Maximum likelihood estimation for social network dynamics

A model for network panel data is discussed, based on the assumption that the observed data are discrete observations of a continuous-time Markov process on the space of all directed graphs on a given node set, in which changes in tie variables are independent conditional on the current graph. The model for tie changes is parametric and designed for applications to social network analysis, where the network dynamics can be interpreted as being generated by choices made by the social actors represented by the nodes of the graph. An algorithm for calculating the Maximum Likelihood estimator is presented, based on data augmentation and stochastic approximation. An application to an evolving friendship network is given and a small simulation study is presented which suggests that for small data sets the Maximum Likelihood estimator is more efficient than the earlier proposed Method of Moments estimator.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS313 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Volatility forecasting

Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. This chapter provides a selective survey of the most important theoretical developments and empirical insights to emerge from this burgeoning literature, with a distinct focus on forecasting applications. Volatility is inherently latent, and Section 1 begins with a brief intuitive account of various key volatility concepts. Section 2 then discusses a series of different economic situations in which volatility plays a crucial role, ranging from the use of volatility forecasts in portfolio allocation to density forecasting in risk management. Sections 3, 4 and 5 present a variety of alternative procedures for univariate volatility modeling and forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses volatility forecast evaluation methods in both univariate and multivariate cases. Section 8 concludes briefly. JEL Klassifikation: C10, C53, G1

Modeling long-range memory with stationary Markovian processes

In this paper we give explicit examples of power-law correlated stationary Markovian processes y(t) where the stationary pdf shows tails which are gaussian or exponential. These processes are obtained by simply performing a coordinate transformation of a specific power-law correlated additive process x(t), already known in the literature, whose pdf shows power-law tails 1/x^a. We give analytical and numerical evidence that although the new processes (i) are Markovian and (ii) have gaussian or exponential tails their autocorrelation function still shows a power-law decay =1/T^b where b grows with a with a law which is compatible with b=a/2-c, where c is a numerical constant. When a<2(1+c) the process y(t), although Markovian, is long-range correlated. Our results help in clarifying that even in the context of Markovian processes long-range dependencies are not necessarily associated to the occurrence of extreme events. Moreover, our results can be relevant in the modeling of complex systems with long memory. In fact, we provide simple processes associated to Langevin equations thus showing that long-memory effects can be modeled in the context of continuous time stationary Markovian processes.Comment: 5 figure

Exact solution of diffusion limited aggregation in a narrow cylindrical geometry

The diffusion limited aggregation model (DLA) and the more general dielectric breakdown model (DBM) are solved exactly in a two dimensional cylindrical geometry with periodic boundary conditions of width 2. Our approach follows the exact evolution of the growing interface, using the evolution matrix E, which is a temporal transfer matrix. The eigenvector of this matrix with an eigenvalue of one represents the system's steady state. This yields an estimate of the fractal dimension for DLA, which is in good agreement with simulations. The same technique is used to calculate the fractal dimension for various values of eta in the more general DBM model. Our exact results are very close to the approximate results found by the fixed scale transformation approach.Comment: 18 pages RevTex, 6 eps figure

On a flexible construction of a negative binomial model

This work presents a construction of stationary Markov models with negative-binomial marginal distributions. A simple closed form expression for the corresponding transition probabilities is given, linking the proposal to well-known classes of birth and death processes and thus revealing interesting characterizations. The advantage of having such closed form expressions is tested on simulated and real data.Comment: Forthcoming in "Statistics & Probability Letters