.Threshold models have a wide variety of applications in economics. Direct applications include models of separating and multiple equilibria. Other applications include empirical sample splitting when the sample split is based on a continuously-distributed variable such as firm size. In addition, threshold models may be used as a parsimonious strategy for nonparametric function estimation. For example, the threshold autoregressive model .TAR is popular in the nonlinear time series literature. Threshold models also emerge as special cases of more complex statistical frameworks, such as mixture models, switching models, Markov switching models, and smooth transition threshold models. It may be important to understand the statistical properties of threshold models as a preliminary step in the development of statistical tools to handle these more complicated structures. Despite the large number of potential applications, the statistical theory of threshold estimation is undeveloped. It is known that threshold estimates are super-consistent, but a distribution theory useful for testing and inference has yet to be provided. This paper develops a statistical theory for threshold estimation in the regression context. We allow for either cross-section or time series observations. Least squares estimation of the regression parameters is considered. An asymptotic distribution theory . for the regression estimates the threshold and the regression slopes is developed. It is found that the distribution of the threshold estimate is nonstandard. A method to construct asymptotic confidence intervals is developed by inverting the likelihood ratio statistic. It is shown that this yields asymptotically conservative confidence regions. Monte Carlo simulations are presented to assess the accuracy of the asymptotic approximations. The empirical relevance of the theory is illustrated through an application to the multiple . equilibria growth model of Durlauf and Johnson 1995 .

/pdf/sample-splitting-and-threshold-estimation-44wnau4n4j.pdf

Sample Splitting and Threshold Estimation

Many econometric testing problems involve nuisance parameters which are not identified under the null hypotheses. This paper studies the asymptotic distribution theory for such tests. The asymptotic distributions of standard test statistics are described as functionals of chi-square processes. In general, the distributions depend upon a large number of unknown parameters. We show that a transformation based upon a conditional probability measure yields an asymptotic distribution free of nuisance parameters, and we show that this transformation can be easily approximated via simulation. The theory is applied to threshold models, with special attention given to the so-called self-exciting threshold autoregressive model. Monte Carlo methods are used to assess the finite sample distributions. The tests are applied to U.S. GNP growth rates, and we find that Potter's (1995) threshold effect in this series can be possibly explained by sampling variation.

/pdf/inference-when-a-nuisance-parameter-is-not-identified-under-1kbhk92q65.pdf

Inference when a nuisance parameter is not identified under the null hypothesis

This paper develops an asymptotic theory of inference for an unrestricted two-regime Ž. threshold autoregressive 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 Ž. Ž allow for the joint consideration of nonlinearity thresholds and nonstationary unit . 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.

/pdf/threshold-autoregression-with-a-unit-root-3nn45abjdy.pdf

Threshold autoregression with a unit root

Threshold autoregressive models in which the process is piecewise linear in the threshold space have received much attention in recent years. In this article I use predictive residuals to construct a test statistic for detecting 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 finite-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. I also apply the proposed procedure to U.S. monthly interest rates and two river flow series of Iceland.

/pdf/testing-and-modeling-multivariate-threshold-models-188ht3it7u.pdf

Testing and modeling multivariate threshold models

This is a survey of the basic properties of strong mixing conditions for sequences of random variables. The focus will be on the “structural” properties of these conditions, and not at all on limit theory. For a discussion of central limit theorems and related results under these conditions, the reader is referred to Peligrad [60] or Iosifescu [50]. This survey will be divided into eight sections, as follows: 

1.

Measures of dependence




2.

Five strong mixing conditions




3.

Mixing conditions for two or more sequences




4.

Mixing conditions for Markov chains




5.

Mixing conditions for Gaussian sequences




6.

Some other special examples




7.

The behavior of the dependence coefficients




8.

Approximation of mixing sequences by other random sequences.

Basic Properties of Strong Mixing Conditions

Some mixing properties of time series models

This paper considers the prediction of the sample mean by extreme order statistics when the population distribution is known. The predictor and its mean square error are found. The problem is studied in details for the normal model.

Tuan D. Pham

Papers

Some mixing properties of time series models

Predicting the sample mean by extreme order statistics