# Exchange rate forecasting using ARIMA

I would like to conduct an ARIMA forecast of an exchange rate with 3030 daily observations. I followed these steps;

1. I looked for the stationary for the original series ($P_t$) and concluded that the series is not stationary,
2. I took the natural log of the series and then differenced so that I came up with return series($r_t = \ln P_t-\ln P_{t-1})$ of the exchange rate.
3. I used R functions auto.arima and arima on $r_t$ and got the same result as ARIMA(0,0,0).

Should I conclude that $r_t$ follows a random walk and the original series ($P_t$) fits ARIMA(0,1,0)?

auto.arima is not tool for testing whether the selected model is somehow "correct" or "true". What it does is tell you that among the candidate models it investigated, the one reported had the lowest AICc value. This suggests the model is the best among them when it comes to forecasting under square loss.
Thus you cannot make statements like $r_t$ follows ARIMA(0,0,0) ($p$-value<0.05); but you can take this as an indication that any ARIMA model tried by auto.arima is unlikely to provide a better forecast than ARIMA(0,0,0) for $r_t$ or ARIMA(0,1,0) for $\text{ln}(P_t)$.
And no, $r_t$ does not follow a random walk; $\text{ln}(P_t)$ does.