Using non stationary independent variables

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Using non stationary independent variables

Dear Time Series/Regression Experts:

Is it econometrically sound/okay to use non-stationary independent variables in a time-series (ols) regression? Here's the following delimma/options using the same starting raw/untransformed original data series:

1. I have a regression whereby the dependent variable and independent variables are all stationary (let's say using log difference transformation for all dependent and independent variables). However, then the R squared value from the regression is very low (below 0.20). Coefficients of all variables are statistically significant and regression residuals are stationary and normally distributed.

2. I can run a regression whereby the dependent variable "is" stationary but independent variables are not stationary. However, the R Squared value is much better (>0.40). Coefficients of all variables are statistically significant and regression residuals are still stationary and normally distributed.

3. I compare the final forecasts of the raw/untransformed data resulting form 1 and 2 and and they are very similar.

Below are my questions:

Is the 2nd model econometrically sound? Are there any published papers to support this? I found an interesting article about "ovedifferencing" by Professor Cochrane which argues that when we overdifference, much of the variation in the data can be thrown out.
https://static1.squarespace.com/static/ ... encing.pdf

Any latest research suggests option 2 is "okay"?

Thanks!