Page 35 - Textos de Matemática Vol. 47

THE TAYLOR PROPERTY IN NON-NEGATIVE AUTOREGRESSIVE AND BILINEAR STOCHASTIC PROCESSES
ESMERALDA GONC¸ALVES AND CRISTINA M. MARTINS
Dedicated to our dear friend Nazar´e
Abstract. Inthiswork,wecomebacktotheanalysisoftheTaylorprop- erty in linear and bilinear models. Limiting the study to non-negative models, we begin by recalling the main theoretical results on its occur- rence in autoregressive and diagonal bilinear models of order one. These results are discussed in models with significantly di↵erent values of the kurtosis coe cient. In both classes, it is possible to relate the presence of the Taylor property and the kurtosis of the model. Moreover, in the autoregressive case the property is present if and only if the generator process is symmetric or right-skewed distributed.
1. Introduction
Taylor e↵ect is a characteristic present in temporal series of diverse nature.
This stylized fact was, for the first time, detected by Taylor (1986) in the returns
of some financial series. Taylor found that the autocorrelations of the absolute
returns were systematically higher than the corresponding autocorrelations of
squared observations. Namely, considering T observations, X1 , X2 , ..., XT , from
aprocessX=(X,t2Z),Taylorfoundthat ⇢b (h)>⇢b 2 (h),h=1,2,..., t |X|X
where ⇢bX denotes the empirical autocorrelation function of X. This fact is now known as Taylor e↵ect.
The corresponding theoretical relation is called the Taylor property, and assuming that the functions ⇢|X| and ⇢X2 are positive, is expressed by the
Accepted: 23 March 2015.
2010 Mathematics Subject Classification. 62M10.
Key words and phrases. Autoregressive and Bilinear models, kurtosis, Taylor property.
This work was partially supported by the Centre for Mathematics of the Univer- sity of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Part- nership Agreement PT2020.
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