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1

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. ...

1

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 ...

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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 ...

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