Daily time series analysis : 1st difference ? I have a time series of daily returns for many years and i have to do preliminary analysis, GARCH modeling and predictions. 
I did all my preliminary analysis, ADF test (there's a unit root) , Jarque-Bera test, acf, ... and GARCH modeling using the logarithm of the data.
My question is : Since the series is non-stationary should i have used the difference of the log(returns) for all my diagnostics ?


*

*The plot of the log(returns) shows that the series is non-stazionary (trend, volatility clustering,..). 

*The ADF test suggest a unit root 

*The fitted garch(1,1) shows leverage effect 

 A: You might want to look closely at 
What are the consequences of not meeting the assumptions for the residuals of ARIMA model? as it discusses the FLAW of using ADF for discrete data ( like small count daily data)
If your values are "large" thus approaching continuous data which can often exhibit some of the following effects and the idea is to identify the important ones .....
Trends, Seasonality, Monthly or Weekly patterns, Level Shifts, Big increases and drops, but not necessarily a trend, Autoregressive behavior (ARIMA), Fixed Day of the month, Seasonal Pulses (Changes in Day of the week effects) , Interventions, Holidays plus before and after and others like day-of-the-month , week-of-the-month. Error variance and/or parameter variance over time can also come into play AND possible response to known external effects .
On average differencing daily data is a very bad idea (although possible) given my long experience with analyzing daily data based upon my consultancy and a number of SE posts on this subject. Search on "user:3382  daily " for a number of comments about daily data
Additionally taking/imposing/assuming the need for a tranformation like logs is very dangerous stuff . See When (and why) should you take the log of a distribution (of numbers)? for when and why to take logs.
A: Take a look at this question. Javlacalle's answer gives you a particularly detailed procedure for how to distinguish stochastic and linear trends, and which procedures to use for each. Make sure you've checked that the ADF assumptions hold for your data.
First difference is one detrending technique, but it's not always appropriate. You should check out the question I linked as it offers tons of resources for choosing a technique. 
