Showing papers on "STAR model published in 2001"

Algorithm 808: ARfit—a matlab package for the estimation of parameters and eigenmodes of multivariate autoregressive models

Abstract: ARfit is a collection of Matlab modules for modeling and analyzing multivariate time series with autoregressive (AR) models. ARfit contains modules to given time series data, for analyzing eigen modes of a fitted model, and for simulating AR processes. ARfit estimates the parameters of AR models from given time series data with a stepwise least squares algorithm that is computationally efficient, in particular when the data are high-dimensional. ARfit modules construct approximate confidence intervals for the estimated parameters and compute statistics with which the adequacy of a fitted model can be assessed. Dynamical characteristics of the modeled time series can be examined by means of a decomposition of a fitted AR model into eigenmodes and associated oscillation periods, damping times, and excitations. The ARfit module that performs the eigendecomposition of a fitted model also constructs approximate confidence intervals for the eigenmodes and their oscillation periods and damping times.

On a mixture autoregressive conditional heteroscedastic model

Abstract: We propose a mixture autoregressive conditional heteroscedastic (MAR-ARCH) model for modeling nonlinear time series. The models consist of a mixture of K autoregressive components with autoregressive conditional heteroscedasticity; that is, the conditional mean of the process variable follows a mixture AR (MAR) process, whereas the conditional variance of the process variable follows a mixture ARCH process. In addition to the advantage of better description of the conditional distributions from the MAR model, the MARARCH model allows a more flexible squared autocorrelation structure. The stationarity conditions, autocorrelation function, and squared autocorrelation function are derived. Construction of multiple step predictive distributions is discussed. The estimation can be easily done through a simple EM algorithm, and the model selection problem is addressed. The shape-changing feature of the conditional distributions makes these models capable of modeling time series with multimodal conditional distr...

A Linear Poisson Autoregressive Model: The Poisson AR(p) Model

Abstract: Time series of event counts are common in political science and other social science applications. Presently, there are few satisfactory methods for identifying the dynamics in such data and accounting for the dynamic processes in event counts regression. We address this issue by building on earlier work for persistent event counts in the Poisson exponentially weighted moving-average model (PEWMA) of Brandt et al. (American Journal of Political Science 44(4):823–843, 2000). We develop an alternative model for stationary mean reverting data, the Poisson autoregressive model of order p, or PAR(p) model. Issues of identification and model selection are also considered. We then evaluate the properties of this model and present both Monte Carlo evidence and applications to illustrate.

Least absolute deviation estimation for all-pass time series models

Abstract: An autoregressive moving average model in which all of the roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa is called an all-pass time series model. All-pass models generate uncorrelated (white noise) time series, but these series are not independent in the non-Gaussian case. An approximation to the likelihood of the model in the case of Laplacian (two-sided exponential) noise yields a modified absolute deviations criterion, which can be used even if the underlying noise is not Laplacian. Asymptotic normality for least absolute deviation estimators of the model parameters is established under general conditions. Behavior of the estimators in finite samples is studied via simulation. The methodology is applied to exchange rate returns to show that linear all-pass models can mimic “nonlinear” behavior, and is applied to stock market volume data to illustrate a two-step procedure for fitting noncausal autoregressions.

On a logistic mixture autoregressive model

Abstract: SUMMARY We generalise the mixture autoregressive, MAR, model to the logistic mixture autoregressive with exogenous variables, LMARX, model for the modelling of nonlinear time series. The models consist of a mixture of two Gaussian transfer function models with the mixing proportions changing over time. The model can also be considered as a generalisation of the self-exciting threshold autoregressive, SETAR, model and the open-loop threshold autoregressive, TARSO, model. The advantages of the LMARX model over other nonlinear time series models include a wider range of shape-changing predictive distributions, the ability to handle cycles and conditional heteroscedasticity in the time series and better point prediction. Estimation is easily done via a simple EM algorithm and the model selection problem is addressed. The models are applied to two real datasets and compared with other competing models.

Estimation for Autoregressive Time Series With a Root Near 1

Abstract: Estimators for the parameters of autoregressive time series are compared, emphasizing processes with a unit root or a root close to 1. The approximate bias of the sum of the autoregressive coefficients is expressed as a function of the test for a unit root. This expression is used to construct an estimator that is nearly unbiased for the parameter of the first-order scalar process. The estimator for the first-order process has a mean squared error that is about 40% of that of ordinary least squares for the process with a unit root and a constant mean, and the mean squared error is smaller than that of ordinary least squares for about half of the parameter space. The maximum loss of efficiency is 6n-1 in the remainder of the parameter space. The estimation procedure is extended to higher-order processes by modifying the estimator of the sum of the autoregressive coefficients. Limiting results are derived for the autoregressive process with a mean that is a linear trend.

Index arbitrage with heterogeneous investors: A smooth transition error correction analysis

Abstract: The traditional index arbitrage model assumes a constant threshold mispricing between the futures and cash prices for all investors. Allowing for heterogeneity in investors' transaction costs, objectives, and capital constraints, we model the intraday mispricing of DJIA futures as a smooth transition autoregressive (STAR) process with the speed of adjustment toward equilibrium varying directly with the mispricing. We show that the observed mean reversion in mispricing changes is induced by heterogeneous arbitrageurs, instead of a statistical illusion-infrequent trading of index portfolio stocks. We further use a STAR error correction model to describe the nonlinear dynamics between the DJIA futures and index. This model describes not only which market is more informationally efficient than the other, but also the legging process – the nonsimultaneous establishing of cash and futures positions.

Bootstrap-After-Bootstrap Prediction Intervals for Autoregressive Models

Abstract: The use of the Bonferroni prediction interval based on the bootstrap-after-bootstrap is proposed for autoregressive (AR) models. Monte Carlo simulations are conducted using a number of AR models including stationary, unit-root, and near-unit-root processes. The major finding is that the bootstrap-after-bootstrap provides a superior small-sample alternative to asymptotic and standard bootstrap prediction intervals. The latter are often too narrow, substantially underestimating future uncertainty, especially when the model has unit roots or near unit roots. Bootstrap-after-bootstrap prediction intervals are found to provide accurate and conservative assessment of future uncertainty under nearly all circumstances considered.

How to estimate autoregressive roots near unity

Abstract: A new model of near integration is formulated in which the local to unity parameter is identifiable and consistently estimable with time series data. The properties of the model are investigated, new functional laws for near integrated time series are obtained that lead to mixed diffusion processes, and consistent estimators of the localizing parameter are constructed. The model provides a more complete interface between I (0) and I (1) models than the traditional local to unity model and leads to autoregressive coefficient estimates with rates of convergence that vary continuously between the O (√n) rate of stationary autoregression, the O ( n ) rate of unit root regression, and the power rate of explosive autoregression. Models with deterministic trends are also considered, least squares trend regression is shown to be efficient, and consistent estimates of the localizing parameter are obtained for this case also. Conventional unit root tests are shown to be consistent against local alternatives in the new class.

A nonlinear time series model of El Niño

Abstract: A smooth transition autoregressive model is estimated for the Southern Oscillation Index, an index commonly used as a measure of El Nino events. Using standard measures there is no indication of nonstationarity in the index. A logistic smooth transition autoregressive model describes the most turbulent periods in the data (these correspond to El Nino events) better than a linear autoregressive model. The estimated nonlinear model passes a battery of diagnostic tests. A generalised impulse response function indicates local instability, but as deterministic extrapolation from the estimated model converges, the nonlinear model may still be useful for forecasting the El Nino Southern Oscillation a few months ahead.

Bootstrap Tests for an Autoregressive Unit Root in the Presence of Weakly Dependent Errors

Abstract: This paper examines bootstrap tests of the null hypothesis of an autoregressive unit root in models that may include a linear rend and/or an intercept and which are driven by innovations that belong to the class of stationary and invertible linear processes. Our approach makes use of a sieve bootstrap procedure based on residual resampling from autoregressive approximations, the order of which increases with the sample size at a suitable rate. We show that the sieve bootstrap provides asymptotically valid tests of the unit-root hypothesis and demonstrate the small-sample effectiveness of the method by means of simulation.

Non-linear autoregressive time series with multivariate Gaussian mixtures as marginal distributions

Abstract: A new form of non-linear autoregressive time series is proposed to model solar radiation data, by specifying joint marginal distributions at low lags to be multivariate Gaussian mixtures. The model is also a type of multiprocess dynamic linear model, but with the advantage that the likelihood has a closed form.

Computationally Efficient Stochastic Realization for Internal Multiscale Autoregressive Models

Abstract: In this paper we develop a stochastic realization theory for multiscale autoregressive (MAR) processes that leads to computationally efficient realization algorithms. The utility of MAR processes has been limited by the fact that the previously known general purpose realization algorithm, based on canonical correlations, leads to model inconsistencies and has complexity quartic in problem size. Our realization theory and algorithms addresses these issues by focusing on the estimation-theoretic concept of predictive efficiency and by exploiting the scale-recursive structure of so-called internal MAR processes. Our realization algorithm has complexity quadratic in problem size and with an approximation we also obtain an algorithm that has complexity linear in problem size.

Bicorrelations and cross‐bicorrelations as non‐linearity tests and tools for exchange rate forecasting

Abstract: This paper proposes and implements a new methodology for forecasting time series, based on bicorrelations and cross-bicorrelations. It is shown that the forecasting technique arises as a natural extension of, and as a complement to, existing univariate and multivariate non-linearity tests. The formulations are essentially modified autoregressive or vector autoregressive models respectively, which can be estimated using ordinary least squares. The techniques are applied to a set of high-frequency exchange rate returns, and their out-of-sample forecasting performance is compared to that of other time series models

On large deviations in the Gaussian autoregressive process: stable, unstable and explosive cases

Abstract: For the Gaussian autoregressive process, the asymptotic behaviour of the Yule‐Walker estimator is totally different in the stable, unstable and explosive cases. We show that, irrespective of this trichotomy, this estimator shares quite similar large deviation properties in the three situations. However, in the explosive case, we obtain an unusual rate function with a discontinuity point at its minimum.

Asymptotic Distributions of Quasi-Maximum Likelihood Estimators for Spatial Econometric Models I: Spatial Autoregressive Processes

Abstract: Asymptotic properties of MLEs and QMLEs of spatial autoregressive processes are investigated. The stochastic rate of convergence of the MLE and QMLE for a spatial autoregressive process may be less than the √ n-rate under some circumstances even though its limiting distribution is asymptotically normal. Implications of the possible low rate of convergence of the estimators on classical statistics such as the likelihood ratio, Wald, and efficient score statistics are analyzed.

Vector autoregressive processes with nonlinear time trends in cointegrating relations

Abstract: We extend the conventional cointegrated VAR model to allow for general nonlinear deterministic trends. These nonlinear trends can be used to model gradual structural changes in the intercept term of the cointegrating relations. A general asymptotic theory of estimation and statistical inference is reviewed and a diagnostic test for the correct specification of an employed nonlinear trend is developed. The methods are applied to Finnish interest-rate data. A smooth level shift of the logistic form between the own-yield of broad money and the short-term money market rate is found appropriate for these data. The level shift is motivated by the deregulation of issuing certificates of deposit and its inclusion in the model solves the puzzle of the “missing cointegration vector” found in a previous study.

Consistent estimation in cointegrated vector autoregressive models with nonlinear time trends in cointegrating relations

Abstract: This paper studies the consistency of the Gaussian maximum likelihood estimator in a cointegrated vector autoregressive model with nonlinear time trends in cointegrating relations. The results are proved in a coordinate free framework that readily allows for general nonlinear parameter restrictions and makes it possible to show the consistency of reduced form parameter estimators without assuming identifiability of underlying structural parameters. Various consistency results for structural parameter estimators can then be obtained by imposing suitable identification conditions for the parameters of interest but not necessarily for nuisance parameters. Orders of consistency are also obtained because they are needed to develop a related asymptotic theory of statistical inference.

EWMA control charts for autoregressive processes

Abstract: Many processes must be monitored by using observations that are correlated. An approach called algorithmic statistical process control can be employed in such situations. This involves fitting an autoregressive/moving average time series model to the data. Forecasts obtained from the model are used for active control, while the forecast errors are monitored by using a control chart. In this paper we consider using an exponentially weighted moving average (EWMA) chart for monitoring the residuals from an autoregressive model. We present a computational method for finding the out-of-control average run length (ARL) for such a control chart when the process mean shifts. As an application, we suggest a procedure and provide an example for finding the control limits of an EWMA chart for monitoring residuals from an autoregressive model that will provide an acceptable out-of-control ARL. A computer program for the needed calculations is provided via the World Wide Web.

Identification and dichotomization of long- and short-run relations of cointegrated vector autoregressive models

Abstract: We show that the usual rank condition is necessary and sufficient to identify a vector autoregressive process whether the variables are I(0) or I(d) for d = 1,2,.... We then use this rank condition to demonstrate the interdependence between the identification of short-run and long-run relations of cointegrated process. We find that both the short-run and long-run relations can be identified without the existence of prior information to identify either relation. But if there exists a set of prior restrictions to identify the short-run relation, then this same set of restrictions is sufficient to identify the corresponding long-run relation. On the other hand, it is in general not possible to identify the long-run relations without information on the complete structure. The relationship between the identification of a vector autoregressive process and a Cowles Commission dynamic simultaneous equations model is also clarified.

Time Series: ARIMA Methods

Abstract: Time series data in business, economics, environment, medicine, and other scientific fields tend to exhibit patterns such as trends, seasonal fluctuations, irregular cycles, and occasional shifts in level or variability. The objectives of analyzing such series are often to extrapolate the dynamic pattern in the data for forecasting future observations, to estimate the effect of known exogenous interventions, and to detect unsuspected interventions. This article sketches some useful methods for achieving these objectives. Serial dependence in time series data can often be adequately approximated by a linear dynamic model. We focus on the development of a class of parsimonious linear models, the mixed autoregressive moving average (ARMA) models, which are widely used to represent stationary and nonstationary series. For unit root nonstationarity, this leads to the class of autoregressive integrated moving average (ARIMA) models. An iterative model building approach proposed by Box and Jenkins, consisting of tentative model specification, efficient estimation, and diagnostic checking, is discussed and illustrated by an actual example. The ARIMA models are extended to represent seasonal patterns in the data, and to include an additive component for the dynamic effect of interventions. The correspondence between the ARMA models and the GARCH models commonly used in analyzing financial time series data with heterogeneous variances is illustrated. Finally, some extensions to nonlinear time series models, in particular the threshold autoregressive models, are presented.

Adaptive parameter estimation in multivariate self-exciting threshold autoregressive models

Abstract: We define multivariate Self–Exciting Threshold Autoregressive (MSETAR) Models and present an adaptive algorithm for the estimation of the AR coefficients. This algorithm has a similar structure as stochastic gradient procedures (so–called LMS algorithms) which have been frequently used in linear models. It is shown that conditions of convergence known from the linear case can be reformulated for the nonlinear MSETAR model.

Random walks and non-linear paths in macroeconomic time series: Some evidence and implications

Abstract: This paper investigates whether the inherent non-stationarity of macroeconomic time series is entirely due to a random walk or also to non-linear components. Applying the numerical tools of the analysis of dynamical systems to long time series for the US, we reject the hypothesis that these series are generated solely by a linear stochastic process. Contrary to the Real Business Cycle theory that attributes the irregular behavior of the system to exogenous random factors, we maintain that the fluctuations in the time series we examined cannot be explained only by means of external shocks plugged into linear autoregressive models. A dynamical and non-linear explanation may be useful for the double aim of describing and forecasting more accurately the evolution of the system. Linear growth models that find empirical verification on linear econometric analysis, are therefore seriously called in question. Conversely non-linear dynamical models may enable us to achieve a more complete information about economic phenomena from the same data sets used in the empirical analysis which are in support of Real Business Cycle Theory. We conclude that Real Business Cycle theory and more in general the unit root autoregressive models are an inadequate device for a satisfactory understanding of economic time series. A theoretical approach grounded on non-linear metric methods, may however allow to identify non-linear structures that endogenously generate fluctuations in macroeconomic time series. (authors' abstract)

Unit root seasonal autoregressive models with a polynomial trend of higher degree

Abstract: Seasonal autoregressive models with a polynomial trend of higher degee are treated In the unit root case, the limiting distribution of the normalized least squares estimator for the autoregressive parameter and that of the corresponding t-statistic are discussed as the length of the sample period tends to infinity In the case where the polynomial trend has the second or third degree, the joint moment generating functions associated with these limiting distributions are derived, and some simulation results are reported The asymptotic behavior of these limiting distributions is discussed when the polynomial degree or the number of seasons tends to infinity