SpringerLink
Log in

Linear Time Series Models with Non-Gaussian Innovations

  • Chapter
  • First Online:
Non-Gaussian Autoregressive-Type Time Series

  • 426 Accesses

Abstract

The time series models with normally distributed innovations generate stationary normal sequences. However, if the innovations are not normal then the stationary marginal distribution may be a member of an entirely different family. This chapter discusses autoregressive models with innovations belonging to various classes of non-Gaussian distributions. Detailed analysis of the models with innovation distributions such as stable, Laplace, heavy-tailed, exponential, gamma, and mixed normal is considered along with the problem of estimation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    See Chap. 1 for some results on explicit forms of ARMA models. More details are available in [11].

  2. 2.

    See the discussions in Sect. 1.2.3.

  3. 3.

    These regularity conditions due to [10] are listed in Theorem 2.2 of Chap. 2.

  4. 4.

    See Sect. 2.7 for details on the Estimating Function methods.

  5. 5.

    See Sect. 2.7 for a list of the regularity conditions and Theorem 2.5 for a complete statement.

  6. 6.

    Billingsley’s regularity conditions for the CAN properties of MLE and the related asymptotic results are stated in Sect. 2.3

  7. 7.

    These conditions are listed in Chap. 2. The main results are stated in Theorem 2.2.

  8. 8.

    See Sect. 2.3.3 for details on MMLE.

  9. 9.

    An AR(1) model with stationary Levy-marginal distribution is discussed in Sect. 3.7.3.

References

  1. B. Mandelbrot, The variation of certain speculative prices. J. Bus. 36, 394– 419 (1963)

    Google Scholar 

  2. B. Mandelbrot, The variation of some other speculative prices. J. Bus. 40, 393–413 (1967)

    Google Scholar 

  3. B. Mandelbrot, H.M. Taylor, On the distribution of stock price difference. Oper. Res. 15, 1057–1062 (1967)

    Google Scholar 

  4. E.F. Fama, The behavior of stock-market prices. J. Bus. 38(1), 34–105 (1965)

    Google Scholar 

  5. G. Schoups, J.A. Vrugt, A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic, and non-gaussian errors. Water Res. Res. 46, 1–17 (2010)

    Article  Google Scholar 

  6. J. Andel, On AR (1) processes with exponential white noise. Commun. Stat. Theory Methods 17(5), 1481–1495 (1988)

    Google Scholar 

  7. C.B. Bell, E.P. Smith, Inference for non-negative autoregressive schemes. Commun. Stat. Theory Methods 15(8), 2267–2293 (1986)

    Article  MATH  Google Scholar 

  8. P. Bondon, Estimation of autoregressive models with epsilon-skew-normal innovations. J. Multivar. Anal. 100(8), 1761–1776 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. W. Hurlimann, On non-Gaussian AR (1) inflation modeling,. J. Stat. Econ. Methods 1(1), 93–101 (2012)

    Google Scholar 

  10. W.K. Li, A.I. McLeod, ARMA modelling with non-Gaussian innovations. J. Time Ser. Anal. 9(2), 155–168 (1988)

    Google Scholar 

  11. G.E.P. Box, G.M. Jenkins, G.C. Reinsel, Time Series Analysis: Forecasting and Control (Prentice Hall, Englewood Cliffs, 1994)

    MATH  Google Scholar 

  12. P.J. Brockwell, R.A. Davis, Time Series Theory and Methods, 2nd edn. (Springer, New York., 2006)

    MATH  Google Scholar 

  13. G. Samorodnitsky, M. Taqqu, Stable Non-Gaussian Random Processes (Chapman Hall, 1994)

    Google Scholar 

  14. S. Davis, R.A. Resnick, Limit theory for the sample covariance and correlation functions of moving averages. Ann. Stat. 14, 533–558 (1986)

    Google Scholar 

  15. J.W. Lin, A.I. McLeod, Portmanteau tests for ARMA models with infinite variance. J. Time Ser. Anal. 29(3), 600–617 (2008)

    Google Scholar 

  16. D.B.H. Cline, P.J. Brockwell, Linear prediction of ARMA processes with infinite variance. Stoch. Process. Appl. 19, 281–296 (1985)

    Google Scholar 

  17. Z. Qiou, N. Ravishanker, Bayesian inference for time series with stable innovations. J. Time Ser. Anal. 19(2), 235–249 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. J.L. Knight, J. Yu, Empirical characteristic function in time series estimation. Economet. Theor. 18(3), 691–721 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. T. Merkouris, Transform martingale estimating functions. Ann. Stat. 35(5), 1975–2000 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Thavaneswaran, N. Ravishanker, Y. Liang, Inference for linear and nonlinear stable error processes via estimating functions. J. Stat. Plann. Inference. 143(4), 827–841 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. B. Andrews, M. Calder, R.A. Davis, Maximum likelihood estimation for \(\alpha -\)stable autoregressive processes. Ann. Stat. 37(4), 1946–1982 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. R.M. Zhang, N.H. Chan, Maximum likelihood estimation for nearly non-stationary stable autoregressive processes. J. Time Ser. Anal. 33, 542–553 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Billingsley, Statistical inference for Markov processes, vol. 2 (University of Chicago Press, 1961)

    Google Scholar 

  24. M. Shao, C.L. Nikias, Signal processing with fractional lower ordermoments: stable processes and their applications. Proc IEEE. 81(7), 986–1010 (1993)

    Google Scholar 

  25. C.M. Gallagher, A method for fitting stable autoregressive models using the autocovariation function. Stat. Probab. Lett. 53, 381–390 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. N. Balakrishna, G. Hareesh, Analysis of autoregressive models with symmetric stable innovations. Statistics 52(2), 288–302 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. N. Balakrishna, G. Hareesh, Statistical signal extraction using stable processes. Stat. Probab. Lett. 79(7), 851–856 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. N. Balakrishna, G. Hareesh, Stable autoregressive models and signal estimation. Commun. Stat. Theory Methods 41(11), 1969–1988 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. J.P. Nolan, N. Ravishanker, Simultaneous prediction intervals for ARMA processes with stable innovations. J. Forecast. 28, 235–246 (2009)

    Google Scholar 

  30. N.L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions (Wiley, New York., 1995)

    MATH  Google Scholar 

  31. N. Balakrishna, C.K. Nampoothiri, Cauchy autoregressive process and its applications. J. Indian Stat. Assoc. 41(2), 143–156 (2003)

    Google Scholar 

  32. J. Andel, T. Barton, A note on threshold AR(1) model with cauchy innovations. J. Time Ser. Anal. 7, 1–5 (1986)

    Google Scholar 

  33. J. Choi, I. Choi, Maximum likelihood estimation of autoregressive models with a near unit root and cauchy errors. Ann. Inst. Stat. Math. 71, 1121–1142 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. E. Damsleth, A.H. El-Shaarawi, ARMA models with double-exponentially distributed noise. J. R. Stat. Soc. B 51(1), 61–69 (1989)

    Google Scholar 

  35. S.Y. Hwang, I.V. Basawa, Godambe estimating functions and asymptotic optimal inference. Stat. Probab. Lett. 81(8), 1121–1127 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. P.H. Diananda, Note on some properties of maximum likelihood estimates. Proc. Camb. Philos. Soc. 45, 536–544 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  37. M.K. Varanasi, B. Aazhang, Parametric generalized gaussian density estimation. J. Acoust. Soc. Am. 86(4), 1404–1415 (1989)

    Article  Google Scholar 

  38. S. Nadarajah, A generalized normal distribution. J. Appl. Stat. 32(7), 685–694 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. G. Agro, Maximum likelihood estimation for the exponential power function parameters. Commun. Stat. Simul. Comput. 24, 523–536 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  40. G.E.P. Box, G.C. Tiao, A further look at robustness via Bayes’s theorem. Biometrika 49, 419–432 (1962)

    Google Scholar 

  41. J. Ledolter, Inference robustness of ARIMA models under non-normality - special application to stock price data. Metrika 26, 43–56 (1979)

    Google Scholar 

  42. R.W. Barnard, A.A. Trindade, R.I.P. Wickamasinghe, Autoregressive moving average models under exponential power distributions. Prob. Stat. Forum 07, 65–77 (2014)

    MathSciNet  MATH  Google Scholar 

  43. N. Balakrishna, C.G. Sri Ranganath, ARMA models with generalized error distributed innovations. J. Indian Stat. Assoc. 53(1), 11–34 (2015)

    Google Scholar 

  44. M.L. Tiku, W.K. Wong, D.C. Vaughan, G. Bian, Time series models in non-normal situations: symmetric innovations. J. Time Ser. Anal. 21(5), 571–596 (2000)

    Google Scholar 

  45. H. Haghbin, A.R. Nematollahi, Likelihood-based inference in autoregressive models with scaled t-distributed innovations by means of em-based algorithms. Commun. Stat. - Simul. Comput. 42(10), 2239–2252 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  46. U.C. Nduka, Em-based algorithms for autoregressive models with t-distributed innovations. Commun. Stat. Simul. Comput. 47(1), 206–228 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  47. M.L. Tiku, R.P. Suresh, A new method of estimation for location and scale parameters. J. Stat. Plann. Inference 30, 281–292 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  48. W. Polasec, J. Pai, Cpo plots for ARMA model selection. CEJOR 12, 211–219 (2004)

    Google Scholar 

  49. A.D. Akkaya, M.L. Tiku, Time series AR(1) model for shorttailed distributions. Stat: A J. Theor. Appl. Stat. 39(2), 117 – 132 (2005)

    Google Scholar 

  50. M.L. Tiku, W.K. Wong, G. Bian, Time series models with asymmetric innovations. Commun. Stat. Theory and Methods 28(6), 1331–1360 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  51. W.K. Wong, G. Bian, Estimating parameters in autoregressive models with asymmetric innovations. Stat. Probab. Lett. 71, 61–70 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  52. T.A. Ula, C. Yozgatligil, Modified maximum-likelihood method for non-normal time series revisited. Commun. Stat. Theory Method. 33(2), 397–417 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  53. N. Sarlac, Annual streamflow modelling with asymmetric distribution function. Hydrol. Process. 22, 3403–3409 (2008)

    Article  Google Scholar 

  54. A. Trindade, Y. Zhu, B. Andrews, Time series models with asymmetric Laplace innovations. J. Stat. Comput. Simul. 80(12), 1317–1333 (2010)

    Google Scholar 

  55. D.S. Mudholkar, A.D. Hutson, The epsilon-skew-normal distribution for analyzing near-normal data. J. Stat. Plann. Inference 83(2), 291–309 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  56. O. Barndorff-Nielsen, Exponentially decreasing distributions for the logarithm of particle size, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 353 (The Royal Society, 1977), pp. 401–419

    Google Scholar 

  57. O. Barndorff-Nielsen, N. Shephard, Non-Gaussian ornstein-uhlenbeck-based models and some of their uses in financial economics (with discussion), J. R. Statist. Soc. B 63, 167–241. The Royal Society (2001)

    Google Scholar 

  58. C. Halgreen, Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Z. Wahrsch. verw. Gebiete 71(1), 13–17 (1979)

    Google Scholar 

  59. S. Ghasami, Z. Khodadadi, M. Maleki. Autoregressive processes with generalized hyperbolic innovations. Commun. Stat. Simul. Comput. 1–14 (2019)

    Google Scholar 

  60. H. Karttunen, An autoregressive model based on the generalized hyperbolic distribution. Scand. J. Stat. 47, 787–816 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  61. O. Barndorff-Nielsen, Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Stat. 24(1), 1–13 (1997)

    Google Scholar 

  62. G.N. Boshnakov, Analytic expressions for predictive distributions in mixture autoregressive models. Statist. Probab. Lett. 79(15), 1704–1709 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  63. N.D. Le, R.D. Martin, A.E. Raftery, Modeling flat stretches, bursts, and outliers in time series using mixture transition distribution models. J. Amer. Statist. Assoc. 91, 1504–1515 (1996)

    MathSciNet  MATH  Google Scholar 

  64. C.S. Wong, W.K. Li, On a mixture autoregressive model. J. Roy. Stat. Soc. B 62(1), 91–115 (2000)

    Article  MathSciNet  Google Scholar 

  65. C.S. Wong, W.S. Chan, Mixture Gaussian time series modeling of long-term market returns. North American Actuarial Journal. 9(4), 83–94 (2013)

    Google Scholar 

  66. C. S. Wong, W. S. Chan, and Kam P. L. A student t-mixture autoregressive model with applications to heavy-tailed financial data. Biometrika., 96:751–760, (2009)

    Google Scholar 

  67. H.D. Nguyena, G.J. McLachlan, J.F.P. Ullmannb, A.L. Janke, Laplace mixture autoregressive models. Stat. Probab. Lett. 110, 18–24 (2016)

    Article  MathSciNet  Google Scholar 

  68. S. **, W.K. Li, Modeling panel time series with mixture autoregressive model. J. Data Sci. 4, 425–446 (2006)

    Article  Google Scholar 

  69. A. Aknouche, Recursive online em estimation of mixture autoregressions. J. Stat. Comput. Simul. 83(2), 370–383 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  70. L. Kalliovirta, M. Meitz, P. Saikkonen, A Gaussian mixture autoregressive model. J. Time Series Anal. 36, 247–266 (2015)

    Google Scholar 

  71. M. Meitz, P. Saikkonen, Testing for observation-dependent regime switching in mixture autoregressive model. J. Econ. 222(1), 601–624 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  72. J.D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2), 357–384 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  73. R.L.G. Latimier, E.L. Bouedec, V. Monbet, Markov switching autoregressive modeling of wind power forecast errors. Electr. Power Syst. Res. 189, 1–7 (2020)

    Google Scholar 

  74. B. Nielsen, N. Shephard, Likelihood analysis of a first-order autoregressive model with exponential innovations. J. Time Ser. Anal. 24, 337–344 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  75. H. Fellag, L. Atil, C. Belkacem, On the stability of estimation of AR(1) coefficient in the presence of contaminated exponential innovations. Afr. Stat. 9, 647–658 (2014)

    Google Scholar 

  76. R.A. Davis, W.P. McCormick, Estimation for first- order autoregressive processes with positive or bounded innovations. Stoch. Process. Appl. 31, 237–250 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  77. P.D. Feigin, S.I. Resnick, Estimation for autoregressive processes with positive innoatins. Commun. Stat. Stoch. Models 8, 479–498 (1999)

    MATH  Google Scholar 

  78. B. Abraham, N. Balakrishna, Inverse Gaussian autoregressive models. J. Time Ser. Anal. 20(6), 605–618 (1999)

    Google Scholar 

  79. Ing Chiang-Kang, Yang Chiao-Yi, Predictor selection for positive autoregressive processes. J. Am. Stat. Assoc. 109(505), 243–253 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  80. S. Datta, W.P. McCormick, Bootstrap inference for a first-order autoregression with positive innovations. J. Am. Stat. Assoc. 90(432), 1289–1300 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  81. Hsian Wei-Cheng, Huang Hao-Yun, Ing Chiang-Kang, Interval estimation for a first order positive autoregressive process. J. Time Ser. Anal. 39, 447–467 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  82. A. Bartlett, W.P. McCormick, Estimation for non-negative time series with heavy-tail innovations. J. Time Ser. Anal. 34, 96–115 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  83. P.D. Feigin, S.I. Resnick, C. Starica, Testing for independence in heavy tailed and positive innovation time series. Commun. Stat. Stoch. Models 11(7), 587–612 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  84. J. Andel, Non-negative autoreressive processes. J. Time Ser. Anal. 10(1), 1–11 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  85. K.H. Lee, J.G. Park, Estimation in autoregressive process with non-negative innovations. Youngnam Stat. Lett. 3(1), 65–78 (1992)

    Google Scholar 

  86. P.D. Feigin, S.I. Resnick, Limit distributions for linear programming time series estimators. Stoch. Process. Their Appl. 51, 135–165 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  87. B. Lagos-Álvarez, G. Ferreira1, E. Porcu, Modified maximum likelihood estimation in autoregressive processes with generalized exponential innovations. Open J. Stat. 4, 620–629 (2014)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Cochin University of Science and Technology, Kochi, Kerala, India

    N. Balakrishna

Authors
  1. N. Balakrishna
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Balakrishna .

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Balakrishna, N. (2021). Linear Time Series Models with Non-Gaussian Innovations. In: Non-Gaussian Autoregressive-Type Time Series. Springer, Singapore. https://doi.org/10.1007/978-981-16-8162-2_6

Download citation

Publish with us

Policies and ethics

Search

Navigation