What are accepted econometric methods to find out if a time series is I1 or not? An important issue in deciding how to specify an time-series econometric model is deciding whether to use series in levels or in differences (or second differences!). However, to decide this, you often have to find out if the series are integrated or order 1 or not (I1). What are the current accepted tests to find out? Are there trade-offs between the tests? It seems hard to know whether interest rates for example are I1 or not!
An I(1) series is also known as a series with a unit root. Therefore, the econometric tests to inquire on the order of integration of a time-series are referred to as "unit-root tests".

There are three widely used unit-root tests: Augmented Dickey-Fuller (ADF), Phillips-Perron (PP) and Kwiatkowski–Phillips–Schmidt–Shin (KPSS). 

The null hypothesis (H0) in the ADF and PP tests assumes the presence of a unit root. For KPPS it assumes no unit root. 

The tests often give different results, and best practice would require to report all three statistics as a robustness check.

There are many drawbacks of using these tests. The major issue lies in the lack of power of PP and ADF to reject H0 when the series is I(0) but close to I(1), that is when the series is highly persistent but stationary. To avoid this low power, other tests have been devised by Elliot, Rothenberg and Stock (1996), as well as Ng and Perron (2001).