Random multiplication versus random sum: auto-regressive-like models with integer-valued random inputs

Abdelhakim Aknouche; Sonia Gouveia; Manuel Scotto

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arXiv · 2312.11137

Random multiplication versus random sum: auto-regressive-like models with integer-valued random inputs

Abdelhakim Aknouche·Sonia Gouveia·Manuel Scotto

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Abstract

A common approach to analyze count time series is to fit models based on random sum operators. As an alternative, this paper introduces time series models based on a random multiplication operator, which is simply the multiplication of a variable operand by an integer-valued random coefficient, whose mean is the constant operand. Such operation is endowed into auto-regressive-like models with integer-valued random inputs, addressed as RMINAR. Two special variants are studied, namely the N0-valued random coefficient auto-regressive model and the N0-valued random coefficient multiplicative error model. Furthermore, Z-valued extensions are considered. The dynamic structure of the proposed models is studied in detail. In particular, their corresponding solutions are everywhere strictly stationary and ergodic, a fact that is not common neither in the literature on integer-valued time series models nor real-valued random coefficient auto-regressive models. Therefore, the parameters of the RMINAR model are estimated using a four-stage weighted least squares estimator, with consistency and asymptotic normality established everywhere in the parameter space. Finally, the new RMINAR models are illustrated with some simulated and empirical examples.

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Abdelhakim Aknouche, Sonia Gouveia, Manuel Scotto. 2023-12-18. Random multiplication versus random sum: auto-regressive-like models with integer-valued random inputs. https://arxiv.org/abs/2312.11137

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