# Is the number of outstanding shares a stationary series?

I'm doing a panel data analysis where the log of the freefloat number of outstanding shares is one of the explanatory variables, but it fails the Augmented Dickey Fuller and Person Phillips unit root tests. But intuitively I suspect it to be stationary but with structural breaks. (I know these unit root tests to be sensitive to breaks.)![nosfhl][1]

The image shows this variable for 108 stocks, and over a time-span of 15 years.

Could you say something useful about the nature of these timeseries?

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I think it is clear that you're looking at a non-stationary time series. –  user2763361 Feb 11 '14 at 10:25
No, it's not clear. –  Cindy88 Feb 11 '14 at 10:28
Some of them you can say are stationary based on the data if you account for the break. But others there is no chance you could conclude this, either through a hypothesis test or by observing charts. I also don't think that it's reasonable theoretically to think that they're stationary. Business circumstances change, balance sheets change, tick sizes change, the economy changes and upper management changes. Sure you could put breakpoints whenever one of these happens but why not just call it non-stationary. –  user2763361 Feb 11 '14 at 11:27
Why not just make everything on a per share basis and avoid the issue? –  John Feb 11 '14 at 14:03
How? I am analyzing stock-liquidity as a dependent variable, number of shares is one of the explanatory variables... –  Cindy88 Feb 11 '14 at 16:48

To quote Wikipedia:

In mathematics, a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when shifted in time. Consequently, parameters such as the mean and variance, if they are present, also do not change over time and do not follow any trends.

Clearly, the underlying distribution can change due to one of the factors user2763361 names. I think even more could be named. Hence, the the number of outstanding shares is not a stationary series. Note that this conclusion can be made a priori and that even if the series passes the test they should be considered non-stationary as user2763361 notes in the comments.

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+1, I think we could even go further and use prior knowledge to say that those series that would pass an ADF test as stationary after accounting for breaks (e.g. those that are almost flat lines with one jump) are actually non-stationary in a population sense. –  user2763361 Feb 11 '14 at 13:35
Could I save it by instead of using number of outstanding shares, the market cap? (Which is price * number of outstanding shares (for the moment I bypass freefloat adjusment). Note that one of the other explanatory variables is price. So I 've got P and P*N as explanatory variables, is that allowed? –  Cindy88 Feb 12 '14 at 18:15
Don't do that it will lead to multicollinearity which is bad. I think a better, new question would be: "How can I use the number of shares outstanding in my panel regression?" –  Bob Jansen Feb 12 '14 at 18:53

I would argue that this is the very definition of a non-stationary process. We know that shares outstanding are incrementally added to or removed from by the company issuing or repurchasing shares. These innovations are added to the previous outstanding share count.

My first instinct would be to model this as:

$$Y_i = Y_{i-1} + dX$$

Or perhaps something jumpier. Definitely not stationary.

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You can't include the levels in OLS. You will get biased coefficient and standard error estimates.

• Look to include this as some sort of ratio with other predictors based on theory, and test the ratio's stationarity. I can't yet imagine how this point would work but think about it.

• Maybe you can test the order of integration and use multiple differencing. It could be applicable if you find evidence for $I(n)$ with $n > 1$ (which it will be if it is in fact integrated). The issue is that you're using daily data and thus do not have much info about the population process. Any test on $n$ will give you wildly varying results (the straight lines will be $I(0)$ for example). If you can find something in the literature to estimate $n$ based on a panel/cross-section, then this is good news for your paper.

• You may also want to look into nonstationary panel data analysis (google this). I've messed around with this a bit and it's tough to get it to give sensible estimates, but maybe I did it wrong. I do not remember if this regression methodology is applicable to your problem and time series. This may require $I(1)$ which I do not think some of your series adheres to (although maybe you can convince a sufficiently bad reviewer that this is the case). From what I can recall you can expect good results with $I(1)$ series entered as levels and small finite samples.

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What is the definition for I(n)? –  user1157 Feb 11 '14 at 17:47
@Anna $n \in \mathbb{N}$ and represents the number of times we need to first difference to achieve a stationary series. –  user2763361 Feb 11 '14 at 18:06
Thanks for thinking along. Now as I simply take the differenced series most of them become a series that is 0 most of the time and now and then makes a sudden jump to come back to 0 (which was expected). This could very well be stationary right? It passed the unit root tests :). But the disadvantage that I'll lose knowledge about the influence of a absolute level of outstanding shares and only examine the effect a change in outstanding shares has. Am I right? –  Cindy88 Feb 12 '14 at 13:06
@Cindy88 You are correct, which is where nonstationary panel analysis comes in to save the day. I feel that it is your only choice if you want to include information about the absolute level. However I am doubtful that a theoretical relationship between the absolute levels can't be detected from the stationary $n$-differences. Still, if I was forced to work on your project I would first focus my attention on nonstationary panel analysis. –  user2763361 Feb 12 '14 at 18:55
@Cindy88 Or, use a different variable that captures the same info (ratio, etc). Or, throw it out altogether if it this variable is not the point of your paper. –  user2763361 Feb 13 '14 at 3:27