[Abstract]

This paper proposes a new test for selecting trigonometric terms when the deterministic trend of a univariate time series is periodic.
The problem of unknown frequencies in such a cyclical trend was first investigated by Anderson (1971) under the assumption of serially uncorrelated error term, but we allow for an autoregressive error term without prior knowledge as to whether the error term is stationary or contains an autoregressive unit root. This is achieved by modifying the test statistic proposed by Perron, Shintani and Yabu (2016) which requires the assumption of a known set of frequencies. We show that limiting distribution of our test statistic can be common between the stationary and unit root cases even if frequencies are unknown. Simulation results confirm that our procedure works well with sample size typically available in practice. We illustrate the usefulness of our method via an application to international data on unemployment rate.

要旨：Until Bengal Province in British India was partitioned in 1947, the region was ruled as British India. This study quantitatively explores the changing population geography in Bengal, with a particular focus on the events of 1947 (Partition of India) and 1971 (Independence of Bangladesh). Based on decadal census data from 1901 to 2001 at the district level, this paper examines how trends in regional population growth evolved with such historical events. Following Redding and Sturm (2008), Differences-in-Differences estimation is also employed here and the events of 1947 and 1971 are taken as the subjects of the test. Estimation results show that there were different shocks on both sides and from both events. In West Bengal, the change in the regional population trends occurred in 1947 and remained similar thereafter. On the other hand, in East Bengal, this did not occur in 1947, but did occur in 1971. Further robustness checks show that the impacts were not uniform with respect to the distance from the border. Overall analyses show that the emergence of the international border in Bengal had asymmetric impacts on both sides of the population geography. The results suggest that changes in the population geography reflect the degree of tensions over the border.