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This chapter covers regression-discontinuity models for analyzing the data from single case designs. Auto-regressive-integrated-moving-average models are also described and illustrated. These statistical methods are useful when there are a large number of observations in the phases of a single case design. These methods are described in detail and illustrated using data from a study of the implementation of an Aggression Replacement Training program implemented in a runaway shelter.

This chapter compares the predictive accuracy of the Michigan and Fair econometric models using the method developed in Ray Fair. These models are compared to each other and to an eighth-order autoregressive model. The method accounts for the four main sources of uncertainty of an economic forecast: uncertainty due to the error terms, the coefficient estimates, the exogenous variables, and the possible misspecification of the model. Because it accounts for these four sources, it can be used to make comparisons across models. The method has been used to compare the Fair model to autoregressive models, vector autoregressive models, Thomas Sargent's classical macroeconomic model, and a small linear model, but this is the first time it has been used to compare two relatively large structural models. The chapter's primary aim is to demonstrate the application of the comparison method to large models.

This chapter discusses the specification and estimation of autoregressive models with individual specific intercepts. It focuses on first order processes, since the main insights generalise in a straightforward way to high-order and multivariate cases. It discusses the role in short panels of assumptions about initial conditions, homoskedasticity, and whether the parameter space includes the possibility of unit roots; alternative representations of restrictions that can be obtained by transformation; and the various aspects of inference with VAR panel data models.

In this chapter, a number of most commonly applied nonlinear time series models are being considered. As opposed to the previous chapter, these models do not generally have their origin in economic theory. Many of the models nest a linear model are therefore relatively easily interpretable. The models include regression models such as the smooth transition, switching regression and Markov switching models. They also include models based on rather general functional forms such as artificial neural network models and polynomial models. More rarely applied models such as bilinear or max‐min models are also mentioned. Models with stochastic coefficients also receive attention. Areas of application of these models to economic time series are briefly mentioned.

This chapter examines the relationship between speculation in the currency markets and aggregate activity in the real economy by drawing on the Structural Slumps theory and the theory of Imperfect Knowledge Economics (IKE). It first considers exchange rate determination in two models, one based on the Rational Expectations Hypothesis (REH) and the other on the theory of IKE, before discussing some general principles for how to structure the observed persistence in the data, and how these principles can be used in the cointegrated vector autoregressive model. The chapter also explains how foreign currency speculation under IKE interacts with a customer market economy where profit shares are adjusting to fluctuations in real exchange rates and where the natural rate of unemployment is a function of nonstationary real long-term interest rates.

This chapter analyzes the risks and rewards of moving from an unfunded defined benefit pension system to a funded plan for civil servants in Germany, allowing for alternative portfolio mixes using a Monte Carlo framework and a Conditional Value at Risk metric. The authors identify an investment strategy for plan assets that will minimize worst-case pension costs; this turns out to be 22 percent in equities, 47 percent in bonds, and 31 percent in real estate. The authors show that moving toward a funded pension system for German civil servants can be beneficial to both taxpayers and civil servants.

This chapter presents the finite sample analysis of the time series models used in economics and finance. It considers the autoregressive model (AR), AR with regressors, and autoregressive moving average models with regressors. The exact and approximate moments, as well as distributions of the estimators of the lag coefficients and regression coefficients were derived and analysed.

Spatial variation has been often considered undesirable noise in ecological studies because many statistical methods used assume random spatial distributions. It is time to change this because spatial structure is the rule in natural systems. Many ecological processes are driven by spatial structures. This chapter introduces the conceptual framework that allows ecologists to understand and analyze these structures. It also shows a basic set of tools for exploratory data analyses and graphs that will make spatial patterns apparent and help readers make analytical decisions. One of the goals of this chapter is to convince readers of the value of spatial analysis of small data sets and subtle spatial variation. Real ecological data sets are used to demonstrate two approaches to adding spatial structure to linear models. The first approach, Generalized Least Squares, directly models the spatial covariance structure with a variance–covariance matrix incorporating the range and sill of a variogram function fitted to the residuals. The second approach, spatial autoregressive models, uses weight matrices that specify a priori which observations are considered neighbors, as well as the strength of interaction between them. Autoregressive models can model spatial patterns in ecological data as imposed by unknown factors that are either external (e.g. environmental) or endogenous (e.g. competitive or social interactions) to the variable. In brief, there is a spatial world out there to be explored.

This chapter discusses theoretical autocovariance, autocorrelation functions of autoregressive models of orders 1 and 2, and autocorrelation function-derived timescale. The autocorrelation function of a scalar time series is a prime tool for estimating the characteristic timescale separating successive independent realizations in the time series. Any process other than completely random noise has some serial correlations. Even variables as volatile and nondeterministic as measures of stock market performance, when valuated close enough, are unlikely to vary appreciably. Thus, a time series of the index at 1-second intervals likely contains significant redundancy; it can be almost as representative and contain almost as much information if degraded to valuation intervals of T > 1 second. Identifying an acceptable characteristic timescale T separating successive independent realization in the time series that balances the need to retain maximum information while minimizing storage and transmission burdens is a key role of the autocorrelation function.

This chapter illustrates vector autoregressive VAR models, with a particular focus on estimation and hypothesis testing. It discusses estimation of parameters, deterministic components, VAR order selection, Granger causality, forecasting with multivariate models, and multivariate spectral density. Exercises are provided at the end of the chapter.

This chapter provides a self-contained review of Bayesian inference and decision making. It begins with a discussion of Bayesian inference for a simple autoregressive (AR) model, which takes the form of a Gaussian linear regression. For this model, the posterior distribution can be characterized analytically and closed-form expressions for its moments are readily available. The chapter also examines how to turn posterior distributions into point estimates, interval estimates, forecasts, and how to solve general decision problems. The chapter shows how in a Bayesian setting, the calculus of probability is used to characterize and update an individual's state of knowledge or degree of beliefs with respect to quantities such as model parameters or future observations.

Chapter Six explains time series analysis with one or more independent variables. The dependent variable is the monthly violent crime rates and the independent variables are unemployment rates and inflation. This chapter discusses several topics related to the robustness of estimated models, such as how to prewhiten a time series, how to deal with trends and seasonal components, how to deal with autoregressive residuals, and how to discern changes of the dependent variable caused by independent variables from its simple continuity. This chapter also discusses the concepts of co-integration and long-memory effect and related topics such as error correction models and autoregressive distributive lags models.

Dynamic economic models typically arise as a characterization of the path of the economy around its long run equilibrium (steady states), and involve modelling expectations, learning, and adjustment costs. A variety of dynamic specifications used in applied time series econometrics exist. This chapter reviews a number of single-equation specifications suggested by econometric literature to represent dynamics in regression models. It provides a preliminary introduction to distributed lag models, autoregressive distributed lag models, partial adjustment models, error-correction models, and adaptive, and rational expectations models. Exercises are provided at the end of the chapter.

This chapter focuses on large N aggregation. It first briefly reviews the main aggregation problems studied in the literature. It then presents a general framework for micro/disaggregate behavioural relationships and develops a forecasting approach to derive the optimal aggregate function. This approach is applied to a large cross section aggregation of panel autoregressive distributed lag (ARDL) models and to the case of large factor augmented vector autoregressive (VAR) models in N cross section units, where each micro unit is potentially related to all other micro units, and where micro innovations are allowed to be cross-sectionally dependent. The optimal aggregate function is used to examine the relationship between micro and macro parameters to show which distributional features of micro parameters can be identified from the aggregate model. The chapter also derives and contrasts impulse response functions for the aggregate variables, distinguishing between the effects of composite macro and aggregated idiosyncratic shocks. Some of these findings are illustrated by Monte Carlo experiments and two applications are presented. The first application investigates the aggregation of life-cycle consumption decision rules under habit formation. The second application investigates the sources of persistence of consumer price inflation in Germany, France and Italy, and re-examines the extent to which ‘observed’ inflation persistence at the aggregate level is due to aggregation and/or common unobserved factors. Exercises are provided at the end of the chapter.

The non-equilibrium dynamics of biological invasions has revived the debate on the balance of nature and mechanisms for population regulation. Invasive species in novel environments, whether experiencing the quasi-equilibrium of halted expansion during the lag phase or undergoing fast expansion to fulfil the opportunity niche, face a multitude of stabilizing and destabilizing forces that affect their population dynamics—from positive (Allee effects) and negative density dependence to rapidly shifting niches during invasion; these have major effects on population variability and spatial dynamics. The temporal and spatial dynamics are also intertwined, which means that the spatial structures of species distributions are potentially indicative of the invasiveness and population trends of target species. Time-series analysis for depicting the temporal population dynamics can be extended to capture the spatial synchrony dynamics of multiple local populations. Spatial autoregressive models further advance species distribution models to unveil the drivers of non-equilibrium dynamics of invasive species.

The chapter considers first-order autoregressive models with individual-specific intercepts. Both the within estimator and OLS estimation of an equation in first-differences have notable deficiencies when the time-series are short. More relevant are estimators utilizing instrumental variables (IV) in levels for the equation in differences, or estimators using the opposite configuration. Procedures using orthogonal forward deviations are also considered. Versions of GMM are discussed, and extensions to handle disturbance memory are explained. For models with random effects, Maximum Likelihood (ML) and stepwise procedures are considered for equations containing both two-dimensional and individual-specific explanatory variables.