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The aim of this chapter is to (a) provide a broad didactical treatment of the first-order stochastic differential equation model—also known as the continuous-time (CT) first-order vector autoregressive (VAR(1)) model—and (b) argue for and illustrate the potential of this model for the study of psychological processes using intensive longitudinal data. We begin by describing what the CT-VAR(1) model is and how it relates to the more commonly used discrete-time VAR(1) model. Assuming no prior knowledge on the part of the reader, we introduce important concepts for the analysis of dynamic systems, such as stability and fixed points. In addition we examine why applied researchers should take a continuous-time approach to psychological phenomena, focusing on both the practical and conceptual benefits of this approach. Finally, we elucidate how researchers can interpret CT models, describing the direct interpretation of CT model parameters as well as tools such as impulse response functions, vector fields, and lagged parameter plots. To illustrate this methodology, we reanalyze a single-subject experience-sampling dataset with the R package ctsem; for didactical purposes, R code for this analysis is included, and the dataset itself is publicly available.

The aim of this chapter is to (a) provide a broad didactical treatment of the first-order stochastic differential equation model—also known as the continuous-time (CT) first-order vector autoregressive (VAR(1)) model—and (b) argue for and illustrate the potential of this model for the study of psychological processes using intensive longitudinal data. We begin by describing what the CT-VAR(1) model is and how it relates to the more commonly used discrete-time VAR(1) model. Assuming no prior knowledge on the part of the reader, we introduce important concepts for the analysis of dynamic systems, such as stability and fixed points. In addition we examine why applied researchers should take a continuous-time approach to psychological phenomena, focusing on both the practical and conceptual benefits of this approach. Finally, we elucidate how researchers can interpret CT models, describing the direct interpretation of CT model parameters as well as tools such as impulse response functions, vector fields, and lagged parameter plots. To illustrate this methodology, we reanalyze a single-subject experience-sampling dataset with the R package ctsem; for didactical purposes, R code for this analysis is included, and the dataset itself is publicly available.

In some panel data studies, respondents are nested in social contexts, like classrooms or organizations. The social relations, such as friendship or collaboration, between the individuals in such contexts can be represented by social networks. Like individual outcomes (e.g., behavior or performance), the social relations between individuals are not static. Social networks and individual outcomes change over time and can mutually affect each other. In this chapter, we present a statistical model to study the interdependent dynamics (or coevolution) of network structure and individual outcomes. We assume panel observations of networks and individual attributes to be the discrete-time realizations of an underlying continuous-time process, which is modeled by a stochastic differential equation for the attribute dynamics and a Markov chain model for the network dynamics. We illustrate the proposed method by a study of the coevolution of friendship ties and mathematics grades among 1160 students in 39 classrooms.

This paper deals with a factor analysis model. Usually one assumes that the model has normally distributed variables. If these variables are not normally distributed, the well-known estimators using only variances and covariances of the observed variables may lead to biased estimates of the factor loadings. This paper uses characteristic functions to derive an approximately consistent BGLS-estimator of the factor loadings in a model with both normal and non-normal variables. An example with generated data and an empirical example will evaluate the estimator. Furthermore, the estimator will be compared to well-known estimators using only second order moments, consistent estimators using third order moments and estimators using transform ations to get normal variables. It turns out that our method gives good estimates of the unknown model parameters for sample sizes larger than 50. Our estimation method also avoids the disadvantage of the above mentioned estimation methods.

This paper offers both a conceptual and an operationalmodel that aims to map out the causes and implicationsof ICT perceptions and views of urban policy makersand/or administrative officials (denoted as urban frontliners). This is followed by the presentation of an opera-tional path model, viz. a linear structural equations model(Lisrel). The model serves to describe and test the rela-tionships between perceptions of the city, policy makers'beliefs about ICT and the associated urban ICT policy.According to the model, respondents that perceive theircity as having many urban functions (e.g., commercialcentre, service centre, higher education centre) have moreawareness to various ICT tools and are likely to consider amultiplicity of ICT measures as relevant to their city. Re-spondents that consider their city as having severe bottle-necks (e.g., traffic congestion, housing shortage) are lesslikely to think of ICT measures and ICT -related goals asrelevant to their city, nor that the municipality impactssignificantly on ICT in the city. Furthermore, respondentsthat perceive their city as suffering from many socio-economic problems (unemployment, ageing population,industrial decline and so on), are likely to consider manyICT tools as relevant to their city, although they have a lowawareness of the specific tools to be deployed. Finally,respondents who believe that ICT will affect significantly(and positively) the city and its administration, tend to at-tach a high municipal influence on ICT, and consider manyICT initiatives as relevant to their city.

We show that a sufficient condition for the identification ofall parameters of the censored regression model with astochastic and unobserved threshold is that the errors are jointlynormally distributed. Exclusion restrictions are not needed.

We argue that many models for multivariate longitudinal and cross-sectional data analysis have a common ancestry. They all are based on the qualitative idea that if we knew the actual state of the world, the relations between the observed quantities would be truly simple. This is shown to lead directly to factor analysis, IRT, state space models, mixture densities, latent Markov chains, MIMIC, LISREL, and various other common models and techniques. We show how our approach provides a convenient framework for looking at these models. The EM algorithm can be used to estimate the unknown parameters. An additional advantage of our approach is that it can incorporate continuous as well as interval, ordinal and categorical variables .

This chapter provides a survey of methods of continuous time modelling based on an exact discrete time representation. It begins by highlighting the techniques involved with the derivation of an exact discrete time representation of an underlying continuous time model, providing specific details for a second-order linear system of stochastic differential equations. Issues of parameter identification, Granger causality, nonstationarity, and mixed frequency data are addressed, all being important considerations in applications in economics and other disciplines. Although the focus is on Gaussian estimation of the exact discrete time model, alternative time domain (state space) and frequency domain approaches are also discussed. Computational issues are explored and two new empirical applications are included along with a discussion of applications in the field of macroeconometric modelling.