Abstract
A fast method for estimating the parameters of a stable-APARCH not requiring likelihood or iteration is proposed. Several powerful tests for the (asymmetric) stable Paretian distribution with tail index 1 < α < 2 are used for assessing the appropriateness of the stable assumption as the innovations process in stable-GARCH-type models for daily stock returns. Overall, there is strong evidence against the stable as the correct innovations assumption for all stocks and time periods, though for many stocks and windows of data, the stable hypothesis is not rejected.
Highlights
Since the seminal work of Mandelbrot [1] and Fama [2], the stable Paretian distribution for modeling financial asset returns has generated increasing interest, as computer power required for its computation became commonplace, and given its appealing theoretical properties of (i) summability; (ii) its heavy tails and asymmetry that are ubiquitously characteristic of financial asset returns; and (iii) it being the limiting distribution of sums of various independent identically distributed (i.i.d.) random variables, i.e., such sums are in the domain of attraction of a stable law; see, e.g., Geluk and De Haan [6] and the references therein
Serving as a heuristic for judging the overall assessment of stability in the (APARCH-filtered innovations of the percentage returns of the) stocks, we show the p-values for all stocks and (i) all windows of length T “ 500 and (ii) based on the entire length of the return sequence, as well as looking at the mean rejection rate of the likelihood ratio test (LRT) test of size 0.05
A fast new method for estimating the parameters of the Sα,β -asymmetric power ARCH (APARCH) is developed, and this is used to obtain the filtered innovation sequences when the model is applied to daily financial asset returns data
Summary
Since the seminal work of Mandelbrot [1] and Fama [2], the stable Paretian distribution for modeling financial asset returns has generated increasing interest, as computer power required for its computation became commonplace, and given its appealing theoretical properties of (i) summability (as can be used for portfolio analysis; see [3,4,5], and the references therein); (ii) its heavy tails and asymmetry that are ubiquitously characteristic of financial asset returns; and (iii) it being the limiting distribution of sums of various independent identically distributed (i.i.d.) random variables, i.e., such sums are in the domain of attraction of a stable law; see, e.g., Geluk and De Haan [6] and the references therein. We attempt to counteract this phenomenon by (i) using short windows of data; (ii) fit the rather flexible stable-APARCH model, as opposed to appealing to quasi-maximum likelihood, which uses a normality assumption on the error term in the GARCH process; and (iii) investigate results when imposing a fixed APARCH structure which is not close to the IGARCH border; see Section 5.3 below, and, in particular, the discussion of the use of (18) In this way, the (in general, highly relevant) findings of Sun and Zhou [37] are less applicable to our setting.
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