First-order random coefficient autoregressive (RCA(1)) model: Joint Whittle estimation and information

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1 ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 9, Number, June 05 Available online at First-order random coefficient autoregressive RCA model: Joint Whittle estimation and information Mahendran Shitan, A. Thavaneswaran, and Turaj Vazifedan Abstract. Random coefficient autoregressive model, RCAp, has been discussed widely as a suitable model for nonlinear time series. The conditional least squares and likelihood parameter estimation of RCAp model has also been discussed in [3]. The statistical inference of RCA model has been presented in [4] while the conditional least square estimates for nonstationary processes is studied in [7]. The optimal estimation for nonlinear time series using estimating equations has been investigated in [6]. Recently there has been interest in joint prediction based on spectral density of popular nonlinear time series models such as RCAp models. Another way of estimating the parameters of the RCA model is to do Whittle s estimation. In this paper the Whittle estimates of the parameters of an RCAp model are studied. It is shown that the Whittle information of the autoregressive parameter in an RCAp model is larger than the corresponding information in an autoregressive AR model.. Introduction As a result of growth of interest in dynamic systems such as financial, environmental and biological science, lots of studies have been done to investigate an appropriate model for such systems. In the statistical literature, the random coefficient autoregressive model RCAp has been discussed widely as a suitable model for nonlinear time series see [3]. The RCAp model is simply a generalisation of the standard autoregressive AR models where the coefficients are allowed to vary over time. Specifically, the RCA Received July 9, Mathematics Subject Classification. 6M0. Key words and phrases. Non-linear time series, RCA model, Whittle s estimation and information. 3

2 4 MAHENDRAN SHITAN, A. THAVANESWARAN, AND TURAJ VAZIFEDAN model is defined as Y t φ + b t Y t + u t, where φ is constant coefficient, {b t } is a sequence of independent and identically distributed i.i.d. random variables with mean zero and constant variance σb, {u t} is a white noise process with mean zero and variance σu and {b t } and {u t } are mutually independent. Note that if σb 0, then the RCA model reduces to the standard AR model. The AR model has a constant coefficient while the RCA model includes the extra random variables {b t } which vary over the time, thus the RCAp model is more general and appropriate for nonlinear data sets. The conditional least squares and likelihood parameter estimation of RCAp model is discussed in [3]. The statistical inference of RCA model has been presented in [4] while the conditional least square estimates for nonstationary processes is studied in [7]. The optimal estimation for nonlinear time series using estimating equations has been investigated in [6] while the optimal estimation functions for RCAp model have been studied in []. The moment properties and the corresponding squared RCAp process have been investigated in []. It has been illustrated in [] that the autocorrelation structure of the RCAp model is the same as for the ARp model, while they have different marginal variances. Furthermore, it is also shown that in [] the joint prediction of mean and volatility of the RCA model and the spectral density function of the RCA model is given as provided φ + σ b fλ σ u φ π φ σ b < see []. φe iλ Another way of estimating the parameters of the RCA model is to do Whittle s estimation. The Whittle estimates are obtained by minimizing the function see [5] n n ln π + j0 n ln fω j + j0 Iω j fω j, where n in the number of observations, ω j πj n, and Iω j is given by Iω j n πn Y t exp iω j t, j 0,,,..., n. t0 The Whittle estimates of the parameters of an RCA model have not been fully investigated and hence in this paper the Whittle estimates of the parameters of an RCA model are studied. It will be shown in Section that the Whittle information of the autoregressive parameter in an RCA,

3 FIRST-ORDER RANDOM COEFFICIENT AUTOREGRESSIVE RCA MODEL 5 model is larger than the corresponding information in an AR model. Finally, the conclusions are drawn in Section 3.. Whittle s information matrix Proposition. For the RCA model as defined in, the Whittle information matrix is n φ σ 4 b φ φ σ b + φσ b φ φ σ b φσ b σ u φ φ σ b φ φσb φ φ σb φ σb σu φ σb φσb σu φ φ σb σu φ σb σu 4 Proof. The spectral density function of RCA model is given as σ fλ u φ π φ σb φe iλ while the Whittle information matrix is where σ u φ π φ σ b [ φ cos λ + φ ], n W W W 3 W W W 3 W 3 W 3 W 33 W W ln fλ dλ, φ ln fλ dλ, σ b W 33 ln fλ σu dλ, W W ln fλ ln fλ φ σb W 3 W 3 ln fλ ln fλ φ σu and W 3 W 3 ln fλ ln fλ σb σu The natural logarithm of fλ is., 3 dλ, dλ, dλ. ln fλ ln σ u + ln φ lnπ ln φ σ b ln φ cos λ + φ

4 6 MAHENDRAN SHITAN, A. THAVANESWARAN, AND TURAJ VAZIFEDAN and the partial derivative of ln fλ with respect to φ is ln fλ φ φ φ + φ φ σ b + cos λ φ φ cos λ + φ φ + φ3 + φσb + φ φ3 cos λ φ φ φ σb + φ cos λ + φ φσ b φ φ σ b + cos λ φ φ cos λ + φ. Next we evaluate the integral [ ] ln fλ [ φσ ] b dλ φ φ φ σb + cos λ φ φ cos λ + φ dλ + + 4φ σ 4 b φ φ σb 8φσb φ φ σb cos λ φ φ cos λ + φ dλ dλ 4cos λ φ φ cos λ + φ dλ. 4 The above integrals in 4 could be evaluated as follows: 4φ σ 4 b φ φ σ b dλ 8πφ σ 4 b φ φ σ b, [ 8φσb ] φ φ σb cos λ φ φ cos λ + φ dλ 8φσb φ φ σb cos λ φ φ cos λ + φ dλ. However, according to proof of Lemma 5 [5], Hence, we have [ cos λ φ φ cos λ + φ dλ 0. 8φσ b φ φ σ b cos λ φ φ cos λ + φ In addition, based on the proof of Lemma [5], 4cos λ φ φ cos λ + φ dλ φ. ] dλ 0.

5 FIRST-ORDER RANDOM COEFFICIENT AUTOREGRESSIVE RCA MODEL 7 Thus we have [ ] ln fλ dλ φ and, consequently, W is π [ ] ln fλ [ dλ φ 8πφ σ 4 b φ φ σ b + φ. φ σ 4 b φ φ σ b + φ The partial derivative of ln fλ with respect to σ b is and thus ln fλ σ b W φ σ b ln fλ σb dλ φ σb φ σb π φ σb φ σb. dλ The partial derivative of ln fλ with respect to σ u is and we have ln fλ σ u σ u W 33 ln fλ σu dλ σu 4 dλ π σu 4 dλ π σu 4 σu 4. dλ ].

6 8 MAHENDRAN SHITAN, A. THAVANESWARAN, AND TURAJ VAZIFEDAN Next we compute W W as follows: W W + ln fλ ln fλ φ σb dλ φσb φ φ σb φ σb dλ cos λ φ φ cos λ + φ φ σb dλ φσ b φ φ σ b. Similarly, W 3 W 3 is computed as W 3 W 3 + ln fλ ln fλ φ σu dλ φσb φ φ σb σu dλ cos λ φ φ cos λ + φ σu dλ φσ b σ u φ φ σ b. Finally, W 3 W 3 is computed as follows: W 3 W 3 ln fλ ln fλ σb σu dλ σ u φ σ b dλ σ u φ σ b. Upon substituting the expressions of W, W, W 33, W, W, W 3, W 3, W 3, W 3 into 3 we obtain. Remark. If we assume that the parameters σb and σ u are known values, then Whittle s information associated with φ for the RCA model is [ φ σ 4 ] b n φ φ σb + φ, 5 while it is important to note that the information associated with φ in the AR model is [ ] n φ. 6

7 3 FIRST-ORDER RANDOM COEFFICIENT AUTOREGRESSIVE RCA MODEL 9 Some numerical calculations for the information associated with φ using equations 5 and 6 are listed down in Table. Table. Numerical values for the Information associated with φ for the RCA and for the AR models φ σb 0.3 σb 0.6 RCA AR RCA AR Equation 5 Equation 6 Equation 5 Equation Clearly we can see that the Whittle information for the RCA model is larger than the AR model when σb 0. Of course when σ b 0, the RCA model reduces to the standard AR model. Since the information is larger for the RCA model, this would give rise to estimators with a smaller variance. 3. Conclusion The objective of this paper was to study the Whittle estimates for the RCA model. It was shown that the Whittle information of the autoregressive parameter in an RCA model is larger than the corresponding information in an AR model. The implication of this study is that since the information is larger for the RCA model, this would give rise to estimators with a smaller variance, and hence more accurate estimates. Acknowledgement The authors wish to thank the referees for their valuable comments and suggestions given by them, in order to improve the quality of this paper. References [] S. Chandra and M. Taniguchi, Estimating functions for non-linear time series models, Ann. Inst. Statist. Math , 5 4.

8 0 MAHENDRAN SHITAN, A. THAVANESWARAN, AND TURAJ VAZIFEDAN [] Y. Liang, A. Thavaneswaran, and N. Ravishanker, RCA models: joint prediction of mean and volatility, Statist. Probab. Lett , [3] D. F. Nicholls and B. G. Quinn, Random Coefficient Autoregressive Models: an Introduction, Springer-Verlag Inc., New York Berlin, 98. [4] P. M. Robinson, Statistical inference for random coefficient autoregressive model, Scand. J. Statist , [5] M. Shitan, M. and S. Peiris, Approximate asymptotic variance-covariance matrix for the Whittle estimators of GAR parameters, Comm. Statist. Theory Methods 45 03, [6] A. Thavaneswaran and B. Abraham, Estimation for non-linear time series models using estimating equations, J. Time Ser. Anal , [7] D. Tjφstheim, Estimation in nonlinear time series models, Stoch. Proc. Appl. 986, Laboratory of Computational Statistics and Operations Research, Institute for Mathematical Research, Universiti Putra Malaysia, Malaysia; Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Malaysia address: sarasmahen@gmail com Department of Statistics, University of Manitoba, Canada address: thavane@cc.umanitoba.ca Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Malaysia address: turaj386@gmail.com