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14
Testing for Linearity
 Journal of Economic Surveys
, 1999
"... Abstract. The problem of testing for linearity and the number of regimes in the context of selfexciting threshold autoregressive (SETAR) models is reviewed. We describe leastsquares methods of estimation and inference. The primary complication is that the testing problem is nonstandard, due to th ..."
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Cited by 89 (1 self)
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Abstract. The problem of testing for linearity and the number of regimes in the context of selfexciting threshold autoregressive (SETAR) models is reviewed. We describe leastsquares methods of estimation and inference. The primary complication is that the testing problem is nonstandard, due to the presence of parameters which are only defined under the alternative, so the asymptotic distribution of the test statistics is nonstandard. Simulation methods to calculate asymptotic and bootstrap distributions are presented. As the sampling distributions are quite sensitive to conditional heteroskedasticity in the error, careful modeling of the conditional variance is necessary for accurate inference on the conditional mean. We illustrate these methods with two applications Ð annual sunspot means and monthly U.S. industrial production. We find that annual sunspots and monthly industrial production are SETAR(2) processes. Keywords. SETAR models; Thresholds; Nonstandard asymptotic theory; Bootstrap
Functionalcoefficient Regression Models for Nonlinear Time Series
 Journal of the American Statistical Association
, 1998
"... We apply the local linear regression technique for estimation of functionalcoefficient regression models for time series data. The models include threshold autoregressive models (Tong 1990) and functionalcoefficient autoregressive models (Chen and Tsay 1993) as special cases but with the added adv ..."
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Cited by 81 (15 self)
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We apply the local linear regression technique for estimation of functionalcoefficient regression models for time series data. The models include threshold autoregressive models (Tong 1990) and functionalcoefficient autoregressive models (Chen and Tsay 1993) as special cases but with the added advantages such as depicting finer structure of the underlying dynamics and better postsample forecasting performance. We have also proposed a new bootstrap test for the goodness of fit of models and a bandwidth selector based on newly defined crossvalidatory estimation for the expected forecasting errors. The proposed methodology is dataanalytic and is of appreciable flexibility to analyze complex and multivariate nonlinear structures without suffering from the "curse of dimensionality". The asymptotic properties of the proposed estimators are investigated under the ffmixing condition. Both simulated and real data examples are used for illustration. Key Words: ffmixing; Asymptotic normali...
On Selecting Models for Nonlinear Time Series
 Physica D
, 1995
"... Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintainin ..."
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Cited by 64 (14 self)
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Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintaining good robustness against noise. The models that are built are close to the simplest possible according to a description length criterion. The method will deliver a linear model if that has shorter description length than a nonlinear model. We show how our models can be used for prediction, smoothing and interpolation in the usual way. We also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series. 1 The Model Selection Problem As our understanding of chaotic and other nonlinear phenomena has grown, it has become apparent that linear models are inadequate to model most dynamical processes. Nevertheless, linear models...
Building neural network models for time series: a statistical approach
 J. Forecasting
"... This paper is concerned with modelling time series by single hidden layer feedforward neural network models. A coherent modelling strategy based on statistical inference is presented. Variable selection is carried out using existing techniques. The problem of selecting the number of hidden units is ..."
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Cited by 40 (13 self)
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This paper is concerned with modelling time series by single hidden layer feedforward neural network models. A coherent modelling strategy based on statistical inference is presented. Variable selection is carried out using existing techniques. The problem of selecting the number of hidden units is solved by sequentially applying Lagrange multiplier type tests, with the aim of avoiding the estimation of unidenti ed models. Misspecication tests are derived for evaluating an estimated neural network model. A smallsample simulation experiment is carried out to show how the proposed modelling strategy works and how the misspecication tests behave in small samples. Two applications to real time series, one univariate and the other multivariate, are considered as well. Sets of onestepahead forecasts are constructed and forecast accuracy is compared with that of other nonlinear models applied to the same series.
A Monte Carlo study of the forecasting performance of empirical SETAR models
, 1997
"... In this paper we investigate the multiperiod forecast performance of a number of empirical selfexciting threshold autoregressive (SETAR) models that have been proposed in the literature for modelling exchange rates and GNP, amongst other variables. We take each of the empirical SETAR models in turn ..."
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Cited by 39 (4 self)
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In this paper we investigate the multiperiod forecast performance of a number of empirical selfexciting threshold autoregressive (SETAR) models that have been proposed in the literature for modelling exchange rates and GNP, amongst other variables. We take each of the empirical SETAR models in turn as the DGP to ensure that the `nonlinearity' characterises the future, and compare the forecast performance of SETAR and linear autoregressive models on a number of quantitative and qualitative criteria. Our results indicate that nonlinear models have an edge in certain states of nature but not in others, and that this can be highlighted by evaluating forecasts conditional upon the regime.
Leeper (2006a): “Endogenous Monetary Policy Regime Change
 in NBER International Seminar on Macroeconomics 2006
"... at Indiana University Bloomington. CAEPR can be found on the Internet at: ..."
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Cited by 18 (4 self)
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at Indiana University Bloomington. CAEPR can be found on the Internet at:
Determining the Number of Regimes in a Threshold Autoregressive Model Using Smooth Transition Autoregressions
, 2003
"... In this paper we propose a method for determining the number of regimes in threshold autoregressive models using smooth transition autoregression as a tool. As the smooth transition model is just an approximation to the threshold autoregressive one, no asymptotic properties are claimed for the propo ..."
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Cited by 9 (1 self)
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In this paper we propose a method for determining the number of regimes in threshold autoregressive models using smooth transition autoregression as a tool. As the smooth transition model is just an approximation to the threshold autoregressive one, no asymptotic properties are claimed for the proposed method. Tests available for testing the adequacy of a smooth transition autoregressive model are applied sequentially to determine the number of regimes. A simulation study is performed in order to find out the finitesample properties of the procedure and to compare it with two other procedures available in the literature. We find that our method works reasonably well for both single and multiple threshold models.
Continuous time threshold autoregressive models
, 1992
"... This thesis considers continuous time autoregressive processes defined by stochastic differential equations and develops some methods for modelling time series data by such processes. The first part of the thesis looks at continuous time linear autoregressive (CAR) processes defined by linear stocha ..."
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Cited by 7 (3 self)
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This thesis considers continuous time autoregressive processes defined by stochastic differential equations and develops some methods for modelling time series data by such processes. The first part of the thesis looks at continuous time linear autoregressive (CAR) processes defined by linear stochastic differential equations. These processes are wellunderstood and there is a large body of literature devoted to their study. I summarise some of the relevant material and develop some further results. In particular, I propose a new and very fast method of estimation using an approach analogous to the Yule–Walker estimates for discrete time autoregressive processes. The models so estimated may be used for preliminary analysis of the appropriate model structure and as a starting point for maximum likelihood estimation. A natural extension of CAR processes is the class of continuous time threshold autoregressive (CTAR) processes defined by piecewise linear stochastic differential
Spline Estimation of Functional Coefficient Regression Models for Nonlinear Time Series with Correlated Errors
"... Many data in applications exhibit nonlinear features such as nonlinearity between lagged variables and heteroscedasticity. They require nonlinear models to describe the data. Parametric nonlinear time series models provide powerful tools for analyzing nonlinear time series data when the models are ..."
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Cited by 1 (0 self)
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Many data in applications exhibit nonlinear features such as nonlinearity between lagged variables and heteroscedasticity. They require nonlinear models to describe the data. Parametric nonlinear time series models provide powerful tools for analyzing nonlinear time series data when the models are correctly specified, thus the choice for the form of a parametric model is very critical. A natural alternative is to use nonparametric methods. One of the interesting nonparametric models to fit nonlinear time series is the well known functional coefficient regression (FCR) model. There are in literature some works related to this model, with different approaches of estimation (e.g., kernel estimation, spline estimation). A very common supposition of the model is related to the errors, where they are natural to be supposed independent. In this work we will study the estimation of FCR model by splines, with dependent errors. The comparison of the rate of convergence between the models with correlated and independent errors will be done. Moreover, a real application will illustrate the method by fitting a model and performing forecasts. We will compare our results with others used in literature.
Bayesian Analysis of Threshold Autoregressive Moving Average Models
"... In this paper we consider a Bayesian analysis for threshold autoregressive moving average models. Two different methods are used for the special case of two regimes. In the first we consider a hierarchical prior and obtain posterior distributions in closed form. In the second, a rearranged procedure ..."
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Cited by 1 (0 self)
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In this paper we consider a Bayesian analysis for threshold autoregressive moving average models. Two different methods are used for the special case of two regimes. In the first we consider a hierarchical prior and obtain posterior distributions in closed form. In the second, a rearranged procedure due Tsay (1989) is used, in conjunction with the Gibbs sampler and MetropolisHastings algorithm. Applications are given for a simulated series and for the sunspot data.