. 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...

A declared need is around for geoinformatic surveillance statistical science and software infrastructure for spatial and spatiotemporal hotspot detection. Hotspot means something unusual, anomaly, aberration, outbreak, elevated cluster, critical resource area, etc. The declared need may be for monitoring, etiology, management, or early warning. The responsible factors may be natural, accidental, or intentional. This proof-of-concept paper suggests methods and tools for hotspot detection across geographic regions and across networks. The investigation proposes development of statistical methods and tools that have immediate potential for use in critical societal areas, such as public health and disease surveillance, ecosystem health, water resources and water services, transportation networks, persistent poverty typologies and trajectories, environmental justice, biosurveillance and biosecurity, among others. We introduce, for multidisciplinary use, an innovation of the health-area-popular circle-based spatial and spatiotemporal scan statistic. Our innovation employs the notion of an upper level set, and is accordingly called the upper level set scan statistic, pointing to a sophisticated analytical and computational system as the next generation of the present day popular SaTScan. Success of surveillance rests on potential elevated cluster detection capability. But the clusters can be of any shape, and cannot be captured only by circles. This is likely to give more of false alarms and more of false sense of security. What we need is capability to detect arbitrarily shaped clusters. The proposed upper level set scan statistic innovation is expected to ®ll this need

COWLES FOUNDATION DISCUSSION PAPER NO. 1229

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

This paper aims to bridge the gap between processes where shocks are permanent and those with transitory shocks by formulating a process in which the long run impact of each innovation is time varying and stochastic. Frequent transitory shocks are supplemented by occasional permanent shifts. The stochastic permanent breaks (STOPBREAK) process is based on the premise that a shock is more likely to be permanent if it is large than if it is small. This formulation is motivated by a class of processes that undergo random structural breaks. Consistency and asymptotic normality of quasi maximum likelihood estimates is established and locally best hypothesis tests of the null of a random walk are developed. The model is applied to relative prices of pairs of stocks and significant test statistics result. KEYWORDS: Structural breaks, nonlinear moving average, unit roots, quasi maximum likelihood estimation, Neyman-Pearson testing, locally best test, temporary cointegration. 1. INTRODUCTION Time series analysts tend to draw a sharp line between processes where shocks have a permanent effect and those where they do not. The most notable example of this is the distinction between stationary AR(1) processes, where all shocks are transitory, and the random walk. As the autoregressive root approaches one, the rate at which shocks are expected to decay decreases, but they remain transitory. This paper aims to bridge the gap between transience and permanence by formulating a process in which the long run impact of each observation is time varying and stochastic. At one extreme all innovations are transitory and at the other, all shocks are permanent. 2

This paper develops an asymptotic theory of inference for an unrestricted two-regime threshold autoregressive Ž TAR. model with an autoregressive unit root. We find that the asymptotic null distribution of Wald tests for a threshold are nonstandard and different from the stationary case, and suggest basing inference on a bootstrap approximation. We also study the asymptotic null distributions of tests for an autoregressive unit root, and find that they are nonstandard and dependent on the presence of a threshold effect. We propose both asymptotic and bootstrap-based tests. These tests and distribution theory allow for the joint consideration of nonlinearity Ž thresholds. and nonstationary Žunit roots.. Our limit theory is based on a new set of tools that combine unit root asymptotics with empirical process methods. We work with a particular two-parameter empirical process that converges weakly to a two-parameter Brownian motion. Our limit distributions involve stochastic integrals with respect to this two-parameter process. This theory is entirely new and may find applications in other contexts. We illustrate the methods with an application to the U.S. monthly unemployment rate. We find strong evidence of a threshold effect. The point estimates suggest that the threshold effect is in the short-run dynamics, rather than in the dominate root. While the conventional ADF test for a unit root is insignificant, our TAR unit root tests are arguably significant. The evidence is quite strong that the unemployment rate is not a unit root process, and there is considerable evidence that the series is a stationary TAR process.

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This paper tests for the existence of "geographic poverty traps", such that characteristics of a household's area of residence ) it's "geographic capital" ) entail that the household's consumption cannot rise over time, while an otherwise identical household living in a better There are various administrative and other restrictions on migration, including registration and residency requirements. For example, it appears to be rare for a rural worker who moves to an urban area to be allowed to enrol his or her children in the urban schools. 5 endowed area enjoys a rising standard of living. The paper also tries to identify the factors which may lead to the emergence of such poverty traps. If borne out by empirical evidence, geographic poverty traps suggest both efficiency and equity arguments for investing in poor areas, such as by developing local infrastructure or by assisting labor export to better endowed areas. The setting for our empirical work is post-reform rural China and we study the determinants of consumption growth for farm households. We can rule out potential endogeneity due to people choosing their locations because there was little or no geographic mobility of labor in rural China at the time. Governmental restrictions on migration within China are part of the reason

The authors thank three anonymous referees and the Editor Badi Baltagi for several useful comments. This work was supported by the Bank of Canada and by grants from the Canadian Network of Centres of Excellence [program on Mathematics