This is a question linked to an earlier discussion (What subset of ARMA(1,1) processes can be represented as AR(1) - a query about the logic in this derivation) .
I am looking at the possibility of MA(1) = MA(1) = white noise, with a view to then considering MA(1) = MA(1) + MA(1), (and the first needed for the second). However I am unsure that my logic is correct - I have seen elsewhere a reference to a "solution", but I am only able to identify a potential range of parameters for the equivalent model.
I started with the simple case of an MA(1) representing a different MA(1) plus white noise as follows:
Is there an $\mathbf{MA}\left(1\right)$ process, $Z_t$ that can arise from the sum of a different $\mathbf{MA}\left(1\right)$ process, $X_t$ and $\textit{white noise}$, denoted $\xi_t$?
To begin we have: \begin{equation} X_t = \left(1+\alpha B\right)\varepsilon_t \quad \mbox{and} \quad Y_t = \xi_t \quad \mbox{so} \quad Z_t = \left(1+\theta B\right)\zeta_t = X_t + Y_t = \left(1+\alpha B\right)\varepsilon_t + \xi_t \end{equation} so we can say for the process $Z_t$: \begin{equation} \mathbf{E}\left[Z_t\right] = 0, \quad \mathbf{Var}\left[Z_t\right] = \gamma_0 = \left(1+\theta^2\right)\sigma_{\zeta}^2 = \left(1+\alpha^2\right)\sigma_{\varepsilon}^2 + \sigma_{\xi}^2, \quad \mbox{and} \quad \gamma_1 = \theta\sigma_{\zeta}^2 = \alpha\sigma_{\varepsilon}^2 \end{equation} For the series $Z_t$ to be equivalent to $X_t + Y_t$, then: \begin{equation} \rho_1 = \frac{\theta}{\left(1+\theta^2\right)} = \frac{\alpha\sigma_{\varepsilon}^2}{\left(1+\alpha^2\right)\sigma_{\varepsilon}^2 + \sigma_{\xi}^2} \end{equation} For a given series, we would be able to determine $\rho_1$ and therefore $\theta$, but for this to be equivalent to another $\mathbf{MA}\left(1\right)$ series - $X_t$ - plus white noise - $\xi_t$ - then: \begin{equation} \sigma_{\xi}^2 = \left[\frac{\alpha\left(1+\theta^2\right)}{\theta} - \left(1+\alpha^2\right)\right]\sigma_{\varepsilon}^2 \quad \mbox{or} \quad \sigma_{\xi}^2 = \left[\frac{\alpha}{\rho_1} - \left(1+\alpha^2\right)\right]\sigma_{\varepsilon}^2 \end{equation} Given $\sigma_{\varepsilon}^2,\,\sigma_{\xi}^2 > 0$, then we have a condition linking $\alpha$ and $\theta$ to $\rho_1$: \begin{equation} \left(1+\alpha^2\right) < \frac{\alpha\left(1+\theta^2\right)}{\theta} \quad \mbox{and} \quad \frac{\alpha\left(1+\theta^2\right)}{\theta} = \frac{\alpha}{\rho_1} \end{equation} For both $\mathbf{MA}\left(1\right)$ processes to be stationary, then $-1<\theta,\,\alpha<1$, which constrains the lag-1 autocorrelation coefficient thus $-0.5<\rho_1<0.5$. We can further consider the relationship between $\alpha$, $\sigma_{\varepsilon}^2$, $\sigma_{\xi}^2$ as constrained by $\rho_1$, as:\begin{equation} \alpha^2 - \frac{\alpha}{\rho_1} + \left(1 + \frac{\sigma_{\xi}^2}{\sigma_{\varepsilon}^2}\right) = 0 \quad \mbox{so} \quad \alpha = \frac{1}{2\rho_1}\left(1\pm\sqrt{1-4\rho_1^2\left(\frac{\sigma_{\xi}^2}{\sigma_{\varepsilon}^2}+1\right)}\right) \end{equation}
so, given that $-1<\alpha<1$ for the model to be stationary, and that $\alpha$ must be real then:
\begin{equation} 4\rho_1^2\left(\frac{\sigma_{\xi}^2}{\sigma_{\varepsilon}^2}+1\right) < 1 \quad \mbox{such that} \quad \sigma_{\xi}^2 < \sigma_{\varepsilon}^2\left[\left(\frac{1}{2\rho_1}\right)^2 - 1\right] \end{equation}
So, I can estimate a set of parameters $\alpha$, $\sigma_{\varepsilon}^2$ and $\sigma_{\xi}^2$ that will meet the necessary condition. Is this the complete resolution to the question, or is there other information that I have missed?
