# KPSS: Difference between level stationary and trend stationary

Can anyone please clarify for me the differences between level stationary and trend stationary in KPSS test? I run the KPSS test with trend and level on same time series and the results are:

H0: level stationary vs. H1: Unit root. Test statistic: 0.1691746

p-value: 0.245586 Upper tail percentiles: 10% 5% 2.5% 1% Critical value 0.347 0.463 0.573 0.739

## from above I infer level stationary at 5% as p-value>0.05 and test stat < 0.463.

H0: trend stationary vs. H1: Unit root. Test statistic: 0.08118887

p-value: 0.2409267 Upper tail percentiles: 10% 5% 2.5% 1% Critical value 0.119 0.146 0.176 0.216

## Here I accept the H0 and infer trend stationary at 5% (stationary around a deterministic trend)

1)I am not sure how to explain these two results? Is it stationary or should I de-trend my time series to make it stationary? 2) How come the second test accepts trend stationary when the p-value from MK test (mann kendall, where H0 is monotonic trend) is less than 0.01? I appreciate your advise.

## 1 Answer

I think you have a missunderstanding of the interpretation of a statistical test. In detail, you cannot conclude anything if the null hypothesis is not rejected. Especially you never accept $$H_0$$. You are just only able to "fail to reject $$H_0$$".

In your case I would recommend an (augmented) Dickey-Fuller test. This test has a (trend) stationary time series as alternative hypothesis. If you can reject the null hypothesis of a unit-root this gives inference about the stationarity properties of your time series

• Thank you for correcting me regarding interpreting the results: I fail to reject the H0 of trend stationary in favor of alternative of having a unit root. Regarding the adf with trend I tried it and again it rejects the H0 of presents of unit root, no trend and no drift. please see below: Value of test-statistic is: -22.591 170.1188 255.1772 Critical values for test statistics: 1pct 5pct 10pct tau3 -3.96 -3.41 -3.12 phi2 6.09 4.68 4.03 phi3 8.27 6.25 5.34 – Saraz Jun 2 at 0:55
• I interpreted it as at least one of them (unit root, drift or trend) is not zero. (So it can be trend stationary?). The ACF plot shows non-stationary (not sure how to paste an image in comments) and I can see weekly seasonality from the plot. – Saraz Jun 2 at 1:14
• @Saraz you are welcome. I don't really get what you mean with "at least one of them is not zero". The first interpretation of the test would be that your time series contains no unit root and is therefore trend stationary on a 1 % significance level (since $-22.591<-3.96$), i.e. is stationary except a deterministic trend. Additional you might want to test if your time series is even level stationary. In order to do so, you apply an ADF test with level and without trend. If you still can reject the null hypothesis of a unit root, you conclude stationarity. – FeynmanBestMan Jun 2 at 9:13
• If this helped you, you might want to consider accepting the answer by clicking the check button – FeynmanBestMan Jun 3 at 13:25
• Thank you for reply. Regarding the comparison, I thought that I should compare the absolute values (|-22.591|>|-3.69|).Failing to reject phi2 implies unit root and no time trend term and no drift term. Rejecting this null implies that one, two, OR all three of these terms was NOT zero. Looking at phi2 test statistic (170.1188) and critical values (6.09 4.68 4.03) at least one of the unit root, trend or drift is not zero. I tested the adf with type "none" and still reject the H0. – Saraz Jun 3 at 22:11