• Bernoulli
  • Volume 25, Number 3 (2019), 2359-2408.

Bootstrapping INAR models

Carsten Jentsch and Christian H. Weiß

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Integer-valued autoregressive (INAR) models form a very useful class of processes to deal with time series of counts. Statistical inference in these models is commonly based on asymptotic theory, which is available only under additional parametric conditions and further restrictions on the model order. For general INAR models, such results are not available and might be cumbersome to derive. Hence, we investigate how the INAR model structure and, in particular, its similarity to classical autoregressive (AR) processes can be exploited to develop an asymptotically valid bootstrap procedure for INAR models. Although, in a common formulation, INAR models share the autocorrelation structure with AR models, it turns out that (a) consistent estimation of the INAR coefficients is not sufficient to compute proper ‘INAR residuals’ to formulate a valid model-based bootstrap scheme, and (b) a naïve application of an AR bootstrap will generally fail. Instead, we propose a general INAR-type bootstrap procedure and discuss parametric as well as semi-parametric implementations. The latter approach is based on a joint semi-parametric estimator of the INAR coefficients and the innovations’ distribution. Under mild regularity conditions, we prove bootstrap consistency of our procedure for statistics belonging to the class of functions of generalized means. In an extensive simulation study, we provide numerical evidence of our theoretical findings and illustrate the superiority of the proposed INAR bootstrap over some obvious competitors. We illustrate our method by an application to a real data set about iceberg orders for the Lufthansa stock.

Article information

Bernoulli, Volume 25, Number 3 (2019), 2359-2408.

Received: September 2017
Revised: March 2018
First available in Project Euclid: 12 June 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

bootstrap consistency functions of generalized means INAR residuals parametric bootstrap semi-parametric bootstrap semi-parametric estimation time series of counts


Jentsch, Carsten; Weiß, Christian H. Bootstrapping INAR models. Bernoulli 25 (2019), no. 3, 2359--2408. doi:10.3150/18-BEJ1057.

Export citation


  • [1] Al-Osh, M.A. and Alzaid, A.A. (1987). First-order integer-valued autoregressive ($\operatorname{INAR}(1)$) process. J. Time Ser. Anal. 8 261–275.
  • [2] Alzaid, A.A. and Al-Osh, M. (1990). An integer-valued $p$th-order autoregressive structure ($\operatorname{INAR}(p)$) process. J. Appl. Probab. 27 314–324.
  • [3] Basawa, I.V., Green, T.A., McCormick, W.P. and Taylor, R.L. (1990). Asymptotic bootstrap validity for finite Markov chains. Comm. Statist. Theory Methods 19 1493–1510.
  • [4] Bisaglia, L.B. and Gerolimetto, M. (2016). Estimation of $\operatorname{INAR}(p)$ models using bootstrap. Working Paper Series 06/2016, Univ. Padova.
  • [5] Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [6] Bu, R., McCabe, B. and Hadri, K. (2008). Maximum likelihood estimation of higher-order integer-valued autoregressive processes. J. Time Ser. Anal. 29 973–994.
  • [7] Bühlmann, P. (1995). Bootstraps for time series. Technical Report 431, Dept. Statistics, Univ. California, Berkeley.
  • [8] Bühlmann, P. (1997). Sieve bootstrap for time series. Bernoulli 3 123–148.
  • [9] Cardinal, M., Roy, R. and Lambert, J. (1999). On the application of integer-valued time series models for the analysis of disease incidence. Stat. Med. 18 2025–2039.
  • [10] De Schepper, A. and Heijnen, B. (1995). General restrictions on tail probabilities. J. Comput. Appl. Math. 64 177–188.
  • [11] Doukhan, P., Fokianos, K. and Li, X. (2012). On weak dependence conditions: The case of discrete valued processes. Statist. Probab. Lett. 82 1941–1948.
  • [12] Doukhan, P., Fokianos, K. and Li, X. (2013). Corrigendum to “On weak dependence conditions: The case of discrete valued processes” [Statist. Probab. Lett. 82 (2012) 1941–1948] [MR2970296]. Statist. Probab. Lett. 83 674–675.
  • [13] Drost, F.C., van den Akker, R. and Werker, B.J.M. (2009). Efficient estimation of auto-regression parameters and innovation distributions for semiparametric integer-valued $\operatorname{AR}(p)$ models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71 467–485.
  • [14] Du, J.G. and Li, Y. (1991). The integer-valued autoregressive $(\operatorname{INAR}(p))$ model. J. Time Ser. Anal. 12 129–142.
  • [15] Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1–26.
  • [16] Fink, T. and Kreiß, J.-P. (2013). Bootstrap for random coefficient autoregressive models. J. Time Series Anal. 34 646–667.
  • [17] Freeland, R.K. and McCabe, B. (2005). Asymptotic properties of CLS estimators in the Poisson $\operatorname{AR}(1)$ model. Statist. Probab. Lett. 73 147–153.
  • [18] Freeland, R.K. and McCabe, B.P.M. (2004). Analysis of low count time series data by Poisson autoregression. J. Time Ser. Anal. 25 701–722.
  • [19] Ibragimov, I.A. (1962). Some limit theorems for stationary processes. Theory Probab. Appl. 7 349–382.
  • [20] Jazi, M.A., Jones, G. and Lai, C.-D. (2012). First-order integer valued AR processes with zero inflated Poisson innovations. J. Time Series Anal. 33 954–963.
  • [21] Jentsch, C. and Leucht, A. (2016). Bootstrapping sample quantiles of discrete data. Ann. Inst. Statist. Math. 68 491–539.
  • [22] Jentsch, C. and Politis, D.N. (2013). Valid resampling of higher-order statistics using the linear process bootstrap and autoregressive sieve bootstrap. Comm. Statist. Theory Methods 42 1277–1293.
  • [23] Jentsch, C. and Weiß, C.H. (2019). Supplement to “Bootstrapping INAR models.” DOI:10.3150/18-BEJ1057SUPP.
  • [24] Johnson, N.L., Kemp, A.W. and Kotz, S. (2005). Univariate Discrete Distributions, 3rd ed. Hoboken, NJ: John Wiley & Sons.
  • [25] Jung, R.C., McCabe, B.P.M. and Tremayne, A.R. (2016). Model validation and diagnostics. In Handbook of Discrete-Valued Time Series. Chapman & Hall/CRC Handb. Mod. Stat. Methods 189–218. Boca Raton, FL: CRC Press.
  • [26] Jung, R.C. and Tremayne, A.R. (2006). Coherent forecasting in integer time series models. Int. J. Forecast. 22 223–238.
  • [27] Jung, R.C. and Tremayne, A.R. (2011). Convolution-closed models for count time series with applications. J. Time Series Anal. 32 268–280.
  • [28] Kim, H.-Y. and Park, Y. (2006). Bootstrap confidence intervals for the $\operatorname{INAR}(p)$ process. Korean Commun. Stat. 13 343–358.
  • [29] Kim, H.-Y. and Park, Y. (2008). A non-stationary integer-valued autoregressive model. Statist. Papers 49 485–502.
  • [30] Kreiß, J.-P. (1988). Asymptotical inference for a class of stochastic processes. Habilitationsschrift, Univ. Hamburg.
  • [31] Kreiß, J.-P. (1992). Bootstrap procedures for $\operatorname{AR}(\infty)$-processes. In Bootstrapping and Related Techniques (Trier, 1990). Lecture Notes in Econom. and Math. Systems 376 107–113. Berlin: Springer.
  • [32] Kreiß, J.-P. (1997). Asymptotical properties of residual bootstrap for autoregression. Preprint, TU Braunschweig.
  • [33] Kreiß, J.-P. and Paparoditis, E. (2011). Bootstrap methods for dependent data: A review. J. Korean Statist. Soc. 40 357–378.
  • [34] 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.
  • [35] Künsch, H.R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217–1241.
  • [36] Latour, A. (1998). Existence and stochastic structure of a non-negative integer-valued autoregressive process. J. Time Ser. Anal. 19 439–455.
  • [37] McKenzie, E. (1985). Some simple models for discrete variate time series. Water Resour. Bull. 21 645–650.
  • [38] Meintanis, S.G. and Karlis, D. (2014). Validation tests for the innovation distribution in INAR time series models. Comput. Statist. 29 1221–1241.
  • [39] Meyer, M., Jentsch, C. and Kreiß, J.-P. (2015). Baxter’s inequality and sieve bootstrap for random fields. Working Paper, Univ. Mannheim.
  • [40] Meyer, M., Jentsch, C. and Kreiß, J.-P. (2017). Baxter’s inequality and sieve bootstrap for random fields. Bernoulli 23 2988–3020.
  • [41] Meyer, M. and Kreiß, J.-P. (2015). On the vector autoregressive sieve bootstrap. J. Time Series Anal. 36 377–397.
  • [42] Park, Y. and Kim, H.-Y. (2012). Diagnostic checks for integer-valued autoregressive models using expected residuals. Statist. Papers 53 951–970.
  • [43] Patton, A., Politis, D.N. and White, H. (2009). Correction to “Automatic block-length selection for the dependent bootstrap” by D. Politis and H. White [MR2041534]. Econometric Rev. 28 372–375.
  • [44] Pavlopoulos, H. and Karlis, D. (2008). $\operatorname{INAR}(1)$ modeling of overdispersed count series with an environmental application. Environmetrics 19 369–393.
  • [45] Politis, D.N. and White, H. (2004). Automatic block-length selection for the dependent bootstrap. Econometric Rev. 23 53–70.
  • [46] Schweer, S. (2016). A goodness-of-fit test for integer-valued autoregressive processes. J. Time Series Anal. 37 77–98.
  • [47] Schweer, S. and Weiß, C.H. (2014). Compound Poisson $\operatorname{INAR}(1)$ processes: Stochastic properties and testing for overdispersion. Comput. Statist. Data Anal. 77 267–284.
  • [48] Silva, I. and Silva, M.E. (2006). Asymptotic distribution of the Yule–Walker estimator for $\operatorname{INAR}(p)$ processes. Statist. Probab. Lett. 76 1655–1663.
  • [49] Steutel, F.W. and van Harn, K. (1979). Discrete analogues of self-decomposability and stability. Ann. Probab. 7 893–899.
  • [50] Tsay, R.S. (1992). Model checking via parametric bootstraps in time series analysis. Appl. Stat. 41 1–15.
  • [51] Weiß, C.H. (2012). Process capability analysis for serially dependent processes of Poisson counts. J. Stat. Comput. Simul. 82 383–404.
  • [52] Weiß, C.H. (2018). An Introduction to Discrete-Valued Time Series. Chichester: Wiley.
  • [53] Weiß, C.H., Puig, P. and Homburg, A. (2016). Testing for zero inflation in $\operatorname{INAR}(1)$ models. Statist. Papers. To appear. DOI:10.1007/s00362-016-0851-y.
  • [54] Weiß, C.H. and Schweer, S. (2016). Bias corrections for moment estimators in Poisson $\operatorname{INAR}(1)$ and $\operatorname{INARCH}(1)$ processes. Statist. Probab. Lett. 112 124–130.
  • [55] Yokoyama, R. (1980). Moment bounds for stationary mixing sequences. Z. Wahrsch. Verw. Gebiete 52 45–57.

Supplemental materials

  • Supplement to “Bootstrapping INAR models”. We provide the full simulation results for Section 4.