Abstract: In the first half of this century, special attention was given to two features of the business cycle: the comovement of many individual economic series and the different behavior of the economy during expansions and contractions. Recent theoretical and empirical research has revived interest in each attribute separately, and we survey this work. Notable empirical contributions are dynamic factor models that have a single common macroeconomic factor and nonlinear regime-switching models of a macroeconomic aggregate. We conduct an empirical synthesis that incorporates both of these features. It is desirable to know the facts before attempting to explain them; hence, the attractiveness of organizing business-cycle regularities within a model-free framework. During the first half of this century, much research was devoted to obtaining just such an empirical characterization of the business cycle. The most prominent example of this work

. The nonparametric and the nuisance parameter approaches to consistently testing statistical models are both attempts to estimate topological measures of distance between a parametric and a nonparametric fit, and neither dominates in experiments. This topological unification allows us to greatly extend the nuisance parameter approach. How and why the nuisance parameter approach works and how it can be extended bears closely on recent developments in artificial neural networks. Statistical content is provided by viewing specification tests with nuisance parameters as tests of hypotheses about Banach-valued random elements and applying the Banach Central Limit Theorem and Law of Iterated Logarithm, leading to simple procedures that can be used as a guide to when computationally more elaborate procedures may be warranted. 1. Introduction In testing whether or not a parametric statistical model is correctly specified, there are a number of apparently distinct approaches one might take. T...

While there has been a great deal of interest in the modelling of non-linearities in economic time series, there is no clear consensus regarding the forecasting abilities of non-linear time series models. We evaluate the performance of two leading non-linear models in forecasting post-war US GNP, the self-exciting threshold autoregressive model and the Markov-switching autoregressive model. Two methods of analysis are employed: an empirical forecast accuracy comparison of the two models, and a Monte Carlo study. The latter allows us to control for factors that may otherwise undermine the performance of the non-linear models. 1 Introduction In recent years there has been a great deal of interest in the modelling of non-linearities in economic time series. While the usefulness of linear time-series models in the tradition of Box and Jenkins (1970) is usually gauged by their predictive ability, there does not appear to be a clear consensus as to whether allowing for non-linearities has l...

Threshold autoregressive models in which the process is piecewise linear in the threshold space have received much attention in recent years. In this paper, we use predictive residuals to construct a test statistic to detect threshold nonlinearity in a vector time series and propose a procedure for building a multivariate threshold model. The thresholds and the model are selected jointly based on the Akaike information criterion. The nite-sample performance of the proposed test is studied by simulation. The modeling procedure is then used to study arbitrage in security markets and results in a threshold cointegration between logarithms of future contracts and spot prices of a security after adjusting for the cost-of-carrying the contracts. In this particular application, thresholds are determined in part by the transaction costs. We also apply the proposed procedure to U.S. monthly interest rates and two river ow series of Iceland.

COWLES FOUNDATION DISCUSSION PAPER NO. 1229

In this paper we investigate the multi-period 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 `non-linearity' 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 non-linear 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.

Abstract. The problem of testing for linearity and the number of regimes in the context of self-exciting threshold autoregressive (SETAR) models is reviewed. We describe least-squares methods of estimation and inference. The primary complication is that the testing problem is non-standard, due to the presence of parameters which are only defined under the alternative, so the asymptotic distribution of the test statistics is non-standard. 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; Non-standard asymptotic theory; Bootstrap

We examine dynamic asymmetries in U.S unemployment using nonlinear time series models and Bayesian methods. We#ndstrong statistical evidence in favor of a two-regime threshold autoregressive model. Empirical results indicate that, once wetakeinto account both parameter and model uncertainty, there are economically interesting asymmetries in the unemployment rate. One #nding of particular interest is that shocks whichlower the unemployment rate tend to have a smaller e#ect than shocks which raise the unemployment rate. This #nding is consistent with unemploymentrises being sudden and falls gradual. Keywords: Nonlinearity, Threshold Autoregression, Bayesian, Unemployment. JEL: C11,C22,C52,E24. # Financial support from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged. y Financial support from the NSF under grant SES 9211726 and the Center for Computable Economics at UCLA is gratefully acknowledged. 1 Introduction The vast majority of reduced fo...

paper was also partly written at the Centre de recherche en Économie et Statistique (INSEE, Paris) and the Technische

We compare a number of methods that have been proposed in the literature for obtaining h- step ahead minimum mean square error forecasts for SETAR models. These forecasts are compared to those from an AR model. The comparison of forecasting methods is made using Monte Carlo simulation. The Monte Carlo method of calculating SETAR forecasts is generally at least as good as that of the other methods we consider. An exception is when the disturbances in the SETAR model come from a highly asymmetric distribution, when a Bootstrap method is to be preferred. An empirical application calculates multi-period forecasts from a SETAR model of US GNP using a number of the forecasting methods. We find that whether there are improvements in forecast performance relative to a linear AR model depends on the historical epoch we select, and whether forecasts are evaluated conditional on the regime the process was in at the time the forecast was made. Keywords: Threshold model, forecasting, simulations. 1.