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).
Some commonly used tests to check for the presence of a unit root (a characteristic of $I(1)$ time series) are: Augmented Dickey Fuller (ADF), Phillips–Perron (PP), Dickey Fuller Generalized Least Squares (DF-GLS) and Kwiatkowski–Phillips–Schmidt–Shin (KPSS).
The main problem with all these tests is that none of them is very powerful and often give conflicting results. Also, not differencing a series if there is a unit root has serious consequences but differencing the series if there is no unit root just causes a minor loss of efficiency. If unit root test results are equivocal, assume there is a unit root, difference the series and correct for nonspherical errors. The minor loss in efficiency can easily be ignored for that extra bit of unbiasedness and robustness in light of John Cochrane's brilliant quip on empirical economics and finance,
I can think of no case in which the application of a clever statistical model to wring the last ounce of efficiency out of a data set, changing t statistics from 1.5 to 2.5, substantially changed the way people think about an issue. Cochrane(2001)