Page 36 - Textos de Matemática Vol. 47

26 E. GONC¸ALVES AND C. M. MARTINS
condition
⇢|X|(h)>⇢X2 (h), h2N,
with ⇢X denoting the autocorrelation function of X.
The presence of this property in a certain class of models for time series can
help in the selection of a more appropriate model to describe the dynamics of the series of interest. However, this study requires the knowledge of moments of X of order higher than 2 which makes such analysis theoretically di cult.
Since the Taylor e↵ect was initially detected in financial time series, it is nat- ural that the first theoretical studies have focused on models used in the analysis of these series, specifically the conditionally heteroskedastic models, as can be found in He and Ter¨asvirta (1999), Gon¸calves, Leite and Mendes-Lopes (2009), Haas (2009) and Leite (2013), whose studies involve the class of Generalized Threshold Autoregressive Conditionally Heteroskedastic (GTARCH) models.
Interestingly, these studies have shown that the presence of the Taylor prop- erty appears to have a stronger relationship with the weight of the tails of the process than with the characteristic of heteroskedasticity. This finding leads us to question the presence of the Taylor property in models without conditional heteroskedasticity but still suitable for financial time series analysis.
The modeling of non-negative time series has been widely regarded in the literature (Zang and Tong, 2001) and in Tsay and Chan (2007) the interest of non-negative ARMA processes in the context of stochastic modeling of fi- nancial time series is referred. Additionally, the recent finding that the Taylor e↵ect is clearly present in some non-negative physical time series (Gonc¸alves et al., 2014) also leads us to question the presence of the property in bilinear models, which reveal themselves useful in the analysis of time series recorded, for instance, in seismology, hydrology and astronomy.
So, this paper focuses on non-negative stochastic models and, given the complexity of the calculations, we limit our study to the first-order AR and bilinear models.
Returning to Gonc¸alves, Martins and Mendes-Lopes (2014, 2015) we begin by recalling the fundamental results on the occurrence of the Taylor property in non-negative autoregressive and bilinear models of order 1. We point out that the property is analyzed for all lag h in the case of linear models. On the other hand, in the case of the bilinear model our study is limited to h = 1, due to the di culty associated with the calculations of the autocorrelations. The presence of the Taylor property is then assured in the linear case when the error process is symmetrically or right-skewed distributed and new examples are developed considering autoregressive and bilinear models with error processes presenting di↵erent values of kurtosis, being notorious the importance of this parameter on the presence of the Taylor property in the model.