To verify whether the maximum or the minimum Fukui function site is better for protonation reactions or an
altogether different local reactivity descriptor, viz., the charge is necessary, we calculate the Fukui functions
(using a finite-difference approximation as well as a frozen-core approximation) and charges (Mulliken,
Hirshfeld, and natural population analysis schemes) of several hydroxylamine derivatives, their sulfur-containing
variants, and amino acids using B3LYP/6-311G(d,p) technique. While the Fukui functions provide the wrong
selectivity criterion for hard−hard interactions, the charges are found to be more reliable, vindicating Klopman's
idea. It is transparent from the present results that the hard−hard interactions are better explained in terms of
charges, whereas the Fukui functions can properly account for soft−soft interactions known to be frontier-controlled.
The assumption that daily stock returns are normally distributed has long been disputed by the data. In this article we test (and clearly reject) the normality assumption using time series of daily stock returns for thirteen European securities markets. More importantly, we fit to the data four alternative specifications, find overall support for the scaled-t distribution (and partial support for a mixture of two Normal distributions), and quantify the magnitude of the error that stems from predicting the probability of obtaining returns in specified intervals by using the Normal distribution. We conclude by arguing that normality may be a plausible assumption for monthly (but not for daily) stock returns.
Abstract. Since the seminal paper by Dickey and Fuller in 1979, unit-root tests have conditioned the standard approaches to analysing time series with strong serial dependence in mean behaviour, the focus being placed on the detection of eventual unit roots in an autoregressive model fitted to the series. In this paper, we propose a completely different method to test for the type of long-wave patterns observed not only in unit-root time series but also in series following more complex data-generating mechanisms. To this end, our testing device analyses the unit-root persistence exhibited by the data while imposing very few constraints on the generating mechanism. We call our device the range unit-root (RUR) test since it is constructed from the running ranges of the series from which we derive its limit distribution. These nonparametric statistics endow the test with a number of desirable properties, the invariance to monotonic transformations of the series and the robustness to the presence of important parameter shifts. Moreover, the RUR test outperforms the power of standard unit-root tests on near-unit-root stationary time series; it is invariant with respect to the innovations distribution and asymptotically immune to noise. An extension of the RUR test, called the forward backward range unit-root (FB-RUR) improves the check in the presence of additive outliers. Finally, we illustrate the performances of both range tests and their discrepancies with the Dickey Fuller unit-root test on exchange rate series.
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