1. MOTIVATION

Among scalar stochastic processes of the autoregressive moving average (ARMA) type, three are of particular interest: complex ARMA or (as we shall say) CARMA(p,q) processes, the real parts of such processes, which we shall call Re CARMA(p,q) processes, and the familiar real-valued ARMA(p′,q′) processes. The main problem considered here is whether, given a weakly stationary Re CARMA(p,q) process, we can find a spectrally equivalent ARMA(p′,q′) process: that is, an ARMA process with the same autocovariance function (ACF) and spectrum. To this question we return a positive answer.

Motives for studying the relationship between these three types of model are as follows. (i) The complex numbers form the largest (in a sense explained by Bell, 1945, pp. 254–255) number field, where theoretical results and concepts often take simple forms not apparent in the real case. The fundamental theorem of algebra, important in the study of the roots of ARMA processes, is one example. Another is the ability of the function ψ^t, where ψ is a complex parameter and t is time, to model exponential growth (or decay), or cyclical behavior, or a combination of the two, according to how the modulus and phase of ψ are chosen. This allows an integrated treatment of cyclicality and growth, whereas the approach using only real parameters tends to try to model them separately. For such reasons, possible maps between the CARMA and ARMA worlds offer to help us bridge the demands of theory and practice. Whether real-part mappings from CARMA processes yield exactly the ARMA class (or, e.g., a larger one) is therefore an interesting and natural question. (ii) More specifically, the ability of a Re CARMA formulation to model cyclical behavior in an illuminating way is evident from one well-known special case (discussed subsequently). In this instance at least the Re CARMA model has an alternative and equivalent but somewhat opaque ARMA reformulation.

These abstract and specific observations both raise the question of whether every Re CARMA has an ARMA equivalent. An answer is given in Theorem 1 in Section 3, which states that every weakly stationary Re CARMA(p,q) process has a weakly stationary ARMA(2p,p + q) representation, or a simpler one. The theorem identifies two possible types of simplification, root cancellation (affecting the AR and MA parts of the model equally) and “MA simplification” (affecting only the MA part). Root cancellation and/or MA simplification in the ARMA process can occur even when there is none in the parent CARMA process.

The special case of a CARMA or Re CARMA model mentioned earlier is the stochastic cycle model (SCM) described, for instance, by Harvey (1989, p. 39). This can be defined as the vector AR(1) process Z_t = ΦZ_t−1 + ε_t, where

,

, 0 ≤ ρ < 1,

and the ε_t are all zero-mean, are uncorrelated across time, and have the time-invariant variance matrix σ²I₂ (as discussed in the next section). The SCM is most naturally viewed as a CARMA process. Explicitly, let Z_t ≡ X_t + iY_t, ε_t ≡ ζ_t + iη_t, and φ ≡ ρe^iω. Then Z_t = φZ_t−1 + ε_t, where φ is a complex parameter and {ε_t} is a sequence of complex innovations. So Z_t follows a CARMA(1,0) or CAR(1) process. In the SCM, X_t but not Y_t is observed; thus interest centers on X_t, which follows a Re CAR(1) process. Seen from the Re CAR(1) point of view, the ACF of X_t is easy to obtain. First we find the ACF of Z_t, which turns out to be γ_Z(k) ≡ E(Z_t Z_t−k) = [2σ²/(1 − φφ)]φ^k when k ≥ 0, γ_Z(−k) = γ_Z(k). The ACF of X_t = Re Z_t is then given by equation (3) in Section 2, yielding γ_X(k) = σ²[ρ^|k|/(1 − ρ²)] cos kω, a damped cosine wave, evaluated at discrete intervals. The simplicity of this form of ACF indicates the theoretical importance of the SCM or Re CAR(1) model and illustrates its usefulness in modeling cyclicality. One should note in particular the close relation between the autoregressive parameter φ and the factor ρ^k cos kω = Re(φ^k) in the ACF. Harvey (1989, p. 69) points out that X_t follows a real-coefficient ARMA(2,1) process. The real formulation is however less natural than the Re CAR(1) formulation, because it uses a parametrization that renders mysterious the striking decaying-cosine nature of the ACF. This confirms that complex parameters can be more appropriate than real ones when cycles are present.

2. MOMENT SPECIFICATION

One important aspect of the SCM requires comment, namely, the special form σ²I₂ it assumes for the variance matrix Σ of (ζ_t,η_t). We explain this assumption with respect to weakly stationary complex linear processes generally. If we are interested in the ACF and spectrum of Z_t only, Σ need not be specified in detail. All that matters is the quantity E(ε_tε_t) = tr Σ, a single real parameter. However if (as in the SCM and other Re CARMA models) we require the spectrum of Re Z_t, then we need all the entries in Σ. To see that this is so consider the real n × 1 vector linear process

where E(ε_t) = 0_n and E(ε_sε_t′) = 0_n×n if s ≠ t. The ACF of Z_t is given by

, and γ_X(k) (where X_t is the first entry in Z_t) is the top-left entry in Γ(k). The expression for Γ(k) is not particularly tractable. It becomes more so if the matrices involved commute. One method of making them do so is obviously to set n = 1 and choose all parameters real (thus in particular Ψ_j = ψ_j, a scalar). A variant of this approach, commonly adopted in theoretical work (see, e.g., Brockwell and Davis, 1987, p. 112), is (still with n = 1) initially to allow the ψ_j and ε_t to be complex and replace the third equation in (1) by E(ε_tε_t) = σ² but to confine the analysis to situations where Z_t is real-valued. Thus Z_t and Re Z_t, and of course their ACFs and spectra, coincide. This simplicity comes at a price, because the requirement for the ψ_jε_t−j to be real implies that the parameters and innovations must either both be purely real or both be purely imaginary, excluding most of the richness that a complex parametrization can yield.

The method used in the SCM to gain full access to the fertile possibilities of a complex parametrization is to set n = 2 and rely on the isomorphism between complex numbers a + ib, where a and b are real, and matrices of the form

. This isomorphism occurs because the operation of multiplying any complex number x + iy by the fixed number a + ib can be associated with the linear transformation that maps the point (x,y) on the Argand diagram to (ax − by, bx + ay). If the 2 × 2 matrices Ψ_j and Σ have this special structure, allowing us to associate them bijectively with the scalars ψ_j and σ², respectively, then Γ(k) is mapped to the complex scalar

. Because Σ is symmetric, σ² must in this case be real, and hence we must have Σ = σ²I₂, explaining the form chosen in the SCM. To obtain the SCM from (1), one sets

, corresponding to the complex scalar (ρe^iω)^j.

The assumption Σ = σ²I₂ is therefore a natural one to impose on complex linear processes, and we shall impose it in defining all CARMA and therefore Re CARMA models. This decision yields the striking simplification that if the complex innovation ε_t = ζ_t + iη_t, where ζ_t and η_t are real, then E(ε_t²) = E(ζ_t² − η_t² + 2iζ_tη_t) = 0; thus E(ε_sε_t) = 0 whether or not s = t. Consequently for any complex linear process obeying Σ = σ²I₂, we have

whether or not k = 0. From this result we can quickly derive the ACF of X_t ≡ Re Z_t = ½(Z_t + Z_t), using γ_X(k) = ¼E [(Z_t + Z_t)(Z_t−k + Z_t−k)] = ¼E [Z_t Z_t−k + Z_t Z_t−k + Z_t Z_t−k + Z_t Z_t−k] . The first and last terms vanish. Because γ_Z(k) = γ_Z(−k) we obtain

By similar means we can show that if Y_t = Im Z_t, then γ_Y(k) = γ_X(k). Thus X_t and Y_t have the same ACF and spectrum. More generally, the ACF and spectrum of the real-part process are not affected by subjecting all the Z_t to a constant rotation about the origin: if, for all t, W_t = Re(αZ_t), where α lies on the unit circle, then γ_W(k) = γ_X(k).

4. PROOF OF THEOREM 1

To prove Theorem 1 we must show that there is an ARMA process with the same spectrum as {Re Z_t}. The demonstration is constructive. Let Z_t follow the normal CARMA(p,q) process defined by (Φ,Θ,σ²). The ACF γ_Z of Z_t is defined by γ_Z(k) ≡ E(Z_t Z_t−k) = γ_Z(−k) and its spectrum by

where τ = e^−iλ and λ is real. We have used

, with a corresponding result for Φ, and the fact that E(ε_tε_t) = 2σ². Let X_t be the real part of Z_t, with ACF γ_X(k) ≡ E(X_t X_t−k). Using (3), the spectrum of {X_t} is

where the denominator and numerator terms are

Because by assumption Φ and Θ lack unit roots, (6) shows that N(1) = 2|Φ(1)|²|Θ(1)|² is real and positive.

To show that X_t has an ARMA(2p − c,p + q − v − c) representation, we first show that D(τ) and N(τ) can be written in the forms A₁(τ)A₁(1/τ) and (apart from a scaling constant) B₁(τ)B₁(1/τ), respectively, where A₁ and B₁ are real-coefficient standardized spectral polynomials of degree 2p and p + q − v, respectively, where 0 ≤ v ≤ p + q. Then we show that the c common roots of A₁ and B₁ are just the common roots of Φ and Φ. Finally we construct the required real-coefficient spectral polynomials A and B, yielding the required normal representation (A,B,σ′²), where A is of degree 2p − c, B is of degree p + q − v − c, and A, B are coprime. This will complete the proof.

Beginning then with the denominator D(τ), define A₁(τ) ≡ Φ(τ)Φ(τ). Then D(τ) = A₁(τ)A₁(1/τ), where A₁(τ) is of degree 2p, satisfies A₁(0) = 1 because Φ(0) = 1, and has real coefficients because

. Its roots are the same as those of Φ, together with their complex conjugates; therefore they all lie outside the unit circle Ω.

We now turn to N(τ). We need to construct a real-coefficient, standardized polynomial B₁, and a real scalar κ > 0, such that N(τ) = κB₁(τ)B₁(1/τ). The degree of B₁ depends on N as follows. Evidently N(1/τ) = N(τ) if τ ≠ 0. Thus N(τ) is of the form

. Degeneracy arises if one or more leading coefficients n_p+q, n_p+q−1,… in this expression are zero. The term N cannot vanish identically, because we showed earlier that N(1) is real and positive. Consequently there must be a smallest nonnegative integer v for which n_p+q−v is nonzero; thus v is the largest integer such that τ^p+q−vN(τ) is a polynomial. Define P(τ) ≡ τ^p+q−vN(τ). Then P is of degree 2(p + q − v) and P(0) = n_p+q−v ≠ 0. Because N(τ) = N(1/τ) for τ ≠ 0, it follows that if r is any root of P, then so is its reciprocal 1/r. (For this reason, polynomials such as P are known as reciprocal polynomials. They are also known as palindromic polynomials, because the coefficient of τ^j equals the coefficient of τ^{2(p+q−v)−j}.) The polynomial P must have equal numbers of roots inside and outside Ω and no root at 0. It cannot have roots on Ω, for if there were such a root r, the property P(r) = 0 = r^p+q−vN(r) would imply, because r ≠ 0, N(r) = 0. An r on Ω would satisfy 1/r = r; hence (6) would yield

Because both terms would be nonnegative, they would both have to be zero. In particular the first term would yield |Φ(r)|²|Θ(r)|² = 0, implying that either Φ(r) = 0 or Θ(r) = 0. Thus either Φ or Θ would have a root on Ω, contradicting our assumption that they are both spectral. Thus the roots of P can be partitioned into the p + q − v that lie outside Ω, which we call r₁,…, r_p+q−v (unless v = p + q, when P is a constant), and the p + q − v inside, namely, 1/r₁,…,1/r_p+q−v. Define

(or B₁ ≡ 1 if p + q = v: we adopt the usual convention that empty products have the value 1) so that B₁ is standardized and spectral. It follows that

where

, and so N(τ) = κB₁(τ)B₁(1/τ) as required, and κ ≠ 0.

Next we show that the polynomial B₁ has real coefficients. From (6), N(τ) = Φ(τ)Φ(1/τ)Θ(1/τ)Θ(τ) + Φ(1/τ)Φ(τ)Θ(τ)Θ(1/τ) = N(τ) for all τ ≠ 0, implying that N, and therefore P, has real coefficients. A polynomial with real coefficients can be expressed as a product of linear and quadratic factors, all with real coefficients. Let r_j be a root of B₁ and therefore of P. If it is the root of one of the linear factors of P, it must be real. If it is a root of one of the quadratic factors, it must either be real or can be paired, if it is nonreal, with its complex conjugate r_j, which has the same magnitude. Because this magnitude exceeds 1, r_j must also be a root of B₁. Thus all roots of B₁ are either real or come paired with their complex conjugates in the definition of B₁. It follows that B₁ is, like P, a product of linear and quadratic factors, each with real coefficients, and therefore that B₁ has real coefficients. To evaluate κ note that for τ = 1 we obtain κ = N(1)/B₁(1)², which we now know to be well defined because B₁(1) is real and nonzero. Because N(1) is real and positive, it follows that κ is also real and positive. Because from (4) we require

we obtain

for the innovation variance of the ARMA process.

Now we discuss the possibility of root cancellation. First note that this can happen even though Φ and Θ are coprime; for instance we saw that it does so in the SCM if φ is real. In fact (returning to the general case) the c common roots r₁,…, r_c of A₁ and B₁ are the common roots of N and D exceeding 1 in magnitude, and these in turn are just the common roots of Φ and Φ. To prove this latter statement, note first that if r is a common root of Φ and Φ then |r| > 1 and r is a root of both D and N, as we can see from (5) and (6). Conversely, if |r| > 1 and r is a root of both D and N, then it is a root of Φ or Φ (not of Φ(1/τ)Φ(1/τ), because of its magnitude). If it is a root of Φ then from (6) it is a root of the first term of N and hence must be a root of the second term. It cannot be a root of Θ(1/τ)Φ(1/τ), because of its magnitude, or of Θ(τ), because this violates the assumption that Φ and Θ are coprime. Hence it must be a root of Φ also. Similarly, if it is a root of Φ it must be a root of Φ.

Using these c common roots, define

and B(τ) = B₁(τ)/C(τ). Because each root of C is a root of both Φ and Φ, either it must be real or it can be paired with another root that is its complex conjugate. Consequently the polynomial C has real coefficients. The polynomials A and B inherit from A₁ and B₁ the properties of being standardized, spectral, and coprime, and they have real coefficients because A₁, B₁, and C have real coefficients. We have shown that the spectrum of Re Z_t is that of the normal ARMA(2p − c,p + q − v − c) process A(L)X_t = B(L)ζ_t, where ζ_t is real white noise with variance σ′². This completes the proof of Theorem 1. █

5. POSSIBLE APPLICATIONS

The CARMA-Re CARMA-ARMA framework suggests a natural generalization of the stochastic cycle approach to situations where there is more than one cycle. If, for instance, the sample spectrum showed the presence of two distinct cycles with known frequencies ω₁ and ω₂, one could try to model the process in question as the real part of Z_t = W_1,t + W_2,t, where W_1,t and W_2,t are independent and follow CAR(1) processes, with autoregressive parameters ρ₁ e^iω₁ and ρ₂ e^iω₂, respectively. According to the results of Granger and Morris (1976), which seem to extend straightforwardly to CARMA processes, Z_t would then be CARMA(2,1). Hence according to Theorem 1, the real-part process X_t would except in degenerate cases be ARMA(4,3). More generally, a sum of s cycles would be ARMA(2s,2s − 1); however if the ω_j were known we would need to estimate only the s moduli ρ_j and the s innovation variances of the component cycles.

In cases where cycles form only part of the model, and where real ARMA models and their usual extensions are used to capture the noncyclical components, Theorem 1 seems to suggest as one possibility setting up Re CARMA-type models in which some but not all complex parameters are restricted to be real. As the theorem states, this can be achieved by placing restrictions on the AR polynomial Φ in the form of assigning Φ some real roots or some conjugate pairs of complex roots. The consideration of such an approach to stationary and nonstationary time series modeling is outside the scope of this note.