Appendix

Since the empirical data are quarterly, the seasonal component of each time series, s_i, is the sum of two cyclical components, s_it=s_i1t+s_i2t, which are evaluated at the seasonal frequencies, λ₁=π/2, and λ₂=π. According to (4), the dynamics of the first cyclical component is

(A1) ( s i 1 t s i 1 t * ) = ( cos π / 2 sin π / 2 − sin π / 2 cos π / 2 ) ( s i 2 t − 1 s i 2 t − 1 * ) + ( ξ i 1 t ξ i 1 t * ) .  (A1)

Using cosπ/2=0 and sinπ/2=1 and rearranging terms, one can obtains

(A2) ( s i 1 t s i 1 t * ) = 1 1 + L 2 ( 1 L − L 1 ) ( ξ i 1 t ξ i 1 t * ) ,  (A2)

which implies that (1+L²)s_i1t=ζ_i1t, where ζi1t=ξi1t+ξi1t−1*. Ifvar(ξi1t)=var(ξi1t*)=σξi12, ∀t, then var(ζi1t)=2σξi12.

Similarly, the dynamics of the second cyclical component can be obtained from

(A3) ( s i 2 t s i 2 t * ) = ( cos π sin π − sin π cos π ) ( s i 2 t − 1 s i 2 t − 1 * ) + ( ξ i 2 t ξ i 2 t * ) ,  (A3)

which, using cosπ=–1 and sinπ=0, leads to

(A4) ( s i 2 t s i 2 t * ) = 1 ( 1 + L ) 2 ( 1 + L 0 0 1 + L ) ( ξ i 2 t ξ i 2 t * ) .  (A4)

This expression implies that (1+L)s_i2t=ζ_i2t, where ζ_i2t=ξ_i2t and var(ζi2t)=σξi22. Let us additionally assume that σξi12=σξi22=σξi2.

Accordingly, the seasonal component of each time series can be expressed as

(A5) s i t = ζ i 1 t ( 1 + L ) + ζ i 2 t ( 1 + L 2 ) ( 1 + L 2 ) ( 1 + L ) ,  (A5)

or

(A6) ( 1 + L + L 2 + L 3 ) s i t = ζ i 1 t ( 1 + L ) + ζ i 2 t ( 1 + L 2 ) .  (A6)

Since the greatest polynomial of the two terms from the right-hand side is of power two, the resulting polynomial (the result of summation) is of power two as well

(A7) ( 1 + L + L 2 + L 3 ) s i t = ( 1 + α L + β L 2 ) ζ i t .  (A7)

To find the unknown coefficients α and β, we derive the spectra of right-hand sides of both expressions. On the one hand, the spectrum of right-hand side of (A6) is

(A8) ( 1 + e i ω ) ( 1 + e − i ω ) σ ζ i 1 2 + ( 1 + e 2 i ω ) ( 1 + e − 2 i ω ) σ ζ i 2 2 = ( 2 + 2 cos ω ) σ ζ i 1 2 + ( 2 + 2 cos 2 ω ) σ ζ i 2 2 = ( 6 + 4 cos ω + 2 cos 2 ω ) σ ξ i 2 .  (A8)

The first equality follows from the fact that e^iλ+e^–iλ=2 cos λ, ∀λ. The last equation uses σζi12=2σξi2 and σζi22=σξi2. On the other hand, the spectrum of right-hand side of (A7) is

(A9) ( 1 + α e i ω + β e 2 i ω ) ( 1 + α e − i ω + β e − 2 i ω ) σ ζ i 2 = ( ( 1 + α 2 + β 2 ) + 2 ( α + α β ) cos ω + 2 β cos 2 ω ) σ ζ i 2 .  (A9)

Since the two spectra must represent the same dynamics, one can use the system of three equations with three unknowns 6σξi2=(1+α2+β2)σζi2,4σξi2=2(α+αβ)σζi2 and 2σξi2=2βσζi2 to obtain α⁴–4α³+12α²–16α+4=0. The real solutions of this equation are α₁=0.3187 and α₂=1.6813, and using again the system of equations, it is easy to obtain that they correspond to values β₁=0.1869 and β₂=5.2745. The first pair of solutions (α=0.3187, β=0.1869) produces invertible MA polynomial in (A7), opposite, the second pair of solutions results in non-invertible (A7). Using the first pair of real solutions we find σζi2=5.3505σξi2 from the last equation of the system. In this way, the seasonal component for the series i is given by:

( 1 + L + L 2 + L 3 ) s i t = ( 1 + 0.3187 L + 0.1869 L 2 ) ζ i t .