This paper examines the interaction between non-linear deterministic trends and long run dependence by means of employing Chebyshev time polynomials and assuming that the detrended series displays long memory with the pole or singularity in the spectrum occurring at one or more possibly non-zero frequencies. The combination of the non-linear structure with the long memory framework produces a model which is linear in parameters and therefore it permits the estimation of the deterministic terms by standard OLS-GLS methods. Moreover, the orthogonality property of Chebyshev’s polynomials makes them especially attractive to approximate non-linear structures of data. We present a procedure which allows us to test (possibly fractional) orders of integration at various frequencies in the presence of the Chebyshev trends with no effect on the standard limit distribution of the method. Several Monte Carlo experiments are conducted and the results indicate that the method performs well. An empirical application, using data of real exchange rates is also carried out at the end of the article.
This study explores the asymmetric exchange rate exposure of stock returns building upon the capital asset pricing model (CAPM) framework, using monthly returns of Chinese industry indices. We are interested in estimating long run and short run relationships as well as asymmetric effects. In order to do so, we estimate nonlinear autoregressive distributed lags models to (1) obtain the long run or cointegrated effects and dynamics, (2) be able to mix I(1) and I(0) variables and (3) to split the effect of positive and negative changes in the variables, i.e. asymmetries. In accordance with the existing literature, industry returns are subject to lagged exposure effects, but the asymmetries vary across industries, which could be due to the discrepancies in, amongst others, trade balance and ownership of certain industries. Furthermore, the dynamic multipliers depict that industry returns quickly respond to changes in the exchange rate and correct the disequilibrium within a short time, making the long run exposure to be symmetric or very small. The remaining shocks are mainly explained by the return of market portfolios. This implies that the ongoing restrictions on the RMB daily trading band do indeed protect the Chinese stock market against the effects of currency movements.