# Contradiction in the ADF (Augmented Dickey-Fuller) and KPSS (Kwiatkowski–Phillips–Schmidt–Shin) tests for financial time series

I use the ADF and KPSS to test for stationarity / non-stationarity of price increments in financial time series.

The two test applied provide different results for low lags, but the same result for higher lags:

Lag = 1: KPSS rejects H(0) is stationary, however ADF rejects H(0) is a unit root in favour for H(1) is stationarity.

Lag = 2: same results as for lag 1.

Lag = 3: both tests indicate stationarity for price increments.

Lag = 4: both tests indicate stationarity for price increments.

Lag = 5: both tests indicate stationarity for price increments.

1. Are the increments of prices of this financial time seris stationary or not?

2. The estimation of the generalized Hurst exponent (I use the algorithm provided by Tomaso Aste (2003)) requires that price increments are stationary. What do the results above mean with respect to this requirement?

• Welcome to Cross Validated! Could you please add the full reference for Aste's paper? May 31, 2016 at 20:09
• The Hurst exponent algorithm is found in MATLAB: es.mathworks.com/matlabcentral/fileexchange/… For examples, some papers: link link Jun 1, 2016 at 0:17
• Thanks! But I just meant edit the question by adding something like this at the bottom: Aste et al. (2003), "A new algorithm ... Hurst exponent", J. Whichever, 5, 7. (Probably not much relevant to answering this question, but just as a general point.) Jun 1, 2016 at 13:06