The difference of a difference for a control variable in a regression?

Question

I am conducting a time series analysis. My dependent variable is income inequality, which has been logged and then differenced. In other words, my Y variable is now the difference in log income inequality.

The independent variable of interest for this study is house prices, which have been transformed in the same fashion. The main X variable is the difference in log house prices.

Log and difference transforming these two variables (and the majority of others) have made them stationary. This is verified by plotting the variable and conducting an Augmented Dickey-Fuller test both before and after.

I am additionally considering the autocorrelation of the variables over time by plotting the correlation of the lags and seeing if they are within a predetermined significance level. The transformations primarily solve an autocorrelation over time.

However, for one of the control variables, taking the difference in the log does not make it stationary. This is for the control variable of Age Dependency, which is the ratio of those over 65 and under 15 compared to the rest of the population. It has been shown to be a determinant of income inequality in the previous literature.

When I conduct an Augmented Dicky-Fuller test of the difference in log age dependency ratio, it is statistically insignificant. Additionally, the plot clearly still has a trend in it, as shown below:

When I take the difference of the difference of log age dependency, it is stationary.

My question is how am I best supposed to proceed now? I believe I cannot include a control variable in the regression that is clearly non-stationary (ie, differenced once).

However, it is also my understanding that I cannot include two different variables that are integrated into different orders. This would mean because most variables are only differenced once and this variable is differenced twice, it would bias the regression results if both were included.

Additionally, I cannot drop the variable because it has been shown as a predictor of income inequality.

Any guidance on this issue would be greatly appreciated. Ideally, if you could point me in the direction of any literature that supports what is the best economic practice, that would be great.

Please let me know if you require any additional information.

Answer 1

My question is how am I best supposed to proceed now? I believe I cannot include a control variable in the regression that is clearly non-stationary (ie, differenced once).

This is correct when we talk about standard regression. Of course, there are some models which would allow for non-stationary variables if there is cointegration but more tests have to be done for that and in addition you should have some reason to believe there is some long term equilibrium relationship between cointegrated variables.

However, it is also my understanding that I cannot include two different variables that are integrated into different orders. This would mean because most variables are only differenced once and this variable is differenced twice, it would bias the regression results if both were included.

No this is not correct. Once you difference variable you change its order of integration. Twice differenced variable which was originally I(2) will now be I(0) the same way as once differenced variable that was I(1) will be I(0). As long as all variables (after transformations) are I(0) there isn't an issue of having them together in regular time series regression.

Rather the problem is that now you significantly changed the meaning of regression. You are no longer controlling for age dependency, nor for change in age dependency, but for acceleration in change in age dependency. However, if you are ok with this then there is no problem in having it in an regression.

PS:

It looks like you only have 25 observations. You should be aware that as a rule of thumb you need about 30 observations per independent regressor to satisfy the asymptotic properties of OLS (see Verbeek A Guide to Modern Econometrics). If your regression has 25 observation total over which you run your OLS you have much greater problems than simply non-stationarity. In that case you should not be estimating parametric models at all.

PPS:

A Dickey-Fuller test is known to have low power in small samples. Its possible you cannot reject the null of non-stationarity just because you have $n=25$. I would recommend to retest the variable with some alternative test with switched null. For example, KPSS test would make more sense when your sample is so incredibly small.