This article examines the interaction between fractional integration and nonlinear structures by using for the latter the Chebyshev polynomials in time that can be taken as an alternative, less abrupt way of modelling breaks in time series data. A Lagrange multiplier test, developed for testing the order of integration in the context of nonlinear deterministic trends, is implemented in three well-known and previously studied time series data: the Nile river data, the temperatures in the Northern hemisphere and CO₂ emissions in the US. The results suggest that the second and especially the third time series display nonlinear behaviour still with fractional degrees of differentiation.

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