I am kinda new to time-series analysis, I want model CEE (EUR/HUF, EUR/PLN, EUR/CZK, EUR/CHF) exchange rates with ARIMA. I understand that according to Box-Jenkins modeling, I should first check if my dataset is stationary. I ran the ADF, and KPSS test, for the exchange rates and I got the results that with a drift term and without trend my data is not stationary, however, when I ran the ADF test with drift and trend term the null hypotesis was rejected. As far as I understand this means there is deterministic trend within my data, so it is trend-stationary. The KPSS test in most of the case accepts stationarity (p value around 0.1), however again when I check for example for EUR/HUF (2013-2020), the ADF test with drift and trend suggests it is stationary, but the KPSS with the same terms (drift and trend) says that the p<5% so meaning non stationary. Also, the EUR/PLN exchange rate is very interesting because the ADF (both with trend and drift; and with drift no trend suggest that the data is stationary (<0.01).
Also, this changes with the amount of data I use for checking (for example 2000-2020 dataset, in almost all of the exchange rates rejects stationarity, but in case of the EUR/HUF it still suggests that there is a deterministic trend, same like before with trend and constant the null hypotesis is rejected).
My question is, even if there is deterministic trend, can I still use log-differencing (aka log-returns) to make it stationary or I need to fit the original dataset to an lm model and use the residual (aka detrending method as follows in R):
Using the following code in case of the EUR/HUF exchange rate the KPSS test still rejects with drift and trend that our data is stationary.
Also, does having less data for model building (2013-2020) means I couldn't check the stationarity of a longer dataset (2000-2020)? I wouldn't use the ma smoothing method, as I have daily frequency, and I wouldn't like to lose data.
In case of differencing (1) of course every test says that our data is stationary, my worry is that it would be misleading not to detrend and use the arima model with such data.