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Stationary Processes

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Introduction to Time Series and Forecasting

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Abstract

A key role in time series analysis is played by processes whose properties, or some of them, do not vary with time. If we wish to make predictions, then clearly we must assume that something does not vary with time. In extrapolating deterministic functions it is common practice to assume that either the function itself or one of its derivatives is constant. The assumption of a constant first derivative leads to linear extrapolation as a means of prediction. In time series analysis our goal is to predict a series that typically is not deterministic but contains a random component. If this random component is stationary, in the sense of Definition 1.4.2, then we can develop powerful techniques to forecast its future values. These techniques will be developed and discussed in this and subsequent chapters.

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References

  • Akaike, H.(1969). Fitting autoregressive models for prediction. Annals of the Institute of Statistical Mathematics, 21, 243–247.

    Article  MathSciNet  MATH  Google Scholar 

  • Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In B. N. Petrov & F. Csaki (Eds.), 2nd International Symposium on Information Theory (pp. 267–281). Budapest: Akademiai Kiado.

    Google Scholar 

  • Akaike, H. (1978). Time series analysis and control through parametric models. In D. F. Findley (Ed.), Applied time series analysis. New York: Academic.

    Google Scholar 

  • Andersen, T. G., & Benzoni, L. (2009). Realized volatility. In T. G. Andersen, R. A. Davis, J.-P. Kreiss, & T. V. Mikosch (Eds.), Handbook of financial time series (pp. 555–576). Berlin, Heidelberg: Springer.

    Chapter  Google Scholar 

  • Andersen, T. G., Davis, R. A., Kreiss, J.-P., & Mikosch, T. V. (Eds.) (2009). Handbook of financial time series. Berlin: Springer.

    Google Scholar 

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

    MATH  Google Scholar 

  • Anderson, T. W. (1980). Maximum likelihood estimation for vector autoregressive moving-average models. In D. R. Brillinger & G. C. Tiao (Eds.), Directions in time series (pp. 80–111). Beachwood: Institute of Mathematical Statistics.

    Google Scholar 

  • Ansley, C. F. (1979). An algorithm for the exact likelihood of a mixed autoregressive-moving-average process. Biometrika, 66, 59–65.

    Article  MathSciNet  MATH  Google Scholar 

  • Ansley, C. F., & Kohn, R. (1985). On the estimation of ARIMA models with missing values. In E. Parzen (Ed.), Time series analysis of irregularly observed data. Springer lecture notes in statistics (Vol. 25, pp. 9–37), Springer-Verlag, Berlin, Heidelberg, New York.

    Google Scholar 

  • Aoki, M. (1987). State space modeling of time series. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Applebaum, D. Lévy processes and stochastic calculus. Cambridge: Cambridge University Press.

    Google Scholar 

  • Atkins, S. M. (1979). Case study on the use of intervention analysis applied to traffic accidents. Journal of the Operations Research Society, 30(7), 651–659.

    Article  MATH  Google Scholar 

  • Bachelier, L. (1900). Théorie de la spéculation. Annales de lÉcole Normale Supérieure, 17, 21–86.

    MathSciNet  MATH  Google Scholar 

  • Baillie, R. T., Bollerslev, T., & Mikkelsen, H. O. (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 74, 3–30.

    Article  MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen, O. E. (1978). Information and exponential families in statistical theory. New York: Wiley.

    MATH  Google Scholar 

  • Barndorff-Niesen, O. E., & Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck based models and some of their uses in financial economics (with discussion). Journal of the Royal Statistical Society Series B, 63, 167–241.

    Article  MathSciNet  MATH  Google Scholar 

  • Bertoin, J. (1996). Lévy processes. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Bhattacharyya, M. N., & Layton, A. P. (1979). Effectiveness of seat belt legislation on the Queensland road toll—An Australian case study in intervention analysis. Journal of the American Statistical Association, 74, 596–603.

    Google Scholar 

  • Billingsley, P. (1995). Probability and measure (3rd ed.). New York: Wiley.

    MATH  Google Scholar 

  • Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654.

    Article  MathSciNet  MATH  Google Scholar 

  • Bloomfield, P. (2000). Fourier analysis of time series: An introduction (2nd ed.). New York: Wiley.

    Book  MATH  Google Scholar 

  • Bollerslev, T. (1986), Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307–327.

    Article  MathSciNet  MATH  Google Scholar 

  • Bollerslev, T., & Mikkelsen, H. O. (1996). Modeling and pricing long memory in stock market volatility. Journal of Econometrics, 73, 151–184.

    Article  MathSciNet  MATH  Google Scholar 

  • Box, G. E. P., & Cox, D. R. (1964). An analysis of transformations (with discussion). Journal of the Royal Statistical Society B, 26, 211–252.

    MathSciNet  MATH  Google Scholar 

  • Box, G. E. P., & Jenkins, G. M. (1976). Time series analysis: Forecasting and control (revised edition). San Francisco: Holden-Day.

    MATH  Google Scholar 

  • Box, G. E. P., & Pierce, D. A. (1970). Distribution of residual autocorrelations in autoregressive-integrated moving-average time series models. Journal of the American Statistical Association, 65, 1509–1526.

    Article  MathSciNet  MATH  Google Scholar 

  • Box, G. E. P., & Tiao, G. C. (1975). Intervention analysis with applications to economic and environmental problems. Journal of the American Statistical Association, 70, 70–79.

    Article  MathSciNet  MATH  Google Scholar 

  • Breidt, F. J., & Davis, R. A. (1992). Time reversibility, identifiability and independence of innovations for stationary time series. Journal of Time Series Analysis, 13, 377–390.

    Article  MathSciNet  MATH  Google Scholar 

  • Brockwell, P.J. (2014), Recent results in the theory and applications of CARMA processes, Ann. Inst. Stat. Math. 66, 637–685.

    MathSciNet  MATH  Google Scholar 

  • Brockwell, P. J., Chadraa, E., & Lindner, A. (2006). Continuous-time GARCH processes. Annals of Applied Probability, 16, 790–826.

    Article  MathSciNet  MATH  Google Scholar 

  • Brockwell, P. J., & Davis, R. A. (1988). Applications of innovation representations in time series analysis. In J. N. Srivastava (Ed.), Probability and statistics, essays in honor of Franklin A. Graybill (pp. 61–84). Amsterdam: Elsevier.

    Google Scholar 

  • Brockwell, P. J., & Davis, R. A. (1991). Time series: Theory and methods (2nd ed.). New York: Springer.

    Book  MATH  Google Scholar 

  • Brockwell, P. J., & Lindner, A. (2009). Existence and uniqueness of stationary Lévy-driven CARMA processes. Stochastic Processes and Their Applications, 119, 2660–2681.

    Article  MathSciNet  MATH  Google Scholar 

  • Brockwell, P. J., & Lindner, A. (2012). Integration of CARMA processes and spot volatility modelling. Journal of Time Series Analysis, 34, 156–167.

    Article  MathSciNet  MATH  Google Scholar 

  • Campbell, J., Lo, A., & McKinlay, C. (1996). The econometrics of financial markets. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Chan, K. S., & Ledolter, J. (1995). Monte Carlo EM estimation for time series models involving counts. Journal of the American Statistical Association, 90, 242–252.

    Article  MathSciNet  MATH  Google Scholar 

  • Chan, K. S., & Tong, H. (1987). A note on embedding a discrete parameter ARMA model in a continuous parameter ARMA model. Journal of Time Series Analysis, 8, 277–281.

    Article  MathSciNet  MATH  Google Scholar 

  • Cochran, D., & Orcutt, G. H. (1949). Applications of least squares regression to relationships containing autocorrelated errors. Journal of the American Statistical Association, 44, 32–61.

    Google Scholar 

  • Davis, M., & Etheridge, A. (2006). Louis Bachelier’s theory of speculation: The origins of modern finance. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Davis, M. H. A., & Vinter, R. B. (1985). Stochastic modelling and control. London: Chapman and Hall.

    Book  MATH  Google Scholar 

  • Davis, R. A., Chen, M., & Dunsmuir, W. T. M. (1995). Inference for MA(1) processes with a root on or near the unit circle. Probability and Mathematical Statistics, 15, 227–242.

    MathSciNet  MATH  Google Scholar 

  • Davis, R. A., Chen, M., & Dunsmuir, W. T. M. (1996). Inference for seasonal moving-average models with a unit root. In Athens conference on applied probability and time series, volume 2: Time series analysis. Lecture notes in statistics (Vol. 115, pp. 160–176). Berlin: Springer.

    Google Scholar 

  • Davis, R. A., & Dunsmuir, W. T. M. (1996). Maximum likelihood estimation for MA(1) processes with a root on or near the unit circle. Econometric Theory, 12, 1–29.

    Article  MathSciNet  Google Scholar 

  • Davis, R. A., & Mikosch, T. V. (2009). Probabilistic properties of stochastic volatility models. In T. G. Andersen, R. A. Davis, J.-P. Kreiss, & T. V. Mikosch (Eds.), Handbook of financial time series (pp. 255–268). Berlin: Springer.

    Chapter  Google Scholar 

  • de Gooijer, J. G., Abraham, B., Gould, A., & Robinson, L. (1985). Methods of determining the order of an autoregressive-moving-average process: A survey. International Statistical Review, 53, 301–329.

    Article  MathSciNet  MATH  Google Scholar 

  • Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society B, 39, 1–38.

    MathSciNet  MATH  Google Scholar 

  • Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of American Statistical Association, 74, 427–431.

    MathSciNet  MATH  Google Scholar 

  • Douc, R., Roueff, F., & Soulier, P. (2008). On the existence of some ARCH(∞) processes. Stochastic Processes and Their Applications, 118, 755–761.

    Article  MathSciNet  MATH  Google Scholar 

  • Duong, Q. P. (1984). On the choice of the order of autoregressive models: a ranking and selection approach. Journal of Time Series Analysis, 5, 145–157.

    Article  MathSciNet  MATH  Google Scholar 

  • Eberlein, E. (2009). Jump-type Lévy processes. In T. G. Andersen, R. A. Davis, J.-P. Kreiss, & T. V. Mikosch (Eds.), Handbook of financial time series (pp. 439–456). Berlin: Springer.

    Chapter  Google Scholar 

  • Eller, J. (1987). On functions of companion matrices. Linear Algebra and Applications, 96, 191–210.

    Article  MathSciNet  MATH  Google Scholar 

  • Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation. Econometrica, 50, 987–1007.

    Article  MathSciNet  MATH  Google Scholar 

  • Engle, R. F. (1995). ARCH: Selected readings. Advanced texts in econometrics. Oxford: Oxford University Press.

    Google Scholar 

  • Engle, R. F., & Bollerslev, T. (1986). Modelling the persistence of conditional variances. Economic Review, 5, 1–50.

    Article  MathSciNet  MATH  Google Scholar 

  • Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation and testing. Econometrica, 55, 251–276.

    Article  MathSciNet  MATH  Google Scholar 

  • Engle, R. F., & Granger, C. W. J. (1991). Long-run economic relationships. Advanced texts in econometrics. Oxford: Oxford University Press.

    Google Scholar 

  • Francq, C., & Zakoian, J.-M. (2010). GARCH models: Structure, statistical inference and financial applications. New York: Wiley.

    Book  MATH  Google Scholar 

  • Fuller, W. A. (1976). Introduction to statistical time series. New York: Wiley.

    MATH  Google Scholar 

  • Gourieroux, C. (1997). ARCH models and financial applications. New York: Springer.

    Book  MATH  Google Scholar 

  • Granger, C. W. J. (1981). Some properties of time series data and their use in econometric model specification. Journal of Econometrics, 16, 121–130.

    Article  Google Scholar 

  • Gray, H. L., Kelley, G. D., & McIntire, D. D. (1978). A new approach to ARMA modeling. Communications in Statistics, B7, 1–77.

    Article  MathSciNet  MATH  Google Scholar 

  • Graybill, F. A. (1983). Matrices with applications in statistics. Belmont, CA: Wadsworth.

    MATH  Google Scholar 

  • Grunwald, G. K., Hyndman, R. J., & Hamza, K. (1994). Some Properties and Generalizations of Nonnegative Bayesian Time Series Models, Technical Report. Statistics Dept., Melbourne University, Parkville, Australia.

    Google Scholar 

  • Grunwald, G. K., Raftery, A. E., & Guttorp, P. (1993). Prediction rule for exponential family state space models. Journal of the Royal Statistical Society B, 55, 937–943.

    MATH  Google Scholar 

  • Hannan, E. J. (1980). The estimation of the order of an ARMA process. Annals of Applied Statistics, 8, 1071–1081.

    Article  MathSciNet  MATH  Google Scholar 

  • Hannan, E. J., & Deistler, M. (1988). The statistical theory of linear systems. New York: Wiley.

    MATH  Google Scholar 

  • Hannan, E. J., & Rissanen, J. (1982). Recursive estimation of mixed autoregressive moving-average order. Biometrika, 69, 81–94.

    Article  MathSciNet  MATH  Google Scholar 

  • Harvey, A. C. (1990). Forecasting, structural time series models and the Kalman filter. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  • Harvey, A. C., & Fernandes, C. (1989). Time Series models for count data of qualitative observations. Journal of Business and Economic Statistics, 7, 407–422.

    Google Scholar 

  • Holt, C. C. (1957). Forecasting seasonals and trends by exponentially weighted moving averages. ONR research memorandum (Vol. 52). Pittsburgh, PA: Carnegie Institute of Technology.

    Google Scholar 

  • Hurvich, C. M., & Tsai, C. L. (1989). Regression and time series model selection in small samples. Biometrika, 76, 297–307.

    Article  MathSciNet  MATH  Google Scholar 

  • Iacus, S.M. and Mercuri, L. (2015), Implementation of Lévy CARMA model in Yuima package, Comput. Stat., 30, 1111–1141.

    Article  MathSciNet  MATH  Google Scholar 

  • Jarque, C. M., & Bera, A. K. (1980). Efficient tests for normality, heteroscedasticity and serial independence of regression residuals. Economics Letters, 6, 255–259.

    Article  MathSciNet  Google Scholar 

  • Jones, R. H. (1975). Fitting autoregressions, Journal of American Statistical Association, 70, 590–592.

    MATH  Google Scholar 

  • Jones, R. H. (1978). Multivariate autoregression estimation using residuals. In D. F. Findley (Ed.), Applied time series analysis (pp. 139–162). New York: Academic.

    Chapter  Google Scholar 

  • Jones, R. H. (1980). Maximum likelihood fitting of ARMA models to time series with missing observations. Technometrics, 22, 389–395.

    Article  MathSciNet  MATH  Google Scholar 

  • Kendall, M. G., & Stuart, A. (1976). The advanced theory of statistics (Vol. 3). London: Griffin.

    MATH  Google Scholar 

  • Kitagawa, G. (1987). Non-Gaussian state-space modeling of non-stationary time series. Journal of the American Statistical Association, 82 (with discussion), 1032–1063.

    Google Scholar 

  • Klebaner, F. (2005). Introduction to stochastic calculus with applications. London: Imperial College Press.

    Book  MATH  Google Scholar 

  • Klüppelberg, C., Lindner, A., & Maller, R. (2004). A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour. Journal of Applied Probability, 41, 601–622.

    Article  MathSciNet  MATH  Google Scholar 

  • Kuk, A. Y. C., & Cheng, Y. W. (1994). The Monte Carlo Newton-Raphson Algorithm, Technical Report S94-10. Department of Statistics, U. New South Wales, Sydney, Australia.

    Google Scholar 

  • Lehmann, E. L. (1983). Theory of point estimation. New York: Wiley.

    Book  MATH  Google Scholar 

  • Lehmann, E. L. (1986). Testing statistical hypotheses (2nd ed.). New York: Wiley.

    Book  MATH  Google Scholar 

  • Lindner, A. (2009). Stationarity, mixing, distributional properties and moments of GARCH(p,q)-processes. In T. G. Andersen, R. A. Davis, J.-P. Kreiss, & T. V. Mikosch (Eds.), Handbook of financial time series (pp. 233–254). Berlin: Springer.

    Google Scholar 

  • Liu, J. & Brockwell, P. J. (1988). The general bilinear time series model. Journal of Applied Probability, 25, 553–564.

    Article  MathSciNet  MATH  Google Scholar 

  • Ljung, G. M., & Box, G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika, 65, 297–303.

    Article  MATH  Google Scholar 

  • Lütkepohl, H. (1993). Introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

    Book  MATH  Google Scholar 

  • Mage, D. T. (1982). An objective graphical method for testing normal distributional assumptions using probability plots. American Statistician, 36, 116–120.

    Google Scholar 

  • Makridakis, S., Andersen, A., Carbone, R., Fildes, R., Hibon, M., Lewandowski, R., Newton, J., Parzen, E., & Winkler, R. (1984). The forecasting accuracy of major time series methods. New York: Wiley.

    Google Scholar 

  • Makridakis, S., Wheelwright, S. C., & Hyndman, R. J. (1997). Forecasting: Methods and applications. New York: Wiley.

    Google Scholar 

  • May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261, 459–467.

    Article  Google Scholar 

  • McCullagh, P., & Nelder, J. A. (1989). Generalized linear models (2nd ed.). London: Chapman and Hall.

    Book  MATH  Google Scholar 

  • McLeod, A. I., & Li, W. K. (1983). Diagnostic checking ARMA time series models using squared-residual autocorrelations. Journal of Time Series Analysis, 4, 269–273.

    Article  MathSciNet  MATH  Google Scholar 

  • Mendenhall, W., Wackerly, D. D., and Scheaffer, D. L. (1990). Mathematical statistics with applications (4th ed.). Belmont: Duxbury.

    MATH  Google Scholar 

  • Merton, R. (1973). The theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183.

    Article  MathSciNet  MATH  Google Scholar 

  • Mikosch, T. (1998), Elementary stochastic calculus with finance in view. Singapore: World Scientific.

    Book  MATH  Google Scholar 

  • Mood, A.M., Graybill, F.A. and Boes, D.C. (1974), Introduction to the Theory of Statistics, McGraw-Hill, New York.

    MATH  Google Scholar 

  • Nelson, D. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59, 347–370.

    Article  MathSciNet  MATH  Google Scholar 

  • Newton, H. J., & Parzen, E. (1984). Forecasting and time series model types of 111 economic time series. In S. Makridakis, et al. (Eds.), The forecasting accuracy of major time series methods. New York: Wiley.

    Google Scholar 

  • Nicholls, D. F., & Quinn, B. G. (1982). Random coefficient autoregressive models: An introduction. Springer lecture notes in statistics (Vol. 11), Springer-Verlag, Berlin, Heidelberg, New York.

    Google Scholar 

  • Oksendal, B. (2013). Stochastic differential equations: An introduction with applications (6th ed.). New York: Springer.

    MATH  Google Scholar 

  • Pantula, S. (1991). Asymptotic distributions of unit-root tests when the process is nearly stationary. Journal of Business and Economic Statistics, 9, 63–71.

    Google Scholar 

  • Parzen, E. (1982), ARARMA models for time series analysis and forecasting. Journal of Forecasting, 1, 67–82.

    Article  MathSciNet  Google Scholar 

  • Pole, A., West, M., & Harrison, J. (1994). Applied Bayesian forecasting and time series analysis. New York: Chapman and Hall.

    Book  MATH  Google Scholar 

  • Priestley, M. B. (1988). Non-linear and non-stationary time series analysis. London: Academic.

    MATH  Google Scholar 

  • Protter, P. E. (2010). Stochastic integration and differential equations (2nd ed.). New York: Springer.

    Google Scholar 

  • Rosenblatt, M. (1985). Stationary sequences and random fields. Boston: Birkhäuser.

    Book  MATH  Google Scholar 

  • Said, S. E., & Dickey, D. A. (1984). Testing for unit roots in autoregressive moving-average models with unknown order. Biometrika, 71, 599–607.

    Article  MathSciNet  MATH  Google Scholar 

  • Sakai, H., & Tokumaru, H. (1980). Autocorrelations of a certain chaos. In IEEE Transactions on Acoustics, Speech and Signal Processing (Vol. 28, pp. 588–590).

    Google Scholar 

  • Samuelson, P. A. (1965). Rational theory of warrant pricing. Industrial Management Review, 6, 13–31.

    Google Scholar 

  • Sato, K. (1999). Lévy processes and infinitely divisible distributions. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Schoutens, W. (2003). Lévy processes in finance. New York: Wiley.

    Book  Google Scholar 

  • Schwert, G. W. (1987). Effects of model specification on tests for unit roots in macroeconomic data. Journal of Monetary Economics, 20, 73–103.

    Article  Google Scholar 

  • Shapiro, S. S., & Francia, R. S. (1972). An approximate analysis of variance test for normality. Journal of the American Statistical Association, 67, 215–216.

    Article  Google Scholar 

  • Shephard, N. (1996). Statistical aspects of ARCH and stochastic volatility. In D. R. Cox, D. V. Hinkley, & O. E. Barndorff-Nielsen (Eds.), Time series models in econometrics, finance and other fields (pp. 1–67). London: Chapman and Hall.

    Google Scholar 

  • Shephard, N., & Andersen, T. G. (2009). Stochastic volatility: Origins and overview. In T. G. Andersen, R. A. Davis, J.-P. Kreiss, & T. V. Mikosch (Eds.), Handbook of financial time series (pp. 233–254). Berlin, Heidelberg: Springer.

    Chapter  Google Scholar 

  • Shibata, R. (1976), Selection of the order of an autoregressive model by Akaike’s information criterion. Biometrika, 63, 117–126.

    Article  MathSciNet  MATH  Google Scholar 

  • Shibata, R. (1980), Asymptotically efficient selection of the order of the model for estimating parameters of a linear process. Annals of Statistics, 8, 147–164.

    Article  MathSciNet  MATH  Google Scholar 

  • Silvey, S. D. (1975). Statistical inference. New York: Halsted.

    MATH  Google Scholar 

  • Smith, J. Q. (1979). A generalization of the Bayesian steady forecasting model. Journal of the Royal Statistical Society B, 41, 375–387.

    MathSciNet  MATH  Google Scholar 

  • Sorenson, H. W., & Alspach, D. L. (1971). Recursive Bayesian estimation using Gaussian sums. Automatica, 7, 465–479.

    Article  MathSciNet  MATH  Google Scholar 

  • Subba-Rao, T., & Gabr, M. M. (1984). An introduction to bispectral analysis and bilinear time series models. Springer lecture notes in statistics (Vol. 24), Springer-Verlag, Berlin, Heidelberg, New York.

    Google Scholar 

  • Tam, W. K., & Reinsel, G. C. (1997). Tests for seasonal moving-average unit root in ARIMA models. Journal of the American Statistical Association, 92, 725–738.

    Article  MathSciNet  MATH  Google Scholar 

  • Tanaka, K. (1990). Testing for a moving-average unit root. Econometric Theory, 9, 433–444.

    Article  MathSciNet  Google Scholar 

  • Taylor, S. J. (1982). Financial returns modelled by the product of two stochastic processes-a study of the daily sugar prices 1961–75. Time Series Analysis: Theory and Practice, 1, 203–226.

    Google Scholar 

  • Taylor, S. J (1986). Modelling financial time series. New York: Wiley.

    MATH  Google Scholar 

  • Tong, H. (1990). Non-linear time series: A dynamical systems approach. Oxford: Oxford University Press.

    MATH  Google Scholar 

  • Venables, W. N., Ripley, B. D. (2003). Modern applied statistics with S (4th ed.). New York: Springer.

    MATH  Google Scholar 

  • Weigt, G. (2015), ITSM-R Reference Manual. The manual can be downloaded from http://eigenmath.sourceforge.net/itsmr-refman.pdf.

  • Weiss, A. A. (1986). Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory, 2, 107–131.

    Article  Google Scholar 

  • West, M., & Harrison, P. J. (1989). Bayesian forecasting and dynamic models. New York: Springer.

    Book  MATH  Google Scholar 

  • Whittle, P. (1963). On the fitting of multivariate autoregressions and the approximate canonical factorization of a spectral density matrix. Biometrika, 40, 129–134.

    Article  MathSciNet  MATH  Google Scholar 

  • Wichern, D., & Jones, R. H. (1978). Assessing the impact of market disturbances using intervention analysis. Management Science, 24, 320–337.

    MATH  Google Scholar 

  • Wu, C. F. J. (1983). On the convergence of the EM algorithm. Annals of Statistics, 11, 95–103.

    Article  MathSciNet  MATH  Google Scholar 

  • Zeger, S. L. (1988). A regression model for time series of counts. Biometrika, 75, 621–629.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Statistics, Colorado State University, Fort Collins, CO, USA

    Peter J. Brockwell

  2. Department of Statistics, Columbia University, New York, NY, USA

    Richard A. Davis

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Brockwell, P.J., Davis, R.A. (2016). Stationary Processes. In: Introduction to Time Series and Forecasting. Springer Texts in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-29854-2_2

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