Bernoulli

  • Bernoulli
  • Volume 24, Number 2 (2018), 1202-1232.

Baxter’s inequality for finite predictor coefficients of multivariate long-memory stationary processes

Akihiko Inoue, Yukio Kasahara, and Mohsen Pourahmadi

Full-text: Open access

Abstract

For a multivariate stationary process, we develop explicit representations for the finite predictor coefficient matrices, the finite prediction error covariance matrices and the partial autocorrelation function (PACF) in terms of the Fourier coefficients of its phase function in the spectral domain. The derivation is based on a novel alternating projection technique and the use of the forward and backward innovations corresponding to predictions based on the infinite past and future, respectively. We show that such representations are ideal for studying the rates of convergence of the finite predictor coefficients, prediction error covariances, and the PACF as well as for proving a multivariate version of Baxter’s inequality for a multivariate FARIMA process with a common fractional differencing order for all components of the process.

Article information

Source
Bernoulli, Volume 24, Number 2 (2018), 1202-1232.

Dates
Received: January 2016
Revised: May 2016
First available in Project Euclid: 21 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1505980894

Digital Object Identifier
doi:10.3150/16-BEJ897

Mathematical Reviews number (MathSciNet)
MR3706792

Zentralblatt MATH identifier
06778363

Keywords
Baxter’s inequality long memory multivariate stationary processes partial autocorrelation functions phase functions predictor coefficients

Citation

Inoue, Akihiko; Kasahara, Yukio; Pourahmadi, Mohsen. Baxter’s inequality for finite predictor coefficients of multivariate long-memory stationary processes. Bernoulli 24 (2018), no. 2, 1202--1232. doi:10.3150/16-BEJ897. https://projecteuclid.org/euclid.bj/1505980894


Export citation

References

  • [1] Baillie, R.T. and Kapetanios, G. (2013). Estimation and inference for impulse response functions from univariate strongly persistent processes. Econom. J. 16 373–399.
  • [2] Baxter, G. (1962). An asymptotic result for the finite predictor. Math. Scand. 10 137–144.
  • [3] Berk, K.N. (1974). Consistent autoregressive spectral estimates. Ann. Statist. 2 489–502.
  • [4] Bhatia, R. (1997). Matrix Analysis. New York: Springer.
  • [5] Bingham, N.H. (2012). Multivariate prediction and matrix Szegö theory. Probab. Surv. 9 325–339.
  • [6] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1989). Regular Variation. Cambridge: Cambridge Univ. Press.
  • [7] Bingham, N.H., Inoue, A. and Kasahara, Y. (2012). An explicit representation of Verblunsky coefficients. Statist. Probab. Lett. 82 403–410.
  • [8] Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods, 2nd ed. New York: Springer.
  • [9] Bühlmann, P. (1995). Moving-average representation of autoregressive approximations. Stochastic Process. Appl. 60 331–342.
  • [10] Bühlmann, P. (1997). Sieve bootstrap for time series. Bernoulli 3 123–148.
  • [11] Cheng, R. and Pourahmadi, M. (1993). Baxter’s inequality and convergence of finite predictors of multivariate stochastic processes. Probab. Theory Related Fields 95 115–124.
  • [12] Chung, C.-F. (2001). Calculating and analyzing impulse responses for the vector ARFIMA model. Econom. Lett. 71 17–25.
  • [13] Damanik, D., Pushnitski, A. and Simon, B. (2008). The analytic theory of matrix orthogonal polynomials. Surv. Approx. Theory 4 1–85.
  • [14] Dègerine, S. (1990). Canonical partial autocorrelation function of a multivariate time series. Ann. Statist. 18 961–971.
  • [15] Ginovian, M.S. (1999). Asymptotic behavior of the prediction error for stationary random sequences. Izv. Nats. Akad. Nauk Armenii Mat. 34 18–36.
  • [16] Granger, C.W.J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Series Anal. 1 15–29.
  • [17] Hannan, E.J. and Deistler, M. (1988). The Statistical Theory of Linear Systems. New York: Wiley.
  • [18] Helson, H. and Lowdenslager, D. (1961). Prediction theory and Fourier series in several variables. II. Acta Math. 106 175–213.
  • [19] Hosking, J.R.M. (1981). Fractional differencing. Biometrika 68 165–176.
  • [20] Ibragimov, I.A. and Solev, V.N. (1968). Asymptotic behavoir of the prediction error of a stationary sequence with a spectral density of special form. Theory Probab. Appl. 13 703–707.
  • [21] Ing, C.-K., Chiou, H.-T. and Guo, M. (2016). Estimation of inverse autocovariance matrices for long memory processes. Bernoulli 22 1301–1330.
  • [22] Inoue, A. (2000). Asymptotics for the partial autocorrelation function of a stationary process. J. Anal. Math. 81 65–109.
  • [23] Inoue, A. (2002). Asymptotic behavior for partial autocorrelation functions of fractional ARIMA processes. Ann. Appl. Probab. 12 1471–1491.
  • [24] Inoue, A. (2008). AR and MA representation of partial autocorrelation functions, with applications. Probab. Theory Related Fields 140 523–551.
  • [25] Inoue, A. and Kasahara, Y. (2004). Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing. J. Multivariate Anal. 89 135–147.
  • [26] Inoue, A. and Kasahara, Y. (2006). Explicit representation of finite predictor coefficients and its applications. Ann. Statist. 34 973–993.
  • [27] Inoue, A., Kasahara, Y. and Pourahmadi, M. (2016). The intersection of past and future for multivariate stationary processes. Proc. Amer. Math. Soc. 144 1779–1786.
  • [28] Kasahara, Y. and Bingham, N.H. (2014). Verblunsky coefficients and Nehari sequences. Trans. Amer. Math. Soc. 366 1363–1378.
  • [29] Kasahara, Y., Inoue, A. and Pourahmadi, M. (2016). Rigidity for matrix-valued Hardy functions. Integral Equations Operator Theory 84 289–300.
  • [30] Katsnelson, V.E. and Kirstein, B. (1997). On the theory of matrix-valued functions belonging to the Smirnov class. In Topics in Interpolation Theory (Leipzig, 1994). Oper. Theory Adv. Appl. 95 299–350. Basel: Birkhäuser.
  • [31] Kreiss, J.-P., Paparoditis, E. and Politis, D.N. (2011). On the range of validity of the autoregressive sieve bootstrap. Ann. Statist. 39 2103–2130.
  • [32] Masani, P. (1960). The prediction theory of multivariate stochastic processes. III. Unbounded spectral densities. Acta Math. 104 141–162.
  • [33] Meyer, M., Jentsch, C. and Kreiss, J.-P. (2017). Baxter’s inequality and sieve bootstrap for random fields. Bernoulli 23 2988–3020.
  • [34] Meyer, M., McMurry, T. and Politis, D. (2015). Baxter’s inequality for triangular arrays. Math. Methods Statist. 24 135–146.
  • [35] Peller, V.V. (2003). Hankel Operators and Their Applications. New York: Springer.
  • [36] Poskitt, D.S., Grose, S.D. and Martin, G.M. (2015). Higher-order improvements of the sieve bootstrap for fractionally integrated processes. J. Econometrics 188 94–110.
  • [37] Pourahmadi, M. (2001). Foundations of Time Series Analysis and Prediction Theory. New York: Wiley.
  • [38] Rozanov, Yu.A. (1967). Stationary Random Processes. San Francisco: Holden-Day.
  • [39] Rupasinghe, M. and Samaranayake, V.A. (2012). Asymptotic properties of sieve bootstrap prediction intervals for FARIMA processes. Statist. Probab. Lett. 82 2108–2114.
  • [40] Sarason, D. (1978). Function Theory on the Unit Circle. Blacksburg, VA: Virginia Polytechnic Institute and State Univ. Notes for lectures given at a Conference at Virginia Polytechnic Institute and State University, Blacksburg, Va., June 19–23, 1978.