Showing papers on "STAR model published in 1999"

An Autoregressive Distributed-Lag Modelling Approach to Cointegration Analysis

Abstract: This paper examines the use of autoregressive distributed lag (ARDL) models for the analysis of long-run relations when the underlying variables are I(1). It shows that after appropriate augmentation of the order of the ARDL model, the OLS estimators of the short-run parameters are p T -consistent with the asymptotically singular covariance matrix, and the ARDL-based estimators of the long-run coe¢cients are super-consistent, and valid inferences on the long-run parameters can be made using standard normal asymptotic theory. The paper also examines the relationship between the ARDL procedure and the fully modi…ed OLS approach of Phillips and Hansen to estimation of cointegrating relations, and compares the small sample performance of these two approaches via Monte Carlo experiments. These results provide strong evidence in favour of a rehabilitation of the traditional ARDL approach to time series econometric modelling. The ARDL approach has the additional advantage of yielding consistent estimates of the long-run coe¢cients that are asymptotically normal irrespective of whether the underlying regressors are I(1) or I(0). JEL Classi…cations: C12, C13, C15, C22. Key Words: Autoregressive distributed lag model, Cointegration, I(1) and I(0) regressors, Model selection, Monte Carlo simulation. ¤This is a revised version of a paper presented at the Symposium at the Centennial of Ragnar Frisch, The Norwegian Academy of Science and Letters, Oslo, March 3-5, 1995. We are grateful to Peter Boswijk, Clive Granger, Alberto Holly, Kyung So Im, Brendan McCabe, Steve Satchell, Richard Smith, Ron Smith and an anonymous referee for helpful comments. Partial …nancial support from the ESRC (Grant No. R000233608) and the Isaac Newton Trust of Trinity College, Cambridge is gratefully acknowledged.

Modelling asymmetries and moving equilibria in unemployment rates

Abstract: The paper discusses a simple univariate nonlinear parametric time-series model for unemployment rates, focusing on the asymmetry observed in many OECD unemployment rate series. The model is based on a standard logistic smooth transition autoregressive (LSTAR) model for the first difference of unemployment, but it also includes a lagged level term. This model allows for asymmetric behaviour by permitting 'local' nonstationarity in a globally stable model. Linearity tests are performed for a number of quarterly, seasonally unadjusted, unemployment series from OECD countries, and linearity is rejected for a number of them. For a number of series, nonlinearity found by testing can be modelled satisfactorily by use of our smooth transition autoregressive model. The properties of the estimated models, including persistence of the shocks according to them, are illustrated in various ways and discussed. Possible existence of moving equilibria in series not showing asymmetry is investigated and modelled with another STAR model.

Modelling Multiple Regimes in the Business Cycle

Abstract: textThe interest in business cycle asymmetry has been steadily increasing over the last fifteen years. Most research has focused on the different behaviour of macroeconomic variables during expansions and contractions, which by now is well documented. Recent evidence suggests that such a two-phase characterization of the business cycle might be too restrictive. In particular, it might be worthwhile to decompose the recovery phase in a high-growth phase (immediately following the trough of a cycle) and a subsequent moderate-growth phase. In this paper, the issue of multiple regimes is addressed using Smooth Transition AutoRegressive [STAR] models. A possible limitation of STAR models as they are currently used is that essentially they deal with only two regimes. We propose a generalization of the STAR model such that more than two regimes can be accommodated. It is demonstrated that the class of Multiple Regime STAR [MRSTAR] models can be obtained from the two-regime model in an elegant way. The main properties of the MRSTAR model and several issues which might be relevant for empirical specification are discussed in detail. In particular, a Lagrange Multiplier-type test is derived which can be used to determine the appropriate number of regimes. Application of the new model class to US real GNP and US unemployment rate provides evidence in favor of the existence of multiple business cycle phases.

On Single-Index Coefficient Regression Models

Abstract: In this article we investigate a class of single-index coefficient regression models under dependence. This includes many existing models, such as the smooth transition threshold autoregressive (STAR) model of Chan and Tong, the functional-coefficient autoregressive (FAR) model of Chen and Tsay, and the single-index model of Ichimura. Compared to the varying-coefficient model of Hastie and Tibshirani, our model can avoid the curse of dimensionality in multivariate nonparametric estimations. Another advantage of this model is that a threshold variable is chosen automatically. An estimation method is proposed, and the corresponding estimators are shown to be consistent and asymptotically normal. Some simulations and applications are also reported.

A Generalized Gamma Autoregressive Conditional Duration Model

Abstract: I extend the ACD model of Engle and Russell (1998) to generalized gamma durations with a conditional mean that depends on the exponential of the explanatory variables. This allows for a non-monotonic hazard function taking U-shaped or inverted U-shaped forms. The extension implies that the trading intensity persistence is reduced considerably, and that the overall fit of the model is enhanced compared to the ACD model. As a further extension of the model it is shown how to include time-varying covariates in a fully parametric framework. We analyze how transaction rates are affected by the posting of price-quotes and their changes. Besides, a model of the time between price-changes is estimated. This model is, as shown by Engle and Russell (1998), closely linked to the volatility of the stock price, and hence showing why price durations are important for intra-day prediction of volatility. The transaction volume and functions of this are used as regressors in this model and are found to be important. The datasets used in the paper consist of a random sample from the fifty stocks at the NYSE with the highest capitalization value on December 13, 1996.

Subsampling intervals in autoregressive models with linear time trend

Abstract: A new method is proposed for constructing confidence intervals in autoregressive models with linear time trend. Interest focuses on the sum of the autoregressive coefficients because this parameter provides a useful scalar measure of the long-run persistence properties of an economic time series. Since the type of the limiting distribution of the corresponding OLS estimator, as well as the rate of its convergence, depend in a discontinuous fashion upon whether the true parameter is less than one or equal to one (that is, trend-stationary case or unit root case), the construction of confidence intervals is notoriously difficult. The crux of our method is to recompute the OLS estimator on smaller blocks of the observed data, according to the general subsampling idea of Politis and Romano (1994a), although some extensions of the standard theory are needed. The method is more general than previous approaches in that it works for arbitrary parameter values, but also because it allows the innovations to be a martingale difference sequence rather than i.i.d. Some simulation studies examine the finite sample performance.

Priors and component structures in autoregressive time series models

Abstract: New approaches to prior specification and structuring in autoregressive time series models are introduced and developed. We focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series. These new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends, quasi-periodic subprocesses and high frequency residual noise components of observed series. The class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters. Analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series, as it does for predictions of future values. Posterior analysis involves easily implemented iterative simulation methods, developed and described here. One motivating field of application is climatology, where the evaluation of latent structure, especially quasi-periodic structure, is of critical importance in connection with issues of global climatic variability. We explore the analysis of data from the southern oscillation index, one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators.

A Spatial Prior for Bayesian Vector Autoregressive Models

Abstract: In this paper we develop a Bayesian prior motivated by cross-sectional spatial autoregressive models for use in time-series vector autoregressive forecasting involving spatial variables. We compare forecast accuracy of the proposed spatial prior to that from a vector autoregressive model relying on the Minnesota prior and find a significant improvement. In addition to a spatially motivated prior variance as in LeSage and Pan (1995) we develop a set of prior means based on spatial contiguity. A Theil-Goldberger estimator may be used for the proposed model making it easy to implement.

Cauchy estimators for autoregressive processes with applications to unit root tests and confidence intervals

Abstract: For autoregressive processes, we propose new estimators whose pivotal statistics have the standard normal limiting distribution for all ranges of the autoregressive parameters. The proposed estimators are approximately median unbiased. For seasonal time series, the new estimators give us unit root tests that have limiting normal distribution regardless of period of the seasonality. Using the estimators, confidence intervals of the autoregressive parameters are constructed. A Monte-Carlo simulation for first-order autoregressions shows that the proposed tests for unit roots are locally more powerful than the tests based on the ordinary least squares estimators. It also shows that the proposed confidence intervals have shorter average lengths than those of Andrews (1993, Econometrica 61, 139–165) based on the ordinary least squares estimators when the autoregressive coefficient is close to one.

Two-dimensional autoregressive (2-D AR) model order estimation

Abstract: Much research has been devoted to the area of one-dimensional autoregressive (1-D AR) and autoregressive moving average (ARMA) model order selection. The most well-known solutions for this problem are the Akaike information criterion (AIC), MDL, and the minimum eigenvalue (MEV) criteria. On the other hand, all works in the 2-D case have focused on the problem of parameter estimation. In this correspondence, we extend the previous criteria to the 2-D AR model order determination. The model is assumed causal, stable, and spatially invariant with p/sub 1//spl times/p/sub 2/ quarter-plane (QP) support. Numerical examples are given to illustrate the effectiveness of each method.

Inverse Gaussian Autoregressive Models

Abstract: A first-order autoregressive process with one-dimensional inverse Gaussian marginals is introduced. The innovation distributions are obtained in certain special cases. The unknown parameters are estimated using different methods and these estimators are shown to be consistent and asymptotically normal. Performance of the estimators is discussed using simulation experiments.

Markov switching time series models with application to a daily runoff series

Abstract: We consider a class of Bayesian dynamic models that involve switching among various regimes. As an example we produce a model for a runoff time series exhibiting pulsatile behavior. This model is a mixture of three autoregressive models which accommodate “rising,” “falling,” and “normal” states in the runoff process. The mechanism for switching among regimes is given by a three-state Markov chain whose transition probabilities are modeled on the basis both of past runoff values and of a time series of rainfall data. We adopt the Bayesian approach and use the Gibbs sampler in the numerical analyses. A study of a daily runoff series from Lake Taupo, New Zealand, is given.

Nonlinear time series modelling with the radial basis function-based state-dependent autoregressive model

Abstract: This paper investigates nonlinear time series modelling using the general state-dependent autoregressive model. To achieve the estimate of the model, an attempt is made to approximate the state-dependent parameter by employing the Gaussian radial basis function for its universal approximation capability. As a result, a radial basis function-based autoregressive (RBF-AR) model is derived which has a form similar to a generalized exponential autoregressive model. To reach the applicability of the RBF-AR model, the evolutionary programming algorithm is employed to select a suitable set of radial basis function centres. By applying the resulting model to some complex data, it is shown that the RBF-AR model can not only reconstruct the dynamics of given nonlinear time series effectively, but also give much better fitting to complextime series than the approach of directly RBF neural network modelling. Therefore, as a paradigm combining a statistical model and neural network, the RBF-AR model has better perform...

Smooth Transition Models: Extensions and Outlier Robust Inference

Abstract: textThe dynamic properties of many economic time series variables can be characterised as state-dependent or regime-switching. A popular model to describe this type of non-linear behaviour is the smooth transition model, which accommodates two regimes facilitating a gradual transition from one regime to the other. The first part of this thesis considers three extensions of the basic smooth transition model. Models are developed which allow for more than two regimes, for time-varying properties in conjunction with regime-switching behaviour, and for modeling several time series jointly. Particular emphasis is placed on the inter-related issues of specification and inference in such models. The second part of the thesis concerns the influence of atypical observations on testing procedures for smooth transition non-linearity and on the estimation of smooth transition models. Traditional methods that are used for these purposes are found to be very sensitive to such outliers. Therefore, outlier robust testing procedures and estimation methods are developed

Bayesian Inference on Periodicities and Component Spectral Structure in Time Series

Abstract: We detail and illustrate time series analysis and spectral inference in autoregressive models with a focus on the underlying latent structure and time series decompositions. A novel class of priors on parameters of latent components leads to a new class of smoothness priors on autoregressive coefficients, provides for formal inference on model order, including very high order models, and leads to the incorporation of uncertainty about model order into summary inferences. The class of prior models also allows for subsets of unit roots, and hence leads to inference on sustained though stochastically time-varying periodicities in time series. Applications to analysis of the frequency composition of time series, in both time and spectral domains, is illustrated in a study of a time series from astronomy. This analysis demonstrates the impact and utility of the new class of priors in addressing model order uncertainty and in allowing for unit root structure. Time-domain decomposition of a time series into estimated latent components provides an important alternative view of the component spectral characteristics of a series. In addition, our data analysis illustrates the utility of the smoothness prior and allowance for unit root structure in inference about spectral densities. In particular, the framework overcomes supposed problems in spectral estimation with autoregressive models using more traditional model-fitting methods.

Detecting periodicity in experimental data using linear modeling techniques

Abstract: Fourier spectral estimates and, to a lesser extent, the autocorrelation function are the primary tools to detect periodicities in experimental data in the physical and biological sciences. We propose a method which is more reliable than traditional techniques, and is able to make clear identification of periodic behavior when traditional techniques do not. This technique is based on an information theoretic reduction of linear (autoregressive) models so that only the essential features of an autoregressive model are retained. These models we call reduced autoregressive models (RARM). The essential features of reduced autoregressive models include any periodicity present in the data. We provide theoretical and numerical evidence from both experimental and artificial data to demonstrate that this technique will reliably detect periodicities if and only if they are present in the data. There are strong information theoretic arguments to support the statement that RARM detects periodicities if they are present. Surrogate data techniques are used to ensure the converse. Furthermore, our calculations demonstrate that RARM is more robust, more accurate, and more sensitive than traditional spectral techniques.

A unified view of linear AR(1) models

Abstract: We review and synthesize the wide range of non-Gaussian first order linear autoregressive models that have appeared in the literature Models are organized into broad classes to clarify similarities and differences and facilitate application in particular situations General properties for process mean, variance and correlation are derived, unifying many separate results appearing in the literature Examples illustrate the wide range of properties that can appear even under the autoregressive assumption These results are used in analysing a variety of real data sets, illustrating general methods of estimation, model diagnostics and model selection

Reconstructing bifurcation diagrams from noisy time series using nonlinear autoregressive models.

Abstract: We introduce a formalism for the reconstruction of bifurcation diagrams from noisy time series. The method consists in finding a parametrized predictor function whose bifurcation structure is similar to that of the given system. The reconstruction algorithm is composed of two stages: model selection and bifurcation parameter identification. In the first stage, an appropriate model that best represents all the given time series is selected. A nonlinear autoregressive model with polynomial terms is employed in this study. The identification of the bifurcation parameters from among the many model parameters is done in the second stage. The algorithm works well even for a limited number of time series.

A rank test based approach to order estimation. I. 2-D AR models application

Abstract: In system identification and parametric spectral estimation by two-dimensional (2-D) autoregressive (AR) and 2-D autoregressive moving average (ARMA) models, the order selection problem is often required. In this correspondence, we show that the information of 2-D AR model order is implicitly contained in a correlation matrix. An algorithm for 2-D quarter-plane AR model order determination is proposed. Numerical simulations are presented to show the efficiency of the proposed singular value decomposition (SVD) based algorithm.

On the Finite-Sample Accuracy of Nonparametric Resampling Algorithms for Economic Time Series

Abstract: In recent years, there has been increasing interest in nonparametric bootstrap inference for economic time series. Nonparametric resampling techniques help protect against overly optimistic inference in time series models of unknown structure. They are particularly useful for evaluating the fit of dynamic economic models in terms of their spectra, impulse responses, and related statistics, because they do not require a correctly specified economic model. Notwithstanding the potential advantages of nonparametric bootstrap methods, their reliability in small samples is questionable. In this paper, we provide a benchmark for the relative accuracy of several nonparametric resampling algorithms based on ARMA representations of four macroeconomic time series. For each algorithm, we evaluate the effective coverage accuracy of impulse response and spectral density bootstrap confidence intervals for standard sample sizes. We find that the autoregressive sieve approach based on the encompassing model is most accurate. However, care must be exercised in selecting the lag order of the autoregressive approximation.

Modeling Nonlinearity of Business Cycles: Choosing Between the CDR and STAR Models

Abstract: Nonlinear modeling has become popular in applied macroeconomics. Successful attempts include Beaudry and Koop's CDR (current depth of the recession) model of real GNP, and various STAR (smooth transition autoregression) models of industrial production. However, these models have not been directly compared. We compare CDR and STAR models of U.S. real GNP and industrial production. We find (i) within sample, the CDR model fits slightly better than the STAR model; (ii) out of sample, the CDR model forecasts better than the STAR model; and (iii) the CDR model generates very different dynamics than the STAR model.

Temporal and contemporaneous disaggregation of multiple economic time series

Abstract: A method is proposed for estimating unobserved values of multiple time series whose temporal and contemporaneous aggregates are known. The resulting estimates are obtained from a model-based procedure in which the models employed are indicated by the data alone. This procedure is empirically supported by a discrepancy measure here derived. Even though the problem can be cast into a state-space formulation, the usual assumptions underlying Kalman filtering are not fulfilled and such an approach cannot be applied directly. Some simulated examples are provided to validate the method numerically and an application with real data serves to illustrate its use in practice.

Forecasting with periodic autoregressive time series models

Abstract: textThis paper is concerned with forecasting univariate seasonal time series data using periodic autoregressive models. We show how one should account for unit roots and deterministic terms when generating out-of-sample forecasts. We illustrate the models for various quarterly UK consumption series.

Rank-Based Autoregressive Order Identification

Abstract: Optimal rank-based procedures have been derived for testing arbitrary linear restrictions on the parameters of autoregressive moving average (ARMA) models with unspecified innovation densities. The finite-sample performances of these procedures are investigated here in the context of AR order identification and compared to those of classical (partial correlograms and Lagrange multipliers) methods. The results achieved by rank-based methods are quite comparable, in the Gaussian case, to those achieved by the traditional ones, which, under Gaussian assumptions, are asymptotically optimal. However, under non-Gaussian innovation densities, especially heavy-tailed or nonsymmetric, or when outliers are present, the percentages of correct order selection based on rank methods are strikingly better than those resulting from traditional approaches, even in the case of very short (n = 25) series. These empirical findings confirm the often ignored theoretical fact that the Gaussian case, in the ARMA context...

Higher-Order Residual Analysis for Simple Bilinear and Threshold Autoregressive Models with the TR Test

Abstract: Linear models have traditionally dominated the time series analysis of macroeconomic and financial data. Evidence of nonlinear behavior in the dynamic behavior of such data would imply that the conventional linear models are misspecified. As a result of such model misspecification, appropriately specified nonlinear forecasts would generally be superior to the optimal linear forecast.

Miscellanea. Random coefficient autoregressive models for longitudinal data

Abstract: SUMMARY Stable autoregressive models with exogenous covariates can be used to mimic nonstationary, stabilising growth profiles in longitudinal data. If different individuals have very different trajectories, the autoregressive parameters can be interpreted as random variables varying from one individual to the next. If all random effects are assumed Gaussian, the model will be quite easy to estimate although the marginal distributions of the observations will no longer be Gaussian. However, this kind of model can only be used when the observations are equally spaced.