= Solution
For a series with constant mean $\mu$, write $X_t=x_t-\mu$. An <autoregressive moving-average process> has the form
$$
X_t=\sum_{j=1}^p a_jX_{t-j}+w_t+\sum_{j=1}^q b_jw_{t-j},
$$
where $w_t$ is zero-mean <white noise> with <variance> $\sigma_w^2>0$. The <backshift operator> is $BX_t=X_{t-1}$. Define the <autoregressive polynomial> and the <moving-average polynomial> by
$$
\Phi(z)=1-\sum_{j=1}^pa_jz^j,\qquad
\Theta(z)=1+\sum_{j=1}^qb_jz^j.
$$
The <autoregressive operator> and <moving-average operator> give the concise equation
$$
\boxed{\Phi(B)X_t=\Theta(B)w_t.}
$$
Orders are normally specified minimally, with nonzero last coefficients and no common polynomial factor. A displayed unreduced equation can have larger orders than the process actually needs.
Causality means that the stationary process can be recovered from present and past noise:
$$
X_t=\sum_{j\ge0}\psi_jw_{t-j},\qquad\sum_j|\psi_j|<\infty.
$$
Invertibility means that the noise can be recovered from present and past observations:
$$
w_t=\sum_{j\ge0}\pi_jX_{t-j},\qquad\sum_j|\pi_j|<\infty.
$$
The <causality and invertibility root criteria for an ARMA model>, for the reduced representation, are
$$
\boxed{\Phi(z)\ne0\ (|z|\le1)\quad\text{for causality},\qquad
\Theta(z)\ne0\ (|z|\le1)\quad\text{for invertibility}.}
$$
Indeed the transfer series are $\sum\psi_jz^j=\Theta(z)/\Phi(z)$ and $\sum\pi_jz^j=\Phi(z)/\Theta(z)$. When all denominator roots lie outside the closed unit disc, each rational function is analytic on a larger disc, so its Taylor coefficients decay geometrically and are absolutely summable. Conversely, a genuine denominator zero in the closed unit disc gives a pole incompatible with such a convergent filter. The reduced-representation proviso prevents a cancelled root from being mistaken for an obstruction.
For <order identification by autocorrelation cutoffs>, first estimate the <sample autocorrelation function> and <sample partial autocorrelation function> from a suitably stationary, centred series. The <autocorrelation function> is $\rho(h)=\operatorname{Cov}(X_t,X_{t-h})/\operatorname{Var}(X_t)$. The lag-$h$ <partial autocorrelation function> is the last coefficient in the best <linear regression> of $X_t$ on $X_{t-1},\ldots,X_{t-h}$, equivalently the correlation left after removing the intervening lags.
For a pure <moving-average model> of order $q$, observations more than $q$ lags apart share no noise, so the population ACF is zero beyond $q$, while the PACF usually tails off. For a pure <autoregressive model> of order $p$, the PACF is zero beyond $p$, while the ACF usually decays, possibly with damped oscillation. Thus a clear sample ACF cutoff suggests an MA order and a clear PACF cutoff suggests an AR order. For a mixed <ARMA> process both usually tail off, so these plots guide a small set of candidate orders rather than determine $p,q$ uniquely. Sampling variation blurs cutoffs; compare fitted candidates and check that their residual ACF resembles <white noise>. Common-factor cancellation can also disguise the orders in a displayed equation.
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