Autoregressive conditional proportion: A multiplicative-error model for (0,1)-valued time series

Aknouche, Abdelhakim and Dimitrakopoulos, Stefanos (2021): Autoregressive conditional proportion: A multiplicative-error model for (0,1)-valued time series.

Abstract

We propose a multiplicative autoregressive conditional proportion (ARCP) model for (0,1)-valued time series, in the spirit of GARCH (generalized autoregressive conditional heteroscedastic) and ACD (autoregressive conditional duration) models. In particular, our underlying process is defined as the product of a (0,1)-valued iid sequence and the inverted conditional mean, which, in turn, depends on past reciprocal observations in such a way that is larger than unity. The probability structure of the model is studied in the context of the stochastic recurrence equation theory, while estimation of the model parameters is performed by the exponential quasi-maximum likelihood estimator (EQMLE). The consistency and asymptotic normality of the EQMLE are both established under general regularity assumptions. Finally, the usefulness of our proposed model is illustrated with simulated and two real datasets.

• Item Type: MPRA Paper
• Language: English
• Keywords: Proportional time series data, Beta-ARMA model, Simplex ARMA, Autoregressive conditional duration, Exponential QMLE.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C25 - Discrete Regression and Qualitative Choice Models ; Discrete Regressors ; Proportions ; Probabilities; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C46 - Specific Distributions ; Specific Statistics; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C51 - Model Construction and Estimation; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C58 - Financial Econometrics

References

Aknouche, A. and Francq, C. (2021a). Count and duration time series with equal conditional stochastic and mean orders. Econometric Theory 37, 248--280.

Aknouche, A. and Francq, C. (2021b). Two-stage weighted least squares estimator of the conditional mean of observation-driven time series models. Journal of Econometrics, forthcoming.

Aknouche, A. and Francq, C. (2021c). Stationarity and ergodicity of Markov-switching positive conditional mean models. Journal of Time Series Analysis, forthcoming.

Aknouche, A., Almohaimeed, B. and Dimitrakopoulos, S. (2021). Periodic autoregressive conditional duration. Journal of Time Series Analysis, forthcoming.

Aknouche, A., Bendjeddou, S. and Touche, N. (2018). Negative binomial quasi-likelihood inference for general integer-valued time series models. Journal of Time Series Analysis 39, 192--211.

Baker, S.R., Bloom, N. Davis, S.J. and Kost, K. (2019). Policy news and stock market volatility. Working paper, Federal Reserve Bank of St. Louis.

Barndorff-Nielsen, O.E. and Jorgensen, B. (1991). Some parametric models on the Simplex. Journal of Multivariate Analysis 39, 106--116.

Bayer, F.M., Cintra, R.J. and Cribari-Neto, F. (2018). Beta seasonal autoregressive moving average models. Journal of Statistical Computation and Simulation 88, 2961-2981.

Benjamin, M.A., Rigby, R.A. and Stasinopoulos, D.M. (2003). Generalized autoregressive moving average models. Journal of the American Statistical Association 98, 214--223.

Bonat, W.H., Petterle, R.R., Hinde, J. and Demétrio, C.G.B. (2019). Flexible quasi-Beta regression models for continuous bounded data. Statistical Modeling 19, 617--633.

Bougerol, P. (1993). Kalman filtering with random coefficients and contractions. SIAM Journal on Control and Optimization 31, 942--959.

Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Annals of Probability, 20, 1714-1730.

Box, G.E.P., Jenkins, G.M. and Reinsel, G.C. (1994) Time Series Analysis; Forecasting and Control. 3rd Edition, Prentice Hall, Englewood Cliff, New Jersey.

Brockwell, P.J. and Davis, R.A. (1991). Time series: theory and methods. Springer Science & Business Media.

Cipolini, F. and Gallo, G.M. (2021). Multiplicative Error Models: 20 years on. ArXiv working paper, arXiv:2107.05923.

Creal, D., Koopman, S.J. and Lucas, A. (2013). Generalized autoregressive score models with applications. Journal of Applied Econometrics 28, 777--795.

Da Silva, C.Q. and Migon, H.S. (2016). Hierarchical dynamic beta model. REVSTAT -- Statistical Journal 14, 49--73.

Davis, R.A. and Liu, H. (2016). Theory and inference for a class of nonlinear models with application to time series of counts. Statistica Sinica 26, 1673--1707.

Engle, R. (2002). New frontiers for ARCH models. Journal of Applied Econometrics 17, 425--446.

Engle, R. and Gallo. G.M. (2006). A Multiple Indicators Model for Volatility Using Intra-Daily Data. Journal of Econometrics 131, 3--27.

Engle, R. and Russell, J. (1998). Autoregressive conditional duration: A new model for irregular spaced transaction data. Econometrica 66, 1127--1162.

Espinheira, P.L. and Silva, A.O.S. (2020). Residual and influence analysis to a general class of Simplex regression. Test 29, 523--552.

Ferrari, S.L.P. and Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics 31 799--815.

Ferland, R., Latour, A., and Oraichi, D. (2006). Integer-valued GARCH process. Journal of Time Series Analysis 27, 923--942.

Francq, C., and Thieu, L. (2019). QML inference for volatility models with covariates. Econometric Theory 35, 37--72.

Francq, C. and Zakoian, J.-M. (2019). GARCH Models: Structure, Statistical Inference and Financial Applications, 2nd edition, John Wiley & Sons.

Fabrizio, C., Engle, R.F. and Gallo. G.M. (2013). Semiparametric Vector MEM. Journal of Applied Econometrics 28, 1067--1086.

Grassia, A. (1977). On a family of distributions with argument between 0 and 1 obtained by transformation of the Gamma distribution and derived compound distributions. Australian Journal of Statistics 19, 108--14.

Grunwald, G., Hyndman, R., Tedesco, L., and Tweedie, R. (2000). Non-Gaussian conditional linear AR(1) models. Australian and New Zealand Journal of Statistics 42, 479--495.

Gorgi, P. and Koopman, S.J. (2021). Beta observation-driven models with exogenous regressors: A joint analysis of realized correlation and leverage effects. Journal of Econometrics, forthcoming.

Guolo, A. and Varin, C. (2014). Beta regression for time series analysis of bounded data, with application to Canada google flu trends. Annals of Applied Statistics, 8, 74--88.

Grunwald, G., Hyndman, R., Tedesco, L., and Tweedie, R. (2000). Non-Gaussian conditional linear AR(1) models. Australian and New Zealand Journal of Statistics 42, 479--495.

Hamilton J.D. (1994). Time series analysis. Princeton University Press, Princeton, NJ.

Han, H. and Kristensen, D. (2014). Asymptotic theory for the QMLE in GARCH-X models with stationary and nonstationary covariates. Journal of Business & Economic Statistics 32, 416--429.

Hautsch, N. (2012). Econometrics of financial high-frequency data. Springer-Verlag.

Heinen, A. (2003). Modelling time series count data: an autoregressive conditional Poisson model. Available at SSRN 1117187.

Jorgensen, B. (1997). The theory of dispersion models. Chapman and Hall, London.

Kieschnick, R. and McCullough, B.D. (2003). Regression analysis of variates observed on (0,1): Percentages, proportions and fractions. Statistical Modelling 3, 193--213.

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology 46, 79--88.

Li, W.K. (1994). Time series models based on generalized linear models: Some further results. Biometrics 50, 506--511.

McKenzie, E. (2003). Discrete variate time series, in Handbook of statistics. Amsterdam: Elsevier Science.

McKenzie, E. (1985) An autoregressive process for beta random variables. Management Science 31, 988--97.

McCullagh, P. and Nelder, J.A. (1989). Generalized Linear Models (2nd edn). London: Chapman and Hall.

Mitnik, P.A. and Baek, S. (2013). The Kumaraswamy distribution: Median-dispersion re-parameterizations for regression modeling and simulation-based estimation. Statistical Papers 54, 177--92.

Paolino, P. (2001). Maximum likelihood estimation of models with Beta-distributed dependent variables. Political Analysis 9, 325--346.

Patton, A.J. (2011). Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics 160, 246--256.

Pumi, G., Valk, M., Bisognin, C., Bayer, F.M., and Prass, T.S. (2019). Beta autoregressive fractionally integrated moving average models. Journal of Statistical Planning and Inference 200, 196--212.

Ristić, M.M., Weiss, C.H. and Janjić, A.D. (2016). A binomial integer-valued ARCH model. International Journal of Biostatistics 12, 1-21.

Rocha, A.V. and Cribari-Neto, F. (2009). Beta autoregressive moving average models. TEST 18, 529--545.

Wooldridge, J.M. (1999). Quasi-Likelihood Methods for Count Data. In M.H. Pesaran and P. Schmidt (ed.), Handbook of Applied Econometrics, Volume 2: Microeconomics, (pp. 35--406). Oxford: Blackwell.

Zeger, S.L., and Qaqish, B. (1988). Markov regression models for time series: A quasi-likelihood approach. Biometrics 44, 1019--1031.

Zhang, P., Qiu, Z. and Shi, C. (2016). Simplexreg: An R package for regression analysis of proportional data using the Simplex distribution. Journal of Statistical Software 71, 1--21.

Zheng, T., Xiao, H. and Chen, R. (2015). Generalized ARMA with martingale difference errors. Journal of Econometrics 189, 492--506.