Two-stage weighted least squares estimator of the conditional mean of observation-driven time series models

Aknouche, Abdelhakim and Francq, Christian (2019): Two-stage weighted least squares estimator of the conditional mean of observation-driven time series models.

Abstract

General parametric forms are assumed for the conditional mean λ_{t}(θ₀) and variance υ_{t}(ξ₀) of a time series. These conditional moments can for instance be derived from count time series, Autoregressive Conditional Duration (ACD) or Generalized Autoregressive Score (GAS) models. In this paper, our aim is to estimate the conditional mean parameter θ₀, trying to be as agnostic as possible about the conditional distribution of the observations. Quasi-Maximum Likelihood Estimators (QMLEs) based on the linear exponential family fulfill this goal, but they may be inefficient and have complicated asymptotic distributions when θ₀ contains zero coefficients. We thus study alternative weighted least square estimators (WLSEs), which enjoy the same consistency property as the QMLEs when the conditional distribution is misspecified, but have simpler asymptotic distributions when components of θ₀ are null and gain in efficiency when υ_{t} is well specified. We compare the asymptotic properties of the QMLEs and WLSEs, and determine a data driven strategy for finding an asymptotically optimal WLSE. Simulation experiments and illustrations on realized volatility forecasting are presented.

• Item Type: MPRA Paper
• Language: English
• Keywords: Autoregressive Conditional Duration model; Exponential, Poisson, Negative Binomial QMLE; INteger-valued AR; INteger-valued GARCH; Weighted LSE.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C18 - Methodological Issues: General; 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 > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C58 - Financial Econometrics

References

Ahmad, A. and Francq, C. (2016). Poisson qmle of count time series models. Journal of Time Series analysis, 37, 291-314.

Aknouche, A., Bendjeddou, S. and Touche, N. (2018). Negative Binomial Quasi-Likelihood Inference for General Integeral Integer-Valued Time Series Models. Journal of Time Series Analysis, 39, 192-211.

Aknouche, A. and Francq, C. (2019). Count and duration time series with equal conditional stochastic and mean orders. MPRA Working paper No. 90838.

Al-Osh, M.A. and Alzaid, A.A. (1987). First-order integer-valued autoregressive (INAR(1)) process. Journal of Time Series Analysis, 8, 261-275.

Alzaid, A.A. and Al-Osh, M.A. (1990). An integer-valued pth-order autoregressive structure (INAR(p)) process. Journal of Applied Probability, 27, 314-324.

Barndorff-Nielsen, O.E., Shephard, N. (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society, 64, 253-280.

Bernardi, M., and Catania, L. (2014). The model confidence set package for R. arXiv preprint arXiv:1410.8504v1.

Blasques, F., Gorgi, P., Koopman, S.J., and Wintenberger, O. (2018). Feasible invertibility conditions and maximum likelihood estimation for observation-driven models. Electronic Journal of Statistics, 12, 1019-1052.

Blasques, F., Koopman, S.J. and Lucas, A. (2015). Information-theoretic optimality of observation-driven time series models for continuous responses. Biometrika, 102, 325-343.

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

Cameron, A.C. and Trivedi P.K. (1986). Econometric models based on count data: comparisons and applications of some estimators and tests. Journal of Applied Econometrics, 1, 29-53.

Cameron, C. and Trivedi, P. (1998). Regression analysis of count data. Cambridge University Press, New York.

Christou, V. and Fokianos, K. (2014). Quasi-likelihood inference for negative binomial time series models. Journal of Time Series Analysis, 35, 55-78.

Cox, D.R. (1981). Statistical analysis of time series: Some recent developments. Scandinavian Journal of Statistics, 8, 93-115.

Creal, D., Koopman, S.J. and Lucas, A. (2011). A dynamic multivariate heavy-tailed model for time-varying volatilities and correlations. Journal of Business and Economic Statistics, 29, 552-563.

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

Csörgö, M. (1968). On the strong law of large numbers and the central limit theorem for martingales. Transactions of the American Mathematical Society, 131, 259-275.

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.

Du, J.-G. and Li, Y. (1991). The integer-valued autoregressive (INAR(p)) model. Journal of Time Series Analysis, 12, 130-142.

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

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

Francq, C. and J-M. Zakoian (2019). GARCH models: structure, statistical inference and financial applications. Chichester: John Wiley, second edition.

Hansen, P. R., Lunde, A., and Nason, J. M. (2011). The model confidence set. Econometrica, 79, 453-497.

Harvey, A.C. (2013). Dynamic models for Volatility and Heavy Tails. Econometric Society Monograph. Cambridge University Press.

Harvey, A.C. and Chakravarty, T. (2008). Beta-t-(E)GARCH. University of Cambridge, Faculty of Economics.

Heinen, A. (2003). Modelling time series count data: an autoregressive conditional poisson model. CORE Discussion Paper 2003/62, Université Catholique de Louvain.

Gouriéroux, C., Monfort, A. and Trognon, A. (1984). Pseudo maximum likelihood methods: Theory. Econometrica, 52, 681-700.

Morris, C. (1982). Natural Exponential Families with Quadratic Variance Functions. The Annals of Statistics, 10, 65-80.

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

Steutel, F.W. and Van Harn, K. (1979). Discrete analogues of self-decomposability and stability. The Annals of Probability, 5, 893-899.

Straumann, D., and Mikosch, T. (2006). Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equations approach. The Annals of Statistics, 34, 2449-2495.

Wedderburn, R.W. (1974). Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika, 61, 439-447.

Zheng, H., Basawa, I.V., and Datta, S. (2006). Inference for pth-order random coefficient integer valued autoregressive processes. Journal of Time Series Analysis, 27, 411-440.