This paper considers a new class of first order moving average type time series model with index δ (\u3e 0) to describe some hidden features of a time series. It is shown that this class of models provides a valid, simple solution to a new direction of time series modelling. In particular, for suitably chosen parameters (coefficient β and index δ) this type of models could be used to describe data with low or high frequency components. Various new results associated with this class are given in a general form. A simulation study is carried out to justify the theory. We justify the importance of this class of models in practice using a set of real time series data

Allen, David E.

Peiris, Shelton

Thavaneswaran, Aerambamoorthy

English

Research Online @ ECU

Edith Cowan University Research Online ECU Publications Pre. 2011 2002 Generalized moving average models and applications in high frequency data Shelton Peiris David E. Allen Aerambamoorthy Thavaneswaran Follow this and additional works at: https://ro.ecu.edu.au/ecuworks  Part of the Statistics and Probability Commons Peiris, S., Allen, D., & Thavaneswaran, A. (2002). Generalized moving average models and applications in high frequency data. Perth, Australia: Edith Cowan University. This Other is posted at Research Online. https://ro.ecu.edu.au/ecuworks/7107      Generalized Moving Average Models and Applications in High Frequency Data   By  Shelton Peiris The University of Sydney, Australia  David E. Allen Edith Cowan University, Australia  and  A. Thavaneswaran The University of Manitoba, Canada        School of Accounting, Finance and Economics Working Paper Series Edith Cowan University August 2002 Working Paper 0205      Correspondence author and address:  Professor David E. Allen School of Accounting, Finance and Economics Faculty of Business and Public Management Edith Cowan University 100 Joondalup Drive Joondalup WA 6027 Phone: 61+ 08 9400 5471 Fax:  61+ 08 9400 5271 Email: d.allen@ecu.edu.au         Abstract  This paper considers a new class of first order moving average type time series model with index δ (> 0) to describe some hidden features of a time series.  It is shown that this class of models provides a valid, simple solution to a new direction of time series modelling.  In particular, for suitably chosen parameters (coefficient β and index δ) this type of models could be used to describe data with low or high frequency components.  Various new results associated with this class are given in a general form.  A simulation study is carried out to justify the theory.  We justify the importance of this class of models in practice using a set of real time series data  Keywords:  Time series, Misclassification, High frequency, Spectrum, Estimation, Financial data, Moving average, Correlation, Index, Periodogram      1  INTRODUCTION   It is known that there are some problems arise in time series modelling in practice for data with high frequency components, for example, financial data.  Although ARMA type models could be used in practice, there is no systematic approach or a suitable class of time series models available in literature to accommodate, analyze and forecast of time series with changing frequency behaviour via a direct method.  This paper attempts to introduce a family of first order moving average (MA(1)) type models (Generalized MA(1) or GEMA(1)) to describe some hidden properties of time series data.   We first consider the family of standardMA(1) processes (see, for instance,Box and Jenkins (1976), Priestley (1989), Brockwell and Davis (1991)) gener-ated byXt = Zt − βZt−1 , (1.1)where |β| < 1 and {Zt} is a sequence of uncorrelated random variables (notnecessarily independent) with zero mean and variance σ2 , known as whitenoise, WN(0, σ2) .Using the backshift operator, B (i.e. BjXt = Xt−j, j ≥ 0) and the identityoperator I = B0 , (1.1) can be written asXt = (I − βB)Zt . (1.2)The process in (1.2) have the following properties:(i) The autocorrelation function (acf), ρk satisfiesρ1 = −β/(1 + β2) and ρk = 0 for k ≥ 2,(ii) The partial autocorrelation function (pacf), φk satisfies,φk = −βk(1− β2)/(1− β2(k+1)),(iii) The spectral density function (sdf), fX(ω) isfX(ω) =σ22pi(1− 2β Cosω + β2); −pi ≤ ω ≤ pi. (1.3)It is clear from the above that for 0 < β < 1 , ρ1 is negative and theseries fluctuate rapidly about its mean. The spectrum is dominated by highfrequencies and this behaviour is represent by a large peak of the spectrum nearthe frequency ω = pi. In other words the power (of the process) is concentratedat high frequencies (the maximum value of fX(ω) occurs at ω = pi and thisis given by f sX(0) =σ22pi(1 + β)2, where s stands for standard). Also note that|φk| < β, and the pacf is dominated by a damped exponential.It is well known that for any time series data set, the density of crossings ata certain level xt = x may vary. We have noticed that this property (say, thevariation in the degree of frequency) is very common in many time series data2sets (see the graphs 1 to 4 in the Appendix I). However, these series cannotdiscriminate from each other using the standard time series techniques. Inother words the acf, the pacf, and the spectrum are similar to each otherin some cases and one may propose the same model for all of these cases.Although these models provide a valid basic solution, a complete picture ofthe underlying process can not be achieved by using these standard models.This motivates us to answer the following question: ‘How can we indicate thedegree of frequency in time series modelling’? This leads us to introduce ageneralized version of (1.2) with an additional parameter (or index) δ(> 0)(to control the degree of frequency) satisfyingXt = (I − βB)δZt; 0 < β < 1; δ > 0 . (1.4)This class of models covers the traditional MA(1) family given in (1.1) whenδ = 1 . It is interesting to note that the frequency of data can be controlledby this additional parameter δ and hence (1.4) constitutes a wide variety ofimportant processes in practice. There are a large number of real world datasets with varying frequencies (especially in finance there are data with highfrequency) and (1.4) can be applied easily using existing techniques with somemodifications. The class of models generated by (1.4) is called ‘generalizedMA(1)’ or ‘GEMA(1)’. Although (1.4) can easily be extended to generalARMA type models, this paper considers the family of GEMA(1) given in(1.4) with 0 < β < 1 and δ > 0, where δ is a real number.With that view in mind Section 2 reports some properties of the underlyingprocess in (1.4).2 Properties of GEMA(1) ProcessesLet(I − βB)δ =∞∑j=0ψjβjBj , (2.1)3where 0 < β < 1, δ ∈ R+; B0 = I, ψ0 = 1 andψj = (−1)j(δj)=(−δ)(−δ + 1) · · · (−δ + j − 1)j!; j ≥ 1If δ is a positive integer, then ψj = 0 for j ≥ δ + 1 . For any non-integralδ > 0 , it is known that,ψj =Γ(j − δ)Γ(j + 1)Γ(−δ) , (2.2)where Γ(·) is the gamma function given byΓ(x) =∫∞0 tx−1e−tdt ; x > 0∞ ; x = 0x−1Γ(x+ 1) ; x < 0 .It is easy to see that the series∑∞j=0 ψjβj converges for all δ since |β| < 1,and in particular, the process Xt in (1.4) is equivalent to a valid MA(∞)process of the form Xt =∑∞j=0 ψjβjZt−j with∑ |ψjβj|2 <∞. (2.3)Now we state and prove the following theorem for a stationary solution of(1.4).Theorem 2.1: Let {Zt} ∼ WN(0, σ2) . Then for all δ > 0 and |β| < 1 ,the infinite seriesXt =∞∑j=0ψjβjZt−j, (2.4)converges absolutely with probability one, where Ψj is given in (2.2).Proof: Using the facts E(∑∞j=0 |ψjβjZt−j|)2=∑∞j=0 |ψjβj|2 E{|Zt−j|2} .and∑∞j=0 |ψjβj|2 <∞ , the result follows.Thus (2.4) gives a stationary solution for the GEMA(1) model in (1.4).Note: For β = 1, (2.4) converges for all δ in 0 < δ < 1/2.4It is clear from (1.4) that {Xt} is equivalent to a valid AR(∞) process ofthe form∞∑j=0pij βj Xt−j = Zt, (2.5)where pij = (−1)j(−δj)=Γ(j + δ)Γ(j + 1)Γ(δ); j ≥ 0.Let γk = Cov(Xt, Xt−k) = E(XtXt−k) be the autocovariance function (at lagk) of {Xt} satisfying the conditions of theorem 2.1. It is clear from (2.5)that {γk} satisfy a Yule-Walker type recursion∞∑j=0pij βj γk−j = 0 ; k > 0 (2.6)and the corresponding autocorrelation function (acf), ρk , at lag k is givenby∞∑j=0pij βjρk−j = 0 ; k > 0 , (2.7)The spectrum (sdf) of {Xt} in (1.4) isfX(ω) = |1− β e−iω|2δ σ22pi; −pi ≤ ω ≤ pi= (1− 2β Cosω + β2)δ σ22pi. (2.8)Note: In a neighbourhood of ω = pi,f gX ∼σ22pi(1 + β)2δ,where g stands for generalized. For δ > 1 and 0 < β < 1, it is clear (from(1.3) and (2.8)) that f gX > fsX . Thus a set of high frequency data satisfying anMA(1) can be replaced by an GEMA(1) model with a suitably chosen indexδ.In Section 3 we obtain the exact form of γk (or ρk) using the sdf in (2.8).That is,γk =∫ pi−pieikω fX(ω)dω=σ2pi∫ pi0Cos(kω)(1− 2β Cosω + β2)δ dω. (2.9)5We first evaluate the integral in (2.9) for k = 0 and then for other values ofk ≥ 1.3 Main ResultsTheorem 3.1:γ0 = V ar(Xt) = σ2F (−δ, −δ; 1; β2) (3.1)where F (θ1, θ2; θ3; θ) is the hypergeometric function given byF (θ1, θ2; θ3; θ) =∞∑j=0Γ(θ1 + j) Γ(θ2 + j) Γ(θ3) θjΓ(θ1) Γ(θ2)Γ(θ3 + j) Γ(j + 1). (3.2)Note that the right hand side of (3.2) terminates if θ1 or θ2 is equal to anegative integer (see also Gradsteyn and Ryzhik (GR)(1965), p.1039).Proof: From GR, p.384 (3.665:2),∫ pi0dω(1− 2β Cosω + β2)δ′ = B(12,12)F (δ′, δ′; 1; β2), (3.3)where B(x, y) = Γ(x) Γ(y)Γ(x+y)is the Beta function.Since B(12, 12) = pi , the result follows since δ′ = −δ.Using (3.2), it is easy to see that for δ′ = −1 ,F (−1,−1; 1; β2) = (1 + β2).Hence (3.1) confirms the corresponding well known result for the variance ofan MA(1) (standard) process satisfying (1.2) with δ = 1. That is,V ar(Xt) = σ2(1 + β2), |β| < 1.As it is not easy to evaluate the integral in (2.10) for k 6= 0 , we find anexpression for γk via ,γk = E(XtXt−k)= σ2∞∑j=0ψj ψj+k βk+2j .6An explicit form of γk is given in Theorem 3.2 .Theorem 3.2:γk =σ2 βkΓ(k − δ)F (−δ, k − δ; k + 1; β2)Γ(−δ) Γ(k + 1) ; k ≥ 0 . (3.4)Proof: Since Xt =∑∞j=0 ψj βj Zt−j ,γk = σ2 βk∞∑j=0ψj ψj+k β2j,= σ2βk∞∑j=0Γ(j − δ) Γ(j + k − δ)(β2)jΓ2(−δ)Γ(j + 1)Γ(j + k + 1) .From p.556 of Abramovitz & Stegun (1965), we have∞∑j=0Γ(−δ + j) Γ(k − δ + j)(β2)jΓ(k + 1 + j) Γ(j + 1)=Γ(−δ) Γ(k − δ)F (−δ, k − δ; k + 1; β2)Γ(k + 1)and hence (3.4) follows.Note: When k = 0 , Theorem 3.2 reduces to Theorem 3.1 .The autocorrelation function plays an important role in the analysis of theunderlying process. The Corollary 3.1 below gives ρk for any k ≥ 0 .Corollary 3.1 : The autocorrelation function of the GEMA(1) process in(1.4) isρk = βk Γ(k − δ)F (−δ, k − δ; k + 1; β2)Γ(−δ) Γ(k + 1)F (−δ,−δ; 1; β2) (3.5)Note: It is interesting to note that (3.5) reduces to the acf of a standardAR(1) process satisfying Xt = αXt−1 + Zt when δ = −1 (see Peiris (2002)),sinceF (1, k + 1; k + 1; α2) = F (1, 1; 1; α2) = (1− α2)−1(see also GR p.1040).Thus the results of these two theorems provide a new set of formulae ingeneral form. Obviously, our new result for γk in (3.4) supersede all existingacf for standard AR(1) MA(1) and also for fractionally differenced whitenoise processes.7Remark : An important consequence of our new result of Theorems 3.2 yield∫ pi0Cos k ω dω(1− 2β Cosω + β2)δ′ =piβk Γ(k + δ′)F (δ′, k + δ′; k + 1; β2)Γ(δ′) Γ(k + 1)(3.6)When k = 0 the equation (3.6) reduces to (3.3) in Theorem 3.1 . (also see GRp 384).Note : The new result in equation (3.6) is, particularly, useful in many theo-retical developments of generalized ARMA (GARMA) processes with indicesand this will be discussed inna future paper.Now in Section 4, we discuss a method of estimating parameters β and δappeared in (1.4).4 Estimation of ParametersConsider the ratio of ρk+1/ρk.From Corollary 3.1 (equation 3.7), we haveρk+1ρk=α(k − δ)F (−δ, k + 1− δ; k + 2; β2)(k + 1)F (−δ, k − δ; k + 1; β2) . (4.1)From the properties of F (θ1, θ2; θ3; θ) given in equation (3.2), it is not difficultto show that the right hand side (rhs) of (4.1) is approximately equal to β forlarge k.Thus an estimate of β is obtained byβˆ =rk+1rk, (4.2)where rk is the lag k sample acf.However, the following approximations for ρ1 and ρ2 will produce themethod of moment (MOM) estimates for both β and δ. that is,ρ1 = −βδ, (4.3)andρ2 =−δ(1− δ)β22. (4.4)8The corresponding MOM are given byβˆ = 2r2/r1 − r1, δˆ = −r1/βˆ.A better approximation for ρ1 is given by ρ1 = −βδ(1 − β2δ2). The max-imum likelihood estimation (MLE) is also possible and will be discussed in afuture paper.Now one can modify the usual regression approach (via periodogram analy-sis) as in the long memory case (see Geweke and Porter-Hudak (1983), Brock-well and Davis (1991), Peiris and Court (1993), Chen et. al. (1994), Hunt et.al. (2001)) to estimate δ.From (2.8) we havelnfX(ω) = C − δ ln(|1− βe−iω|2), (4.5)where C = ln(σ2/2pi). Suppose we have T observations X1, X2, · · · , XT .The corresponding sample periodogram isIT,X(ωj) =12piT−1∑h=−(T−1)γˆhe−ihωj ,where γˆh =1T∑T−|h|j=1 (Xj+|h| − X¯)(Xj − X¯) and ωj = 2pij/T .Now the equation in (4.6) reduces to the linear regression equationyj = a− δxj + εj, j = 1, · · · , KT ,where yj = ln{IT,X(ωj)}, xj = ln(|1− βˆe−iωj |2), a = C, εj = ln( IT,X(ωj)fX(ωj) )and KT is a constant chosen so that 2piKT/T is small. In practice we selectKT = Tη, 0 < η < 1.Thus an estimator for δ isδˆp = −KT∑j=1(xj − x¯)yj/KT∑j=1(xj − x¯)2 . (4.6)Since var(εj) =pi26, the asymptotic distribution of δˆ isδˆ ∼ N(δ, pi26∑(xj − x¯)2 ).9Peiris and Court (1993), Chen et al (1994), and Hunt et al (2001) suggestedreplacing the periodogram in the above expression by a general scaled lagwindow estimator for fX(ω) . Suppose that κ(x) is a real valued, boundedsymmetric function defined and continuous on x ∈ [−1, 1] , and 0 elsewhere.Let {RT} be a sequence of integers such that RT →∞ and RT/T → 0as T → ∞ . The general form of a scaled lag window estimator for fX(ω)isfˆX(ω) =12pi∑|r|<Tκ(rRT)Cr cos(ωr), (4.7)whereCr =1T − |r|T−|r|∑t=1(Xt − µ)(Xt+|r| − µ), |r| = 0, 1, · · · , T − 1is an unbiased estimator of the autocovariance function of {Xt}, γr , calcu-lated knowing the true mean µ . The lag window estimator for d is obtainedby substituting fˆX for IT,X in (4.4). Denote the resulting estimator by δˆL.That is,δˆL = −KT∑j=1(xj − x¯)yj/KT∑j=1(xj − x¯)2 , (4.8)where yj = lnfˆX(ωj) .Further, letC∗r =1TT−|r|∑t=1(Xt − X¯)(Xt+|r| − X¯), |r| = 0, · · · , T − 1be an estimator of γr calculated using the sample mean X¯ . The corre-sponding spectral density estimate using C∗r is denoted byfˆ ∗X(ω) =12pi∑|r|<RTκ(rRT)C∗r cos(ωr). (4.9)Let δˆ∗L denote the estimator obtained by substituting fˆ∗X for IT,X in (4.4).Chen, Abraham, and Peiris (1994) investigated the bias and mean squarederror of δˆL via some simulation studies using various lag window spectral10density estimators. They noted that while δˆL had smaller mean squared errorcompared with δˆp the estimator did typically have larger bias. In Section 5we apply this theory to a real data set given in Abraham and Ledolter (1983).5 An ApplicationConsider the time series of 197 readings from a chemical process concentration(in every two hours) given in series A of Box and Jenkins (1976). The originalseries is nonstationary. The time series plot, the acf, the pacf and the spectrumindicate that the differenced data has a high frequency component. Since theoriginal series is nonstationary, Box and Jenkins (1976) fitted the followingMA(1) model for the differenced data:xt = zt − 0.7zt−1, (5.1)where xt represents the differenced data.However, we fit a generalized MA(1) model for the data in order to explainthe high frequency behaviour using the methods described in Section 4. Theresults are:βˆ = 0.414, δˆ = 0.620, and σ2 = 0.138. The fitted model isxt = (I − 0.414)0.620zt, (5.2)where var(zt) = 0.138.The first three forecasts from the time origin at t = 193 and the corre-sponding 95% confidence intervals for xt are:0.02 and 0.02± 0.52-0.05 and −0.05± 0.300.005 and 0.005± 0.38The true (observed) values for the last three readings on xt are -0.10,-0.50, and 0.20 respectively.11The corresponding results due to standard MA modelling are:-0.10 and −0.10± 0.790.00 and 0.00± 0.870.00 and 0.00± 0.87Our results are closer to the true values than in the traditional MA (orARMA) modelling. Also note that our new results provide shorter confidenceintervals in all three cases above (see Appendix II for comparison).Note: All calculations and simulations reported here are carried out usingSplus.ReferencesAbramowitz, M. and Segun, I. (1968). Handbook of Mathematical Functions,Dover, New York.Beran, J. (1994). Statistics for Long Memory Processes. Chapman and Hall,New York.Box, G.E.P. and Jenkins, G.M. (1976).Time Series: Forecasting and ControlHolden-Day, San Francisco.Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and MethodsSpringer-Verlag, New York.Chen, G., Abraham, B. and Peiris, S. (1994). Lag window estimation of thedegree of differencing in fractionally integrated time series models, J. TimeSeries Anal. 15, 473-487.Geweke, J. and Porter-Hudak, S. (1983). The estimation and application oflong memory time series models, J. Time Series Anal., 4, 231-238.Gradshteyn, I.S. and Ryzhik, I.M. (1980). Tables of Integrals, Series, andProducts, 4th edn., academic Press, New York.Granger, C.W.J. and Joyeux, R. (1980). An introduction to long memory timeseries models and fractional differencing, J. Time Series Anal., 1, 15-29.Peiris, M.S. and Court, J.R. (1993). A note on the estimation of degree of12differencing in long memory time series analysis, Probab. and Math. Statist.,14, 223-229.Peiris, S. (2002). Correlation Properties of Generalized ARMA Processes.Submitted for PublicationPriestly, M.B. (1989). Spectral Analysis and time Series. Academic Press, SanDiego.13Time0 50 100 150 200 250 300-4-2024Time Series Plot of an MA(1)beta = 0.8delta = 1.0Time0 50 100 150 200 250 300-3-2-10123Time Series Plot of a GEMA(1)beta = 0.8delta = 0.8Time0 50 100 150 200 250 300-2024Time Series Plot of a GEMA(1)beta = 0.8delta = 1.2Time0 50 100 150 200 250 300-4-2024Time Series Plot of a GEMA(1)beta = 0.8delta = 1.5Time0 50 100 150 200-1.0-0.50.00.51.0Time Series Plot of Diff. Series ATime0 50 100 150 200 250 300-3-2-1012Time Series Plot of a GEMA(1)beta = 0.414delta = 0.620

Generalized moving average models and applications in high frequency data

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Generalized moving average models and applications in high frequency data

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