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The discrete Fourier transform (dft) of a fractional process is studied. An exact representation of the dft is given in terms of the component data, leading to the frequency domain form of the model for a fractional process. This representation is particularly useful in analyzing the asymptotic behavior of the dft and periodogram in the nonstationary case when the memory parameter d≥12. Various asymptotic approximations are established including some new hypergeometric function representations that are of independent interest. It is shown that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d < 1. When d = 1, the spectral estimates are inconsistent and converge weakly to random variates. Applications of the theory to log periodogram regression and local Whittle estimation of the memory parameter are discussed and some modified versions of these procedures are suggested for nonstationary cases.

This chapter proposes M-estimators of a fractional response model with an endogenous count variable under the presence of time-constant unobserved heterogeneity. To address the endogeneity of the right-hand-side count variable, I use instrumental variables and a two-step procedure estimation approach. Two methods of estimation are employed: quasi-maximum likelihood (QML) and nonlinear least squares (NLS). Using these methods, I estimate the average partial effects, which are shown to be comparable across linear and nonlinear models. Monte Carlo simulations verify that the QML and NLS estimators perform better than other standard estimators. For illustration, these estimators are used in a model of female labor supply with an endogenous number of children. The results show that the marginal reduction in women's working hours per week is less as women have one additional kid. In addition, the effect of the number of children on the fraction of hours that a woman spends working per week is statistically significant and more significant than the estimates in all other linear and nonlinear models considered in the chapter.

In this chapter we demonstrate the construction of inverse test confidence intervals for the turning-points in estimated nonlinear relationships by the use of the marginal or first derivative function. First, we outline the inverse test confidence interval approach. Then we examine the relationship between the traditional confidence intervals based on the Wald test for the turning-points for a cubic, a quartic, and fractional polynomials estimated via regression analysis and the inverse test intervals. We show that the confidence interval plots of the marginal function can be used to estimate confidence intervals for the turning-points that are equivalent to the inverse test. We also provide a method for the interpretation of the confidence intervals for the second derivative function to draw inferences for the characteristics of the turning-point.

This method is applied to the examination of the turning-points found when estimating a quartic and a fractional polynomial from data used for the estimation of an Environmental Kuznets Curve. The Stata do files used to generate these examples are listed in Appendix A along with the data.

This chapter focuses on the impact of national economic conditions and voters’ attitudes on the positioning of European national political parties with regard to the European Union (EU). We provide an empirical analysis based on data gathered through the Chapel Hill Expert Survey (CHES) covering parties from 14 European countries observed over the 1999–2010 time span. We perform a regression analysis where the dependent variable measures the position of political parties vis-à-vis EU integration and explanatory variables include a number of measures of national economic conditions, features of the national political and institutional framework and voters’ Euroscepticism. Fixed effect, ordered logit and fractional logit estimates provide the following main results. Compared with other parties, non-mainstream political parties and those acting in established economies are more prone to mirror citizens’ Eurosceptic sentiments. National economic conditions such as inflation as well as gross domestic product (GDP) growth affect mainstream party support for the EU. Smaller and ideologically extreme parties are, on average, less supportive of European integration.

This study employs a new time series representation of persistence in conditional mean and variance to test for the existence of the long memory property in the currency futures market. Empirical results indicate that there exists a fractional exponent in the differencing process for foreign currency futures prices. The series of returns for these currencies displays long-term positive dependence. A hedging strategy for long memory in volatility is also discussed in this article to help the investors hedge for the exchange rate risk by using currency futures.

In this chapter we explore different ways to obtain decompositions of rank-dependent indices of socioeconomic inequality of health, such as the Concentration Index. Our focus is on the regression-based type of decomposition. Depending on whether the regression explains the health variable, or the socioeconomic variable, or both, a different decomposition formula is generated. We illustrate the differences using data from the Ethiopia 2011 Demographic and Health Survey (DHS).

This chapter derives asymptotic properties of the least squares (LS) estimator of the autoregressive (AR) parameter in local to unity processes with errors being fractional Gaussian noise (FGN) with the Hurst parameter H∈(0,1). It is shown that the estimator is consistent for all values of H∈(0,1). Moreover, the rate of convergence is n−1 when H∈[0.5,1). The rate of convergence is n−2H when H∈(0,0.5). Furthermore, the limiting distribution of the centered LS estimator depends on H. When H=0.5, the limiting distribution is the same as that obtained in Phillips (1987a) for the local to unity model with errors for which the standard functional central limit theorem is applicable. When H > 0.5 or when H < 0.5, the limiting distributions are new to the literature. The asymptotic properties of the LS estimator with fitted intercept are also derived. Simulation studies are performed to check the reliability of the asymptotic approximation for different values of sample size.

A large literature over several decades reveals both extensive concern with the question of time-varying betas and an emerging consensus that betas are in fact time-varying, leading to the prominence of the conditional CAPM. Set against that background, we assess the dynamics in realized betas, vis-à-vis the dynamics in the underlying realized market variance and individual equity covariances with the market. Working in the recently popularized framework of realized volatility, we are led to a framework of nonlinear fractional cointegration: although realized variances and covariances are very highly persistent and well approximated as fractionally integrated, realized betas, which are simple nonlinear functions of those realized variances and covariances, are less persistent and arguably best modeled as stationary I(0) processes. We conclude by drawing implications for asset pricing and portfolio management.