Can someone teach me on how to find the trend and cycle of my series and also the seasonality of my series to check if I will use SARIMA instead of arima?
We are not satisfied with just graphs to show it that's why I also want the test or some statistical idea you use to explain it.
The auto.arima() function in the forecast package for R, which I personally consider the gold standard in automatic ARIMA modeling, uses the OCSB test. See here for the reference and further information.
Alternatively, you can use a holdout sample, fit both ARIMA and SARIMA to the rest of the data, forecast into the holdout sample and see which method gives lower errors (e.g., Mean Squared Errors). This is simple and easy to explain to the consumers of the forecast.
To detect seasonality you could take the first difference and plot the autocorrelation function and see if there is some seasonal pattern. To test for trend I guess you can just fit a regression line and see if differs from zero. If it does you need to determine if the trend is stochastic or deterministic, this can be done with the augmented dickey fuller test.
