RCA models with correlated errors

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1 Applied Mathematics Letters 19 (006) RCA models with correlated errors S.S. Appadoo a,a.thavaneswaran a,,jagbir Singh b a Department of Statistics, The University of Manitoba, Winnipeg, Manitoba, Canada R3T N b Department of Statistics, Fox School of Business and Management, Temple University, Philadelphia, PA, USA Received 14 February 005; received in revised form November 005; accepted 10 November 005 Abstract Financial time series data cannot be adequately modelled by a normal distribution and empirical evidence on the non-normality assumption is very well documented in the financial literature; see R.F. Engle, Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation, Econometrica 50 (198) and T. Bollerslev, Generalized autoregressive conditional heteroscedasticity, J. Econometrics 31 (1986) for details. The kurtosis of various classes of RCA models has been the subject of a study by Appadoo et al. S.S. Appadoo, M. Gharahmani, A. Thavaneswaran, Moment properties of some volatility models, Math. Sci. 30 (005) and Thavaneswaran et al. A. Thavaneswaran, S.S. Appadoo, M. Samanta, Random coefficient GARCH models, Math. Comput. Modelling 41 (005) In this work we derive the kurtosis of the correlated RCA model as well as the normal GARCH model under the assumption that the errors are correlated. c 005 Elsevier Ltd. All rights reserved. Keywords: Moments properties; Kurtosis; Correlated RCA models; Correlated GARCH models 1. Introduction Rapid developments of time series models and methods addressing nonlinearity in computational finance have recently been reported in the literature. These theories either extend and complement existing time series methodology by introducing more general structures or provide an alternative framework. Volatility modelling has attracted attention in recent years and the quest for heavy-tailed distributions is still an ongoing process. In this regard, we derive the kurtosis, which characterizes the heavy-tail properties, of the correlated RCA model of Nicholls and Quinn 5. Many financial series, such as returns on stocks and foreign exchange rates, exhibit leptokurtosis and volatility (varying in time). Kurtosis, measured by the moment ratio K µ 4,givesan estimate of the peakedness of unimodal curves. µ Estimation for RCA models had been studied in 6andnew predictors have been derived in 7.. Moment properties Random coefficient autoregressive time series were introduced by Nicholls and Quinn 5 andsomeof their properties have been studied recently by Appadoo et al. 4. RCA models exhibiting long memory properties have Corresponding author. Tel.: ; fax: address: thavane@ccu.umanitoba.ca (A. Thavaneswaran) /$ - see front matter c 005 Elsevier Ltd. All rights reserved. doi: /j.aml

2 S.S. Appadoo et al. / Applied Mathematics Letters 19 (006) been considered in 8. A sequence of random variables {y t } is called an RCA( time series if it satisfies the equations y t (φ + b t )y t 1 + e t t, where denotes the set of integers and ( ) (( ) ( )) (i) bt e t 0 0, σ b 0 0 σe, (ii) φ + σb < 1. The sequences {b t } and {e t } respectively, are the errors in the model. According to Nicholls and Quinn 5, (ii) is a necessary and sufficient condition for the second-order stationarity of {y t }.So,together with (i), it also ensures strict stationarity. Moreover, Feigin and Tweedie 9 showedthateyt k < for some k 1ifthemoments of the noise sequences satisfy Eet k < and E(φ + b t ) k < 1, for the same k. Theorem.1. Let {y t } be an RCA( time series satisfying conditions (i) and (ii), and let γ y be its covariance function. Then, σ e (a) Ey t 0, Eyt,thekthlagautocovariance for y 1 φ σb t is given by γ y (k) and the autocorrelation 1 φ σb for y t is ρ k φ k for all k. That is, the usual AR( process has same autocorrelation as the RCA(. (b) If {b t } and {e t } are normally distributed random variables and if e t and b t are correlated with correlation coefficient ρ, thenthe kurtosis K (y) of the RCA process {y t } is given by K (y) 6(σ b + φ 1 )1 φ3 1 3φ 1σb +7φ3 1 ρ σb + 31 (φ 1 + σ b )1 φ3 1 3φ 1σb 1 φ1 3 3φ 1σb 1 6φ 1 σ b φ4 1 3σ b 4 1 (φ1 + σ b ) (. and for an AR( process K (y) reduces to 3 and when ρ 0,the kurtosis reduces to the one observed by Appadoo et al. 4. Proof. The proof of part (a) of Theorem.1 is given in Appadoo et al. 4. Proof of part (b) is as follows: y t (φ 1 + b t )y t 1 + e t y t 1 φ 1 + y t 1 b t + e t y t y t 1 φ 1 + y t 1 φ 1b t + y t 1 φ 1 e t + y t 1 b t + y t 1 b t e t + e t. Under the condition of φ 1 +σb < 1, the second moment of y t and, hence, the variance is given by Eyt The third moment of the process is given by φk σ e σ ɛ 1 (φ 1 +σ b ). y 3 t φ 3 1 y3 t 1 + 3φ 1 y3 t 1 b t + 3φ 1 y t 1 e t + 3φ 1 y 3 t 1 b t + 6φ 1 y t 1 b te t + 3φ 1 y t 1 e t + b 3 t y3 t 1 + 3b t y t 1 e t + 3b t y t 1 e t + e 3 t E(y 3 t ) E(φ3 1 y3 t 1 + 3φ 1y 3 t 1 b t + 6φ 1 y t 1 b te t ) The fourth moment of the process is given by 6φ 1 ρσ 3 e σ b (1 σ b φ 1 )1 φ3 1 3φ 1σ b. E(yt 4 ) 6φ 1 E(y4 t 1 b t ) + 6E(b t y t 1 e t ) + φ4 1 E(y4 t 1 ) + E(b4 t y4 t 1 ) + E(e4 t ) + 1φ1 E(y3 t 1 b te t ) + 6φ1 E(y t 1 e t ) 6φ1 σ b E(y4 t 1 ) + 6σ e σ b E(y t 1 ) + φ4 1 E(y4 t 1 ) + 3σ b 4 E(y4 t 1 ) + 3σe 4 + 1φ 1 σ eσ b ρ E(yt 1 3 ) + 6φ 1 σ e E(y t 1 ) 1 6φ1 σ b φ4 1 3σ b 4 E(y4 t ) 6σ e σ b E(y t 1 ) + 3σ e 4 + 1φ 1 σ eσ b ρ E(yt 1 3 ) + 6φ 1 σ e E(y t 1 ) 1 6φ1 σ b φ4 1 3σ b 4 E(y4 t ) 6σ e E(y t 1 )σ b + φ 1 +3σ e 4 + 1φ 1 σ eσ b ρ E(yt 1 3 ) 1 6φ 1 σ b φ4 1 3σ 4 b E(y4 t 1 ) 6σ 4 e (σ b + φ 1 ) 1 (φ 1 + σ b ) + 3σ 4 e + 7φ 3 1 ρ σ b σ 4 e 1 (φ 1 + σ b )1 φ3 1 3φ 1σ b.

3 86 S.S. Appadoo et al. / Applied Mathematics Letters 19 (006) By substitution, E(y 4 t 1 ) 6σ 4 e (σ b + φ 1 )1 φ3 1 3φ 1σ b +7φ3 1 ρ σ b σ 4 e + 3σ 4 e 1 (φ 1 + σ b )1 φ3 1 3φ 1σ b 1 (φ 1 + σ b )1 φ3 1 3φ 1σ b 1 6φ 1 σ b φ4 1 3σ 4 b when ρ 0, E(y 4 t 1 ) 3σ 4 e σ b + φ (φ 1 + σ b )1 6φ 1 σ b φ4 1 3σ 4 b. K (y) 6σ e 4(σ b + φ 1 )1 φ3 1 3φ 1σb +7φ3 1 ρ σb σ e 4 + 3σ e 41 (φ 1 + σ b )1 φ3 1 3φ 1σb 1 (φ1 + σ b )1 φ3 1 3φ 1σb 1 6φ 1 σ b φ4 1 3σ b 4 1 (φ 1 + σb ) σ 4 e 6(σ b + φ 1 )1 φ3 1 3φ 1σb +7φ3 1 ρ σb + 31 (φ 1 + σ b )1 φ3 1 3φ 1σb 1 φ1 3 3φ 1σb 1 6φ 1 σ b φ4 1 3σ b 4 1 (φ1 + σ b ). When ρ 0, our results converge to the one reported in the literature: K (y) 31 + (σ b + φ 1 )1 (φ 1 + σ b ) 31 (φ1 1 6φ1 σ b φ4 1 3σ b 4 + σ b ) 1 (φ φ 1 σ b + 3σ b 4). The kurtosis of the RCA model is a special case of Theorem.1.Thecorrelated RCA model has a higher kurtosis than its uncorrelated counterpart and easy computation leads to the following inequality for the kurtosis for the different classes of RCA models: K (y) AR K (y) RCA K (y) CRCA..1. Random coefficient ARCH( model In analogy with the RCA models, we introduce a class of RCA versions of GARCH models. Consider the class of ARCH( model for the time series y t,where y t h t t h t ω + (α 1 + b t 1 )y t 1 (.) and t is asequence of independent, identically distributed random variables with zero mean and unit variance. Let u t y t h t be the martingale difference and let σ u be the variance of u t.onwriting the model as y t ω + (α 1 + b t 1 )y t 1 + u t, (.3) the minimum mean square error forecast is optimal for yt ;however, for the random coefficient ARCH( model introduced in 1 givenby(.), the minimum mean square error forecast of y t is not optimal (see 7 formore details). Lemma.1. For the GARCH model considered in y t h t t,h t ω 0 +(α 1 +b t 1 )yt 1,where t N(0,σ ) and b t N(0,σa ),thekurtosis is given by K (y) 31 α 1 σ 4 1 3σ 4(α 1 +σ b ). Proof. y t h t t and Ey t Eh tσ. We will make use of the above relationship to find the expected value of E(h t ): Eh t ω 0 + α 1 Ey t 1 ω 0 + α 1 Eh t 1 σ ω 0 1 α 1 σ

4 S.S. Appadoo et al. / Applied Mathematics Letters 19 (006) Eh t ω 0 + ω 0α 1 σ Eh t 1+3α1 σ 4 Eh t 1 +3σ 4 Eh t 1 σ b ω 0 + ω 0α 1 σ Eh t 1 1 3α1 σ 3σ 4σ b (1 α 1σ )ω 0 + ω 0 α 1σ (1 3σ 4(α 1 + σ b )) ω 0 (1 + α 1σ ) 1 3σ 4(α 1 + σ b ) Eyt Eh tσ σ ω 0, 1 α 1 σ ω Eyt 4 E t 4 h t E t 4 Eh t 3σ 4 Eh t 3σ 4 0 (1 + α 1 σ ) 1 3σ (α 1 + σ b ). (1 α 1 σ 4 ) Hence, K (y) 3 1 3σ (α 1 +σ b )... RCA GARCH models Consider the following random coefficient model: y t h t t, h t ω 0 + (α 1 + b t 1 )y t 1 + β 1h t 1 t N(0,σ ) and b t N(0,σ b ). Lemma.. Under suitable stationary conditions, the kurtosis of y t is given by K (y) 31 (α1 σ + β 1 α 1 β 1 σ 3α 1 σ 4 3σ b σ 4 β 1. Proof. y t h t t (.4) h t ω 0 + (α 1 + b t 1 )yt 1 + β 1h t 1. (.5) Thus, we have yt h t t (.6) Eyt Eh t t Eh t σz. (.7) The mean of h t is given by Eh t Eω 0 + (α 1 + b t 1 )yt 1 + β 1h t 1 E(ω 0 ) + E(α 1 yt 1 ) + E(b t 1yt 1 ) + E(β 1h t 1 ) ω 0 + α 1 σ E(h t + β 1 E(h t 1 ) Eh t α 1 σ E(h t β 1 E(h t 1 ) ω 0 Eh t (1 α 1 σ β ω 0 Eh t ω 0 1 α 1 σ β 1. (.8) Computation of E(h t ): h t (ω 0 + (α 1 + b t 1 )yt 1 + β 1h t 1 ) h t (ω 0 + (α 1 + b t 1 )y t 1 + β 1h t 1 ) ω 0 + ω 0y t 1 α 1 + ω 0 y t 1 b t 1 + ω 0 β 1 h t 1 + y 4 t 1 α 1 + y 4 t 1 α 1b t 1 + y t 1 α 1β 1 h t 1 + y 4 t 1 b t 1 + y t 1 b t 1β 1 h t 1 + β 1 h t 1 Eh t ω 0 + ω 0α 1 σ Eh t 1+ω 0 β 1 Eh t 1 +3σ 4 α 1 Eh t 1 + σ α 1β 1 Eh t 1 +3σ 4 σ b Eh t 1 +β 1 Eh t 1. (.9)

5 88 S.S. Appadoo et al. / Applied Mathematics Letters 19 (006) Simple computation leads us to Eh t 1 σ α 1β 1 3σ 4 (α 1 + σ b ) β 1 ω0 + ω 0(α 1 σ + β Eh t 1 ω Eh t 1 σ α 1β 1 3σ 4 (α 1 + σ b ) β 1 ω0 + 0 (α 1 σ + β 1 α 1 σ β. 1 Thus, ω Eh 0 + ω0 (α 1σ +β t 1 α 1 σ β 1 1 σ α 1 β 1 3σ 4(α 1 + σ b ) 1 β ω ( 0 1 α1 σ β ) 1 + ω 0 (α 1 σ + β 1 σ α 1 β 1 3σ 4(α 1 + σ b ) β 1 1 α1 σ β 1 ω0 (1 + α 1σ + β 1 σ α 1 β 1 3σ 4(α 1 + σ b ) β 1 1 α1 σ β. (.10) 1 The kurtosis of the process is given by K (y) Ey4 t E yt Eh t E t 4 E h t t 3σ 4 Eh t ( Eht σz ) 3σ 4 Eh t σz 4 (Eh t ) 3Eh t (Eh t ) ω0 3 (1+α 1σ +β 1 σ α 1 β 1 3σ 4 (α 1 +σ b ) β 1 1 α1 σ β 1 ω 0 1 α 1 σ β 1 3ω0 (1+α 1σ +β 1 σ α 1 β 1 3σ 4(α 1 +σ b ) β 1 1 α1 σ β 1 ω0 ( 1 α1 σ β ) 1 ( 1 α1 σ β 3ω0 (1 + α 1σ + β 1 σ α 1 β 1 3σ 4(α 1 + σ b ) β 1 1 α1 σ β 1 ω0 3(1 + α1 σ + β ( 1 α 1 σ β 3(1 (α 1 σ 1 σ α 1 β 1 3σ 4(α 1 + σ b ) 1 + β ) β 1 σ α 1 β 1 3σ 4(α 1 + σ b ) 1 β. (.1 For a GARCH(1, process, we have 3(1 K (y) (α1 + β 1 ) ) 1 α 1 β 1 3α1. β 1 (. 3. Conclusions In this work, the kurtosis of the correlated RCA model and that of the random coefficient ARCH(, GARCH(1,, are derived. The correlated random coefficient GARCH model may be viewed as a special case of a state space model for yt and the parameter process θ t,andinferences for these processes may be studied as in 6. The kurtosis for these processes, important in finance, has applications including in comparative simulations. The results for random coefficient GARCH given in 3 arealsoextended to RCGARCH with correlated errors. References 1 R.F. Engle, Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation, Econometrica 50 (198)

6 S.S. Appadoo et al. / Applied Mathematics Letters 19 (006) T. Bollerslev, Generalized autoregressive conditional heteroscedasticity, J. Econometrics 31 (1986) A. Thavaneswaran, S.S. Appadoo, M. Samanta, Random coefficient GARCH models, Math. Comput. Modelling 41 (005) S.S. Appadoo, M. Gharahmani, A. Thavaneswaran, Moment properties of some volatility models, Math. Sci. 30 (005) D.F. Nicholls, B.G. Quinn, Random Coefficient Autoregressive Models: An Introduction, in: Lecture Notes in Statistics, vol. 11, Springer, New York, A. Thavaneswaran, B. Abraham, Estimation of nonlinear time series models using estimating equations, J. Time Ser. Anal. 9 (1988) A. Thavaneswaran, C.C. Heyde, Prediction via estimating functions, J. Statist. Plann. Inference 77 (1999) R. Leipus, D. Surgallis, Random coefficient autoregression, regime switching and long memory, Adv. Appl. Probab. 35 (003) P.D. Feigin, R.L. Tweedie, Random coefficient autoregressive processes: a Markov chain analysis of stationarity and finiteness of moments, J. Time Ser. Anal. 6 (1985) 1 14.