This paper extends the cross‐sectionally augmented panel unit‐root test (CIPS) developed by Pesaran et al. (2013, Journal of Econometrics, Vol. 175, pp. 94–115) to allow for smoothing structural changes in deterministic terms modelled by a Fourier function. The proposed statistic is called the break augmented CIPS (BCIPS) statistic. We show that the non‐standard limiting distribution of the (truncated) BCIPS statistic exists and tabulate its critical values. Monte‐Carlo experiments point out that the sizes and powers of the BCIPS statistic are generally satisfactory as long as the number of time periods, T, is not less than fifty. The BCIPS test is then applied to examine the validity of long‐run purchasing power parity.

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This paper highlights the crucial role of a time-varying threshold effect of public debt on economic growth. Our contribution is two-fold. First, we extend the constant-threshold regression kink model of Hansen (2017) by allowing for a time-varying, state-dependent threshold. Second, we apply our model to investigate the effect of debt on growth, using data from the U.S. over the period of 1791-2009. Our empirical results clearly support a nonlinear debt-threshold effect and the threshold is time-varying and state-dependent.

Classical threshold models assume that threshold values are constant and stable, which appears overly restrictive and unrealistic. In this article, we extend Hansen's (2000) constant threshold regression model by allowing for a time‐varying threshold which is approximated by a Fourier function. Least‐square estimation of regression slopes and the time‐varying threshold is proposed, and test statistics for the existence of threshold effect and threshold constancy are constructed. We also develop the asymptotic distribution theory for the time‐varying threshold estimator. Through Monte Carlo simulations, we show that the proposed estimation and testing procedures work reasonably well in finite samples, and there is little efficiency loss by the allowance for Fourier approximation in the estimation procedure even when there is no time‐varying feature in the threshold. On the contrary, the slope estimates are seriously biased when the threshold is time‐varying but being treated as a constant. The model is illustrated with an empirical application to a nonlinear Taylor rule for the United States.

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