What do I do if I want to use Johansen test for cointegration, but the time series I use are not non-stationary? Can I search for a time-span that is? For instance, when I examine the 1900-2012 period, the series is stationary. The same is true when I look at the 1920-2012 period. But if I choose 1921-2012, the test cannot reject the null of unit root. Or should I motivate the choice of period historically? Also, unit root tests have low power. Does this mean I should use 10% rather than 5% as significance level?
I think that there are a few items that need clarification here.
1. When I think of different definitions of stationarity I think of two - stationary in distribution and 2nd order stationary (also known as strong and weak stationarity). A variable is trend stationary if after temoving a trend the remainder is stationary. There are variables that are stationary in distribution that are not second order stationary. Note that the definition refers to the unconditional distribution of a variable and not to the conditional distribution (AR Arch or Garch processes have non constant conditional distributions and subject to certain conditions on their parameters be stationary).
2. There is no such thing as long run or short run stationarity. A variable is either stationary or not stationary. In applying ADF tests things can go wrong in two ways. The type 1 error occurs when you conclude that the series is stationary when it is non-stationary. You try to control for this kind of error using the 5% test size. The type 2 error is when you conclude that a series is non- stationary when it is in effect stationary. The probability of a tye 2 error is greater for a smaller sample. Thus in smaller samples you are more likely to get the wrong result implying non-stationarity when the series is stationary. This is a problem of insufficient data. This is why I like to take theory into account when I am looking at unit roots and cointegration. There are also known size distortions as well as low power in unit root tests.
3. You should work with the log of the ratio of your variables. About 25 years ago there were several articles on the stationarity of the great ratios. (Share of labour income, savings ratio etc. ). When variables are considered to grow by a percentage their lpgs are generally better behaved and log(x/y)=log(x)-log(y) leading to a natural unit root / cointegration test. Are log(x) and log(y) I(1)? Are they cointegrated? Is log(x)-log(y) stationary? Using logs conforms better with statistical theory and removes the limits on your ratio.
In economics most time series are non-stationary, and the phenomena usually evolve in cycles. I cannot understand what you might be studying and how many variables, but if it's stationary for such a long period, then it's either too long or too short a time-series. the differences between significance levels are comparable. Integration tests, take first differences though.. what software are you into?
What kind of data are you studying? There are many stationary time-series as well - did you try to treat it as a state-space model? Did you do information criteria test? Did you test each variable for Normality (eg Jarque Bera Test)? Just wondering if you are really sure your time series is a) non-stationary and b) can not be brought to a stationary form.
Very often nonstationarity is a result of long-term and periodical factors influence. So at first if you have to check the time series on trend and seasonal component's existence - there are numerous methods for doing this. If they are attended you have to eliminate them, and just AFTER that you can check your time series on nonstationarity.
Please try what Dr Sandra said.
Do your data treatment first. ensure your series are normal. Even from the descriptive statistic you can begin to make inferences before running test for stationary
I agree with Jerry, descriptive statistics are your first step, that is useful as you get to know your data - also consider your context - in how far is it relevant? Consider your theory - which framework or hypothesis are you trying to verify? Consider your methodology - is it sound? Is it logical? Can you back your approach up from other scolars or from well known methods or do can you justify what you do?
The very idea of cointegration means that the individual times series are non-stationary but their combination is stationary in the long run. Thus to test for cointegration with stationary time series is a needless task and would gain nothing, as you can use them directly in any autoregressive form. A possible problem can only arise, if your variables are non-stationary and their combination is non-stationary as well. In this case some variables would be integrated of degree higher than one, indicating that a VEC might be too simple to capture the data, at least when used in first differences: switching successively to higher differences might solve the problem, but is a tedious procedure.
I agree with Frank, and consider the Jarque Bera test which tells, if a time series is NV or not - which means, you can use a model or not. Additionally the information criteria show you, which lag could be used, additionally, there is a great way to see, which time series type is appropriate: it is identified in Shumway and Stoffer 2000 looking at the behaviour of ACF and PACF of each variable in the long run: is it decliining? is it a damped sinus wave? etc. Check that source and you will see which model could fit.
From your question it implies you are working with stationary data. The question that is important is to know what you are trying to establish and this will help whether you should estimate your model in levels (with logs were necessary) with a VAR or state space methods.
Yes, I conducted IC tests, and I use Gretl. Does a data set have to be normally distributed to be of use for time series analysis? I know it's good if a VAR passes a test for normality, but if it doesn't you can just adjust the significance level tro be more strict. And for cointegration, normality is not necessary. That's what I've learnt at lest.
The null in a unit root test is that the series is non-stationary. Before applying a unit root test you should have some theory that says that the series that you are testing has or possibly has a unit root. Some description of your series and your theories might aid in explaining what is going on,
In your case a single observation can decide if a series is stationary or not. This may not be the 1920 observation. If you are using IC to decide lag length to be included in an adf style test (e.g. adf-gls) from 1920 to 2012 and the max lag level being considered is say 11 all IC estimates should use the period 1931-2012 +lags. You need to examine this process carefully to determine which observation is causing the problem and why it is a problem.
To use Johansen you need non-stationary series. Stationary series can only enter the short-run and not the equilibrium series. From a practical point of view if my theory indicated strongly that one of my series should be non-stationary but unit root tests indicated that it was marginally stationary I might include it as an I(1) variable in my analysis. In such a case the outcome might be that it had little effect on the long-run. I would ,of course, give the full results of my marginal (and other) unit root tests in my report.
Interesting! But isn't it possible that a serie is non-stationary in the medium-run, and stationary in the long-run or very long-run? I look on inequality, measured both as functional income inequality and the top income share of incomes. I the very long-run it could be argued that those series are stationary, because in order for capitalists to enjoy higher relative incomes, they need to realize their capital in sales to the workers. Therefore, income inequality cannot prevail in the very long-run under capitalism, at least not in a closed economy. On the other hand, Piketty and others have clearly demonstrated that inequality is non-stationary over more than 100 years. An other example is bank debt-to-GDP. This variable can neither grow indefinitely, cause debt has to be paid out of GDP sooner or later. Even so, it is'nt stationary over the course of a 100 years or more. The beginning and the end of the 19th century both experienced high peaks, while the middle of the century experienced lows. Cointegration provides an exellent tool for analysing the link bewteen inequality and band debt/GDP, which is what I am doing. Over the very long-run however, both variables should be stationary in my opinion.
Lars,
First, have a look at the data, plot it, whether there are any structural brakes, because this very often biases results. Second, think about whether it makes sense that the series is non-stationary, as you partly did. Remeber: ratios which are bounded between two numbers cannot be I(1) per definition and so on. You can still use Johansen when you have a stationary variable among some other non-stationary ones, but then you have to consider this when identifying the number of cointegrationg vectors, i.e. [0 0 . . . 0 1 0 . . . 0] is then per definition a cointegrating vector. If you have a cointegrating system where there are one or more variables which are stationary, then you will have to make restricitons, to identify the system. Then, things get messy.
Remeber also, that sometimes the processes do not really match the structure of the data, that is the data generating processes in reality are much more complicated than what is permitted in the ADF-test. This may of course also bias the results.
Finally, check the link below. The slides on Stationarity, Cointegration, Multivariate Cointegration and Identification should help. Prof. Hall used to work a lot with the data in practice and his slides should help solve your problem.
http://www2.le.ac.uk/departments/economics/people/shall/teaching/lectures
Thank you Balint. There are a lot of slides to go through. Are you saying that I can't perform a Johansen test for cointegration of inequalty, without getting into "messy" things? Secondly, I learnt that in order to perform a Johansen test, the variables invloved have to be non-stationary. Is that wrong you say? Thirdly, how do I impose restrictions? I have allready imposed a restriction in the form of normalization of the (assumed) dependent variable. Fourth, if I want to analyse the long-run relationship between functional income inequality and disposable income inequality, can't I do that? Both ate bouned between 0 and 1, but still demonstrate long swings over the course of history. There must be some way to capture the cointegration between these long swings.
Lars,
1. You should go through the slides, invest the time,it will help a lot.
2. "Are you saying that I can't perform a Johansen test for cointegration of inequalty, without getting into "messy" things?" No. You can perform the Johanssen test, it will tell you how many cointegrating vectors you have, right? So if you are sure, that one of the variables is stationary and the system is cointegrated, then the Johanssen test should say that you have at least 2 cointegrating vectors. It can be also three, or whatever. But not one! Because as I said, if you have a stationary variable in the system, then "[0 0 . . . 0 1 0 . . . 0] is per definition a cointegrating vector"
3. "Secondly, I learnt that in order to perform a Johansen test, the variables invloved have to be non-stationary. Is that wrong you say?" Actually it is the "optimal" case that all variables are non-stationary. There should be also some literature on the behaviour of the Johansen test if not all variables are I(1), because this issue was already raised. So it is not true that you cannot perform a Johansen test if there are also stationary variables in the system, but you have to be very careful.
4. Lets assume you have 5 variables, one of them is stationary and the other four are I(1). Assume that the Johansen test tells you that there are 2 cointegrating vectors. Then you know that the system is cointegrated and the second cointegrating vector is [0 0 . . . 0 1 0 . . . 0] by definition. However to identify the entire system and recover all the coefficients you will need to impose some more restrictions and test the system! Go through the slides. And there is a whole bunch of literature on the identification of the system if you have multiple cointegrating vectors!
5. "Fourth, if I want to analyse the long-run relationship between functional income inequality and disposable income inequality, can't I do that? Both ate bouned between 0 and 1, but still demonstrate long swings over the course of history. There must be some way to capture the cointegration between these long swings" - How many variables do you have in the system? As you deal here with ratios, I would say that these variables cannot be I(1). Go through the properties of I(1) series! They are not bounded, to say the least, so ratios which are captured between 0 and 1 cannot be in my view I(1). It can be though that the persistence of the series is very high. As far as I can remember in this case it very often happens that you cannot reject the test, but this does not mean anything at all, this happens very often in practice.
6. Maybe do the following: run a simple OLS first and see what comes out. Then assume that both of your series are I(1) (for whatever reasons - to me this does not make much sense from a mathematical point of view, but anyway) and do a simple Engle-Granger Two-Step procedure and see whether the residuals are stationary, but take the right critical values, not those from the ADF test. See also slides and each normal econometrics/time series book.
I hope this helps a bit.
Interesting stuff. I will go through the slides. A question however: There are different kinds of stationarity, for instance trend stationarity. Isn't it true that most variables are stationary over the very long run? Over time, most fluctuations will disappear and the many series will create a trend. And another thing: Would you say that the debt-to-GDP ratio is stationary by nature too? I mean, it can't grow forever?
I think that there are a few items that need clarification here.
1. When I think of different definitions of stationarity I think of two - stationary in distribution and 2nd order stationary (also known as strong and weak stationarity). A variable is trend stationary if after temoving a trend the remainder is stationary. There are variables that are stationary in distribution that are not second order stationary. Note that the definition refers to the unconditional distribution of a variable and not to the conditional distribution (AR Arch or Garch processes have non constant conditional distributions and subject to certain conditions on their parameters be stationary).
2. There is no such thing as long run or short run stationarity. A variable is either stationary or not stationary. In applying ADF tests things can go wrong in two ways. The type 1 error occurs when you conclude that the series is stationary when it is non-stationary. You try to control for this kind of error using the 5% test size. The type 2 error is when you conclude that a series is non- stationary when it is in effect stationary. The probability of a tye 2 error is greater for a smaller sample. Thus in smaller samples you are more likely to get the wrong result implying non-stationarity when the series is stationary. This is a problem of insufficient data. This is why I like to take theory into account when I am looking at unit roots and cointegration. There are also known size distortions as well as low power in unit root tests.
3. You should work with the log of the ratio of your variables. About 25 years ago there were several articles on the stationarity of the great ratios. (Share of labour income, savings ratio etc. ). When variables are considered to grow by a percentage their lpgs are generally better behaved and log(x/y)=log(x)-log(y) leading to a natural unit root / cointegration test. Are log(x) and log(y) I(1)? Are they cointegrated? Is log(x)-log(y) stationary? Using logs conforms better with statistical theory and removes the limits on your ratio.
Work following Perron looks at (1) difference stationary vs. (2) trend stationary with a break in the trend. In particular, Zivot and Andrews in Cowles WP 811 (also JBES 1992 10(3)) follow up with a test that determines the best break point. They are concerned with difference-stationarity that seems to go away with a sample break. The test used in this applied paper may help with the situation described in the question, along with numerous points made by others already. Moreover, the paper looks at the issue of heavy-tailedness in the shocks, which I comes up if the distribution of the residuals appears non-Gaussian. This approach keeps numerous possibilities in mind at once, including nonstationarity, heavy-tailedness in the shocks, heteroscedasticity, structural break, etc. It may be that any or all of these issues turn out to be relevant.
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