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`\tilde \epsilon_t=u_t-\theta_1*\tilde \epsilon_{t+1}-...-\theta_q*\tilde \epsilon_{t+q} `

with \tilde \epsilon_{T+i}= for i>0. This allows computing

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`{\tilde \epsilon_{0},...,\tilde \epsilon_{-(q-1)}`

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`\hat \epsilon_t=u_t-\theta_1*\hat \epsilon_{t-1}-...-\theta_q*\hat \epsilon_{t-q} `

i.e. the \tilde \epsilon_{i}, i<1 are used as the \hat \epsilon_{i}. But the reference is the the Box/Jenkins (1970) book. In its 2016 edition, Chapter 7.1.4, the procedure seems to be quite different as the forward and backward model deliver two different, distinct sets of innovations. They therefore do not allow using the innovations from the forward model in the backward model. To solve this problem, they backcast the u_t for t={0,..,q-1} using the backcasts for \tilde \epsilon_t and employ that \tilde \epsilon_{i}=0,i<1 due to them being independent from u_t. These u_t, t<1 are then used in the backward model to run a forward recursion with \hat \epsilon_{i}=0 for i<q-1 due to independence. This then allows computing {\hat \epsilon_{0},...,\hat \epsilon_{-(q-1)}.

So what exactly does Eviews do here?