It is known that volatility plays a central role in financial modelling problems. This paper studies, in detail, a class of discrete time stochastic volatility (SV) models driven by ARMA models with innovations having a stochastic variances. The auto- correlation function of this class of models is derived and methods of identification of such processes are described. An example is added to illustrate the development of the theory over the standard methods

Allen, David E.

Bhar, Ramprasad

Peiris, Shelton

English

Research Online @ ECU

Edith Cowan University Research Online ECU Publications Pre. 2011 2006 Analysis and applications of autoregressive moving average models with stochastic variance Shelton Peiris Ramprasad Bhar David E. Allen Follow this and additional works at: https://ro.ecu.edu.au/ecuworks  Part of the Statistics and Probability Commons Peiris, S., Bhar, R., & Allen, D. (2006). Analysis and applications of autoregressive moving average models with stochastic variance. Perth, Australia: Edith Cowan University. This Other is posted at Research Online. https://ro.ecu.edu.au/ecuworks/7102 Analysis and Applications of Autoregressive Moving AverageModels with Stochastic VarianceShelton Peiris1 Ramprasad Bhar2 David Allen3AbstractIt is known that volatility plays a central role in financial modelling problems. Thispaper studies, in detail, a class of discrete time stochastic volatility (SV) modelsdriven by ARMA models with innovations having a stochastic variances. The auto-correlation function of this class of models is derived and methods of identification ofsuch processes are described. An example is added to illustrate the development ofthe theory over the standard methods.Keywords: GARCH models, Volatility, Stochastic variance, Innovations, Heteroscedas-ticity, Random, Conditional expectation, Autocorrelation, Estimation.1 IntroductionThe class of autoregressive moving average (ARMA) models has been used in manyapplications related to time series observations. This class of ARMA moedsl of order(p,q) or ARMA(p,q) is given byXt = C + α1Xt−1 + α2Xt−2 + · · ·+ αpXt−p + Zt + β1Zt−1 + · · ·+ βqZt−q, (1.1)where C is a constant, αj’s and βj’s are not all zeros for all j, {Zt} is a sequenceof uncorrelated random variables (not necessarily independent) with mean zero andconstant variance, σ2 (known as white noise, WN(0, σ2)).1School of Mathematics and Statistics The University of Sydney, NSW, Australia2School of Banking and Finance, The University of NSW3School of Accounting, Finance and Economics, Edith Cowan University, Perth, WA1Under stationarity conditions (ie. the roots of 1 − α1ω − α2ω2 − · · · − αpωp = 0 lieout side the unit circle), one has the stationary solution to (1.1) given byXt = µ+∞∑j=0ψjZt−j, (1.2)where µ = E(Xt) is the unconditional mean of Xt and ψ(B) =∑∞j=0 ψjBj suchthat ψ(B)α(B) = β(B) (α(B) = I − α1B − α2B2 − · · · − αpBp; β(B) = I + β1B +β2B2 + · · · + βqBq, and B is the backshift operator satisfying BjXt = Xt−j withB0Xt = IXt = Xt).Under stationary conditions, the unconditional mean of the process isµ = E(Xt) =C1− α1 − α2 − · · · − αp; 1− α1 − α2 − · · · − αp 6= 0 (1.3)and the corresponding constant variance isV ar(Xt) = σ2∞∑j=0ψ2j , (1.4)where∑∞j=0 ψ2j <∞.However, the conditional mean and variance of the process, respectively areEt|t−1 = E(Xt|Ft−1) = α1Xt−1 + α2Xt−2 + · · ·+ αpXt−p + β1Zt−1 + · · ·+ βqZt−qand Vt|t−1 = V ar(Xt|Ft−1) = σ2, where Ft−1 is the history of the process upto timet−1. Therefore, standard ARMA models are useful in modelling the conditional meanbut not suitable for modelling the conditional variance.In many practical problems in finance, it has been noticed that the models assum-ing constant conditional variance may not produce good forecast values. Further ithas been recognised that many financial time series data are uncorrelated while thesquared values are highly correlated. Engle (1982) exploited this idea and put forwarda class of Autoregressive Conditional Heteroskedastic (ARCH) models to describe theconditional variance which proved to be extremely useful in many financial time series2applications. Bollerslev (1986) generalised the class of ARCH models to incorporatethe temporal dependence in conditional variances for skewness and excess kurtosis.The class of (p, q)th order generalised ARCH or GARCH is given byXt =√htζt,ht = α0 +p∑i=1αiX2t−i +q∑j=1βjht−j, (1.5)where ζt is a sequence of independent and identically distributed (iid) random vari-ables with zero mean and unit variance, α0 > 0, αi ≥ 0, βj ≥ 0 and∑pi=1 αi +∑qj=1 βj < 1.Notes:1. When βj = 0, j = 1, 2, · · · , q, equation (1.5) reduces to the ARCH(p) model. Theconditions on α′s and β ′s ensure that V ar(Xt) > 0, while∑pi=1 αi +∑qj=1 βj < 1 isrequired for wide sense stationarity of {Xt}.2. It is clear that the unconditional variance of Xt is greater than the conditionalvariance of Xt provided the past realizations of X2t >α01−Ppi=1αi−Pqj=1βj.Remark:When the original time series has significant serial correlations, first we fit an ARMAmodel and and then investigate the residuals for any heteroscedasticity.A shortcoming of the GARCH model in (1.5) is that the sign of Xt does notinfluence the conditional variance, which may contradict the observed dynamics ofthe process. Therefore, the main concern of this paper is to further develop the theorywhich may turn out to be useful in modelling the statistical properties of financialdata. With that view in mind, the next section considers the class of ARMA modelswith a stochastic variance specification.32 ARMA Models with Stochastic VarianceSuppose that {Xt} in (1.1) represents the mean corrected return of a stock given byXt = Yt − µ, where µ = E(Yt). The word volatility in finance literature is frequentlyassociated with the V ar(Xt) and the changes of volatility occur in almost all classesof assets and have been extensively studied and reported in the past. This paperconsiders the changes of volatility by allowing the variance of the noise process in(1.1) to be σ2t . For example, set Zt such that Zt ∼ NID(0, σ2t ). The assumption ofnormality of Zt is not necessary, but can be used to simplify many related results.There are two interesting cases arise in practice, where σ2t is(i) a deterministic function of time and(ii) a stochastic process.These two cases can be analysed to emphasize different related practical issues. How-ever, in each case we need additional assumptions relating to σ2t such as(a) 0 < m < σ2t < M <∞ in (i) and(b) stationarity in (ii)for stable solutions. Case (i) has been considered by many authors (see, for example,Peiris and Singh (1987), Peiris (1991) and Singh and Peiris (1997) and referencesthere in for details).This paper considers the case (ii) with the following specification for ht = log σ2tto describe a class of stochastic volatility (SV) models. Suppose that {ht} follows anARMA(p′, q′) model satisfyinght = C′ + η1ht−1 + η2ht−2 + · · ·+ ηp′ht−p′ + ν0Vt + ν1Vt−1 + · · ·+ νq′Vt−q′ , (2.1)where C ′ is a positive constant, η′js and ν′js are (constants) not all zeros for all j and{Vt} is assumed to be a sequence of serially uncorrelated random variables which aremutually uncorrelated with {Zt}. Further assume that the mean and the variance of{Vt} are zero and σ2V respectively(ie.{Vt} ∼ WN(0, σ2V )). Without loss of generality4(w.l.o.g.) we may take σ2V = 1.It is known that the volatility logarithm given in (2.1) follows a stationary ARMA(p′, q′)process when the zeros of η(ω) = 1 − η1ω − η2ω2 − · · · − ηp′ωp′ are outside the unitcircle. In this case C ′ = (1−∑p′i=1 ηi) δ, where δ = E(ht) is the unconditional meanof ht.Notes:1. The class of SV models driven by (1.1) with Zt ∼ NID(0, σ2t ) and ht = ln(σ2t ),where ht follows (2.1) is called the class of ARMA− ARMASV (p, q, p′, q′).2. It is convenient to set Zt = σtUt, where σt and Ut are two independent variablesand {Ut} ∼ NID(0, 1). In this case, it is clear thatE(Zjt ) = MjE(σjt ), (2.2)where Mj = E(Ujt ) =djdtj(exp(t2/2))|t=0 and exp(t2/2) is the moment generatingfunction (mgf) of an N(0, 1) distribution. For example, M1 = 0,M2 = 1,M3 = 0 andM4 = 3.3. When σt and Ut are not independent, we may consider a more general case suchthat the pairs (Vt, Ut) are iid and corr(Vt, Ut) 6= 0. For example, (Vt, Ut) are iid andbivariate normal with the correlation matrix(1 δ′δ′ 1).Let ν(ω) = ν0 + ν1ω+ ν2ω2 + · · ·+ νq′ωq′ and let the sequence of square summableconstants {ψ′i} is obtained by Ψ′(B)η(B) = ν(B), where Ψ′(B) =∑∞j=0 ψ′jBj. In thiscaseht = δ +∞∑j=0ψ′jVt−j (2.3)is a valid solution to (2.1). The assumption on the zeros of η(ω) ensures that theprocess {ht} is weakly stationary with mean δ and variance, ξ2 =∑∞j=0 ψ′2j . Clearly,with the assumption of normality, the unconditional distribution of Xt is a lognormalmixture of normal distributions. See, for instance, Taylor (1994).5In many applications of time series with the specification in (2.1), one needs themoments of {Xt}. The next section derives the moments , E(Xjt ) for j, 1 ≤ j ≤ 4.3 Moments and KurtosisWhen {Vt} ∼ NID(0, 1), the unconditional distribution of σ2t is log normal withmean δ and variance ξ2. Since ht ∼ N(δ, ξ2), we haveE[(σ2t )j] = exp(jδ + j2ξ2/2).Now the mean and variance of σ2t respectively are exp(δ + ξ2/2) and exp(2δ +ξ2)(expξ2 − 1). From (1.2) it is obvious that E[(Xt − µ)r] < ∞ for all r > 0 wherethe mean return is E(Xt) = µ and its variance is E[(Xt − µ)2] =∑∞j=0 ψ2jE(σ2t ) =exp(δ+ ξ2/2)∑∞j=0 ψ2j . Further, we obtain a lower bound for E[(Xt−µ)4] is obtainedas follows:E[(Xt − µ)4] = E[(∞∑j=0ψjZt−j)4] (3.1)= E[(∞∑j=0ψjZt−j)2(∞∑j=0ψjZt−j)2] (3.2)≥ (∞∑j=0ψ2j )2E(σ4t ) (3.3)= 3exp(2δ + 2ξ2)(∞∑j=0ψ2j )2.Therefore, the kurtosis of Xt is greater than 3exp(ξ2) and clearly, the distribution ofXt is leptokurtic.Example 1Consider an ARMA − ARMASV (1, 0, 1, 0) given by the following two equations:Xt = C + αXt−1 + Zt, {Zt} ∼ WN(0, σ2t ) (3.4)ht = C′ + ηht−1 + Vt, {Vt} ∼ WN(0, 1), (3.5)6where |α|, |η| < 1.In this case we haveXt = µ+∞∑j=0αjZt−jht = δ +∞∑j=0ηjVt−j,where µ = C1−αand δ = C′1−η.Under the assumption of normalityV ar(Xt) = E[(Xt − µ)2] =exp(δ + ξ2/2)1− α2,where ξ2 = var(ht) =∑∞j=0 η2j = 11−η2.The corresponding kurtosis K1 > 3exp(1/(1− η2)).Example 2Consider an ARMA − ARMASV (1, 0, 1, 1) model given by (3.4) and the followingvolatility equation:ht = C′ + ηht−1 + Vt + νVt−1, {Vt} ∼ WN(0, 1). (3.6)In this case we haveht = δ +∞∑j=0ψ′jVt−j,where ψ′0 = 1 and ψ′j = ηj−1(η + ν) for j ≥ 1.Under the assumption of normalityV ar(Xt) = E[(Xt − µ)2] =exp(δ + ξ′2/2)1− α2,where ξ2 = var(ht) =∑∞j=0 ψ′2j = 1+2ην+ν21−η2.The corresponding kurtosis K2 > 3exp(1+2ην+ν21−η2).Next section considers the modelling of excess equity returns in the US(Jan 1971to Sept 2004) as an application of ARMA models with stochastic variance (SV).74 An application of SV modellingWe obtained the data for equity index from Morgan Stanley Capital Internationaland the short-term interest rate data from the International Financial Statistics ofthe International Monetary Fund. Excess return is calculated by subtracting theinterest rate from the return computed from the equity index. These are all annualisedfigures. The time series plot, acf and the pacf of data show that there is no significantserial correlation (see, Appendix 1). The analysis shows that the squared values areautocorrelated (see Appendix 2) and therefore we fit AR(1) and ARMA(1,1) modelsto these squared values. The following tables report the corresponding results:Table 1: Results for Excess Equity Return (USA)Estimated Model ParametersModel α0 α1 θ1 σ2vAR -0.1099 0.9207 0.2130(0.0475) (0.0313) (0.0577)ARMA -0.1110 0.9199 0.5016 0.0204(0.0504) (0.0331) (0.0044) (0.0105)Model parameters are estimated by the quasi-maximum likelihood method usingnumerical maximization algorithm in GaussTM . The robust standard errors (se) aregiven in parentheses.Entries are p-values for the respective statistics. These diagnostics are computedfrom the recursive residual (standardised) of the measurement equation. The nullhypothesis in the portmanteau test is that the residuals are serially uncorrelated.The ARCH test checks for no serial correlations in the squared residual up to lag 26.This is applicable to recursive residuals as explained in Wells (1996, p 27). MNR is8Table 2: Residual Diagnostics and Model Adequacy TestsEstimated Model ParametersModel Portmanteau ARCH MNR Recursice TAR 0.443 0.320 0.019 0.946ARMA 0.469 0.642 0.116 0.933the modified Von Neumann ratio test using a recursive residual for model adequacy(see, Harvey, 1990, Chapter 5). Similarly, if the model is correctly specified then therecursive T has a Student’s t-distribution (see, Harvey, 1990, p 157).9 Figure 1Estimated Variance from the SV Model (AR Structure)  Figure 2Estimated Variance from the SV Model (ARMA Structure)  10References1. Bolleslev, T. (1986), Generalised Autoregressive Conditional heteroscedasticity,Journal of Econometrics, 31, 307-327.2. Engle. R. F, (1982), Autoregressive conditional heteroskedasticity with esti-mates of the variance of U.K. inflation, Econometrica, 50, 987-1008.3. Harvey, A.C. (1990), The Econometric Analysis of Time Series, MIT Press,cambridge.4. Peiris, M.S. & Singh, N. (1987), A note on the properties of some nonstationaryARMA processes, Stochastic Processes and their Applications, 24, 151-155.5. Peiris, M.S. (1990), Analysis of multivariate ARMA processes with nonsta-tionary innovations, Communications in Statistics: Theory and Methods, 19,2847-2852.6. Singh, N. & Peiris, M.S. (1997), A simulation study on vector arma processeswith nonstationary innovation: a new approach to identification, Journal ofStatistical Computation and Simulation, 58, 37-58.7. Taylor,S. (1994), Modelling Stochastic Volatility: A Review and ComparativeStudy, Mathematical Finance, 4, 183-204.8. Wells, C. (1996), The Kalman Filter in Finance, Kluwer Academic Publications,Amsterdam.11Appendix1Time0 100 200 300 400−2−1012Excess Return − Jan 71 to Sept 04Lag0 5 10 15 20 250.00.20.40.60.81.0 Series : MC05Lag0 5 10 15 20 25−0.10−0.050.00.050.10 Series : MC0512Appendix2Time0 100 200 300 4000246Squared Excess Return − Jan 71 to Sept 04Lag0 5 10 15 20 250.00.20.40.60.81.0 Series : MC05^2Lag0 5 10 15 20 25−0.10−0.050.00.050.10 Series : MC05^213

Analysis and applications of autoregressive moving average models with stochastic variance

https://ro.ecu.edu.au/cgi/viewcontent.cgi?article=8103&amp;context=ecuworks

Analysis and applications of autoregressive moving average models with stochastic variance

Abstract

Similar works

Full text

Available Versions

Research Online @ ECU