Does nonlinearity help resolve the Fisher effect puzzle?

Even though there is little evidence for linear cointegration, Christopoulos and Leon-Ledesma (2007) recently have found nonlinear cointegrating relations between the US quarterly nominal interest rate and CPI inflation rate. Through Monte Carlo simulations, they also show that the nonlinear relations are responsible for the 'Fisher effect puzzle' of less than a proportional coefficient of inflation in the linear Fisher relation. We provide, in this study, an explanation of their simulation results on the Fisher effect puzzle and further examine if nonlinear relations really help resolving the puzzle.