I was estimating a long-run relationship of exchange rate and purchasing power parity. The residual of the long-run relation which should be $I(0)$, but it is only $I(0)$ when I introduce trend in the long-run relationship. Can someone provide the logic or study material that shows introducing trend is not a problem and how to justify it?
The critical values of the unit root test you are using depends if there is a trend or not. For example, the quantiles of the Dickey-Fuller distribution is different when a trend is included from when a trend is not included, hence the critical values for your unit root test are different. The critical values of unit root tests are generated by simulation, with different kinds of deterministic terms to match the series you are considering.