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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.

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 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.

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 a square-integrable time series, let ${\mathcal{H}}_{t-1}$ be the closed linear span of its past. The innovation ${\epsilon }_t=X_t-{\mathrm{proj}}_{{\mathcal{H}}_{t-1}}{X}_t$ is orthogonal to every past linear observation. A causal invertible autoregressive moving-average model is driven by these innovations, which are weak white noise for a weakly stationary process.

A linear filter of a stationary time series with real coefficients $(a_j{)}_{j\in \mathbb{Z}}$ satisfying ${\sum }_j|a_j|<\infty$ is $Y_t={\sum }_j{a}_j{X}_{t-j}$. Its filter generating function is the Laurent series $A(z)={\sum }_j{a}_j{z}^j$, absolutely convergent on $|z|=1$, and its filter transfer function is $H_A(\lambda )=A(e^{-i\lambda })$. For a one-sided filter this is a power series; a general two-sided filter need not converge on a larger annulus.

The filter theorem states that if $X$ is second-order stationary, then $Y$ exists in mean square, is second-order stationary, and has mean and covarianceIts spectral measure of a stationary time series is $d{F}_Y(\lambda )=|H_A(\lambda ){|}^{2}d{F}_X(\lambda )$. In particular, if $X$ has a time-series spectral density in the convention ${\gamma }_X(h)={\int }_{-\pi }^{\pi }{e}^{ih\lambda }{f}_X(\lambda )d\lambda$, thenTo prove existence, the $L^2$ norm of a tail of the filter is at most $\Vert X_0{\Vert }_2$ times the sum of the omitted $|a_j|$, which tends to zero. Hence finite partial sums converge in $L^2$. Expectations and covariances pass to the limit, and the covariance double sum is absolutely convergent because $|{\gamma }_X(h)|\le {\gamma }_X(0)$. Its formula depends only on $h$, proving stationarity. Insert the spectral representation of ${\gamma }_X(h-j+k)$ and interchange sums and integration; the bound ${\sum }_{j,k}|a_j{a}_k|F_X([-\pi ,\pi ])<\infty$ justifies the interchange. The resulting factor is ${\sum }_j{a}_j{e}^{-ij\lambda }{\sum }_k{a}_k{e}^{ik\lambda }=|A(e^{-i\lambda }){|}^2$. This proves the spectral statement, including the spectral density transformation under a linear filter without assuming that the input autocovariances are absolutely summable.

For an autoregressive moving-average model, use real polynomials normalized by ${\varphi }_0={\theta }_0=1$ and first work with the reduced noise-driven representation. If $\varphi$ has no zero on the unit circle, the rational function $C(z)=\theta (z)/\varphi (z)$ has a Laurent expansion in an annulus containing that circle, with absolutely summable coefficients. Filtering the white noise with it gives a stationary solution. Its time-series spectral density isConversely, for coprime $\varphi ,\theta$, a unit-circle zero of $\varphi$ would make this ratio nonintegrable, since the numerator does not vanish there; thus there is no finite-variance stationary noise-driven solution. A stationary solution can use a bilateral filter when autoregressive roots lie inside the unit circle. The stronger causal condition, meaning dependence only on present and past noise, requires every zero of $\varphi$ strictly outside the unit disk. Common factors must be handled before applying these conditions; a cancelled unit-root factor can also allow additional deterministic stationary components, which are outside the minimal noise-driven convention.

Identifiability means that different admissible parameters cannot produce the same stationary process law. In this zero-mean Gaussian setting, its law is determined by its covariance, equivalently its spectrum. Unrestricted ARMA coefficients are not identifiable from that law: if a moving-average factor has root $r$ inside the disk, then on $|z|=1$,Reflecting the root outside and multiplying the innovation variance by $|r{|}^{-2}$ preserves the spectrum, and therefore the Gaussian law. Common polynomial factors and arbitrary overall scaling give further redundancy. A regular canonical identifiable parameterization requires constant coefficients one, coprime polynomials of their stated minimal orders, all zeros of both $\varphi$ and $\theta$ outside the closed unit disk, and ${\sigma }^2>0$.

Here is a proof of canonical identifiability of Gaussian ARMA models. Suppose two such models have equal spectra and write their innovation standard deviations as $\sigma ,\tau >0$. The ratio $R(z)=\sigma \theta (z)\tilde{\varphi }(z)/[\tau \tilde{\theta }(z)\varphi (z)]$ and its reciprocal are analytic on a neighbourhood of the closed unit disk, with modulus one on its boundary. The maximum modulus principle applied to both makes $|R|=1$ throughout the disk, so $R$ is constant. Its value at zero is the positive number $\sigma /\tau$, forcing $R=1$ and $\sigma =\tau$. Therefore $\theta \tilde{\varphi }=\tilde{\theta }\varphi$; coprimality and the normalization at zero force equality of both polynomials. This proves the claimed parameter uniqueness within that canonical class.

In this class, $D(z)=\varphi (z)/\theta (z)$ is analytic on a disk of radius greater than one. Its Taylor series coefficients are absolutely summable, soThis is an invertible time-series representation, proved by composition of the stable filters. Causality makes ${\epsilon }_t$ orthogonal to past observations; invertibility makes past noises recoverable from past observations. Their closed past linear spans agree, so this noise is the linear innovation and the one-sided expansion $X_t={\sum }_{j\ge 0}{c}_j{\epsilon }_{t-j}$ is the Wold representation. Mere stationarity without the stated canonical restrictions does not imply invertibility of the originally written noise representation.

For this one-sided canonical representation, $C(z)=\theta (z)/\varphi (z)={\sum }_{j\ge 0}{c}_j{z}^j$. Comparing coefficients gives the ARMA coefficient recursionsThus for exactly the range asked for,The characteristic rates are reciprocals of the zeros of $\varphi$. Partial fractions therefore express the tail as sums of $P_r(k)r^k$, where $|r|<1$ and the degree of $P_r$ is at most one less than the corresponding root multiplicity. Hence the coefficients decay exponentially, with possible polynomial and damped oscillatory factors. If $p=0$, they instead vanish after $q$. For a noncausal or noninvertible original specification, the Wold coefficients refer to the canonical spectral factor; using the original driving noise as the Wold innovation without checking these restrictions would be incorrect.

The autocovariance is $\gamma (h)={\sigma }^2{\sum }_{j\ge 0}{c}_j{c}_{j+|h|}$. Analyticity of $C$ on a disk of radius $R>1$ gives $|c_j|\le M{R}^{-j}$ for a slightly smaller radius than its first pole, and consequently $|\gamma (h)|\le {\sigma }^2{M}^2{R}^{-|h|}/(1-R^{-2})$. Moreover, multiply the ARMA equation at time $t$ by $X_{t-h}$ and take expectations. For $h>q$, all noises ${\epsilon }_{t-s}$ with $0\le s\le q$ are later than $X_{t-h}$ and orthogonal to it. For $h>\max (p-1,q)$ this givesThe covariance tail therefore has the same characteristic exponential rates and multiplicity patterns as the filter coefficients, subject to possible cancellation of amplitudes. Pure moving averages have an exact covariance cutoff.

For the final process, factor the polynomials:The common factor has its zero at $3/2$, so it can be cancelled as a stable filter. Explicitly, let $Y_t=(1+B/3)X_t-{\epsilon }_t$; the original equation gives $Y_t=(2/3)Y_{t-1}$. The second moments of $Y_t$ are uniformly bounded because those of $X_t$ and ${\epsilon }_t$ are. Iterating gives $Y_t=(2/3{)}^m{Y}_{t-m}$, whose $L^2$ norm tends to zero as $m\to \infty$, so $Y_t=0$ in mean square. ThusSumming the covariance series givesThe process is a causal stationary autoregressive process of order one, with alternating-sign geometric correlations, and is invertible. The displayed ARMA(2,1) description is nonminimal because of its common factor; its minimal identifiable class is ARMA(1,0).

For the usual causal, stable autoregressive moving-average model with zero-mean white noise, sufficient conditions for a stationary invertible representation areThe roots of $1-\varphi z$ and $1+\theta z$ then lie outside the closed unit disc. For a minimal representation these are the usual causality and invertibility root criteria for an ARMA model. A genuinely minimal ARMA(1,1) additionally has nonzero coefficients and no cancellation $\varphi +\theta =0$; if a coefficient vanishes the order is lower. Common factors can reduce the process to white noise, so inequalities on an unreduced representation should not be asserted as necessary for every reduced process. A noncausal stationary solution of an unstable autoregressive equation is a different representation, and does not give the requested innovation-based Wold coefficients below.

Using the backshift operator $B$, write $(1-\varphi B)X_t=(1+\theta B){ϵ}_t$. The absolutely summable geometric series givesThus the Wold representation isIts coefficients are square-summable, so the series converges in mean square and defines a weakly stationary process. Invertibility makes the closed spans of past observations and past innovations equal. Since the current driving white noise is orthogonal to all past innovations, it is orthogonal to all past observations and is the linear innovation of this causal invertible process.

The spectral density of an ARMA process follows from the squared frequency response of the linear filter of a stationary time series:This displayed density uses the usual two-sided convention. For the one-sided spectral density used in Question 5, multiply it by two and restrict to $0\le \omega \le \pi$.

Let ${\psi }_0=1$ and ${\psi }_j=(\varphi +\theta ){\varphi }^{j-1}$ for $j\ge 1$. Orthogonality of the driving white noise gives $\gamma (h)={\sigma }^2{\sum }_{j\ge 0}{\psi }_j{\psi }_{j+h}$ for $h\ge 0$. HenceFor $h\ge 1$, separate the $j=0$ term and sum the remaining geometric series:Negative lags satisfy $\gamma (-h)=\gamma (h)$. With $\varphi =0$, interpret the lag-one factor as one and later powers as zero, recovering the MA(1) covariance.

To express prediction entirely through observations, invert the moving-average factor:The best linear predictor from the whole past is thereforeIndeed $X_t-{\hat{X}}_t={ϵ}_t$ is orthogonal to the closed linear span of the past. For any other past-measurable linear predictor $L$, the mean squared error decomposes as $E[(X_t-L{)}^2]={\sigma }^2+E[({\hat{X}}_t-L{)}^2]$, establishing optimality and error variance ${\sigma }^2$. The result only uses second-order properties; identifying this linear predictor with a full conditional expectation requires stronger assumptions such as independent innovations or joint Gaussianity. In the cancellation case $\theta =-\varphi$ within the stable range, $X_t={ϵ}_t$, the off-zero covariances and predictor vanish, and the spectrum is constant.

Define the autoregressive polynomial $\Phi (z)=1-{\sum }_{k=1}^p{\varphi }_k{z}^k$ and the moving-average polynomial $\Theta (z)=1+{\sum }_{k=1}^q{\theta }_k{z}^k$. The standard causality and invertibility root criteria for an ARMA model areIn the usual minimal autoregressive moving-average model, the two polynomials have no common roots. Under that convention, these are the necessary and sufficient root conditions for a causal and invertible representation with the given noise. Without minimality they remain sufficient; shared factors must be considered before claiming necessity.

To see why the conditions work, both ratios $C(z)=\Theta (z)/\Phi (z)$ and $D(z)=\Phi (z)/\Theta (z)$ are analytic on a disc of radius greater than one. Their power series coefficients therefore decay geometrically and are absolutely summable. The backshift operator $B{X}_t=X_{t-1}$ then gives a well-defined mean-square convergent causal time-series representationFiltering white noise by these fixed coefficients gives a weakly stationary process. The identity $\Phi (B)C(B)=\Theta (B)$ verifies the original equation. Multiplication by the absolutely summable inverse of $\Phi (B)$ also shows uniqueness among finite-variance stationary solutions. Similarly $D(B)X_t={ϵ}_t$ gives the invertible time-series representation.

Comparing coefficients in $\Phi (z)C(z)=\Theta (z)$ derives the first of the ARMA coefficient recursions:Here $c_j=0$ for negative $j$, ${\theta }_0=1$, and ${\theta }_j=0$ for $j>q$, so in particular $c_0=1$. For the inverse filter, compare coefficients in $\Theta (z)D(z)=\Phi (z)$. With $a_0=1$, $a_j=-{\varphi }_j$ for $1\le j\le p$, $a_j=0$ for $j>p$, and $d_j=0$ for negative indices, this givesThe minus sign in the inverse recursion comes from moving the nonconstant moving-average terms to the other side.

For the first-order model, the standard uncancelled conditions are $|\varphi |<1$ and $|\theta |<1$. The geometric power series expansions giveThus the ARMA(1,1) causal and inverse coefficients areAt $j=1$ these give $c_1=\varphi +\theta$ and $d_1=-(\varphi +\theta )$, including zero-valued parameters without any negative powers. If $\theta =-\varphi$, the filtered process reduces to $X_t={ϵ}_t$ and all later coefficients vanish. This common factor cancellation in an ARMA model is why the root inequalities should not be advertised as necessary for every nonminimal parameterization. The stated inequalities still ensure uniqueness for the original equation as well as the desired causal and invertible filters.

In the usual causal stationary convention, a centered autoregressive moving-average model satisfiesSet $\Phi (z)=1-{\sum }_{j=1}^p{\varphi }_j{z}^j$ and $\Theta (z)=1+{\sum }_{j=1}^q{\theta }_j{z}^j$. After removal of common factors, the usual causality condition is that all zeros of $\Phi$ lie outside the closed unit disk. Then $\Theta (z)/\Phi (z)$ has an absolutely summable causal coefficient sequence, giving a linear-filter representation of $X$ in terms of white noise. Invertibility of the moving-average part is a separate issue and is not needed for its covariance spectrum.

White noise has one-sided density ${\sigma }^2/\pi$. Applying the filtering result, or equating the spectra of the two filtered sides of the defining equation, givesThe moving-average sign agrees with the positive $\theta$ convention used later. More generally, the same quotient applies to a stationary two-sided solution when the reduced denominator has no unit-circle zeros; causality selects the past-dependent realization. This is the spectral density of an ARMA process.

For an autoregressive moving-average model, the driving-to-output transfer function is $A(z)=\theta (z)/\varphi (z)$, and the inverse transfer function is $A(z{)}^{-1}=\varphi (z)/\theta (z)$. The causality and invertibility root criteria for an ARMA model require these respective rational functions to have power series about zero converging on a disk larger than the unit disk. If the two polynomials have no common factor, the conditions becomeIndeed, outside-disk roots leave a radius of convergence greater than one, so the coefficients decay geometrically and are absolutely summable. Conversely an uncancelled pole inside or on the unit disk prevents the required stable power series. If factors are common, apply the criterion after cancellation, to the noise-driven solution rather than additional homogeneous components. The backshift operator translates these power series into the desired one-sided filters.

The original representation has $\varphi (z)=1-2z$ and $\theta (z)=1+2z$, with roots $1/2$ and $-1/2$. There is no cancellation. Hence it is neither causal nor invertible relative to its specified driving noise. Stationarity is nevertheless possible through a two-sided solution: expanding the autoregressive inverse in negative powers givesThis is an anticausal time series representation with square-summable coefficients.

To establish the alternative representation on the same process, defineThe inverse of $1+\frac{1}{2}B$ is a stable one-sided filter. For $|z|=1$, the identities $|1+2z{|}^2=4|1+z/2{|}^2$ and $|1-2z{|}^2=4|1-z/2{|}^2$ giveThe time-series spectral density of $X$ is ${\sigma }^2|1+2e^{-i\omega }{|}^2/(2\pi |1-2e^{-i\omega }{|}^2)$; therefore the defined $\eta$ has constant time-series spectral density ${\sigma }^2/(2\pi )$. Its mean is zero, its variance is ${\sigma }^2$, and all its nonzero-lag autocovariances vanish. It is thus weak white noise. Its definition directly givesThis root reflection of an ARMA representation has roots $2,-2$, so is causal and invertible. A constant spectrum proves whiteness, not independence of non-Gaussian coordinates; no Gaussian assumption is needed for the required white noise representation.

For best linear prediction from an infinite past, let ${\mathcal{H}}_0$ be the closed linear span of $X_t$ with $t\le 0$. The causal representation expresses every such $X_t$ in present and past $\eta$ values, while invertibility puts every ${\eta }_t$ with $t\le 0$ in ${\mathcal{H}}_0$. In particular ${\eta }_1$ is orthogonal to ${\mathcal{H}}_0$. The representation at time one then gives the orthogonal projectionConsequently the best linear predictor and its error variance areOrthogonality proves optimality among linear predictors in the closed past span, without asserting that the predictor must be the conditional mean for a non-Gaussian process.

Among the supplied candidates, choose ARMA(2,1). It has the smallest Akaike information criterion. Relative to ARMA(2,2), its AIC improvement is $1.71$; the additional second moving-average estimate is only half a standard error from zero and increases the log likelihood function by only about $0.14$. Dropping it is a reasonable parsimony choice, although that AIC gap is small. ARMA(1,1) has AIC larger by $17.55$, a much clearer loss of fit. In the chosen model the second autoregressive term is about seven standard errors from zero, so it should not be dropped merely to obtain order one. The first autoregressive term is less precisely estimated; this does not justify automatically deleting it without fitting and comparing the reduced candidate.

Using the usual positive-sign moving-average convention, the fitted autoregressive moving-average model isThese are plug-in values rounded as given, not exact population parameters. The model has zero mean. Its polynomials are $\varphi (z)=1+0.30z-0.28z^2=(1-0.4z)(1+0.7z)$ and $\theta (z)=1+0.5z$.

With angular frequency $\omega \in [-\pi ,\pi ]$, so that the autocovariance is $\gamma (h)={\int }_{-\pi }^{\pi }{e}^{ih\omega }{f}_X(\omega )d\omega$, the spectral density of a stationary process isThe frequency convention makes the normalization unambiguous.

Two useful features of autoregressive conditional heteroscedasticity are persistent changes in conditional scale and excess unconditional kurtosis. Its time-varying conditional variance can explain volatility clustering, where large absolute returns occur in groups even when signed returns have little autocorrelation. A homoscedastic autoregressive moving-average model has a fixed innovation variance.

Also, a conditional normal distribution with a random scale is a Gaussian scale mixture. Its unconditional kurtosis can exceed $3$, or its fourth moment can be infinite. A Gaussian autoregressive moving-average model remains jointly Gaussian and cannot reproduce this effect. In the particular lag-two ARCH process, persistence of the squared scale occurs within each parity subsequence; the two parity subsequences are independent.

Use the sample autocorrelation function and sample partial autocorrelation function to look for approximate cutoffs. For a minimal causal autoregressive model of order $p$, the partial autocorrelation function is zero beyond $p$, while the autocorrelation function usually tails off, possibly with damped oscillation. For an invertible moving-average model of order $q$, the autocorrelation function is zero beyond $q$, while the partial autocorrelation function tails off. In a mixed autoregressive moving-average model, both generally tail off.

Thus the last visibly nonzero partial correlation suggests an autoregressive order, and the last visibly nonzero autocorrelation suggests a moving-average order. Finite samples do not produce exact zeros. Approximate white-noise reference bands can help flag clearly nonzero lags, but they are not universal confidence intervals for arbitrary correlated processes. Fit a few suggested orders and check residual autocorrelation, rather than treating these plots as a proof of the model order. This is order identification by autocorrelation cutoffs.

For the spectral questions take the unique weakly stationary process solving the equation, as is customary for a stable autoregressive moving-average model. The equation alone also admits nonstationary solutions differing by $C{4}^{-t}$, for which a spectral density of a stationary process need not exist. Let $B$ be the backshift operator. Since the autoregressive root is $4$, the stationary solution is causal andUsing the convention $\gamma (h)={\int }_{-\pi }^{\pi }{e}^{ih\lambda }f(\lambda )d\lambda$, the spectral density of a stationary process isThe stationary mean is zero, since $\mu -\mu /4=0$.

The root-reflected, invertible autoregressive moving-average model isThis follows by multiplying the filter identity for $\epsilon$ by $1-B/3$. Its moving-average root $3$ and autoregressive root $4$ are both outside the unit disk. Thus it is the representation in terms of the innovation process, rather than merely another noise representation with the same spectrum.

For a causal autoregressive model of order $p$, the population partial autocorrelation function cuts off after lag $p$, whereas the autocorrelation function generally decays. For an invertible moving-average model of order $q$, the autocorrelation function cuts off after lag $q$, whereas the partial correlation coefficients generally decay. For a mixed autoregressive moving-average model, both generally decay. Exponential decay may alternate in sign or show damped oscillations. Sample correlation coefficients only approximate these patterns, so isolated crossings of the significance bands are not exact order tests.

In PDF Figure 1, the autocorrelation function alternates sign, with a large negative lag-one correlation coefficient and geometrically decreasing magnitude. The partial autocorrelation function has essentially one substantial spike, negative at lag one and about $-0.8$; later values mostly lie within the displayed bands. Thus a stationary AR(1) with a negative coefficient is the natural first model, with $\hat{\varphi }$ initially near $-0.8$. The trace fluctuates around an approximately constant mean and shows no evident deterministic trend.

Fit that model, compare nearby low-order alternatives using likelihood function and an Akaike information criterion or Bayesian information criterion, and inspect the residual autocorrelation function and residual variance. A few small later PACF spikes are expected in a plot with many lags. The roughly $\pm 1.96/\sqrt{T}$ white noise bands are a guide, rather than simultaneous guarantees for every lag.

The sign alternation also suggests concentration of spectral power near the high-frequency end: the AR(1) denominator $1+{\varphi }^2-2\varphi \cos \lambda$ is smallest near $\lambda =\pi$ when $\varphi <0$. From estimated $\gamma (0)$ and lag-one correlation coefficient one can estimate innovation variance as ${\hat{\sigma }}_{\epsilon }^2\approx \hat{\gamma }(0)(1-{\hat{\varphi }}^2)$. The persistence parameter controls the decay rate and long-run variance of a stationary process. These plots do not establish a normal distribution, independence, or the absence of nonlinear dependence; correlation coefficients of squared residuals can provide a separate variance diagnostic.

For an autoregressive moving-average model of order $(1,1)$, this state has observation row $F=(\varphi ,1,\theta )$ and transition matrixThe state noise is $(0,{\epsilon }_t,0{)}^{\top }$. Its first transition row reconstructs $X_{t-1}$ and its third row shifts the previous white noise value.