Generalized fractional processes with long memory and time dependent volatility revisited

Peiris, M. Shelton; Asai, Manabu Article Generalized fractional processes with long memory and time dependent volatility revisited Econometrics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Peiris, M. Shelton; Asai, Manabu (2016) : Generalized fractional processes with long memory and time dependent volatility revisited, Econometrics, ISSN 2225-1146, MDPI, Basel, Vol. 4, Iss. 3, pp. 1-21, https://doi.org/10.3390/econometrics4030037 This Version is available at: https://hdl.handle.net/10419/171887 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Article Generalized Fractional Processes with Long Memory and Time Dependent Volatility Revisited M. Shelton Peiris 1,∗and Manabu Asai 2 1School of Mathematics and Statistics, The University of Sydney, Sydney 2006, Australia 2Faculty of Economics, Soka University, Tokyo 192-8577, Japan; [email protected] *Correspondence: [email protected]; Tel.: +612-9351-5764 Academic Editor: Kerry Patterson Received: 15 January 2016; Accepted: 15 August 2016; Published: 5 September 2016 Abstract: In recent years, fractionally-differenced processes have received a great deal of attention due to their flexibility in financial applications with long-memory. This paper revisits the class of generalized fractionally-differenced processes generated by Gegenbauer polynomials and the ARMA structure (GARMA) with both the long-memory and time-dependent innovation variance. We establish the existence and uniqueness of second-order solutions. We also extend this family with innovations to follow GARCH and stochastic volatility (SV). Under certain regularity conditions, we give asymptotic results for the approximate maximum likelihood estimator for the GARMA-GARCH model. We discuss a Monte Carlo likelihood method for the GARMA-SV model and investigate finite sample properties via Monte Carlo experiments. Finally, we illustrate the usefulness of this approach using monthly inflation rates for France, Japan and the United States. Keywords: GARMA; GARCH; stochastic volatility; long-memory; fractional differencing JEL Classification: C18, C40, C58 1. Introduction Consider the well-known ARFIMA(p,d,q)model given by: φ(B)Yt=θ(B)et, (1) where Yt= ( 1 −B)dXt , d∈(− 1, 0.5 ) , {et} is a sequence of uncorrelated (not necessarily independent) random variables, such that Var(et) = σ2 , and φ(B) (stationary AR( p )) and θ(B) (invertible MA( q )) polynomials respectively. This standard case of constant variance innovations has been considered in many traditional time series analysis with applications. However, in recent years, there has been a great number of developments based on time-dependent instantaneous innovation variance (or volatility) such that Var(et) = σ2 t. In particular, the following cases have been considered: (i) σ2 tis a deterministic function of tor σ2 t=f(t), (ii) etfollows the family of (G)ARCH process (see, [1,2]), (iii) log(σ2 t)is another stochastic process. These cases (i) to (iii) can be analysed with emphasis on different practical issues. However, in applications, we need additional assumptions on σ2 t, such as: (a) 0 <m<σ2 t<M<∞to ensure Var(Yt) is finite, in (i) (b) stationarity or stability of both {e2 t}and log({e2 t})in (ii) and (iii) Econometrics 2016,4, 37; doi:10.3390/econometrics4030037 www.mdpi.com/journal/econometrics Econometrics 2016,4, 37 2 of 21 Assumption (a) is imposed as it is natural to set bounds for the deterministic function σ2 t=f(t). In fact, Assumption (b) is required when σ2 t is a stochastic process. For case (i), the effect of non-stochastic and time-dependent instantaneous variance when d= 0 was studied by Niemi [ 3 ] under the standard AR and MA regularity conditions on the zeros of φ(B) and θ(B) , respectively. In his work, Peiris [ 4 ] argues that the results of Niemi [ 3 ] can be extended to the ARFIMA family when d∈(− 0.5, 0.5 ) . In (ii), it is known that Xt is strictly stationary if et is strictly stationary, and in particular, Xt is 2 m -th order stationary if et is 2 m -th order stationary. For the Integrated GARCH (IGARCH) model, Nelson [ 5 ] and Bougerol and Picard [ 6 ] have argued that IGARCH is strictly stationary under additional regularity conditions. For case (iii) with stochastic volatility, it is obvious that if log σ2 t is m -th order stationary, then et is 2 m -th order stationary. Therefore, it can be argued that when σ2 t is a stationary stochastic process, the conditional likelihood estimation carried out by taking σ2 t=σ2 cannot be significant since σ2 t is bounded. This would be useful in the estimation of parameters, especially in both cases (ii) and (iii). See the survey papers of McAleer [ 7 ] and Shephard [ 8 ] for the various extensions of (ii) and (iii). An alternative way of modelling time-dependent volatilities has been extensively studied using ARMA models with time-dependent coefficients driven by constant variance innovations. See, for example, [ 4 , 9 – 13 ] and the references therein for details. However, this approach is not very attractive in applications, as it involves too many parameters to estimate. Turning to applications in economic and financial time series, there are many popular directions of modelling and analysis of long-memory. Among others, the analysis of long-memory in inflation has been considered by Backus and Zin [ 14 ], Hassler and Wolters [ 15 ], Baillie, Chung and Tieslau [ 16 ], and Caporale and Gil-Alana [ 17 ]. In their paper, Delgado and Robinson [ 18 ] considered a number of methods for the analysis of long-memory time series using ARFIMA. An alternative and a general approach is to use the ARFIMA family with conditional and stochastic volatility as considered by Baillie et al. [ 19 ], Bollerslev and Mikkelsen [ 20 ], Ling and Li [ 21 ], Breidt et al. [ 22 ], Deo and Hurvich [23] , and Bos et al. [ 24 ]. In their recent paper, Bos et al. [ 24 ] accommodate the stochastic volatility in ARFIMA modelling. Empirical evidence confirms that such models are very satisfactory in practice. Therefore, the aim of this paper is to extend the ARFIMA models with time-varying volatility to a general flexible class of time series models based on Gegenbauer polynomials together with the ARMA structure. The Gegenbauer ARMA (GARMA) model is a generalization of the ARFIMA model. Clearly, the former encompasses the latter as a special case. We will also extend the class of k -factor Gegenbauer process following Woodward et al. [ 25 ], Ferrara and Gueganand [ 26 ], and Caporale and Gil-Alana [ 17 ], by accommodating time-dependent volatility. The organization of the paper is as follows. Section 2reviews the family of GARMA with constant variance (volatility). Section 3shows the existence and uniqueness of second-order solutions for the GARMA model with time-dependent volatility and develops new classes of GARMA-GARCH and GARMA-SV . Section 4presents the asymptotic results for the maximum likelihood estimator for the GARMA-GARCH model and reports a Monte Carlo likelihood method for estimating the GARMA-SV model. Section 5presents an illustrative example via simulation data, while Section 6demonstrate an empirical example using inflation data in France, Japan and the United States. Section 7gives concluding remarks. 2. Basic Results on GARMA with Constant Volatility In this section, we review the family of GARMA processes with constant volatility. Based on the work of Gray et al. [27] and Chung [28], consider the family of time series generated by: φ(B)(1−2uB +B2)dXt=θ(B)et, (2) where the polynomials φ(B) and θ(B) are as defined before, |u| ≤ 1, |d|< 1 are real parameters and et is white noise with zero mean and variance σ2 e. Econometrics 2016,4, 37 3 of 21 This process in (2) is known as Gegenbauer ARMA of order ( p , d , q ) or GARMA ( p , d , q ; u ) and has the following properties: •The power spectrum is given by: fX(ω) = C(ω)×[4(cos ω−u)2]−d,−π<ω<π, (3) where C(ω) = σ2 e 2π θ(e−iω) φ(e−iω) 2and i=√−1. • The process in (2) is stationary with long-memory when |u|< 1 and 0 <d< 1 / 2 or |u|= 1 and d<1/4. The long-memory features are characterized by: –hyperbolic decay of the autocorrelation function (ACF ) superimposed with a sinusoidal, –unbounded spectrum at the Gegenbauer frequency, ω=ωg=cos−1(u). In their recent papers, a modified class of generalized fractional processes has been studied by Shitan and Peiris [29,30]. Now, consider the following special case or the GARMA ( 0, d , 0; u) process and its properties for later reference. That is, when φ(B) = θ(B) = 1, we have: (1−2uB +B2)dXt=et. (4) Suppose that the following regularity conditions are satisfied: R1: AR regularity: |u|<1 and d<1/2 or |u|=1 and d<1/4. R2: MA regularity: |u|<1 and d>−1/2 or |u|=1 and d>−1/4. A Stationary Solution to GARMA(0, d, 0; u)Model Under the regularity conditions in R1, there exists the Wold representation to (4) given by: Xt=ψ(B)et= ∞ ∑ j=0 ψjet−j, (5) where ψ(B) = ( 1 − 2 uB +B2)−d=∑∞ j=0ψjBj with ψ0= 1 and the Gegenbauer coefficients ψj have the explicit representation: ψj= [j/2] ∑ q=0 (−1)q(2u)j−2qΓ(d−q+j) q!(j−2q)!Γ(d) such that ∑∞ j=0ψ2 j<∞ ( Γ( . ) is the Gamma function; see [ 31 ] for details). These coefficients ψj , j≥ 2 are recursively related by: ψj=2ud−1+j jψj−1−2d−2+j jψj−2(6) with initial values ψ0=1 and ψ1=2du. An Invertible Solution to GARMA(0, d, 0; u)Model Under the MA regularity conditions, there exists an invertible solution to (4) given by: et= (1−2uB +B2)dXt= ∞ ∑ j=0 πjXt−j, (7) where πj,j≥0 are obtained from (6) replacing dby −dwith corresponding initial values. The next section develops the class of GARMA(0, d , 0; u ) driven by time-dependent or stochastic innovations for later reference. Econometrics 2016,4, 37 4 of 21 3. GARMA with Time-Dependent Innovations Suppose that Var(et) in (4) is time-dependent. Consider the class of regular (in both AR and MA) GARMA(0, d, 0; u) driven by time-dependent innovations satisfying: et=σtzt,zt∼NID(0, 1), (8) where σtis the time-dependent volatility. Below, we establish the existence and uniqueness of second-order solutions to (4) with innovations in (8) under certain additional regularity conditions. 3.1. Unique Stable Solutions We use the following general approach: Let (Ω , A , P) be a probability space, and let Lr 0(Ω , A , P) be the space of all real-valued random variables on (Ω , A , P) with finite r -th order moments. Suppose that {ζt} is sequence of random variables in Lr 0(Ω , A , P) and Ms(ζ) is the closed linear subspace of Lr 0(Ω , A , P) spanned by the elements ζt , t≤s . Let Lr 0(Ω , A , P) be the closed linear subspace spanned by all of the elements ζt ; t≥ 1 by M(ζ). In the case of r= 2, let ξ and ζ be any two random elements in L2 0(Ω , A , P) such that the inner product and the norm satisfy <ξ,ζ>=Cov(ξ,ζ)and ||ξ||2=E(ξ2) respectively. It is easy to verify that L2 0(Ω,A,P)is a Hilbert space. Intuitively, if the volatility process is stationary, it guarantees the existence of the second moment of et, which enables us to establish the following two lemmas. Lemma 1. Under the AR regularity conditions R1, if the innovation process is stationary, then the solution in (5) belongs to L2 0(.), i.e., Xt∈L2 0(.). Proof. Since the innovation process et is stationary, var(et) = σ2 t is bounded. Hence the solution Xt=ψ(B)et=∑∞ j=0ψjet−j∈L2 0(.). Lemma 2. Under the MA regularity conditions R2, if the innovation process is stationary, then an invertible solution to (7) belongs to L2 0(.)or et∈L2 0(.). The proof is similar to that of Lemma 1. Lemma 3. Under the both AR and MA regularity conditions and stationarity of the innovation process, one has Mt(X) = Mt(e). The proof to this follows from Lemmas 1and 2. Next, we consider two special cases useful in applications. 3.2. Two Special Cases Consider two popular cases where the innovations etfollow GARCH or SV processes. Econometrics 2016,4, 37 5 of 21 3.2.1. GARMA(0, d, 0; u)-GARCH(r,s) Suppose that σ2 tin (8) follows a GARCH(r,s) process, such that σ2 t=α0+ r ∑ i=1 αie2 t−i+ s ∑ j=1 βjσ2 t−j, (9) where r and s are positive integers (both not simultaneously zero), α0> 0, αi≥ 0 and βj≥ 0. It is well-known that an equivalent representation of the GARCH(r,s) process is: e2 t=α0+ r ∑ i=1 αie2 t−i+ s ∑ j=1 βje2 t−j+ηt− s ∑ j=1 βjηt−j, where ηt=e2 t−σ2 t= (z2 t− 1 )σ2 t and are serially uncorrelated with mean zero. Now, we state the following lemma: Lemma 4. Let {Xt}be generated by GARMA(0, d, 0; u)and σ2 tfollows (9). (a) If the AR regularity conditions and ∑r i=1αi+∑s j=1βj< 1are satisfied, then {Xt} is second-order stationary and Xt∈L2 0(.). (b) If the MA regularity conditions are satisfied, then {Xt}is invertible and et∈L2 0(.). Proofs follow from Lemmas 1and 2. The next section develops the class of GARMA(0, d , 0; u ) driven by SV innovations or GARMA-SV. 3.2.2. GARMA(0, d, 0; u)-SV Suppose that ht=ln(σ2 t)satisfies the following recursion ρ(B)ht=κ∗+ν(B)ξt−j, (10) where ρ(B) = 1 −ρ1B−ρ2B2−···−ρlBl and ν(B) = 1 +ν1B+ν2B2+···+νmBm , κ∗ is a constant, ξt∼NID( 0, σ2 ξ) , and the disturbances zt and ξt are mutually independent for all t . Further assume that the roots of ρ(z) = 0 and ν(z) = 0 lie outside the unit circle. Note that σ2 ξ measures the conditional volatility of the log-volatility. Let Ψ∗(B)ρ(B) = ν(B). Then, it is known that there exists a sequence {ψ∗ i}, such that Ψ∗(B) = ∞ ∑ j=0 ψ∗ jBj with ∑∞ j=0(ψ∗ j)2<∞. Now, we have the following lemma: Lemma 5. Under the AR regularity condition on ρ(θ) ,the log-volatility process in (10) has uniquely determined L2 0(.)solution, such that ht=κ+ ∞ ∑ j=0 ψ∗ jξt−j, (11) where κ=E(ht) = κ∗ ρ(1)is the mean of the log-volatility process. It is clear from (11) that the log-volatility process {ht}converges in the mean square to κand: E[(ht−κ)2]→σ2 ξ ∞ ∑ j=0 (ψ0 j)2<∞. Econometrics 2016,4, 37 6 of 21 In their paper, Chesney and Scott [ 32 ] have considered the case where ht follows an AR(1) process, such that ht=κ∗+ρht−1+ξt, (12) where κ∗is a positive constant, and |ρ|<1. Lemma 6. The process in (12) is equivalent to ln(e2 t) = κ+ρln(e2 t−1) + wt, (13) where κ=a( 1 −ρ) + κ∗ , a=E[ln(z2 t)] = − 1.2704, wt=ut−ρut−1+ξt and ut=ln(z2 t)−a . Note that E[wt] = 0; Var[wt] = 4.93(1+ρ2) + σ2 ξ;E(wtwt−1) = −ρ;E(wtwt−j) = 0(j≥2). Proof. Since ln(e2 t) = ln(σ2 t) + ln(z2 t) , we have ln(e2 t) = ht+ln(z2 t) . Substituting for ht=ln(e2 t)−a−ut in (12) and noting that Var[ln(z2 t)] = 4.93, the lemma follows. Remark. Clearly, {wt} in (12) is not a martingale difference series, and hence, it is not useful in applications. However, this can be written as an ARMA(1,1) in the form: ln(e2 t) = κ+ρln(e2 t−1) + vt−θvt−1, (14) where vt∼WN(0, σ2 v);θand σ2 vare given by 1+θ2 θ=(1+ρ2)σ2 u+σ2 ξ ρσ2 u,σ2 u=Var[log(z2 t)] = 4.93. This result can be extended to any general ARMA structure for the log-volatility process. Lemma 7. Suppose that the SV process follows an ARMA( l , m ) model as in (10). Then, the corresponding {ln(e2 t)}process satisfies and ARMA(k,k)in the form: ln(e2 t) = κ+ρ1ln(e2 t−1) + ρ2ln(e2 t−2) + ···+ρkln(e2 t−k) + vt+θ1vt−1+···+θrvt−k, where κ=κ∗+a( 1 −ρ1−ρ2−···−ρp) and vt=ξt−ut , vt−j=θjξt−j−ρjut−j for j= 1, 2, ··· , k with k=max(l,m). Proof. The proof follows from extending the approach in Lemma 6. Section 4discusses the estimation of parameters. 4. Estimation of Parameters This section discusses the estimation for the GARMA ( p , d , q ; u ) model with time-dependent volatility. We divide the section into to two parts, namely, (i) the GARMA-GARCH and (ii) the GARMA-SV. 4.1. GARMA-GARCH Model Suppose that X={X1 , ··· , Xn} is generated by the GARMA ( p , d , q ; u )-GARCH( r , s ) model (2), (8) and (9). Define φ= (φ1 , . . . , φp)0 , θ= (θ1 , . . . , θq)0 , α= (α0 , α1 , . . . , αr) , 0β= (β1 , . . . , βs)0 , γ= (d , φ0 , θ0) and δ= (α0 , β0)0 . Let λ= (u , δ0 , γ0)0 , and let λ0= (u0 , δ0 0 , γ0 0)0 be the true value of λ in the interior of the compact set Λ . The approximate log-likelihood function (excluding the constant) is given by L(λ) = 1 n n ∑ t=1 lt,lt=−1 2log(σ2 t)−e2 t 2σ2 t , Econometrics 2016,4, 37 7 of 21 where et= [θ(B)]−1φ(B)(1−2uB +B2)dXt, σ2 t= [β(B)]−1[α0+α(B)e2 t] and α(B) = α1B+α2B2+···+αrBr,β(B) = 1−β1B−β2B2−···−βsBs. Assuming the initial values of X0 , X−1 , X−2 , . . . are zero, an approximate maximum likelihood estimator ˆ λ of λ in Λ is obtained by maximizing the above function, which is asymptotically equivalent to the maximum likelihood (ML) estimator. For the case of a non-normal distribution, we can still use the same approach with the corresponding quasi-maximum likelihood (QML) estimator. For the ARFIMA ( p , d , q )-GARCH ( r , s ) model, Breidt et al. [ 22 ] suggested the above approach. However, Ling and Li [ 21 ] established the consistency and asymptotic normality of the corresponding ML estimator and showed that the information matrix is block-diagonal. Combining the results of Ling and Li [ 21 ] and Chung [ 28 , 33 ], we can obtain the asymptotic results for the corresponding estimator of the GARMA-GARCH model. Proposition 1. Let ˆ u and (ˆγ , ˆ δ)0 be approximate ML estimators of u and (γ , δ) based on a sample {Xt}n t=1 from a GARMA-GARCH model under the conditions in Lemma 4. Then, ˆ u is asymptotically independent of (ˆγ,ˆ δ)0and: n(ˆ u−u0)L −→ Ksin(ωg) dY0if |u|<1and d 6=0, (15) where K =E(σ−2 t) + 2∑∞ j=1ϕσ(j)E(e2 t−i/σ4 t), Y0≡R1 0˜ W1dW2−R1 0W1d˜ W2 R1 0˜ W2 1(r)dr +R1 0W2 1(r)dr and (˜ W1(t) , ˜ W2(t)) and (W1(t) , W2(t)) are two independent Brownian motions with mean zero and covariance t E(σ2 t)1 1K!. Furthermore, n2(ˆ u∓1)L −→± K 2dY1if u =±1and d 6=0, (16) where Y1is a random variable defined as Y1≡R1 0Rr 0W1(s)dsdW2(r) R1 0Rr 0W1(s)ds2dr . Proof. See Appendix A.2. As discussed in Chung [ 28 , 33 ], the convergence rates of ˆ u are faster than those of the remaining parameters. The off-diagonal blocks in the information matrix (with respect to the parameter u and the remaining parameters) approach zero. Hence, the distribution of ˆ u is asymptotically independent of the remaining parameters. Below, we report the asymptotic result of the remaining parameters. Proposition 2. Based on the sample {Xt}n t=1 from the GARMA-GARCH model and under the conditions in Lemma 4, we have √n ˆγ−γ0 ˆ δ−δ0!L −→N 0,"Ω−1 γO OΩ−1 δ#!, Econometrics 2016,4, 37 8 of 21 where u 6=1and Ωγ=E 1 σ2 t ∂et ∂γ ∂et ∂γ+1 2σ4 t ∂σ2 t ∂γ ∂σ2 t ∂γ!, (17) Ωδ=E 1 2σ4 t ∂σ2 t ∂γ ∂σ2 t ∂γ!. Proof. See Appendix A.3. For the case of uto be one, we can use the asymptotic result of Ling and Li [21]. Following Gray et al. [ 27 ] and Chung [ 28 ], in practice, we use the grid search procedure for different value of u over the range [− 1, 1 ] to minimize the likelihood function. For selecting the order of the GARMA-GARCH model, we can use an information criterion, such as AIC and BIC. Furthermore, we can use the conventional t test for the parameters except for u . For testing the null hypothesis regarding u, we can use the approach of Chung [33] to obtain percentiles via simulations. 4.2. Estimation of GARMA-SV Model Suppose that X={X1 , ··· , Xn} are generated from the GARMA ( p , d , q ; u )-SV model (2), (8) with the SV structure ht=κ+ρ(ht−1−κ) + ξt,|ρ|<1, (18) where ht=ln(σ2 t),{ξt} ∼ NID(0, σ2 ξ)and ξt,ztare independent. From (18), we have E(ht) = κ and Var(ht) = σ2 ξ 1−ρ2 . Further, Var(et) = exp σ2 ξ 2(1−ρ2) . Since {ht} is a latent process, the evaluation of the likelihood function requires integrating it with respect to (h1, . . . , hn). As mentioned earlier, the evaluation of the likelihood function involves high-dimensional integration, which is difficult to calculate. Nevertheless, among others, Danielsson and Richard [34] , Shephard and Pitt [ 35 ], Durbin and Koopman [ 36 , 37 ] and Liesenfeld and Richard [ 38 ] suggested evaluating high-dimensional integrals using simulation methods and then maximizing the corresponding likelihood function. In this case, we use the Monte Carlo likelihood (MCL) estimator of Durbin and Koopman [ 37 ]. In their recent paper, Bos et al. [ 24 ] extended the MCL estimator for the estimation of parameters of ARFIMA-SV parameters. This approach creates a set of realized values for h= (h1 , h2 , ··· , hn)0 by ‘importance sampling’. Conditional on h and using the prediction error decomposition, it is clear that the density is given by: log(X|h,d,φ,θ,κ.ρ,σξ) = −n 2log(2π)−1 2 n ∑ t=1 log(ft)−1 2 n ∑ t=1 a2 t ft, (19) where atis the one-step ahead prediction error, ftis its variance and his from importance sampling. From (19), we evaluate the simulated likelihood function based on the true density and the importance density using the results of Durbin and Koopman [ 37 ]. We also extend the work of Bos et al. [24] , by replacing the autocovariance functions of the ARFIMA by those of the GARMA and use the results of McElroy and Holan [ 39 ] to obtain the ML estimator. As a practical issue, we use the grid search procedure for different values of u over the range [− 1, 1 ] for minimizing the likelihood function. Noting that ∂ht/∂u= 0, ∂ht/∂γ=0 , we can consider that the information matrix of (u , γ0 , δ0 2)0 has a block diagonal structure similar to the GARMA-GARCH case, where δ2= (κ , σξ , ρ)0 . If the MCL approximates the true likelihood accurately, we can use the conventional t test for the parameters except for u . For testing the null hypothesis regarding u , we can use the approach of Chung [ 33 ] to Econometrics 2016,4, 37 15 of 21 where ∂et ∂u= [1−2uB +B2]−1(−2dB)et=−2d ∞ ∑ j=1 ajet−j, ∂et ∂d=log(1−2uB +B2)·et=−2 ∞ ∑ j=1 cos(jωg) jBjet, ∂et ∂φj =−[φ(L)]−1et−j, ∂et ∂θj =−[θ(L)]−1et−j, (A2) ∂σ2 t ∂δ=bt+ s ∑ i=1 βi ∂σ2 t−i ∂δ, ∂σ2 t ∂γ=2 r ∑ i=1 αiet−i ∂et−i ∂γ+ s ∑ i=1 βi ∂σ2 t−i ∂γ, ∂σ2 t ∂u=2 r ∑ i=1 αiet−i ∂et−i ∂u+ s ∑ i=1 βi ∂σ2 t−i ∂u, with aj=     jif u=1, sin(jωg)/ sin(ωg)if |u|<1, (−1)j−1jif u=−1, and bt= ( 1, e2 t−1 , . . . , e2 t−r , σ2 t−1 , . . . , σ2 t−s)0 . Note that only n observations are available. However, et , σ2 t , ∂et/∂u , ∂et/∂γ , ∂σ2 t/∂u , ∂σ2 t/∂γ and ∂σ2 t/∂δ all depends on the theoretically infinite past history of {Xt} or {et} . For simplicity, we assume that the pre-sample values of {Xt} and {et} are zero and choose the pre-sample estimates of σ2 t and e2 t to be ∑n t=1e2 t/n , as in Bollerslev [ 2 ], Baillie et al. [16] , Ling and Li [21] and Weiss [ 44 ]. This will not affect the asymptotic efficiency and other asymptotic properties. We obtain the second derivatives, by directly differentiating ∂lt/∂γ , ∂lt/∂u and ∂lt/∂δ . For using later, we only give: ∂2lt ∂u2=−1 σ2 t∂et ∂u2 −e2 t 2σ6 t∂σ2 t ∂u2 +e2 t σ2 t−1∂ ∂u1 2σ2 t ∂σ2 t ∂u+2et σ4 t ∂et ∂u ∂σ2 t ∂u−et σ2 t ∂2et ∂u2(A3) Let Ft be the past information of et available up to time t . Then, we obtain the conditional expectation of the second derivatives: E∂2lt ∂u2Ft−1=−1 σ2 t∂et ∂u2 −1 2σ4 t∂σ2 t ∂u2 , E∂2lt ∂γ∂γ0Ft−1=−1 σ2 t ∂et ∂γ ∂et ∂γ0−1 2σ4 t ∂σ2 t ∂γ ∂σ2 t ∂γ0, (A4) E∂2lt ∂δ∂δ0Ft−1=−1 2σ4 t ∂σ2 t ∂δ ∂σ2 t ∂δ0, Econometrics 2016,4, 37 16 of 21 and those of the cross partial derivatives, E∂2lt ∂γ∂uFt−1=−1 σ2 t ∂et ∂γ ∂et ∂u−1 2σ4 t ∂σ2 t ∂γ ∂σ2 t ∂u, E∂2lt ∂δ∂uFt−1=−1 2σ4 t ∂σ2 t ∂δ ∂σ2 t ∂u, E∂2lt ∂γ∂δ0Ft−1=−1 2σ4 t ∂σ2 t ∂γ ∂σ2 t ∂δ0. Below, we will deal with u and (γ0 , δ0)0 separately, since their speeds of convergence are different, as shown by Chung [33]. Noting that E"∂et ∂d2#=2π2 3−πωg+ω2 gE(e2 t)<∞, we apply the proof of Theorem 3.1 of Ling and Li (1997) for (γ0 , δ0)0 to obtain the asymptotic properties of the information matrix: −1 n n ∑ t=1 ∂2lt/∂γ∂γ0∂2lt/∂γ∂δ0 ∂2lt/∂δ∂γ0∂2lt/∂δ∂δ0!a.s. −→ ΩγO OΩδ!, (A5) as n→∞ , and Ωγ and Ωδ are positive definite matrices, where Ωγ and Ωδ are defined in (17). We can show that the information matrix is a block diagonal, in the following way. We can express each element of ∂σ2 t/∂δ as the infinite sum e2 t−i(i= 1, 2, . . . , ) , while each element of ∂σ2 t/∂γ has a representation of the infinite sum of et−iet−i−j(j= 1, 2, , . . .) . Thus, we have E[(∂σ2 t/∂γ)(∂σ2 t/∂δ)] = Oif E(e3 t−i) = 0 , which is satisfied by the normality assumption of et . Noting the difference of time of σ2 t and (∂σ2 t/∂γ)(∂σ2 t/∂δ), we can show that −n−1∑n t=1(∂2lt/∂γ∂δ0)−→a.s. O. As in Ling and Li [ 21 ], we can also show that −n−1∑n t=1(∂2lt/∂δ∂u)−→a.s. O . The block diagonal structure of the information matrix with respect to (u , γ0)0 and δ implies that the distribution of (ˆ u , ˆγ0)0 is asymptotically independent of that of ˆ δ . Hence, we focus on the first derivatives and the information matrix regarding (u,γ0)0for deriving the asymptotic properties of ˆ u, in the next subsection. For deriving the asymptotic properties of the information matrix regarding ˆ u , we present the results of Chung [33]. Iu=E"∂et ∂u2#=E(e2 t)×4d2 ∞ ∑ j=1 a2 j, Iud =∂et ∂u∂et ∂d=E(e2 t)×4d ∞ ∑ j=1 cos(jωg)aj j, and the results indicate that when |u|<1 Iu=E(e2 t)×4d2 sin2(ωg) ∞ ∑ j=1 sin2(jωg) = ∞,Iud =E(e2 t)×d(π−2ωg) sin(ωg)<∞, and when u=±1: Iu=E(e2 t)×4d2 ∞ ∑ j=1 j2=∞,Iud =E(e2 t)× ∞ ∑ j=1 (±1)j−1cos(jωg)<∞. Econometrics 2016,4, 37 17 of 21 As discussed in [ 21 ], we also show that every element of Iγu=E[(∂et/∂γ) (∂et/∂u)] is finite. Using a similar approach as in Theorem 3.1 of [ 21 ], we have E[∂2L(λ)/∂γ∂u] = O(1). By the above results for Iu , we show that E[∂2L(λ)/∂u2] does not exist, indicating that the usual asymptotic theory based on Op(n−1/2)convergence will not work as discussed in [33]. Appendix A.2. Proof of Proposition 1 Since the information matrix is block diagonal, we focus on the parts related to (u , γ0)0 in order to prove the proposition. We consider the following Taylor series expansion of the first conditions for the maximization of the approximate likelihood function around the true value of λ0   1 cn ∂L(λ0) ∂u 1 √n ∂L(λ0) ∂γ +  1 c2 n ∂2L(λ0) ∂u21 cn√n ∂2L(λ0) ∂u∂γ0 1 cn√n ∂2L(λ0) ∂γ∂u1 n ∂2L(λ0) ∂γ∂γ0  cn(ˆ u−u0) √n(ˆγ−γ0)!=op(1), where cn=(nwhen |u|<1, n2when |u|=1, are the rates suggested by Chung [33]. If cnis appropriate, we have c−1 nn−1/2∂2L(λ0)/∂γ∂u=op(1), immediately. If we can show 1 cn n ∑ t=1 ∂lt(λ0) ∂u=Op(1),1 c2 n n ∑ t=1 ∂2lt(λ0) ∂u2=Op(1), (A6) then we obtain cn(ˆ u−u) = −"1 c2 n n ∑ t=1 ∂2lt(λ0) ∂u2#−1"1 cn n ∑ t=1 ∂lt(λ0) ∂u#+op(1), (A7) since ˆ u is asymptotically independent of ˆγ . In the remainder of the proof, we derive the limiting distributions of the two series in (A6) and then that of cn(ˆ u−u) in (A7). For this purpose, we use the theory of nonstationary ARMA process with GARCH errors, derived by Ling and Li [45]. By the conditions of Lemma 4, we can write [θ(z)]−1= ∞ ∑ k=0 ϕe(k)zk,α(z)[β(z)]−1= ∞ ∑ k=1 ϕσ(k)zk. Assuming pre-sample values, e0=e1=e2=··· =0 and combining with (A1), we have ∂σ2 t ∂u= ∞ ∑ k=1 ϕσ(k)et−k ∂et−k ∂u= t−1 ∑ k=1 ϕσ(k)et−k ∂et−k ∂u. Let ζt= (1−2uB +B2)−1θ(B)et. Then, from (A2), we have ∂et ∂u=−2d[θ(B)]−1ζt−1=−2d ∞ ∑ k=0 ϕe(k)ζt−k−1=−2d t−1 ∑ k=0 ϕe(k)ζt−k−1. Econometrics 2016,4, 37 18 of 21 Denote ζt= (ζt,ζt−1)0and define ξt= ξ1t ξ2t!=et σ2 t t−1 ∑ k=0 ϕe(k)ζt−k−1−1 σ2 t1 σ2 t−1t−1 ∑ j=1 t−1 ∑ k=0 ϕσ(j)ϕe(k)et−jζt−j−k−1, Ξt= Ξ11,tΞ12,t Ξ21,tΞ22,t! =1 σ2 t t−1 ∑ j=0 t−1 ∑ k=0 ϕe(j)ϕe(k)ζt−j−1ζ0 t−k−1 +2e2 t σ6 t t−1 ∑ j1,j2=1 t−1 ∑ k1,k2=0 ϕe(j1)ϕe(j2)ϕe(k1)ϕe(k2)et−j1et−j2ζt−j1−k1−1ζ0 t−j2−k2−1. From (A1)–(A3), it is easy to verify ∂lt(λ0) ∂u=−2dξ1t,∂2lt(λ0) ∂u2=4d2Ξ11,t+op(1)(A8) Below, we apply the theorems of [ 45 ] to derive their limiting distributions. For this purpose, we separately consider three cases: |u|<1, u=1 and u=−1. Case 1: |u|<1 By Theorem 4.3 of [45], we obtain 1 n n ∑ t=1 ξtL −→ ξ∗ 1 ξ∗ 2!, 1 n2 n ∑ t=1 ΞtL −→ K 4 sin2(ωg)Z1 0 ˜ W2 1(r)dr +Z1 0W2 1(r)dr 1 cos(ωg) sin(ωg)1!, where ξ∗ 1=1 2 sin(ωg)Z1 0 ˜ W1dW2−Z1 0W1d˜ W2, ξ∗ 2=1 2 sin(ωg)cos(ωg)Z1 0 ˜ W1dW2−Z1 0W1d˜ W2 −sin(ωg)Z1 0 ˜ W1d˜ W2+Z1 0W1dW2, and K , (˜ W1(t) , ˜ W2(t)) and (W1(t) , W2(t)) are defined in Proposition 1. Noting (A8), we obtain (15). Case 2: u=1 By Theorem 4.1 of [45], we obtain n ∑ t=1 MnξtL −→ ξ∗ 1 ξ∗ 2!, n ∑ t=1 MnΞtM0 nL −→K Ξ∗ 11 Ξ∗ 12 Ξ∗ 21 Ξ∗ 22 !, Econometrics 2016,4, 37 19 of 21 where Mn= 1/n20 1/n−1/n!, ξ∗ 1=Z1 0Zr 0W1(s)dsdW2(r), ξ∗ 2=Z1 0W1(r)dW2(r), Ξ∗ 11 =Z1 0Zr 0W1(s)ds2 dr, Ξ∗ 12 =Ξ∗ 21 =Z1 0Zr 0W1(s)dsZr 0Zs 0W1(t)dtdsdr, Ξ∗ 22 =Z1 0Zr 0Zs 0W1(t)dtds2 dr. Noting (A8), we obtain (16) for the distribution of n2(ˆ u−1). Case 3: u=−1 By Theorem 4.2 of [45], we obtain n ∑ t=1 MnξtL −→ ξ∗ 1 ξ∗ 2!, n ∑ t=1 MnΞtM0 nL −→K Ξ∗ 11 Ξ∗ 12 Ξ∗ 21 Ξ∗ 22 !, where Mn= 1/n20 −1/n1/n!, ξ∗ 1=−Z1 0Zr 0W1(s)dsdW2(r), ξ∗ 2=−Z1 0W1(r)dW2(r), and Ξ∗ 11,Ξ∗ 12,Ξ∗ 21 and Ξ∗ 22 are the same as in Case 2. Noting (A8), we obtain (16) for the distribution of n2(ˆ u+1). Appendix A.3. Proof of Proposition 2 Applying (A5) and the proof of Theorem 3.2 of [ 21 ], we can show that the conditions provided by Basawa et al. [46] are satisfied. 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A 1976,38, 259–270. c  2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).