In this blog, we examine the issue of identifying unit roots in the presence of structural breaks.
We will use the quarterly US current account to GDP ratio to compare results from a number of unit root test found in the GAUSS tspdlib library including the:
- Zivot-Andrews (1992) unit root test with a single structural break.
- Narayan and Popp (2010) unit root test with two structural breaks
- Lee and Strazicich (2013, 2003) LM tests with one and two structural breaks
- Enders and Lee Fourier (2012) ADF and LM tests
What is stationarity?
Stationary series have a mean and covariance that do not change over time. This implies that a series is mean-reverting and any shock to the series will have a temporary effect.
In the graph above we compare a stationary, non-stationary, and trend stationary AR(1) series. Looking at the blue stationary series we can see that despite the random shocks, the series fluctuates around its zero mean.
The green line is the same series as the stationary series with an added constant time trend. Though this time-series grows over time, it fluctuates around its constant time trend.
Finally, the orange line is a non-stationary series and has a unit root. It does not revert to a mean or trend line and its stochastic shocks have permanent impacts on the series.
How are structural breaks and stationarity related?
The graph above helps demonstrate the impact of structural breaks on stationarity. The series plotted above shows a structural break in the level and clearly does not revert around the same mean across all time.
Though the series is stationary within each section, most standard unit roots will bias towards non-rejection of the unit root for this series.
This is problematic because different modeling techniques should be used for the unit-root series than the series with a structural break.
Testing for unit roots with structural breaks
A number of different unit root tests have emerged from the research surrounding structural breaks and unit roots. These tests vary depending on the number of breaks in the data, whether a trend is present or not, and the null hypothesis that is being tested.
Today we will compare the results from six different tests, all of which can be found in the GAUSS tspdlib.
Preparing for testing
Before running the tests in the GAUSS tspdlib library, we must set-up a number of input parameters. All of the tests, with the exception of the fourier expansion test, require the same inputs:
- Vector, the time-series to tested.
- Scalar, indicates the type of model to be tested.
1 = break in level.
2 = break in level and mean.
- Scalar, Maximum number of lags for Dy. 0 = no lags.
- Scalar, the information criterion used to select lags.
1 = Akaike.
2 = Schwarz.
3 = t-stat significance.
- Scalar, data trimming rate. Not required for the fourier expansion tests.
- Scalar, maximum number of single Fourier frequency. Required only for the fourier expansion tests.