During the study of the Wold decomposition I got stucked on its meaning regarding non-invertible processes. To be clearer, I found the usual example of two MA(1) spanning the same ACF, one with $|\theta|<1$ and one with $|\hat{\theta}|>1$, ( where $\theta=1/\hat{\theta}$ ) namely \begin{align*} X_{t} = \varepsilon_{t} + \theta \varepsilon_{t-1}, \quad \quad Y_{t} = \nu_{t} + \hat{\theta} \nu_{t-1} \end{align*} The Wold theorem states that every covariance-stationary stochastic process can be represented as \begin{align*} X_{t} = \sum_{t=0}^{\infty}\psi_{i}\epsilon_{t-i} \end{align*} where the infinite order polynomial $\psi(L)$ is invertible. So far, I conclude that a Wold decomposition exists for both the processes described above. Assuming that $\theta = 0.77$ and, consequently, $\hat{\theta} = 1.33$, the two Wold decompositions are, respectively, \begin{align*} X_{t} = \epsilon_{t} + 0.77 \epsilon_{t-1}, \quad \quad Y_{t} = \epsilon_{t} + 1.33 \epsilon_{t-1} \end{align*} However, the second one makes no sense to me, as the polynomial $\psi(L)$ is clearly not invertible, violating the statement of the theorem. Here is my question. Up to best of my knowledge, ARMA processes are a way of representing almost every covariance-stationary process. In particular ARMA processes are not unique, in the sense that a bijection between ACFs and ARMA processes does not exist (the two MA(1) above span indeed the same ACF). This bijection does exist between covariance-stationary processes and ACFs. Based on this, it seems to me that the Wold decomposition strictly refers to ACFs, meaning that for each ACF (i.e. for each process) there exists a Wold representation, whose polynomial $\psi(L)$ is invertible. What I don't get is why I'm not able to find the Wold decomposition for the second process, even if it is covariance-stationary.


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Reflecting on my question I think I finally managed to answer it. The Wold decomposition theorem refers (for the formal statement see Hamilton p.109) to covariance stationary processes. From now on I will call them data generating process - or DGP - and I will denote them as $\{x_{t}\}_{-\infty}^{+\infty}$. Accordingly, my question is wrong in the way it is formulated. Indeed, ARMA processes are just a way of representing the DGP, not the true DGP!! The Wold theorem ensures us that $\{x_{t}\}_{-\infty}^{+\infty}$ can be represented as a linear combination of forecast errors but it does not guarantee that such a representation is the true process.

I try to be clearer now. If we take, for example, the non invertible $MA(1)$ I provided in my question as the DGP, for the Wold theorem we know for sure that there must be a way to represent it as a linear combination of forecast errors. Indeed, it's Wold decomposition does exist and it is the other $MA(1)$ I provided. Why? On one hand it spans the same autocovariance function as the DGP (in this sense this ARMA is a good representation of the DGP). On the other hand its polynomial $\theta(L)$ is invertible as the theorem implies. However, sadly, it is not the true process. This is the issue of non-invertibility. An econometrician try to estimate the DGP will always end up estimating it as if it was the invertible $MA(1)$ with huge impacts especially on the computation of impulse response functions.

Up to this point, I am curious to know if there are solutions proposed in literature to tackle this important issue.


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