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For the three-coordinate state realization of an ARMA(1,1) process, the observation row and transition matrix areThe first row of $G{S}_{t-1}$ is $x_{t-1}$ by the previous-time recursion; the second row is zero until the new white noise value is added; and the third row copies ${\epsilon }_{t-1}$. Hence the state equation has noise covariance

We use the causal state model, in which the new driving noise is independent of the previous state. The Gaussian filtering with exact observations form of the Kalman filter alternates prediction through the state equation with conditioning on the next observation. Given the posterior expected value and covariance at time $t-1$, independence of the new white noise value gives predictive expected value $a_t=G{\hat{S}}_{t-1}$ and predictive covarianceProject through the observation row to getThe residual $v_t=x_t-{\hat{x}}_t$ is the innovation. Conditioning the joint Gaussian state-observation vector adjusts the expected value according to this residual and reduces the covariance. In this exact-observation model, the updates can be writtenThus $F{\hat{S}}_t=x_t$ and $F{P}_t{F}^{\top }=0$: the observed linear combination of the state is now known exactly, though other state combinations remain uncertain.

The PDF's stated prediction variance omits $FQ{F}^{\top }={\sigma }^2$. This omission cannot be used in a valid likelihood: for the AR(1) specialization below it would give zero prediction variance after the first observation. The required likelihood calculation uses the corrected innovation variance just derived.

For stationary initialization of an ARMA(1,1) state with $|\varphi |<1$, take ${\hat{S}}_0=0$ and the unconditional stationary covariance. The causal coefficients are $1$ at lag zero and $(\varphi +\theta ){\varphi }^{j-1}$ at lags $j\ge 1$, givingSince $S_0=(x_{-1},{\epsilon }_0,{\epsilon }_{-1}{)}^{\top }$, the cross-covariance with ${\epsilon }_0$ is zero and that with ${\epsilon }_{-1}$ is ${\sigma }^2$. ThereforeDirect substitution verifies $P_0=G{P}_0{G}^{\top }+Q$. If prior state information is available, instead use its conditional expected value and covariance. A diffuse initialization is another option when the initial state is genuinely unknown, but it changes the finite-sample likelihood and should not silently replace the stationary initial law.

By the Gaussian prediction step,The chain rule for probabilities for the joint probability density function therefore givesTaking minus twice its logarithm shows that maximum likelihood estimation is equivalent to minimizingwith each $V_t$ and ${\hat{x}}_t$ evaluated at the candidate parameters and the specified initialization. The corrected $+{\sigma }^2$ is essential in every $V_t$.

For the stationary causal AR(1) case, $\theta =0$ and $|\varphi |<1$. The first observation is $N(0,{\sigma }^2/(1-{\varphi }^2))$, whereas each later conditional observation is $\varphi x_{t-1}+{\epsilon }_t$. HenceThis also gives a direct counterexample to the uncorrected PDF variance: with $\theta =0$, $FG=\varphi F$ and $F{P}_{t-1}{F}^{\top }=0$ after observing $x_{t-1}$, so its printed expression would give $V_t=0$ for $t\ge 2$.

To compare the conditional and stationary AR1 likelihood estimators, putThen its negative twice log-likelihood isFor fixed $\varphi$, $\widehat{{\sigma }^2}=Q_T(\varphi )/T$. Writingthe exact interior score equation isThe stationary initial contribution is order one for a fixed interior parameter, while $C_T$ is order $T$. Discarding this lower-order contribution givesConditioning on $x_1$ makes this ratio the exact unconstrained conditional maximum likelihood estimate. Under the stationary initial law it is only the leading approximation: at an interior optimum,This approximation presumes a nondegenerate sample denominator and a true coefficient fixed away from the unit-root boundary; a finite-sample constrained optimum must respect $|\varphi |<1$.