Modelling nonlinear economic time series / by Timo Teräsvirta, Dag Tjøstheim, and Clive W.J. Granger

Includes bibliographical references (pages 470-536) and indexes

Table Of Contents: List of Figures List of Tables Acronyms and abbreviations 1 Concepts, models, and definitions 1.1 Defining nonlinearity 1.2 Where does nonlinearity come from? 1.3 Stationarity and nonstationarity 1.4 Invertibility 1.5 Trends 1.6 Seasonality 1.7 Conditional distributions 1.8 Wold's representation and Volterra expansion 1.9 Additive models 1.10 Spectral analysis 1.11 Chaos 2 Nonlinear models in economic theory 2.1 Disequilibrium models 2.2 Labour market models 2.2.1 Theory 2.2.2 Practice 2.3 Exchange rates in a target zone 2.3.1 Theory 2.3.2 Practice 2.4 Production theory 3 Parametric nonlinear models 3.1 General considerations 3.2 Switching regression models 3.2.1 Standard switching regression model 3.2.2 Vector threshold autoregressive model 3.3 Markov-switching regression models 3.4 Smooth transition regression models 3.4.1 Standard smooth transition regression model 3.4.2 Additive, multiple, and time-varying STR models 3.4.3 Vector smooth transition autoregressive model 3.5 Polynomial models 3.6 Artificial neural network models 3.7 Min-max models 3.8 Nonlinear moving average models 3.9 Bilinear models 3.10 Time-varying parameters and state space models 3.11 Random coefficient and volatility models 4 The nonparametric approach 4.1 Introduction 4.2 Autocovariance and spectrum 4.3 Density, conditional mean, and conditional variance 4.3.1 Non-Gaussian marginals 4.3.2 Conditional quantities 4.4 Dependence measures for nonlinear processes 4.4.1 Local measures of dependence 4.4.2 Global measures of dependence 4.4.3 Measures based on density and distribution functions 4.4.4 The copula 5 Testing linearity against parametric alternatives 5.1 Introduction 5.2 Consistent misspecification tests 5.3 Lagrange multiplier or score test 5.3.1 Standard case 5.3.2 Test in stages and a heteroskedasticity-robust version 5.3.3 Robustifying against conditional heteroskedasticity 5.4 Locally equivalent alternatives 5.5 Nonlinear model only identified under the alternative 5.5.1 Identification problem 5.5.2 General solution 5.5.3 Lagrange multiplier-type tests 5.5.4 Monte Carlo tests 5.5.5 Giving values to the nuisance parameters 5.6 Testing linearity against unspecified alternatives 5.6.1 Regression Specification Error Test 5.6.2 Tests based on expansions 5.7 Comparing parametric linearity tests using asymptotic relative efficiency 5.7.1 Definition 5.7.2 An example 5.8 Which test to use? 6 Testing parameter constancy 6.1 General considerations 6.2 Generalizing the Chow test 6.2.1 Testing against a single break 6.2.2 Testing against multiple breaks 6.3 Lagrange multiplier type tests 6.3.1 Testing a stationary single-equation model 6.3.2 Testing a stationary vector autoregressive model 6.3.3 Testing a nonstationary vector autoregressive model 6.4 Tests based on recursive estimation of parameters 6.4.1 Cumulative sum tests 6.4.2 Moving sum tests 6.4.3 Fluctuation tests 6.4.4 Tests against stochastic parameters 6.4.5 Testing the constancy of cointegrating relationships 7 Nonparametric specification tests 7.1 Introduction 7.2 Nonparametric linearity tests 7.2.1 Nonparametric tests: the spectral domain 7.2.2 Testing linearity in the conditional mean and conditional variance 7.2.3 Estimation 7.2.4 Asymptotic theory 7.2.5 Finite-sample properties and use of the asymptotics 7.2.6 A bootstrap approach to testing 7.3 Testing for specific functional forms 7.3.1 Tests based on residuals 7.3.2 Conditional mean and conditional variance testing 7.3.3 Continuous time 7.4 Selecting lags 7.5 Testing for additivity and interaction 7.5.1 Testing in additive models 7.5.2 A simulated example 7.6 Tests for partial linearity and semiparametric modelling 7.7 Tests of independence 7.7.1 Traditional tests 7.7.2 Rank correlation 7.7.3 Frequency based tests 7.7.4 BDS test 7.7.5 Distribution based tests of independence 7.7.6 Generalized spectrum and tests of independence 7.7.7 Density based tests of independence 7.7.8 Some examples of independence testing 8 Models of conditional heteroskedasticity 8.1 Autoregressive conditional heteroskedasticity 8.1.1 The ARCH model 8.2 The Generalized ARCH model 8.2.1 Why Generalized ARCH? 8.2.2 Families of univariate GARCH models 8.2.3 Nonlinear GARCH 8.2.4 Time-varying GARCH 8.2.5 Moment structure of first-order GARCH models 8.2.6 Moment structure of higher-order GARCH models 8.2.7 Integrated and fractionally Integrated GARCH 8.2.8 Stylized facts and the GARCH model 8.2.9 Building univariate GARCH models 8.3 Family of Exponential GARCH models 8.3.1 Moment structure of EGARCH model 8.3.2 Stylized facts and the EGARCH model 8.3.3 Building EGARCH models 8.4 The Autoregressive Stochastic Volatility model 8.4.1 Definition 8.4.2 Moment structure of ARSV models 8.4.3 Stylized facts and the stochastic volatility model 8.4.4 Estimation of ARSV models 8.4.5 Comparing the ARSV model with GARCH 8.5 GARCH-in-Mean model 8.6 Realized volatility 8.7 Multivariate GARCH models 8.7.1 General multivariate GARCH model 8.7.2 Link to random coefficient models 8.7.3 Constant Conditional Correlation GARCH 8.7.4 Testing the constant correlation assumption and the Dynamic Conditional Correlation model 8.7.5 Other extensions to the CCC-GARCH model 8.7.6 The BEKK-GARCH model 8.7.7 Factor GARCH models 9 Time-varying parameters and state space models 9.1 Introduction 9.2 Linear state space models 9.3 Time-varying parameter models 9.4 Nonlinear state space models 9.4.1 Extended Kalman filter 9.4.2 Kitagawa's grid approximation 9.4.3 Monte Carlo methods 9.4.4 Particle filters 9.4.5 Approximating with a Gaussian density 9.5 Hidden Markov chains and regimes 9.5.1 Hidden Markov chains 9.5.2 Mixture models 9.6 Estimating parameters 9.6.1 Stationarity 9.6.2 Identification 9.6.3 Estimation in linear models 9.6.4 The nonlinear case 9.6.5 Estimation in hidden Markov and mixture models 10 Nonparametric models 10.1 Additive models 10.1.1 Estimation in purely additive models 10.1.2 Marginal integration 10.1.3 Backfitting and smoothed backfitting 10.1.4 Additive models with interactions 10.1.5 A simulated example 10.1.6 Nonparametric and additive estimation of the conditional variance function 10.2 Some related models 10.2.1 Functional coefficient autoregressive models 10.2.2 Transformation of dependent variables and the ACE algorithm 10.2.3 Regression trees, splines, and MARS 10.2.4 Quantile regression 10.3 Semiparametric models 10.3.1 Index models 10.3.2 Projection pursuit regression 10.3.3 Partially linear models 10.4 Robust and adaptive estimation 11 Nonlinear and nonstationary models 11.1 Long memory models 11.2 Linear unit root models 11.3 Vector autoregressive processes and linear cointegration 11.4 Nonlinear I(1) processes 11.5 Nonlinear error correction models 11.6 Parametric nonlinear regression 11.7 Nonparametric estimation in a nonlinear cointegration type framework 11.8 Stochastic unit root models 12 Algorithms for estimating parametric nonlinear models 12.1 Optimization without derivatives 12.1.1 Grid and line searches 12.1.2 Conjugate directions 12.1.3 Simulated annealing 12.1.4 Evolutionary algorithms 12.2 Methods requiring derivatives 12.2.1 Gradient methods 12.2.2 Variable metric methods 12.3 Other methods 12.3.1 EM algorithm 12.3.2 Sequential estimation for neural networks 13 Basic nonparametric estimates 13.1 Density estimation 13.1.1 Kernel estimation 13.1.2 Bias and variance reduction 13.1.3 Choice of bandwidth 13.1.4 Variable bandwidth and nearest neighbour estimation 13.1.5 Multivariate density estimation 13.2 Nonparametric regression estimation 13.2.1 Kernel regression estimation 13.2.2 Local polynomial estimation 13.2.3 Bias, convolution, and higher-order kernels 13.2.4 Nearest neighbour estimation 13.2.5 Splines and MARS 13.2.6 Series

expansion 13.2.7 Choice of bandwidth for nonparametric regression 14 Forecasting from nonlinear models 14.1 Introduction 14.2 Conditional mean forecasts from parametric models 14.2.1 Analytical point forecasts 14.2.2 Numerical techniques in forecasting 14.3 Forecasting with nonparametric models 14.4 Forecast accuracy 14.5 The usefulness of forecasts from nonlinear models 14.6 Forecasting volatility 14.7 Overview of forecasting from nonlinear models 15 Nonlinear impulse responses 15.1 Generalized impulse response function 15.2 Graphical representation 16 Building nonlinear models 16.1 General considerations 16.2 Nonparametric and semiparametric models 16.3 Building smooth transition regression models 16.3.1 The three stages of the modelling procedure 16.3.2 Specification 16.3.3 Estimation of parameters 16.3.4 Evaluation 16.3.5 Graphical tools for characterizing the dynamic behaviour of the STAR model 16.3.6 Examples 16.4 Building switching regression models 16.4.1 Specification 16.4.2 Estimation and evaluation 16.4.3 Examples 16.5 Building artificial neural network models 16.5.1 Specification 16.5.2 Estimation 16.5.3 Evaluation 16.5.4 Alternative modelling approaches 16.5.5 Examples 16.6 Two forecast comparisons 16.6.1 Forecasting Wolf's annual sunspot numbers 16.6.2 Forecasting the monthly US unemployment rate 17 Other topics 17.1 Aggregation 17.2 Seasonality 17.2.1 Time-varying seasonality 17.2.2 Temporal aggregation and time-varying seasonality 17.2.3 Nonlinear filters in seasonal adjustment 17.3 Outliers and nonlinearity 17.3.1 What is an outlier? 17.3.2 Model-based definitions Bibliography Author Index General Index

A comprehensive assessment of many recent developments in the modelling of time series, this text introduces various nonlinear models and discusses their practical use, encouraging the reader to apply nonlinear models to their practical modelling problems