Why Outstanding Share Counts Are Usually Nonstationary

# Is the number of outstanding shares a stationary series?


# 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?

## Answer by Bob Jansen (score 5, accepted)

https://quant.stackexchange.com/a/10202

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.

## Answer by experquisite (score 1)

https://quant.stackexchange.com/a/10205

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.

## Answer by user2763361 (score 1)

https://quant.stackexchange.com/a/10206

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.