=\varepsilon_{t}+\sum_j^p \theta_i \varepsilon_{t-i} + \sum_k^r \sum_{k'}^s b_{k,k'}X_{t-k}\varepsilon_{t-k'} $$

where the sequence of $\varepsilon$'s is i.i.d with mean zero and variance $\sigma^2$

For the model motivation, properties and derivation please check this.

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