ARMA causality and invertibility root criteria

Suppose a causal representation exists, with $\mathrm{\Psi }(z)={\sum }_{j\ge 0}{\psi }_j{z}^j$ and ${\sum }_j\mathrm{\mid }{\psi }_j\mathrm{\mid }<\mathrm{\infty }$. Applying the finite polynomial $\varphi (B)$ and using the ARMA equation gives $(\varphi (B)\mathrm{\Psi }(B)-\theta (B)){ϵ}_t=0.$ The coefficient sequence $a$ of $\varphi \mathrm{\Psi }-\theta$ is in ${\mathrm{\ell }}^1$ because $\varphi ,\theta$ are finite polynomials and $\psi \in {\mathrm{\ell }}^1$. By [F3], the variance of this zero random variable is ${\sigma }^2{\sum }_{j\ge 0}{a}_j^2$. Since ${\sigma }^2>0$, every $a_j=0$, so $\varphi (z)\mathrm{\Psi }(z)=\theta (z)$ on the closed unit disk; the series for $\mathrm{\Psi }$ converges uniformly there because ${\sum }_j\mathrm{\mid }{\psi }_j\mathrm{\mid }<\mathrm{\infty }$.

Suppose $\varphi$ has degree $m\ge 1$ and roots $r_1,\dots ,r_m$ counted with multiplicity. None is zero because $\varphi (0)=1$. By [F4] and normalization, $\varphi (z)={\prod }_{\mathrm{\ell }=1}^m(1-z\mathrm{/}{r}_{\mathrm{\ell }})\mathrm{.}$ If every $\mathrm{\mid }{r}_{\mathrm{\ell }}\mathrm{\mid }>1$, then for $\mathrm{\mid }z\mathrm{\mid }\le 1$ each factor has the geometric expansion $(1-z\mathrm{/}{r}_{\mathrm{\ell }}{)}^{-1}={\sum }_{k\ge 0}{r}_{\mathrm{\ell }}^{-k}{z}^k\mathrm{.}$ The coefficient of $z^j$ in $1\mathrm{/}\varphi (z)$ has absolute value at most $\left(\genfrac{}{}{0px}{}{j+m-1}{m-1}\right)r_{\ast }^{-j}$, where $r_{\ast }={\min }_{\mathrm{\ell }}\mathrm{\mid }{r}_{\mathrm{\ell }}\mathrm{\mid }>1$, since there are $\left(\genfrac{}{}{0px}{}{j+m-1}{m-1}\right)$ nonnegative $m$-tuples summing to $j$. Choose $1<r<r_{\ast }$. The polynomial factor $\left(\genfrac{}{}{0px}{}{j+m-1}{m-1}\right)(r\mathrm{/}{r}_{\ast }{)}^j$ is bounded in $j$, so these coefficients are bounded by $C{r}^{-j}$. Multiplication by the finite polynomial $\theta$ preserves an exponential bound. If $m=0$, $\varphi =1$ and $\theta \mathrm{/}\varphi$ is finite, which also has an exponential bound after enlarging $C$. Thus $\eta (z):=1\mathrm{/}\varphi (z)$ and $\mathrm{\Psi }(z):=\eta (z)\theta (z)=\theta (z)\mathrm{/}\varphi (z)$ have absolutely summable one-sided coefficients, and those of $\mathrm{\Psi }$ have the stated exponential decay.

Suppose an invertible representation exists, ${ϵ}_t=\pi (B)(X_t-\mu )$ with $\pi \in {\mathrm{\ell }}^1({\mathbb{N}}_0)$. Apply the finite filter $\varphi (B)$ and commute the absolutely summable and polynomial filters by [F3] on the weakly stationary process [F2]: $\varphi (B){ϵ}_t=\varphi (B)\pi (B)(X_t-\mu )=\pi (B)\varphi (B)(X_t-\mu )=\pi (B)\theta (B){ϵ}_t\mathrm{.}$ The coefficient sequence of $\varphi -\pi \theta$ is in ${\mathrm{\ell }}^1$. Its filter of white noise is zero, so [F3] and ${\sigma }^2>0$ imply every coefficient is zero. The absolutely summable series $\mathrm{\Pi }$ converges uniformly on the closed unit disk, so the coefficient identity yields $\varphi (z)=\mathrm{\Pi }(z)\theta (z)$ for $\mathrm{\mid }z\mathrm{\mid }\le 1$.

If $\varphi (z_0)=0$ for some $\mathrm{\mid }{z}_0\mathrm{\mid }\le 1$, step 1.1 gives $\theta (z_0)=\varphi (z_0)\mathrm{\Psi }(z_0)=0$. This is a common root, contrary to the coprime condition in [F1]. Therefore causality implies that $\varphi$ has no zero in the closed unit disk.

Let $\eta (z)=1\mathrm{/}\varphi (z)$ be the absolutely summable causal series from step 1.2. Apply the filter $\eta (B)$ to the given equation. Since $X-\mu$ is weakly stationary, [F2] and [F3] justify composition and $\eta \ast \varphi$ is the constant coefficient sequence $1$. Therefore $X_t-\mu =\eta (B)\varphi (B)(X_t-\mu )=\eta (B)\theta (B){ϵ}_t=\mathrm{\Psi }(B){ϵ}_t\mathrm{.}$ This is the required causal representation of the given solution.

If $\theta (z_0)=0$ for some $\mathrm{\mid }{z}_0\mathrm{\mid }\le 1$, step 1.3 gives $\varphi (z_0)=\mathrm{\Pi }(z_0)\theta (z_0)=0$, contradicting coprimality. Hence invertibility implies that $\theta$ has no zero in the closed unit disk.

If $\theta$ has no zero in $\mathrm{\mid }z\mathrm{\mid }\le 1$, the same geometric-series argument as in step 1.2 applied to its roots gives an absolutely summable causal series $\eta (z)=1\mathrm{/}\theta (z)$. Set $\mathrm{\Pi }(z)=\eta (z)\varphi (z)$; its coefficients are absolutely summable. Applying $\eta (B)$ to the ARMA equation and using filter composition [F3] on the weakly stationary process [F2] gives $\mathrm{\Pi }(B)(X_t-\mu )=\eta (B)\varphi (B)(X_t-\mu )=\eta (B)\theta (B){ϵ}_t={ϵ}_t\mathrm{.}$ Thus the model is invertible.

Steps 1.1, 1.2, 2.1, and 2.2 prove both directions of the causal criterion and identify its coefficients; steps 1.3, 2.3, and 2.4 prove both directions of the invertibility criterion and identify its coefficients. The proof uses no choice axiom.