Scattering phase shifts in quasi-one-dimension

Scattering of an electron in quasi-one dimensional quantum wires have many unusual features, not found in one, two or three dimensions. In this work we analyze the scattering phase shifts due to an impurity in a multi-channel quantum wire with special emphasis on negative slopes in the scattering phase shift versus incident energy curves and the Wigner delay time. Although at first sight, the large number of scattering matrix elements show phase shifts of different character and nature, it is possible to see some pattern and understand these features. The behavior of scattering phase shifts in one-dimension can be seen as a special case of these features observed in quasi-one-dimensions. The negative slopes can occur at any arbitrary energy and Friedel sum rule is completely violated in quasi-one-dimension at any arbitrary energy and any arbitrary regime. This is in contrast to one, two or three dimensions where such negative slopes and violation of Friedel sum rule happen only at low energy where the incident electron feels the potential very strongly (i.e., there is a very well defined regime, the WKB regime, where FSR works very well). There are some novel behavior of scattering phase shifts at the critical energies where $S$-matrix changes dimension.Comment: Minor corrections mad

The Variance Ratio Statistic at Large Horizons

We make three contributions to using the variance ratio statistic at large horizons. Allowing for general heteroscedasticity in the data, we obtain the asymptotic distribution of the statistic when the horizon k is increasing with the sample size n but at a slower rate so that k=n ! 0. The test is shown to be consistent against a variety of relevant mean reverting alternatives when k=n ! 0. This is in contrast to the case when k=n ! – > 0; where the statistic has been recently shown to be inconsistent against such alternatives. Secondly, we provide and justify a simple power transformation of the statistic which yields almost perfectly normally distributed statistics in finite samples, solving the well known right skewness problem. Thirdly, we provide a more powerful way of pooling information from different horizons to test for mean reverting alternatives. Monte Carlo simulations illustrate the theoretical improvements provided. --Mean reversion,Frequency domain,Power transformation

Four Dimensional Supergravity from String Theory

A derivation of N=1 supergravity action from string theory is presented. Starting from a Nambu-Goto bosonic string, matter field is introduced to obtain a superstring in four dimension. The excitation quanta of this string contain graviton and the gravitino. Using the principle of equivalence, the action in curved space time are found and the sum of them is the Deser-Zumino N=1 supergravity action. The energy tensor is Lorentz invariant due to supersymmetry.Comment: 9 page

Estimation of Mis-Specified Long Memory Models

We study the asymptotic behaviour of frequency domain maximum likelihood estimators of mis-specified models of long memory Gaussian series. We show that even if the long memory structure of the time series is correctly specified, mis-specification of the short memory dynamics may result in parameter estimators which are slower than pn consistent. The conditions under which this happens are provided and the asymptotic distribution of the estimators is shown to be non-Gaussian. Conditions under which estimators of the parameters of the mis-specified model have the standard pn consistent and asymptotically normal behaviour are also provided. --

On testing the adequacy of stable processes under conditional heteroscedasticity

We consider a recently proposed method of estimating the tail index and testing the goodness-of-fit of dependent stable processes. Through Monte Carlo simulations, we evaluate the ability of the procedure to distinguish between stable and non-stable processes in the presence of non-linear dependence and to estimate the tail index of the distribution. We then apply the test to black market East European exchange rates, whose distributional and tail behaviour has been analysed previously in the literature. After adjusting for seasonality, we conclude, unlike the earlier analysis, that a stable process cannot be rejected as a model for some of the currencies. Estimates of the tail index for these currencies are also obtained.Statistics Working Papers Serie

Spectral tests of the martingale hypothesis under conditional heteroscedasticity

We study the asymptotic distribution of the sample standardized spectral distribution function when the observed series is a conditionally heteroscedastic martingale difference. We show that the asymptotic distribution is no longer a Brownian bridge but another Gaussian process. Furthermore, this limiting process depends on the covariance structure of the second moments of the series. We show that this causes test statistics based on the sample spectral distribution, such as the CramÃÂér von-Mises statistic, to have heavily right skewed distributions, which will lead to over-rejection of the martingale hypothesis in favour of mean reversion. A non-parametric correction to the test statistics is proposed to account for the conditional heteroscedasticity. We demonstrate that the corrected version of the CramÃÂér von-Mises statistic has the usual limiting distribution which would be obtained in the absence of conditional heteroscedasticity. We also present Monte Carlo results on the finite sample distributions of uncorrected and corrected versions of the CramÃÂér von-Mises statistic. Our simulation results show that this statistic can provide significant gains in power over the Box-Ljung-Pierce statistic against long-memory alternatives. An empirical application to stock returns is also provided.Statistics Working Papers Serie

Forecasting Realised Volatility using a Long Memory Stochastic Volatility Model: Estimation, Prediction and Seasonal Adjustment

We study the modelling of large data sets of high frequency returns using a long memory stochastic volatility (LMSV) model. Issues pertaining to estimation and forecasting of datasets using the LMSV model are studied in detail. Furthermore, a new method of de-seasonalising the volatility in high frequency data is proposed, that allows for slowly varying seasonality. Using both simulated as well as real data, we compare the forecasting performance of the LMSV model for forecasting realised volatility to that of a linear long memory model fit to the log realised volatility. The performance of the new seasonal adjustment is also compared to a recently proposed procedure using real data. --