Analysis of Models with Complex Roots on the Unit Circle

J. Japan Statist. Soc. Vol. 38 No. 1 2008 145–155

ANALYSIS OF MODELS WITH COMPLEX ROOTS ON THE UNIT CIRCLE

Katsuto Tanaka*

This paper deals with nonstationary autoregressive (AR) models with complex roots on the unit circle. We examine the asymptotic properties of the least squares estimators (LSEs) in the model. We also extend the model to the case where the error term follows a stationary linear process. We show that the limiting distribution of the LSE of the unit root parameter has a property comparable to that of the LSE in the standard nonstationary seasonal model with period two. Percent points and moments of the limiting distribution are computed by numerical integration.

Key words and phrases: Complex B-N decomposition, complex roots, limiting distribution, numerical integration, unit root.

1. Introduction This paper deals with nonstationary AR models with two complex conjugate roots on the unit circle. Thus the model has the AR ﬁlter of the form (1−eiθL)(1− e−iθL), where L is the lag operator and θ (0 <θ<π)is a dominant frequency. The simplest model of this type is the AR(2). This was dealt with by Ahtola and Tiao (1987), where the asymptotic distributions of the LSEs were derived. Chan and Wei (1988)obtained the asymptotic distributions in a much simpler way. The purpose of the present paper is to extend the above AR(2)model by allowing the error term to follow a stationary process, and to discuss the asymptotic properties of the LSEs. It seems that there is no theoretical result for that extension. One of the advantages of the present model is that it can deal with any frequency that is dominant in the nonstationary series. In practice, we need to identify an appropriate model for explaining the frequency contribution. For that purpose Akaike’s information criterion (Akaike (1973)) may be useful. In Section 2 we present a basic model, and the model is extended in Section 3. In both sections we derive the asymptotic distributions of the LSEs to ﬁnd a similarity to the LSE in the standard seasonal model. Section 4 is concerned with numerical computation of the limiting distributions, and Section 5 concludes this paper.

2. Simple case In this section we consider a nonstationary AR(2)model with two complex conjugate roots on the unit circle, which is given by iθ −iθ (2.1) (1 − e L)(1 − e L)yj = εj ⇔ yj = φ1yj−1 + φ2yj−2 + εj, Accepted November 13, 2007. *Department of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan. 146 KATSUTO TANAKA where φ1 = 2 cos θ and φ2 = −1. The parameter θ is restricted to be 0 <θ<π, and gives a frequency contribution to the model with period 2π/θ. To generate the process we assume that y−1 = y0 = 0. It is also assumed in this section that 2 {εj}∼i.i.d.(0,σ ). The model in (2.1)was dealt with earlier in Ahtola and Tiao (1987)and Chan and Wei (1988), where the asymptotic properties of the LSEs of φ1 and φ2 were studied. In particular, the limiting expressions for the normalized estimators in the sense of weak convergence were obtained. The former also gives percent points of the limiting distributions as well as those of finite sample distributions by simulations, while the latter gives limiting expressions which are much neater than the former. Since, in the next section, we extend the model to the case where the error process {εj} follows a general stationary process by employing the basic asymptotic arguments associated with the present model, we briefly summarize the arguments here. For this purpose we first obtain ε 1 eiθ e−iθ y = j = − ε j (1 − eiθL)(1 − e−iθL) 2i sin θ 1 − eiθL 1 − e−iθL j

1 iθ i(j−1)θ i(j−2)θ iθ = [e {e ε + e ε + ···+ e ε − + ε } 2i sin θ 1 2 j 1 j −iθ −i(j−1)θ −i(j−2)θ −iθ −e {e ε1 + e ε2 + ···+ e εj−1 + εj}] 1 j ei(j−k+1)θ − e−i(j−k+1)θ 1 j = ε = ε sin(j − k +1)θ sin θ 2i k sin θ k k=1 k=1 1 1 = [x(θ) sin(j +1)θ − y(θ) cos(j +1)θ]= (a(θ)) z (θ), sin θ j j sin θ j j where (θ) j x cos kθ sin(j +1)θ z (θ) = j = ε , a(θ) = . j (θ) sin kθ k j − cos(j +1)θ yj k=1

{ (θ)} It is noticed that zj is a two-dimensional random walk expressed by cos jθ (2.2) z (θ) = z (θ) + v (θ), v (θ) = ε , z (θ) = 0, j j−1 j j sin jθ j 0

{ (θ)} where the error term vj is a sequence of martingale diﬀerences that are independent with heterogeneous variances. { (θ)} ≤ ≤ On the basis of the random walk zj , let us construct, for 0 t 1, the two-dimensional partial sum process   (θ) √ 1 v (2.3) Z (θ)(t)= 2 √ z (θ) +(Tt− [Tt]) √[Tt]+1  , T Tσ [Tt] Tσ COMPLEX ROOTSON THE UNIT CIRCLE 147

(θ) { (θ) } where [x] is the greatest integer not exceeding x. Note that ZT = ZT (t) is a stochastic process in a function space C2 = C[0, 1] × C[0, 1], where C[0, 1] is the space of all real-valued continuous functions deﬁned on [0, 1]. Then the following functional central limit theorem (FCLT)holds (Helland (1982),Chan and Wei (1988)). Theorem 1. { (θ) } For the two-dimensional partial sum process ZT (t) in { (θ)} (2.3) constructed from the random walk zj in (2.2), the following FCLT holds: (θ) ⇒ ZT W , where W = {W (t)} is the two-dimensional standard Brownian motion. More- over the following joint weak convergence of basic quantities holds: 1 1 (2.4) (QT ,RT ) ⇒ W (t)W (t)dt, W (t)dW (t) , 0 0 where T T 2 1 j j Q = z (θ)(z (θ)) = Z (θ) Z (θ) , T T 2σ2 j j T T T T T j=1 j=1 T T 2 j − 1 j R = z (θ) (z (θ) − z (θ) ) = Z (θ) ∆Z (θ) , T Tσ2 j−1 j j−1 T T T T j=1 j=1 (θ) (θ) − (θ) − ∆ZT (j/T)=ZT (j/T) ZT ((j 1)/T ).

Let us return to the model (2.1)and put φ =(φ1,φ2) = (2 cos θ, −1) . Let us denote the LSE of φ based on a sample of size T by φˆ =(φˆ1, φˆ2) . Then we ˆ − −1 have T (φ φ)=AT bT , where T 2 T 1 yj−1 yj−1yj−2 1 yj−1εj (2.5) AT = 2 , bT = . T 2σ2 y − y − y − Tσ2 y − ε j=3 j 1 j 2 j 2 j=3 j 2 j

Let us ﬁrst consider the second moment of {yj}, which is

1 T 1 T y2 = [x(θ) sin(j +1)θ − y(θ) cos(j +1)θ]2 T 2 j 2 2 j j j=1 T sin θ j=1 σ2 T j j 1 = Z (θ) Z (θ) − 2 T T T T 2 2 4T sin θ j=1 2T sin θ T × (θ) 2 − (θ) 2 (θ) (θ) [((xj ) (yj ) )cos 2( j +1)θ +2xj yj sin 2(j +1)θ]. j=1 Since it follows from Chan and Wei (1988)that j ikθ 2 iθ (2.6) sup e wk = op(T ), (e =1) , ≤ ≤ 1 j T k=1 148 KATSUTO TANAKA

(θ) 2 (θ) 2 (θ) (θ) where wk is either (xk ) or (yk ) or xk yk , it holds that T 1 1 1 y2 ⇒ W (t)W (t)dt. T 2σ2 j 2 j=1 4 sin θ 0

Similarly, we have

T T 1 (θ) (θ) − − yj 1yj = 2 [xj−1 sin jθ yj−1 cos jθ] j=2 sin θ j=2 × (θ) − (θ) [xj sin(j +1)θ yj cos(j +1)θ] T 1 (θ) (θ) (θ) (θ) (θ) (θ) − (θ) (θ) = 2 [(xj−1xj + yj−1yj )cos θ +(xj−1yj xj yj−1)sin θ 2 sin θ j=2 − (θ) (θ) − (θ) (θ) (xj−1xj yj−1yj )cos(2 j +1)θ − (θ) (θ) (θ) (θ) (xj−1yj + xj yj−1)sin(2 j +1)θ] Tσ2 cos θ T j j = Z (θ) Z (θ) + o (T 2), 2 T T T T p 4 sin θ j=1 which yields T 1 1 cos θ y − y ⇒ W (t)W (t)dt. T 2σ2 j 1 j 2 j=2 4 sin θ 0 Therefore we have 1 0 W (t)W (t)dt 1 cos θ (2.7) AT ⇒ . 4 sin2 θ cos θ 1 We also have, for h>0,

T T 1 (θ) (θ) y − ε = [x sin(j − h +1)θ − y cos(j − h +1)θ]ε j h j sin θ j−h j−h j j=h+1 j=2 T 1 = (z (θ) ) J (θ)∆z (θ) sin θ j−h h j j=h+1 T 1 = (z (θ) ) J (θ)∆z (θ) + o (T ) sin θ j−1 h j p j=h+1 Tσ2 T j − 1 j = Z (θ) J (θ)∆Z (θ) + o (T ), 2 sin θ T T h T T p j=h+1 where − sin(h − 1)θ cos(h − 1)θ (2.8) J (θ)= . h − cos(h − 1)θ − sin(h − 1)θ COMPLEX ROOTSON THE UNIT CIRCLE 149

Therefore we have 1 1 W (t)J1(θ) (2.9) bT ⇒ dW (t). 2 sin θ 0 W (t)J2(θ) It follows from Theorem 1 and the continuous mapping theorem (CMT)that (2.7)and (2.9)lead us to ˆ − −1 ⇒ Z1 (2.10) T (φ φ)=AT bT , Z2 where 1 1 cos θ sin θ (2.11) Z =2 W (t) dW (t) W (t)W (t)dt, 1 − sin θ cos θ 0 0 1 1 (2.12) Z2 = −2 W (t)dW (t) W (t)W (t)dt. 0 0

It is noticed that Z2, which is the limit in distribution of T (φˆ2 −φ2), does not depend on the seasonal frequency θ, although Z1, which is the limit in distribution of T (φˆ1 − φ1), does. We note in passing that the limiting distribution of T (φˆ2 − φ2)also appears when we deal with a standard nonstationary seasonal model with period m:

(2.13) yj = ρyj−m + εj, (j =1,...,T), where the true value of ρ is unity, and the initial values of yj (j =1− m,...,0) are all set at 0. Letρ ˆ be the LSE of ρ. Then it holds (Tanaka (1996)) that 1 1 N(ˆρ − 1) ⇒ Z = B (t)dB(t) B (t)B(t)dt, 0 0 where N =[T/m] and {B(t)} is the m-dimensional standard Brownian motion. Then it is seen that the distribution of −2Z is the same as that of Z2 when m =2. We shall show in the next section that this last property continues to hold when the present model is extended to the case where the error term {εj} be- comes a stationary process. We shall also indicate in Section 4 how to compute numerically percent points of Z1 and Z2 together with their moments.

3. Extended case In this section we extend the model (2.1)to the following case:

yj = φ1yj−1 + φ2yj−2 + uj, (3.1) ∞ ∞ uj = α(L)εj = αlεj−l, l|αl| < ∞, l=0 l=1

2 where φ1 = 2 cos θ, φ2 = −1, and {εj}∼i.i.d.(0,σ ). 150 KATSUTO TANAKA

The error term {uj} is now not independent, but follows a stationary linear process. To study the asymptotic properties of the LSEs of φ1 and φ2 in the present case, it is convenient to decompose {uj} in the following way:

iθ −iθ (θ) (3.2) uj = α(L)εj =[α(e ) − (1 − e L)˜α (L)]εj iθ −iθ (θ) − (θ) = α(e )εj + e ε˜j−1 ε˜j , where ∞ ∞ (θ) (θ) (θ) (θ) i(k−l)θ ε˜j =˜α (L)εj = α˜l εj−l, α˜l = αke . l=0 k=l+1 Note here that it follows from the last condition in (3.1)that ∞ ∞ ∞ ∞ ∞ ∞ (θ) − (3.3) |α˜ | = α ei(k l)θ ≤ |α | = l|α | < ∞, l k k l l=0 l=0 k=l+1 l=0 k=l+1 l=1

{ (θ)} which implies that the process ε˜j is a complex-valued stationary process. The expression in (3.2)for {uj} is based on an expansion of α(L)around L = eiθ. This expansion is called the complex B-N decomposition (Phillips and Solo (1992)). Usefulness of this decomposition can be seen when dealing with the present model (3.1)with complex unit roots. To see this, consider u 1 y = j = [˜x(θ) sin(j +1)θ − y˜(θ) cos(j +1)θ] j (1 − eiθL)(1 − e−iθL) sin θ j j 1 = (a(θ)) z˜(θ), sin θ j j where (θ) j x˜ cos lθ sin(j +1)θ z˜(θ) = j = u , a(θ) = . j (θ) sin lθ l j − cos(j +1)θ y˜j l=1

(θ) Applying the complex B-N decomposition to ul in the expression for z˜j ,we obtain (θ) (θ) x˜ a(θ) −b(θ) x (3.4) z˜(θ) = j = j + w (θ) = K(θ)z (θ) + w (θ), j (θ) b(θ) a(θ) (θ) j j j y˜j yj where a(θ)= Re[ α(eiθ)], b(θ)= Im[ α(eiθ)] and (θ) (θ) a(θ) −b(θ) Re[˜ε − eijθε˜ ] K(θ)= , w (θ) = 0 j . b(θ) a(θ) j (θ) − ijθ (θ) Im[˜ε0 e ε˜j ]

It is noticed that the quantity z˜(θ) is made up of two terms: one is K(θ)z (θ) which √ j j (θ) (θ) is Op( j), and the other is wj which is Op(1). Thus z˜j which reﬂects the (θ) stationarity assumption on the error term is approximately equal to K(θ)zj , COMPLEX ROOTSON THE UNIT CIRCLE 151

(θ) where zj is the quantity based on the independence assumption on the error term. For later purposes we also deﬁne the spectrum of {uj} by f(ω), which may be expressed as ∞ 2 σ2 σ2 σ2 (3.5) f(ω)= α eilω = (a2(ω)+b2(ω)) = |K(ω)|. 2π l 2π 2π l=0 The above discussion leads us to derive T T 2 2 (3.6) z˜(θ)(z˜(θ)) = K(θ) z (θ)(z (θ)) K (θ)+o (1) T 2σ2 j j T 2σ2 j j p j=1 j=1 T 1 j j = K(θ) Z (θ) Z (θ) K (θ)+o (1) T T T p j=1 1 ⇒ K(θ) W (t)W (t)dtK (θ), 0 T T 2 2 (3.7) (z˜(θ)) z˜(θ) = [(˜x(θ))2 +(˜y(θ))2] T 2σ2 j j T 2σ2 j j j=1 j=1 1 ⇒ tr K(θ) W (t)W (t)dtK (θ) 0 1 2πf(θ) = 2 W (t)W (t)dt. σ 0 T 2 Moreover let us consider RT (h)= j=h+1 yj−hyj/T for a nonnegative integer h. Then we have

1 T R (h)= [˜x(θ) sin(j − h +1)θ − y˜(θ) cos(j − h +1)θ] T 2 2 j−h j−h T sin θ j=1 × (θ) − (θ) [˜xj sin(j +1)θ y˜j cos(j +1)θ] 1 T = [(˜x(θ) x˜(θ) +˜y(θ) y˜(θ))cos hθ 2 2 j−h j j−h j 2T sin θ j=h+1 (θ) (θ) − (θ) (θ) +(˜xj−hy˜j x˜j y˜j−h)sin hθ − (θ) (θ) (θ) (θ) − +( x˜j−hx˜j +˜yj−hy˜j )cos(2 j h +2)θ − (θ) (θ) − (θ) (θ) − +( x˜j−hy˜j x˜j y˜j−h)sin(2 j h +2)θ] cos hθ T = [(˜x(θ))2 +(˜y(θ))2]+o (1). 2 2 j j p 2T sin θ j=h+1 Thus it follows from (3.7)that 1 ⇒ πf(θ)cos hθ RT (h) R(h)= 2 W (t)W (t)dt. 2 sin θ 0 152 KATSUTO TANAKA

As for a quantity related to the Ito integral, we consider T T 1 1 S = z˜(θ) (∆z˜(θ)) = (K(θ)z (θ) + w (θ) )(K(θ)∆z (θ) +∆w (θ)) . T T j−1 j T j−1 j−1 j j j=2 j=2 After some algebra we arrive at  T T 1 S = K(θ) z (θ) (∆z (θ)) K (θ) − K(θ) ∆z (θ)(w (θ)) T T j−1 j j j j=2 j=2  T T (θ) (θ) (θ) (θ)  + wj−1(∆zj ) K (θ)+ wj−1(∆wj ) + op(1), j=2 j=2 which yields   1 ∞ 1 (3.8) S ⇒ S = σ2K(θ) W (t)dW (t)K (θ)+ γ(j)P (θ) , T 2 j 0 j=1 where γ(j)=E(u0uj), and Pj(θ)is the orthogonal matrix defined by cos jθ sin jθ P (θ)= . j − sin jθ cos jθ Using the above result we can discuss the weak convergence of the following quantity for h>0: T T 1 1 (θ) (θ) sin(j − h +1)θ S = y − u = x˜ y˜ u . hT T j h j T sin θ j−h j−h − cos(j − h +1)θ j j=h+1 j=h+1 After some algebra we obtain 1 T ∆˜y(θ) cos(h − 1)θ − ∆˜x(θ) sin(h − 1)θ S = x˜(θ) y˜(θ) j j hT T sin θ j−h j−h − (θ) − − (θ) − j=h+1 ∆˜xj cos(h 1)θ ∆˜yj sin(h 1)θ 1 T = (˜x(θ) ∆˜y(θ) − y˜(θ) ∆˜x(θ))cos( h − 1)θ T sin θ j−1 j j−1 j j=h+1 (θ) (θ) (θ) (θ) − (˜x − ∆˜x +˜y − ∆˜y )sin( h − 1)θ j 1 j j 1 j h−2 + uj uj+l+1−h sin lθ l=1 1 ⇒ {S(1, 2) − S(2, 1)} cos(h − 1)θ −{S(1, 1)+ S(2, 2)} sin(h − 1)θ sin θ h−2 + γ(h − l − 1)sin lθ  l=1  1 ∞ 1 = S = πf(θ) W (t)J (θ)dW (t)+ γ(j)sin( j − h +1)θ , h sin θ h 0 j=h COMPLEX ROOTSON THE UNIT CIRCLE 153 where S(i, j)is the ( i, j)-th element of S defined in (3.8), whereas Jh(θ)is defined in (2.8). Denoting by φ˜ the LSE of φ in the present model, we can now establish that     −1 T 2 T ˜ −  1 yj−1 yj−1yj−2   1 yj−1uj  T (φ φ)= 2 T 2σ2 y − y − y − Tσ2 y − u j=3 j 1 j 2 j 2 j=3 j 2 j − R(0) R(1) 1 S Z˜ ⇒ 1 = 1 , R(1) R(0) S2 Z˜2 where cos θ sin θ ∞ 2[πf(θ) 1 W (t) dW (t)+ sin θ γ(j)cos( j − 1)θ] 0 − sin θ cos θ j=1 Z˜1 = , πf(θ) 1 W (t)W (t)dt 0 −2[ 1 W (t)dW (t)+1− γ(0)/(2πf(θ))] ˜ 0 Z2 = 1 . 0 W (t)W (t)dt

It is seen that the limiting distribution of T (φ˜1−φ1)depends on the frequency θ, as in the independent error case. Moreover, that of T (φ˜2 − φ2)now depends on θ. This is quite comparable to the situation where we deal with the LSE in the following seasonal model with period m: ∞ ∞ (3.9) yj = ρyj−m + uj,uj = αlmεj−lm, l|αlm| < ∞, l=0 l=1 where the true value of ρ is unity. Letρ ˜ be the LSE of ρ based on a sample of size T . Then it is shown (Tanaka (1996)) that 1 m{ − } 0 B (t)dB(t)+ 1 γ(0)/(2πf(0)) − ⇒ ˜ 2 (3.10) N(˜ρ 1) Z = 1 , 0 B (t)B(t)dt where N =[T/m] and {B(t)} is the m-dimensional standard Brownian motion. It is seen that the distribution of −2Z˜ reduces to that of Z˜2 when m = 2 and θ =0.

4. Percent points and moments The limiting distribution can be computed by numerical integration in the following way. Let us consider Z˜ in (3.10), which is of the form 1 B (t)dB(t)+am/2 U X(a, m)= 0 = . 1 V 0 B (t)B(t)dt Then we compute 1 1 ∞ 1 P (X(a, m) ≤ x)=P (xV − U ≥ 0)= + Im[φ(u; x)]du, 2 π 0 u 154 KATSUTO TANAKA where φ(u; x)is the characteristic function of xV − U given by √ √ −m/2 − − sin 2iux φ(u; x)= e iu(1 a) cos 2iux + iu √ . 2iux

Care needs to be taken when computing square roots of complex-valued quantities, as is described in Tanaka (1996). Moments of X(a, m)can also be computed. We have ∞ j 1 − ∂ ψ(u , −u ) j j 1 1 2 E(X (a, m)) = u2 j du2, Γ(j) 0 ∂u 1 u1=0 where ψ(u1,u2)is the joint moment generating function of U and V with √ √ −m/2 u1(1−a) sinh 2u2 ψ(u1, −u2)= e cosh 2u2 − u1 √ . 2u2

On the basis of these results, we can compute percent points and moments of the limiting distributions of the LSEs discussed in this paper. Table 1 gives those of Z2 = −2X(0, 2)deﬁned in (2.12).

Table 1. Percent points and some moments of Z2 in (2.12).

0.01 0.05 0.1 0.5 0.9 0.95 0.99 Mean SD −2.956 −2.028 −1.539 0.775 6.010 8.389 14.115 1.664 3.460

5. Concluding remarks We have discussed the asymptotic properties of the LSEs in extended AR(2) models with two complex conjugate unit roots e±iθ. It was shown that the limiting distribution associated with the complex unit roots is closely related to that of the LSE in standard nonstationary seasonal models with period two. The analysis may also be extended to cases of multiple complex unit roots exp{±iθj} (j =1,...,k), and to cases where the error term follows a stationary long- memory process. Testing for the existence of complex unit roots has been discussed in Hylleberg et al. (1990)for standard nonstationary seasonal models. Their ap- proach rewrites the AR model representation in a convenient way, assuming the seasonal unit roots are known. In the present case, however, the frequency θ is supposed to be unknown. Thus it must be estimated before testing, but their idea may be useful.

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