Autoregressive moving-average model ($\mathrm{ARMA}(p,q)$)

An autoregressive–moving-average model satisfieswhere $B$ is the backshift operator, $\varphi$ and $\theta$ are finite polynomials, and $\epsilon$ is white noise.

For a causal autoregressive moving-average model driven by independent white noise, multiply its recursion by $X_{t-k}$ and take covariance. When $k>q$, all driving white noise values in its moving-average term occur strictly after $t-k$, so their covariances with $X_{t-k}$ vanish. The resulting homogeneous tail recurrence has the same characteristic roots as the autoregressive model recursion. Distinct roots of modulus below one give exponential autocovariance decay.

Normalize the autoregressive and moving-average polynomials to have constant coefficient one, cancel common factors, and require all their zeros to lie outside the closed unit disk. With positive innovation variance these conditions give a causal, invertible, minimal Gaussian process representation. To prove identifiability, compare two such models with the same time-series spectral density. The ratio of their scaled rational transfer functions and its reciprocal are analytic on a neighbourhood of the closed unit disk and have modulus one on its boundary. The maximum modulus principle makes the ratio constant; its positive value at zero forces equality of the innovation variances and equality of the rational functions. Coprimality and the constant-coefficient normalization then force equality of both polynomials. Without the root restrictions, reciprocal root reflection of an ARMA representation and a rescaling of the innovation variance can preserve the spectrum, so the unrestricted parameters are not identifiable.

An ARIMA model applies an autoregressive moving-average model to the differenced process $(1-B{)}^d{X}_t$, where $B$ is the backshift operator. Differencing permits stochastic trends whose levels are not weakly stationary processes. A local random-walk level observed with independent measurement white noise has an ARIMA(0,1,1) representation, or a formal ARMA(1,1) equation with autoregressive coefficient one.

For an autoregressive moving-average model, set $\Phi (z)=1-{\sum }_{k=1}^p{\varphi }_k{z}^k$ and $\Theta (z)=1+{\sum }_{k=1}^q{\theta }_k{z}^k$. If both have no zeros on the closed unit disc, the causal and inverse filters have absolutely summable coefficients $C(z)=\Theta (z)/\Phi (z)={\sum }_{j\ge 0}{c}_j{z}^j$ and $D(z)=\Phi (z)/\Theta (z)={\sum }_{j\ge 0}{d}_j{z}^j$. Comparing power series coefficients gives $c_j={\theta }_j+{\sum }_{k=1}^p{\varphi }_k{c}_{j-k}$ and $d_j=a_j-{\sum }_{k=1}^q{\theta }_k{d}_{j-k}$, where $a_0={\theta }_0=1$, $a_j=-{\varphi }_j$ for $1\le j\le p$, and all out-of-range coefficients vanish. The backshift operator then gives the corresponding time-series filters.

For $|\varphi |<1$ and $|\theta |<1$, the autoregressive moving-average model $(1-\varphi B)X_t=(1+\theta B){ϵ}_t$ has $c_0=d_0=1$ and the displayed coefficients for $j\ge 1$. These follow by multiplying the two geometric power series. If $\varphi +\theta =0$, all later coefficients vanish and the filtered process is white noise. Thus root restrictions on an uncancelled nonminimal representation are sufficient but need not be necessary for the process after common factor cancellation in an ARMA model.

With ${\theta }_0=1$ and $|\varphi |<1$, expand the transfer function $\Theta (z)/(1-\varphi z)$ by its geometric series. Its coefficient of $z^j$ is the displayed expression; after $j=q$ the coefficients follow a geometric tail. Causality gives $\mathrm{Cov}({ϵ}_{t-i},X_{t-k})={\sigma }^2{c}_{i-k}$ for $i\ge k$, and zero otherwise. Hence ${\gamma }_k-\varphi {\gamma }_{k-1}={\sigma }^2{\sum }_{i=k}^q{\theta }_i{c}_{i-k}$ for $0\le k\le q$, with ${\gamma }_{-1}={\gamma }_1$ in the zero-lag equation.

An ARMA equation need not be a minimal-order representation when its autoregressive polynomial and moving-average polynomial have a common factor. For a stationary causal solution, a common stable factor can be cancelled using its convergent inverse filter. For example, $(1-0.5B)X=(1-0.5B)(1-0.9B)W$ reduces to $X=(1-0.9B)W$, an MA(1). Its autocorrelation function vanishes beyond lag one, despite the displayed ARMA(1,2) equation. Root conditions should be applied to a reduced representation when asserting necessary and sufficient causality and invertibility root criteria for an ARMA model.

With positive moving-average sign and $|\varphi |<1$, the noise coefficients are ${\psi }_0=1$ and ${\psi }_j=(\varphi +\theta ){\varphi }^{j-1}$ for $j\ge 1$. Summing their overlap gives ${\gamma }_0={\sigma }^2(1+{\theta }^2+2\varphi \theta )/(1-{\varphi }^2)$ and ${\gamma }_k={\sigma }^2(\varphi +\theta )(1+\varphi \theta ){\varphi }^{k-1}/(1-{\varphi }^2)$ for $k\ge 1$. Negative lags follow by symmetry.

With the convention $\Phi (B)X_t=\Theta (B){\epsilon }_t$, the one-sided spectral density is ${\sigma }^2|\Theta (e^{i\omega }){|}^2/(\pi |\Phi (e^{i\omega }){|}^2)$ whenever the stationary filter is well defined. Stable autoregressive roots give the causal representation; moving-average invertibility is a separate property.

Adding independent white noise of variance $w$ to a process with transfer function $(1+\theta z)/(1-\varphi z)$ and driving variance $v$ gives numerator $A+2C\cos \omega$ in its rational time-series spectral density. Put $P=A+2C$ and $Q=A-2C$. If both are positive, the invertible moving-average factor has coefficient $\alpha =(\sqrt{P}-\sqrt{Q})/(\sqrt{P}+\sqrt{Q})$ and driving variance $\lambda =(\sqrt{P}+\sqrt{Q}{)}^2/4$. Filtering the actual sum by $(1+\alpha B{)}^{-1}(1-\varphi B)$ produces its weak white noise driver.

Reflecting zeros across the unit circle can convert a noncausal or noninvertible ARMA representation to a causal invertible one with the same time-series spectral density. Scale the driving white noise variance according to the modulus factors. Define the new noise as a filter of the actual process; equality of two spectra alone proves equality of second-order structure, not equality of non-Gaussian laws. The resulting noise is weak white noise and need not be independent across time.

For a minimal ARMA representation $\varphi (B)X=\theta (B)ϵ$, causality requires every zero of $\varphi$ to lie strictly outside the unit disk, and invertibility requires the same of $\theta$. This makes the transfer function and its reciprocal analytic on a disk larger than the unit disk, giving geometrically decreasing one-sided filter coefficients. Cancel common factors before applying the criterion to the noise-driven representation.

An invertible time-series representation reconstructs the driving white noise from present and past observations. The stable convention requires absolute summability of the inverse coefficients. A convergent bilateral inverse using future observations does not establish this one-sided property.

For minimal causal and invertible models, the partial autocorrelation function cuts off after the order of a pure autoregressive model, whereas the autocorrelation function cuts off after the order of a pure moving-average model. Estimated cutoffs suggest candidate orders; finite-sample noise and residual diagnostics still matter.

The backshift operator acts on a time series by $B{X}_t=X_{t-1}$ and hence $B^j{X}_t=X_{t-j}$.

An autoregressive model expresses the current value as a linear combination of finitely many past values plus white noise.

For $X_t=\alpha X_{t-1}+{\alpha }^2{X}_{t-2}+{\epsilon }_t$, let $\phi =(1+\sqrt{5})/2$. The autoregressive polynomial factors as $(1-\alpha \phi z)(1+\alpha z/\phi )$. Its roots lie outside the closed unit disk exactly when $|\alpha |<1/\phi$. In that causal region,The coefficients follow from ${\psi }_0=1,{\psi }_1=\alpha$ and ${\psi }_j=\alpha {\psi }_{j-1}+{\alpha }^2{\psi }_{j-2}$. They are absolutely summable in the stated region. Causality makes future white noise orthogonal to the observation past, giving forecasts $\alpha X_T+{\alpha }^2{X}_{T-1}$ and $2{\alpha }^2{X}_T+{\alpha }^3{X}_{T-1}$ at horizons one and two.

For a causal autoregressive model $X_t={\sum }_{j=1}^p{\varphi }_j{X}_{t-j}+{ϵ}_t$ with centered white noise of variance ${\sigma }^2$, multiplying by past values and taking expectations yields $\gamma (h)={\sum }_j{\varphi }_j\gamma (h-j)$ for $h\ge 1$, and $\gamma (0)={\sum }_j{\varphi }_j\gamma (j)+{\sigma }^2$. These Yule-Walker equations determine the autocovariance from the coefficients and noise variance, or estimate coefficients from observed autocovariances. Causality supplies the needed orthogonality of the current noise to past observations.

For independent fair signs, $|X_n|\to \infty$ almost surely from every deterministic initial value. Outside $[-L,L]$ with $L>1/(a-1)$, the sign is preserved and $|X_{n+k}|\ge a^k(|X_n|-1/(a-1))+1/(a-1)$. To reach that region, choose $m$ so $(a^m-1)/(a-1)>L$. At the beginning of each $m$-step block, choose the sign of the current value; a block of that sign pushes the absolute value past $L$ and has conditional probability $2^{-m}$. The chance of remaining inside after $k$ blocks is at most $(1-2^{-m}{)}^k$. This proof avoids any assumption about a density of the limiting random series.

With two fixed initial states and iid N(0,1) errors, the parameter likelihood is the product of conditional transition densities. For $X_0=X_1=0$, it is proportional to $\exp [-\frac{1}{2}{\sum }_{t=0}^{n-2}(x_{t+2}-a{x}_{t+1}-b{x}_t{)}^2]$. The initial observation is a point mass and the first nonzero observation has a parameter-independent likelihood term.

The autoregressive polynomial is $\Phi (z)=1-{\sum }_{j=1}^p{\varphi }_j{z}^j$. Substituting the backshift operator gives the autoregressive filter. Zeros outside the unit disk give the causality root criterion for an autoregressive model; zeros on the unit circle obstruct a nondegenerate stationary innovation-driven solution.

The autoregressive operator acts as $\Phi (B)X_t=X_t-{\sum }_{j=1}^p{a}_j{X}_{t-j}$. Its coefficients are encoded by the autoregressive polynomial. An ARMA equation equates this filtered observation to the noise filtered by the moving-average operator.

A unit root is a zero of an autoregressive polynomial on the unit circle. A zero at $1$ is removed by first differencing. A conjugate pair $e^{\pm i\omega }$ corresponds to the real factor $1-2\cos \omega B+B^2$. Nonzero white-noise excitation at an uncancelled unit root prevents a finite-variance stationary solution.

A large, slowly damping oscillatory sample autocorrelation function suggests autoregressive zeros near a conjugate pair on the unit circle. For period $s$, the pair is near $e^{\pm 2\pi i/s}$, with minimal real filter $1-2\cos (2\pi /s)B+B^2$. Such a plot is evidence, not proof: a finite sample cannot distinguish a near-unit stationary model or a random sinusoid solely from the shape.

A causal time series satisfying $\Phi (B)X_t={\epsilon }_t$ with nondegenerate white noise has a square-summable present-and-past filter. The identity $\Phi (z)\Psi (z)=1$ excludes roots inside the unit disk; a pole of $1/\Phi$ on its boundary also prevents square summability. Thus all roots of $\Phi$ lie strictly outside the unit circle.

The inverse autoregressive polynomial is an analytic function on a disk larger than the unit disk. The Cauchy estimate gives exponentially decaying coefficients of its infinite moving-average representation. Summing their products gives $|\gamma (h)|\le C{s}^{|h|}$ for some $s\in (0,1)$.

A weakly stationary process satisfying an autoregressive model can depend on future driving white noise. For $X_t=\varphi X_{t-1}+{\epsilon }_t$ with $|\varphi |>1$, the convergent in the sense of mean-square convergence solution is $X_t=-{\sum }_{j\ge 1}{\varphi }^{-j}{\epsilon }_{t+j}$. Its autocovariance is ${\sigma }^2{\varphi }^{-|h|}/({\varphi }^2-1)$. This is different from a causal time series.

If an autoregressive model has white noise of positive variance, a unique weakly stationary process solving its equation on all integer times exists precisely when its autoregressive polynomial has no root on the unit circle. In that case $1/A(z)$ has an absolutely summable Laurent series on an annulus containing the unit circle; its coefficients define a bilateral linear filter of a stationary time series. Applying that inverse also proves uniqueness among bounded second-moment solutions. A root on the unit circle forces the candidate spectral density of a stationary process to have a nonintegrable singularity. The stronger causality root criterion for an autoregressive model requires every root outside the closed unit disk.

An autoregressive polynomial with a root at one gives a unit-root model. The basic example $X_t=X_{t-1}+{\epsilon }_t$ is a random walk, whose variance grows with time when the increments have positive variance. Differencing removes this root and recovers its white noise increments.

The Dickey–Fuller test detects a unit-root autoregressive process against a stationary causal alternative. Regress $\Delta X_t$ on $X_{t-1}$, with deterministic terms appropriate to the model. Under the null, the usual regression statistic has a nonnormal Brownian motion functional limit, so ordinary normal critical values are inappropriate.

A centered model $X_t={\varphi }_{\nu }{X}_{t-1}+{\sigma }_{\nu }{\epsilon }_t$, with $\nu =t\mathrm{mod}S$, has season-dependent coefficients and white noise innovations. In the nondegenerate case, the causal stability condition is $|{\prod }_{\nu =1}^S{\varphi }_{\nu }|<1$. Individual coefficients can exceed one in modulus while the product remains stable. Its second-order law is a periodically correlated process.

For unit-variance innovations, orthogonality to the past givesConsequently ${\varphi }_{\nu }={\gamma }_{\nu }(1)/V_{\nu -1}$ and ${\sigma }_{\nu }^2=V_{\nu }-{\gamma }_{\nu }(1{)}^2/V_{\nu -1}$. Replacing autocovariance by matched sample estimates yields seasonal regression estimators. The unit innovation variance is necessary to identify the noise scale separately.

An autoregressive process of order one obeyswhere $\varphi$ is real and ${\epsilon }_t$ is centered white noise of variance ${\sigma }^2>0$. The zero-mean version is $X_t=\varphi X_{t-1}+{\epsilon }_t$; in the affine form $X_t=c+\varphi X_{t-1}+{\epsilon }_t$, the stationary mean satisfies $c=(1-\varphi )\mu$.

For $|\varphi |<1$, its unique weakly stationary process solution on all integer times is the causal time-series representationThe sum converges in the sense of mean-square convergence because ${\sum }_{j\ge 0}{\varphi }^{2j}=1/(1-{\varphi }^2)<\infty$. It is an infinite moving-average representation using only present and past driving noise. Weak white noise suffices for this second-order conclusion; strong white noise also gives strict stationarity. The autocovariance of an AR(1) process is ${\sigma }^2{\varphi }^{|h|}/(1-{\varphi }^2)$, so its variance is ${\sigma }^2/(1-{\varphi }^2)$ and its ACF is ${\varphi }^{|h|}$. See Bartlett, lecture 5: AR(1) representation and causality.

The appeal of an AR(1) is that one coefficient expresses how a deviation persists. For $0<\varphi <1$, disturbances propagate with weights $1,\varphi ,{\varphi }^2,\dots$. For $-1<\varphi <0$, these weights alternate sign. With $\mu =10$, $\varphi =0.7$, and $X_t=12$, the recursive forecasts of an autoregressive process give $11.4$ and $10.98$ one and two steps ahead. If ${\sigma }^2=1$, the stationary variance is $100/51\simeq 1.9608$, while the one-step forecast error variance is only $1$. Under weak white noise these are best linear forecasts; independent driving noise also makes them conditional expectations given the observed past.

The condition $|\varphi |<1$ describes a stationary causal solution, not every finite recursion with those coefficients. Starting from a prescribed value gives the transient in finite initialization of an AR(1) process. Nor is it a condition for all possible two-sided stationary solutions: when $|\varphi |>1$, noncausal stationary autoregression uses future driving noise. See stationary versus causal solution of a two-sided AR(1) equation for the boundary cases.

For the stationary causal AR(1) with $|\varphi |<1$, matching terms in its infinite moving-average representation gives, for $h\ge 0$,Symmetry of the autocovariance extends this to negative lags. Dividing by $\gamma (0)$ gives the autocorrelation $\rho (h)={\varphi }^{|h|}$. The geometric series explains both the enlarged stationary variance and the diminishing memory: all past disturbances contribute to the present, but their squared weights have a finite sum. At $\varphi =0$, the process reduces to shifted white noise and all nonzero-lag autocorrelations vanish. See Bartlett, lecture 4: AR(1) representation and mean-square convergence.

For $t\ge 1$, iterating an AR(1) from a fixed value $X_0=x_0$ gives the displayed finite sum. With centered white noise of variance ${\sigma }^2$ and $|\varphi |<1$,For $h\ge 0$, its covariance is $\mathrm{Cov}(X_{t+h},X_t)={\varphi }^h\mathrm{Var}(X_t)$, which depends on $t$. Even setting $x_0=\mu$ leaves a changing variance. At $\varphi =0.7$, ${\sigma }^2=1$, and $x_0=\mu$, the variances at times zero through three are $0,1,1.49,1.7301$, approaching $100/51$; this finite initialized record is not weakly stationary.

To initialize in weak stationarity, give $X_0$ mean $\mu$, variance ${\sigma }^2/(1-{\varphi }^2)$, and zero covariance with every future driving noise value. The full stationary construction sets $X_0-\mu ={\sum }_{j\ge 0}{\varphi }^j{\epsilon }_{-j}$. For discrete Gaussian white noise, equivalently draw $X_0$ from $N(\mu ,{\sigma }^2/(1-{\varphi }^2))$ independently of future noise. This gives exact strict stationarity from the first plotted point; it is the initialization used in the figure in time series. Matching these first two moments alone does not guarantee strict stationarity for non-Gaussian inputs. See also stationary initialization of a scalar linear state-space model.

For $|\varphi |<1$, the bilateral sum ${\sum }_{k\in \mathbb{Z}}{\varphi }^{|k|}{z}^k$ equals $(1-{\varphi }^2)/((1-\varphi z)(1-\varphi z^{-1}))$ on the unit circle. Multiplying by ${\sigma }^2/(\pi (1-{\varphi }^2))$ gives the one-sided autoregressive spectrum and reads off its covariance coefficients.

For a causal AR(1) plus uncorrelated observation white noise, cross covariances vanish by L2 convergence of the autoregressive noise expansion. Its covariance is ${\sigma }_z^2{\varphi }^{|h|}/(1-{\varphi }^2)+{\sigma }_w^2{1}_{\{h=0\}}$. The additional noise changes only lag zero, attenuating the normalized positive-lag correlations.

Filtering the observations by $1-\varphi B$ gives covariance $A={\sigma }_z^2+(1+{\varphi }^2){\sigma }_w^2$ at zero, $C=-\varphi {\sigma }_w^2$ at lag one and zero elsewhere. Set $\nu =(A+\sqrt{A^2-4C^2})/2$ and $\vartheta =C/\nu$. Then $\nu (1+{\vartheta }^2)=A$, $\nu \vartheta =C$ and $|\vartheta |<1$. Applying $(1+\vartheta B{)}^{-1}$ defines actual white-noise innovations with variance $\nu$, proving an at-most-(1,1) ARMA representation without assuming Gaussianity.

With nondegenerate white noise, a two-sided AR(1) equation has a unique weakly stationary solution exactly when $|\varphi |\ne 1$. The solution is causal for $|\varphi |<1$ and anticausal for $|\varphi |>1$. At $\varphi =\pm 1$, an n-term noise sum has variance $n{\sigma }^2$, while its difference-of-stationary-values representation has variance at most four times the stationary variance, a contradiction.

For a stationary autoregressive process of order one with mean $\mu$, autoregressive coefficient $\varphi$ and innovation variance ${\sigma }^2$, an interior observation conditional on its neighbors hasThe Markov property makes the same law valid when all other observations are also conditioned on.

For $|\varphi |<1$, independent $N(0,{\sigma }^2)$ innovations and stationary initial variance ${\sigma }^2/(1-{\varphi }^2)$, the full likelihood function isConditioning on $X_1$ gives a different likelihood.

Conditioning on the first observation of a centered autoregressive process of order one makes the maximum-likelihood estimator of its coefficient the least-squares ratio above, when the optimum is interior. The stationary initial normal distribution contributes an additional factor $\sqrt{1-{\varphi }^2}$ and quadratic term $(1-{\varphi }^2)X_1^2$. These change the exact maximizer, but for a fixed interior stationary parameter their score is order one while the conditional score has order $T$, giving the same leading estimator.

For $X_t-\mu =\varphi (X_{t-1}-\mu )+{\epsilon }_t$, the autocovariance equation is $\gamma (1)=\varphi \gamma (0)$. Replacing both autocovariances by sample sums divided by the same $n$ yields ${\hat{\varphi }}_{YW}=\hat{\gamma }(1)/\hat{\gamma }(0)$, the lag-one sample autocorrelation function.

For a stationary unit-variance normal distribution autoregression with coefficient $a\in (-1,1)$, its vector of length $T\ge 2$ has precision matrix $J$ that is tridiagonal: the two endpoint diagonal entries are $1/(1-a^2)$, interior entries are $(1+a^2)/(1-a^2)$, and adjacent off-diagonal entries are $-a/(1-a^2)$. An independent observation $r=\lambda Z+\eta$, with covariance $vI$, updates precision and information vector toThe conditional law is multivariate normal distribution $N(Q^{-1}h,Q^{-1})$.

A moving-average model expresses the current value as a finite linear combination of present and past white-noise innovations.

A multivariate moving-average model filters vector white noise through fixed matrix coefficients. If the noise has zero mean and lag-zero covariance matrix $D$, with all nonzero-lag cross-covariances zero, then the filtered process is a weakly stationary process. With the convention $\Gamma (h)=\mathbb{E}[Z_t{Z}_{t+h}^T]$, its covariance matrix is $\Gamma (h)={\sum }_j{\Theta }_{j}D{\Theta }_{j+h}^T$, omitting terms outside the filter range. Positive and negative lags satisfy $\Gamma (-h)=\Gamma (h{)}^T$.

For $Z_t=W_t+\Theta W_{t-1}$ and vector white noise of covariance matrix $D$, the only nonzero covariance lags are $\Gamma (0)=D+\Theta D{\Theta }^T$, $\Gamma (1)=D{\Theta }^T$ and $\Gamma (-1)=\Theta D$. This follows by retaining only matching noise times in $\mathbb{E}[Z_t{Z}_{t+h}^T]$. Its spectral matrix, with this lag convention, is $(I+\Theta e^{i\lambda })D(I+{\Theta }^T{e}^{-i\lambda })/(2\pi )$. In two dimensions, with $D=\mathrm{diag}(a,b)$, its cross-spectrum is $[a{\theta }_{11}{\theta }_{21}+b{\theta }_{12}{\theta }_{22}+a{\theta }_{21}{e}^{-i\lambda }+b{\theta }_{12}{e}^{i\lambda }]/(2\pi )$.

The moving-average polynomial records the coefficients of the finite noise filter. Evaluating it at the backshift operator gives $\Theta (B)w_t=w_t+{\sum }_{j=1}^q{b}_j{w}_{t-j}$. In a reduced ARMA representation, invertibility is equivalent to having no roots in the closed unit disc.

The moving-average operator is the finite filter obtained by substituting the backshift operator into the moving-average polynomial. It maps a white noise sequence to its weighted finite sum of present and past values.

For coefficients ${\theta }_0=1,{\theta }_1,\dots ,{\theta }_q$, the long-run variance of a stationary process sampled every second time is ${\sigma }^2[({\sum }_{j\text{ even}}{\theta }_j{)}^2+({\sum }_{j\text{ odd}}{\theta }_j{)}^2]$. It equals ${\sigma }^2[\Theta (1{)}^2+\Theta (-1{)}^2]/2$.

A causal square-integrable linear time series can be written $X_t=\mu +{\sum }_{j\ge 0}{\psi }_j{\epsilon }_{t-j}$. Square summability of the coefficients ensures mean-square convergence for white noise. An innovation representation additionally identifies its driving noise as the innovation process.

For $X_t={\sum }_{j\ge 0}{\psi }_j{\eta }_{t-j}$ driven by centered unit-variance iid noise with a finite fourth moment, put ${\kappa }_4=E{\eta }^4-3$ and $m_3=E{\eta }^3$. Independence givesThe sum of the two linear-quadratic cross covariances is $m_3{\sum }_{j\ge 0}({\psi }_j^2{\psi }_{j+|h|}+{\psi }_j{\psi }_{j+|h|}^2)$. The cumulant term disappears for Gaussian noise. Merely assuming strong white noise does not justify the Gaussian formula; a fourth moment is needed for the variance of the quadratic transform.

Invertibility permits reconstruction of innovations from present and past observations by a stable filter. For $Y_t-\mu ={\epsilon }_t+\theta {\epsilon }_{t-1}$, $|\theta |<1$ gives ${\epsilon }_t={\sum }_{j\ge 0}(-\theta {)}^j(Y_{t-j}-\mu )$. This is a different property from stationarity: a finite moving-average model is stationary for every finite coefficient.

Reflecting an inside-unit-circle zero across the unit circle produces an invertible moving-average model with the same spectral density of a stationary process, after rescaling the driving white noise variance. For real $|\theta |>1$, $|1-\theta e^{-i\lambda }{|}^2={\theta }^2|1-{\theta }^{-1}{e}^{-i\lambda }{|}^2$. The transformed driving sequence is a linear innovation process; without Gaussianity it need not be strong white noise.

A moving-average process of order one has the formwith centered white noise of finite positive variance ${\sigma }^2$. Each disturbance enters the current observation with weight $1$ and the next observation with weight $\theta$, then leaves the filter. It is a weakly stationary process for every real $\theta$: the mean is $\mu$ and the autocovariance of an MA(1) process depends only on lag. Strong white noise additionally makes it a strictly stationary process. No $|\theta |<1$ restriction is needed for stationarity.

For a worked observation calculation, take $\mu =10$, $\theta =0.7$, and consecutive noise values ${\epsilon }_{t-1}=-1$, ${\epsilon }_t=2$, ${\epsilon }_{t+1}=0$. Then $X_t=11.3$ and $X_{t+1}=11.4$; the same disturbance $2$ appears in both. If ${\sigma }^2=1$, the variance is $1.49$, the lag-one autocovariance is $0.7$, and the lag-one autocorrelation is $70/149\simeq 0.4698$. All higher-lag autocorrelations vanish, as shown in autocorrelation.

An MA(1) is useful for brief dependence caused by overlapping disturbances or measurements. Its name refers to a weighted moving sum of noise; its weights need not be positive or sum to one, and it differs from the data smoother in seasonal extraction by a centred moving average. Under strong white noise, observations more than one interval apart depend on disjoint independent noise families and are independent. Under weak white noise, the same cutoff establishes only uncorrelated random variables.

Invertibility of a moving-average model is a separate question: can the specified driving noise be recovered by a stable linear filter of present and past observations? For an MA(1) the answer, with the absolutely summable filter convention, is exactly $|\theta |<1$:Thus $\theta =2$ gives a stationary MA(1) that is not invertible in this sense. At $\theta =\pm 1$ it is still stationary, but the inverse coefficients are not absolutely summable. The case $\theta =0$ is simply shifted white noise. See Bartlett, lecture 5: MA(1) and stable invertibility.

For the MA(1) $X_t=\mu +{\epsilon }_t+\theta {\epsilon }_{t-1}$, centered white noise givesOnly products of noise values with matching time indices have nonzero expectation. At lag zero there are two squared weights, $1$ and ${\theta }^2$; at lag one there is one shared noise term with weight product $\theta$; at higher lags there are none. The calculation needs zero covariance between distinct noise values, not independence. This tridiagonal finite covariance structure makes the finite-sample innovations of an MA(1) process particularly simple.

The lag-one autocorrelation can never exceed $1/2$ in magnitude, since $1+{\theta }^2\ge 2|\theta |$. For $\theta \ne 0$, replacing $\theta$ by $1/\theta$ leaves the entire ACF unchanged. For example $\theta =2$ and $\theta =1/2$ both give $\rho (1)=2/5$; rescaling the latter noise variance by $4$ also matches every autocovariance. Moving-average root reflection explains this ambiguity and why a stable invertibility convention helps identify a representation. Equality of second-order laws need not imply equality of all joint probability distributions unless the processes are Gaussian processes. See Bartlett, lecture 6: equivalent representations and covariance of ARMA processes.