While working on different papers, I encountered an interesting finding and decided to reach out to this community for some clarification. I normally use multiple tests, namely ADF, PP & KPSS, to find out the order of integration of different series I work with, before performing any targeted analysis.
I have come across scenarios wherein the ADF and PP test results pertaining to a time series at levels indicate rejection of null hypothesis I(1), while the KPSS test results also indicate rejection of null hypothesis I(0). In other words, the results from all the unit root tests are not mutually reinforcing.
I then first-differenced time series hoping for mutually reinforcing results from different unit root tests when applied upon first-differenced time series (As opposed to the original levels). However, I find that the unit root results still persist despite first-differencing. When applied to first-differenced time series, ADF and PP test results still indicate rejection of null hypothesis I(1), while the KPSS test results still indicate rejection null hypothesis I(0).
When we have mixed results, what do we do?
I have come across many publications in finance, wherein the order of integration is reported based on any one unit root test and the authors proceed to targeted tests such as cointegration, VAR, and Seemingly Unrelated Regressions. In all such cases, I wonder if the trajectory of such research papers would be the same if multiple unit root tests that in-turn give mixed results were shown.
Hi! No test is absolutelly perfect, so it is very usual that you obtain mixed results. You must check the advantages and disadvantages of each test and try to choose the one that better fits your needs and your data. Even so, if you run 3 or 4 different tests and the majority of them suggest you to go in one direction, I think you should proceed like this. Unit root tests are commonly a source of controversy, so you have to make a choice and do not expect to have absolutely consistent results across the several alternative tests.
I have the vague memory of the KPSS test being a test for stationarity while the others are tests for non-stationarity.
But overall why does it matter? What do you want to do with those variables? If you want to regress one on the other then it does not matter if they integrated or nonintegrated, as long as they are both the same degree of integration.
To see if they are related, do a cross-plot. And plotting the series themselves will tell you if they are stationary or not --better than any unit root tests. Look, UR tests are know to have many issues. Your data may have structural breaks or outliers in which case you can kiss goodbye to any meaningful result in the UR tests.
In Finance stationarity is referred to as mean reversion. The extent to which reversion to mean is observed in response to the data depends on the number of observations. With 900 a series may be quite persistent, but return to mean quite a lot. I always advise looking at the data and when there is drift and some variation as will arise with a stochastic trend then the non-stationary null may seem likely. The reverse is true when the data is differenced and returns often to mean with the data being differenced such as will occur with daily stock market returns. However, the performance of a parametric test such as the ADF is sensitive to the model within which it is framed. Assuming that there are no breaks in structure and with a stochastic trend it is much more difficult than might be thought to distinguish the natural variation in the process as compared with breaks especially with series that are volatile. The ADF test may be sensitive to initial values and Rob Taylor has suggested recursive demeaning of the data is more appropriate than a simple mean correction. Non-normality may also be important and to this end there is evidence on the wild bootstrap while Beirne, Hunter and Simpson (2007) available from RG suggest that White standard errors may be robust to simple non-iid errors. The ADF critical values are not sensitive to the lag order of the time series model in which they are embedded just the sample so the simplification of the model by excluding intermediate lags does not impact the critical value, but may improve the efficiency of the estimation and alter the subsequent results.
If the ADF and KPSS are compared, then the ADF is usually defined on a first differenced variable whose dynamic will be shorter than is the case for the non-transformed data especially when that is defined by a random walk. While in addition to the null of the KPSS being stationary, the numerator of the test statistic includes squared cumulated residuals. The statistic is consistent when scaled by an appropriate measure of the long-run variance and this relates to a residual that in the simplest form of the test is simply corrects for the mean. However, these residuals may be quite persistent as they are not differenced. Hence, the lag in the ADF test cannot be an indication of the lag truncation in the KPSS test that as an appropriate correction to the lag cumulants may need to be significantly longer than is anticipated. Hence, the test is likely to be incorrect when the lag truncation is too short. This may be further complicated when the series exhibits near integrated volatility as may occur with financial data.
Hadri (2000) provides a panel test that is the average of a number of KPSS tests. He shows the test is normal when a Bartlet correction is applied and the test with a reasonable sample N>50 is appropriately sized and not sensitive to non-normality almost irrespective of the cross section dimension. In fact the correction can be applied to a single KPSS test and again this will follow a normal distribution in the limit. So the Hadri small sample corrected test may be preferred to the KPSS test with moderately dimensioned samples.
Often when semi-parametric corrections are applied as is the case with the Newey-West robust standard errors then the lag truncation is set at T/3. However, applying such a rule coherently with an evolving sample would not be consistent as the truncation on the KPSS test cannot grow at a rate T, but T/3 can be applied to a fixed sample. When the sample is extended, then the truncation lag must evolve more slowly so a third initially, then by a quarter, a fifth etc.
It is also useful to consider what is the most appropriate approximation.
Dear Prof. John Hunter
I thank you very much for a very insightful and comprehensive response. Appreciate your time and attention in this regard.
Contrary to the data generating process for ADF type unit root tests, where high positive autocorrelation of the series is assumed, the starting point for the KPSS stationarity tests is a sum of two components, one is stationary while the other one a random walk. The latter is a constant if the null hypothesis is true. Due to these different models different results in testing stationarity versus nonstationarity can occur.
Hello we have a thesis with title "getting results what you wants from unit root tests" at M Phil Econometrics level at International Islamic University Islamabad Pakistan. If your are interested then I will try my best to send a soft copy
Thanks
Please send the soft copy to me at [email protected]. Thanks in advance.
Dear Muhammad Malik, could you please send the stated thesis to [email protected]. Thanks in advance.
Sorry Rajan for late reply. I will send a soft copy to you people when my friend send to me. I have send a request to him.
Thanks
Dear Muhammad Malik, could you please send the stated thesis to [email protected]. Thanks in advance.
Dear Muhammad Malik, could you please send the stated thesis to
Thanks in advance.
Dr Malik, I am equally interested in the thesis. Please send to:[email protected]. Thanks
Dear Muhammad, please include me in the mailing list.
Regards,
Mariano
Dear Malik, I want to obtain if you send your valuable thesis. I am in favor of the subject. Sincerely yours. [email protected]
Dear Muhammed, can you send me this thesis too? Via a message in Research Gate should be okey. Thank you
Hi Vinodh: I saw your question now. If you did not get answer yet, then my manuscript entitled "Price Transmission in Thai Fish Market: An Analysis along Value Chain and across Species" available on researchgate.net might be helpful to you.
Dear Malik,
Could you send me a soft copy of your thesis?
Thanks, [email protected]
Dear Malik,
Could you send me a soft copy of your thesis?
Thanks [email protected]
Dear Malik,
I have uploaded in my profile a working paper: "Are Copper Prices Mean Reverting? A practitioner’s point of view". In that paper there is a comparison of the different methods for detecting mean reversion for the same series. Also the relation between the different unit root analyisis is explained.
Hope is useful for you.
Regards,
Mariano
Edited: Sorry I did not realize that documents can be uploaded, now is included.
According to Katircioglu, Feridun and Kilinc (2014) and Jafari, Othman and Nor (2012) PP and ADF unit root tests which were designed on the basis of the null hypothesis that a series is stationary have a low power of rejecting the null. It is suggested that KPSS unit root test eliminates a possible low power against stationary unit root that occurs in the ADF and PP ( Katircioglu , Feridun , & Kilinc , 2014; Jafari , Othman, & Nor, 2012). Therefore, if there is a situation in which you obtained a mixed and inconsistent results we should rely on the KPSS unit root test results as it yields more robust results.
according to your statements , there are two issues
1. oorder can be more than one
2. order can be fractional
they can have more than I(1)
when we have mixed results, go for fractional integration
Dear Muhammad Malik, could you please send the stated thesis "getting results what you wants from unit root tests" to [email protected]. Thanks
Dear Dr Muhammad Malik, could you please send the stated thesis. My email is : [email protected] Thanks in advance.
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