# Estimating FARIMA models with uncorrelated but non-independent error terms

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## Abstract

In this paper we derive the asymptotic properties of the least squares estimator (LSE) of fractionally integrated autoregressive moving-average (FARIMA) models under the assumption that the errors are uncorrelated but not necessarily independent nor martingale differences. We relax the independence and even the martingale difference assumptions on the innovation process to extend considerably the range of application of the FARIMA models. We propose a consistent estimator of the asymptotic covariance matrix of the LSE which may be very different from that obtained in the standard framework. A self-normalized approach to confidence interval construction for weak FARIMA model parameters is also presented. All our results are done under a mixing assumption on the noise. Finally, some simulation studies and an application to the daily returns of stock market indices are presented to corroborate our theoretical work.

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## Notes

1. To cite few examples of nonlinear processes, let us mention the self-exciting threshold autoregressive (SETAR), the smooth transition autoregressive (STAR), the exponential autoregressive (EXPAR), the bilinear, the random coefficient autoregressive (RCA), the functional autoregressive (FAR) (see Tong (1990) and Fan and Yao (2008) for references on these nonlinear time series models).

2. Recall that the fractional version of Cesàro’s Lemma states that for $$(h_t)_t$$ a sequence of positive reals, $$\kappa >0$$ and $$c\ge 0$$ we have

\begin{aligned} \lim _{t\rightarrow \infty }h_tt^{1-\kappa }=\left| \kappa \right| c \Rightarrow \lim _{n\rightarrow \infty }\frac{1}{n^{\kappa }}\sum _{t=0}^n h_t=c. \end{aligned}

## References

• Aknouche A, Francq C (2021) Count and duration time series with equal conditional stochastic and mean orders. Economet Theory 37(2):248–280

• Akutowicz EJ (1957) On an explicit formula in linear least squares prediction. Math Scand 5:261–266

• Anderson TW (1971) The statistical analysis of time series. Wiley, New York

• Baillie RT, Chung C-F, Tieslau MA (1996) Analysing inflation by the fractionally integrated ARFIMA-GARCH model. J Appl Economet 11(1):23–40

• Beran J (1995) Maximum likelihood estimation of the differencing parameter for invertible short and long memory autoregressive integrated moving average models. J Roy Stat Soc Ser B 57(4):659–672

• Beran J, Feng Y, Ghosh S, Kulik R (2013) Long-memory processes. Probabilistic properties and statistical methods. Springer, Heidelberg

• Berk KN (1974) Consistent autoregressive spectral estimates. Ann Stat 2:489–502 (collection of articles dedicated to Jerzy Neyman on his 80th birthday)

• Boubacar Maïnassara Y (2012) Selection of weak VARMA models by modified Akaike’s information criteria. J Time Series Anal 33(1):121–130

• Boubacar Mainassara Y, Carbon M, Francq C (2012) Computing and estimating information matrices of weak ARMA models. Comput Stat Data Anal 56(2):345–361

• Boubacar Mainassara Y, Francq C (2011) Estimating structural VARMA models with uncorrelated but non-independent error terms. J Multivar Anal 102(3):496–505

• Boubacar Maïnassara Y, Kokonendji CC (2016) Modified Schwarz and Hannan-Quinn information criteria for weak VARMA models. Stat Inference Stoch Process 19(2):199–217

• Boubacar Maïnassara Y, Saussereau B (2018) Diagnostic checking in multivariate ARMA models with dependent errors using normalized residual autocorrelations. J Am Stat Assoc 113(524):1813–1827

• Brillinger DR (1981) Time series: data analysis and theory, vol 36. SIAM

• Cavaliere G, Nielsen MØ, Taylor AR (2017) Quasi-maximum likelihood estimation and bootstrap inference in fractional time series models with heteroskedasticity of unknown form. J Economet 198(1):165–188

• Chiu S-T (1988) Weighted least squares estimators on the frequency domain for the parameters of a time series. Ann Stat 16(3):1315–1326

• Dahlhaus R (1989) Efficient parameter estimation for self-similar processes. Ann Stat 17(4):1749–1766

• Davidson J (1994) Stochastic limit theory. An introduction for econometricians. Advanced texts in econometrics. The Clarendon Press, Oxford University Press, New York

• Davis RA, Matsui M, Mikosch T, Wan P (2018) Applications of distance correlation to time series. Bernoulli 24(4A):3087–3116

• Ding Z, Granger CW, Engle RF (1993) A long memory property of stock market returns and a new model. J Empir Finance 1:83–106

• Dominguez MA, Lobato IN (2003) Testing the martingale difference hypothesis. Economet Rev 22(4):351–377

• Doukhan P, León J (1989) Cumulants for stationary mixing random sequences and applications to empirical spectral density. Probab Math Stat 10:11–26

• Escanciano JC, Velasco C (2006) Testing the martingale difference hypothesis using integrated regression functions (Nonlinear Modelling and Financial Econometrics). Comput Stat Data Anal 51(4):2278–2294

• Fan J, Yao Q (2008) Nonlinear time series: nonparametric and parametric methods. Springer

• Fox R, Taqqu MS (1986) Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann Stat 14(2):517–532

• Francq C, Roy R, Zakoïan J-M (2005) Diagnostic checking in ARMA models with uncorrelated errors. J Am Stat Assoc 100(470):532–544

• Francq C, Zakoïan J-M (1998) Estimating linear representations of nonlinear processes. J Stat Plann Inference 68(1):145–165

• Francq C, Zakoïan J-M (2005) Recent results for linear time series models with non independent innovations. In: Statistical modeling and analysis for complex data problems, vol 1 of GERAD 25th Anniv Ser. Springer, New York, pp 241–265

• Francq C, Zakoïan J-M (2007) HAC estimation and strong linearity testing in weak ARMA models. J Multivar Anal 98(1):114–144

• Francq C, Zakoïan J-M (2019) GARCH models: structure, statistical inference and financial applications. Wiley

• Giraitis L, Surgailis D (1990) A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle’s estimate. Probab Theory Related Fields 86(1):87–104

• Granger CWJ, Joyeux R (1980) An introduction to long-memory time series models and fractional differencing. J Time Ser Anal 1(1):15–29

• Hallin M, Taniguchi M, Serroukh A, Choy K (1999) Local asymptotic normality for regression models with long-memory disturbance. Ann Stat 27(6):2054–2080

• Hauser M, Kunst R (1998) Fractionally integrated models with arch errors: with an application to the swiss 1-month Euromarket interest rate. Rev Quant Financ Acc 10(1):95–113

• Herrndorf N (1984) A functional central limit theorem for weakly dependent sequences of random variables. Ann Probab 12(1):141–153

• Hosking JRM (1981) Fractional differencing. Biometrika 68(1):165–176

• Hsieh DA (1989) Testing for nonlinear dependence in daily foreign exchange rates. J Bus 62(3):339–368

• Hualde J, Robinson PM (2011) Gaussian pseudo-maximum likelihood estimation of fractional time series models. Ann Stat 39(6):3152–3181

• Keenan DM (1987) Limiting behavior of functionals of higher-order sample cumulant spectra. Ann Stat 15(1):134–151

• Klimko LA, Nelson PI (1978) On conditional least squares estimation for stochastic processes. Ann Stat 6(3):629–642

• Kuan C-M, Lee W-M (2006) Robust $$M$$ tests without consistent estimation of the asymptotic covariance matrix. J Am Stat Assoc. 101(475):1264–1275

• Ling S (2003) Adaptive estimators and tests of stationary and nonstationary short- and long-memory ARFIMA-GARCH models. J Am Stat Assoc 98(464):955–967

• Ling S, Li WK (1997) On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. J Am Stat Assoc 92(439):1184–1194

• Lobato IN (2001) Testing that a dependent process is uncorrelated. J Am Stat Assoc 96(455):1066–1076

• Lobato IN, Nankervis JC, Savin NE (2001) Testing for autocorrelation using a modified box- pierce q test. Inter Econ Rev 42(1):187–205

• Newey WK (1991) Uniform convergence in probability and stochastic equicontinuity. Econometrica 59(4):1161–1167

• Nielsen MØ (2015) Asymptotics for the conditional-sum-of-squares estimator in multivariate fractional time-series models. J Time Ser Anal 36(2):154–188

• Palma W (2007) Long-memory time series. Wiley series in probability and statistics. Theory and methods. Wiley, Hoboken

• Romano JP, Thombs LA (1996) Inference for autocorrelations under weak assumptions. J Am Stat Assoc 91(434):590–600

• Shao X (2010a) Corrigendum: a self-normalized approach to confidence interval construction in time series. J. R. Stat. Soc. Ser. B Stat. Methodol. 72(5):695–696

• Shao X (2010b) Nonstationarity-extended Whittle estimation. Economet Theory 26(4):1060–1087

• Shao X (2010c) A self-normalized approach to confidence interval construction in time series. J R Stat Soc Ser B Stat Methodol 72(3):343–366

• Shao X (2011) Testing for white noise under unknown dependence and its applications to diagnostic checking for time series models. Economet Theory 27(2):312–343

• Shao X (2012) Parametric inference in stationary time series models with dependent errors. Scand J Stat 39(4):772–783

• Shao X (2015) Self-normalization for time series: a review of recent developments. J Am Stat Assoc 110(512):1797–1817

• Sowell F (1992) Maximum likelihood estimation of stationary univariate fractionally integrated time series models. J Economet 53(1–3):165–188

• Székely GJ, Rizzo ML, Bakirov NK (2007) Measuring and testing dependence by correlation of distances. Ann Stat 35(6):2769–2794

• Taniguchi M (1982) On estimation of the integrals of the fourth order cumulant spectral density. Biometrika 69(1):117–122

• Taqqu MS, Teverovsky V (1997) Robustness of Whittle-type estimators for time series with long-range dependence (Heavy tails and highly volatile phenomena). Commun. Statist. Stoch. Models 13(4):723–757

• Tong H (1990) Non-linear time series: a dynamical system approach. Oxford University Press

• Whittle P (1953) Estimation and information in stationary time series. Ark Mat 2:423–434

• Zhu K, Li WK (2015) A bootstrapped spectral test for adequacy in weak ARMA models. J Econometr 187(1):113–130

• Zhu K, Ling S (2011) Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA-GARCH/IGARCH models. Ann Stat 39(4):2131–2163

## Acknowledgements

We sincerely thank the anonymous referees and Editor for helpful remarks.

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Correspondence to Yacouba Boubacar Maïnassara.

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Boubacar Maïnassara, Y., Esstafa, Y. & Saussereau, B. Estimating FARIMA models with uncorrelated but non-independent error terms. Stat Inference Stoch Process 24, 549–608 (2021). https://doi.org/10.1007/s11203-021-09243-7

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• DOI: https://doi.org/10.1007/s11203-021-09243-7