In Wold's decomposition we have $ Y_t = \sum_{j=0}^\infty b_j \varepsilon_{t-j} + \eta_t $, where the variables have definition as in the Wikipedia page.

I'm confused about why the summation goes to infinity. What does this equation mean when $j > t$?

I think what the equation is trying to say is that the summation actually goes to $t$, but that we do this for all our $t$s (at least, that's what I gather from the answer to this question, where the summation of $j$ goes from $-\infty$ to $\infty$). Would it be correct to interpret the decomposition this way? If so, then the decomposition can be thought of as

$ Y_t = \sum_{j=0}^t b_j \varepsilon_{t-j} + \eta_t$

but where we work this same equation through all our $T \in \left(0,1,...,t \right)$.

Is that right, or is there something I'm missing? Thank you.

  • $\begingroup$ If you consider what it means for $Y_t$ to be second-order stationary, you can deduce what the $\varepsilon_{t-j}$ must be for $j\gt t.$ $\endgroup$
    – whuber
    Commented Oct 12, 2021 at 14:06
  • 1
    $\begingroup$ Thanks as always whuber. I see it more clearly now. I was getting tripped up in the transition from the general form to actual implementation. $\endgroup$
    – user221772
    Commented Oct 15, 2021 at 13:18


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