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Many researchers have used a cointegration approach to test for the Fisher effect. This note argues that the cointegration of the nominal interest rate and the inflation rate is consistent with any theory implying a stationary real interest rate and so is not a sufficient condition for ex post the Fisher effect to hold. The sufficient condition is the unpredictability of the inflation forecast error implied by the nominal interest rate and this condition may be tested using the signal extraction framework of Durlauf and Hall (1988, 1989).
Johnson, Paul A., "Is it really the Fisher effect?" (2005). Faculty Research and Reports. 82.