Estimation of weak ARMA models with regime changes

  • Yacouba Boubacar MaïnassaraEmail author
  • Landy Rabehasaina


In this paper we derive the asymptotic properties of the least squares estimator (LSE) of autoregressive moving-average (ARMA) models with regime changes under the assumption that the errors are uncorrelated but not necessarily independent. Relaxing the independence assumption considerably extends the range of application of the class of ARMA models with regime changes. Conditions are given for the consistency and asymptotic normality of the LSE. A particular attention is given to the estimation of the asymptotic covariance matrix, which may be very different from that obtained in the standard framework. The theoretical results are illustrated by means of Monte Carlo experiments.


Least square estimation Random coefficients Weak ARMA models 

Mathematics Subject Classification

Primary 62M10 62F03 62F05 Secondary 91B84 62P05 



We sincerely thank the anonymous reviewers and Editor in Chief for helpful remarks. The authors wish to acknowledge the support from the “Séries temporelles et valeurs extrêmes : théorie et applications en modélisation et estimation des risques” Projet Région grant No OPE-2017-0068.


  1. Amendola A, Francq C (2009) Concepts of and tools for nonlinear time-series modelling, chapter 10. Wiley, Hoboen, pp 377–427Google Scholar
  2. Anderson PL, Meerschaert MM (1997) Periodic moving averages of random variables with regularly varying tails. Ann Statist 25(2):771–785MathSciNetzbMATHGoogle Scholar
  3. Andrews DWK (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59(3):817–858MathSciNetzbMATHGoogle Scholar
  4. Andrews B, Davis RA, Breidt FJ (2006) Maximum likelihood estimation for all-pass time series models. J Multivariate Anal 97(7):1638–1659MathSciNetzbMATHGoogle Scholar
  5. Azrak R, Mélard G (1998) The exact quasi-likelihood of time-dependent ARMA models. J Statist Plann Inference 68(1):31–45MathSciNetzbMATHGoogle Scholar
  6. Azrak R, Mélard G (2006) Asymptotic properties of quasi-maximum likelihood estimators for ARMA models with time-dependent coefficients. Stat Inference Stoch Process 9(3):279–330MathSciNetzbMATHGoogle Scholar
  7. Basawa IV, Lund R (2001) Large sample properties of parameter estimates for periodic ARMA models. J Time Ser Anal 22(6):651–663MathSciNetzbMATHGoogle Scholar
  8. Berk KN (1974) Consistent autoregressive spectral estimates. Ann. Statist. 2:489–502 Collection of articles dedicated to Jerzy Neyman on his 80th birthdayMathSciNetzbMATHGoogle Scholar
  9. Bibi A, Francq C (2003) Consistent and asymptotically normal estimators for cyclically time-dependent linear models. Ann Inst Statist Math 55(1):41–68MathSciNetzbMATHGoogle Scholar
  10. Billio M, Monfort A, Robert CP (1999) Bayesian estimation of switching ARMA models. J Econom 93(2):229–255MathSciNetzbMATHGoogle Scholar
  11. Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econom 31(3):307–327MathSciNetzbMATHGoogle Scholar
  12. Boubacar Mainassara Y (2011) Multivariate portmanteau test for structural VARMA models with uncorrelated but non-independent error terms. J Statist Plann Inference 141(8):2961–2975MathSciNetzbMATHGoogle Scholar
  13. Boubacar Maïnassara Y (2012) Selection of weak VARMA models by modified Akaike’s information criteria. J Time Series Anal 33(1):121–130MathSciNetzbMATHGoogle Scholar
  14. Boubacar Mainassara Y, Carbon M, Francq C (2012) Computing and estimating information matrices of weak ARMA models. Comput Statist Data Anal 56(2):345–361MathSciNetzbMATHGoogle Scholar
  15. Boubacar Maïnassara Y, Kokonendji CC (2016) Modified Schwarz and Hannan-Quinn information criteria for weak VARMA models. Stat Inference Stoch Process 19(2):199–217MathSciNetzbMATHGoogle Scholar
  16. Boubacar Maïnassara Y, Saussereau B (2018) Diagnostic checking in multivariate ARMA models with dependent errors using normalized residual autocorrelations. J Am. Statist Assoc 113(524):1813–1827MathSciNetzbMATHGoogle Scholar
  17. Brandt A (1986) The stochastic equation \(Y_{n+1}=A_nY_n+B_n\) with stationary coefficients. Adv Appl Probab 18(1):211–220MathSciNetGoogle Scholar
  18. Brockwell PJ, Davis RA (1991) Time series: theory and methods. Springer series in statistics, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  19. Dahlhaus R (1997) Fitting time series models to nonstationary processes. Ann Statist 25(1):1–37MathSciNetzbMATHGoogle Scholar
  20. Davidson J (1994) Stochastic limit theory. Advanced texts in econometrics. An introduction for econometricians. The Clarendon Press, Oxford University Press, New YorkGoogle Scholar
  21. Davydov JA (1968) Convergence of distributions generated by stationary stochastic processes. Theor Probab Appl 13(2):691–696zbMATHGoogle Scholar
  22. den Haan WJ, Levin AT (1997) A practitioner’s guide to robust covariance matrix estimation. In: Robust inference, volume 15 of Handbook of statist. North-Holland, Amsterdam, pp 299–342Google Scholar
  23. Engle RF (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50(4):987–1007MathSciNetzbMATHGoogle Scholar
  24. Francq C, Gautier A (2003) Estimation of time-varying ARMA models and applications to series subject to Markovian changes in regime. Accessed 11 July 2019
  25. Francq C, Gautier A (2004a) Estimation of time-varying ARMA models with Markovian changes in regime. Statist Probab Lett 70(4):243–251MathSciNetzbMATHGoogle Scholar
  26. Francq C, Gautier A (2004b) Large sample properties of parameter least squares estimates for time-varying ARMA models. J Time Ser Anal 25(5):765–783MathSciNetzbMATHGoogle Scholar
  27. Francq C, Roussignol M (1997) On white noises driven by hidden Markov chains. J Time Ser Anal 18(6):553–578MathSciNetzbMATHGoogle Scholar
  28. Francq C, Roussignol M (1998) Ergodicity of autoregressive processes with Markov-switching and consistency of the maximum-likelihood estimator. Statistics 32(2):151–173MathSciNetzbMATHGoogle Scholar
  29. Francq C, Zakoïan J-M (1998) Estimating linear representations of nonlinear processes. J Statist Plann Inference 68(1):145–165MathSciNetzbMATHGoogle Scholar
  30. Francq C, Zakoïan J-M (2001) Stationarity of multivariate markov-switching ARMA models. J Econom 102(2):339–364MathSciNetzbMATHGoogle Scholar
  31. Francq C, Zakoïan J-M (2002) Autocovariance structure of powers of switching-regime ARMA processes. ESAIM Probab Statist 6:259–270 New directions in time series analysis (Luminy, 2001)MathSciNetGoogle Scholar
  32. 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, volume 1 of GERAD 25th Anniv. Ser. Springer, New York, pp 241–265Google Scholar
  33. Francq C, Zakoïan J-M (2007) HAC estimation and strong linearity testing in weak ARMA models. J Multivariate Anal 98(1):114–144MathSciNetzbMATHGoogle Scholar
  34. Francq C, Zakoïan J-M (2010) GARCH models. Structure, statistical inference and financial applications. Wiley, ChichesterzbMATHGoogle Scholar
  35. Gautier A (2004) Modèles de séries temporelles à coefficients dépendant du temps. Doctoral thesis. University of Lilles 3Google Scholar
  36. Grenander U, Szegö G (1958) Toeplitz forms and their applications. California monographs in mathematical sciences. University of California Press, Berkeley, Los AngeleszbMATHGoogle Scholar
  37. Hamilton JD (1988) Rational-expectations econometric analysis of changes in regime: an investigation of the term structure of interest rates. J Econom Dyn Control 12(2–3):385–423 Economic time series with random walk and other nonstationary componentsMathSciNetzbMATHGoogle Scholar
  38. Hamilton JD (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2):357–384MathSciNetzbMATHGoogle Scholar
  39. Hamilton JD (1990) Analysis of time series subject to changes in regime. J Econom 45(1–2):39–70MathSciNetzbMATHGoogle Scholar
  40. Hamilton JD (1994) Time series analysis. Princeton University Press, PrincetonzbMATHGoogle Scholar
  41. Hamilton JD, Susmel R (1994) Autoregressive conditional heteroskedasticity and changes in regime. J Econom 64(1):307–333zbMATHGoogle Scholar
  42. Herrndorf N (1984) A functional central limit theorem for weakly dependent sequences of random variables. Ann Probab 12(1):141–153MathSciNetzbMATHGoogle Scholar
  43. Jones GL (2004) On the markov chain central limit theorem. Probab Surv 1:299–320MathSciNetzbMATHGoogle Scholar
  44. Kim C-J, Kim J (2015) Bayesian inference in regime-switching ARMA models with absorbing states: the dynamics of the ex-ante real interest rate under regime shifts. J Bus Econom Statist 33(4):566–578MathSciNetGoogle Scholar
  45. Newey WK, West KD (1987) A simple, positive semidefinite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55(3):703–708MathSciNetzbMATHGoogle Scholar
  46. Nicholls DF, Quinn BG (1982) Random coefficient autoregressive models: an introduction, volume 11 of Lecture Notes in Statistics. Springer, New York, Berlin. Lecture Notes in Physics, 151Google Scholar
  47. Norris JR (1998) Markov chains, volume 2 of Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge (reprint of 1997 original)Google Scholar
  48. Romano JP, Thombs LA (1996) Inference for autocorrelations under weak assumptions. J Am Statist Assoc 91(434):590–600MathSciNetzbMATHGoogle Scholar
  49. Stelzer R (2009) On Markov-switching ARMA processes–stationarity, existence of moments, and geometric ergodicity. Econom Theory 25(1):43–62MathSciNetzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Laboratoire de mathématiques de Besançon, UMR CNRS 6623Université Bourgogne Franche-ComtéBesançonFrance

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