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A distribution theory is developed for least-squares estimates of the threshold in Threshold Autoregressive (TAR) models. We find that if we let the threshold effect (the difference in slopes between the two regimes) become small as the sample size increases, then the asymptotic distribution of the threshold estimator is free of nuisance parameters (up to scale). Similarly, the likelihood ratio statistic for testing hypotheses concerning the unknown threshold is asymptotically free of nuisance parameters. These asymptotic distributions are nonstandard, but are available in closed form, so critical values are readily available. To illustrate this theory, we report an application to the U.S. unemployment rate. We find statistically significant threshold effects.

This paper accomplishes two goals. First, it introduces a powerful nonparametric test of asymmetry to the economics literature, namely, the triples test of Randles et al. (1980). Second, it documents the presence of two specific kinds of asymmetry in U.S. macroeconomic time series. Depth, or asymmetry in the distribution of a (detrended) series, is a feature of numerous economic time series; and steepness, or asymmetry in the distribution of first differences, is a feature of hours, employment, and the unemployment rate, but is absent from real GDP and aggregate industrial production. The pattern of asymmetries found provides guidelines for restricting the set of alternative nonlinear models from which to select in modeling these time series.