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.