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38 J. LEITE
Taylor [20]): (i) absence of autocorrelations in the returns; (ii) positive corre- lation in squared returns and absolute returns; (iii) volatility clustering, since large absolute returns tend to appear in clusters; (iv) leverage e↵ects, meaning that there is an asymmetric impact of past positive and negative values on the current volatility; and (v) heavy-tailed distribution of returns.
The study of stylized facts associated to these series dates back, at least, to 1963, when Mandelbrot [16] identified the volatility clusters. The leverage e↵ects were noted by Black [1], in 1976. These better known stylized facts have very intuitive explanations, as, for example, in the leverage e↵ects case (see [4]), we all now recognize that negative returns (corresponding to price decreases) tend to increase volatility by a larger amount than positive returns (price increases) of the same magnitude.
detected by Taylor in 1986. More precisely, while studying 40 return series, Taylor [19] observes that, for n = 1, 2, ..., 30,
A relatively more recent and less intuitive stylized fact, that seems to emerge from the combination of volatility clusters and high kurtosis, is the Taylor ef- fect, which involves the sample autocorrelations of power-transformed absolute
returns, ⇢ˆ (k) = cdorr ⇣|r |k , |r |k ⌘ with k > 0, so-called because it was first n tt n
⇢ˆn (1) > ⇢ˆn (2) . (1.1)
Ding, Granger and Engle [2], Granger and Ding [10], Granger, Spear and Ding [11] and Taylor [20] extend this initial analysis, finding that almost always, for n = 1, 2, ..., 625,
⇢ˆn (1) > ⇢ˆn (k) , for some values of k between 0 and 3. (1.2) So Granger and Ding [10] define the Taylor e↵ect to be present in a series if
⇢ˆn(1)>⇢ˆn(k), forallnandk6=1. (1.3)
Granger [9] adds this to the list of stylized facts that characterize return se- ries dynamics. More studies with further discussion and evidence are cited in Haas [12]. Additionally, we refer that this e↵ect, more specifically the relation (1.1), has also been detect by Gon¸calves, Mendes-Lopes, Dorotoviˇc, Fernandes and Garcia [8] in physical phenomena time series.
Taylor e↵ect strong empirical evidence raises the question whether or not empirically relevant volatility models, like the ones in the generalized autore- gressive conditional heteroskedastic (GARCH) class, can capture this e↵ect. In literature, two ways have been considered to address this problem: via simu- lation or using a theoretical approach. Simulations have the advantage of not requiring ⇢n (k) theoretical expressions for each model, however only a small number of model parameterizations can be considered. Since, in the GARCH