Bubble Testing under Deterministic Trends

This paper develops the asymptotic theory of the ordinary least squares estimator of the autoregressive (AR) coefficient in various AR models, when data is generated from trend-stationary models in different forms. It is shown that, depending on how the autoregression is specified, the commonly used right-tailed unit root tests may tend to reject the null hypothesis of unit root in favor of the explosive alternative. A new procedure to implement the right-tailed unit root tests is proposed. It is shown that when the data generating process is trend-stationary, the test statistics based on the proposed procedure cannot find evidence of explosiveness. Whereas, when the data generating process is mildly explosive, the unit root tests find evidence of explosiveness. Hence, the proposed procedure enables robust bubble testing under deterministic trends. Empirical implementation of the proposed procedure using data from the stock and the real estate markets in the US reveals some interesting findings. While our proposed procedure flags the same number of bubbles episodes in the stock data as the method developed in Phillips, Shi and Yu (2015a, PSY), the estimated termination dates by the proposed procedure match better with the data. For real estate data, all negative bubble episodes flagged by PSY are no longer regarded as bubbles by the proposed procedure.