## How to interpret the coefficients of DLOG(X1) and DLOG(X2,2)? Alternative way to address spurious regressions?

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### How to interpret the coefficients of DLOG(X1) and DLOG(X2,2)? Alternative way to address spurious regressions?

I'm working on a time series estimation and forecasting project that involves investigating if air carriers' operating costs are sensitive to fluctuations in oil prices. If the answer is yes, I will forecast air carriers' operating costs based on the model specified in the first step.

I ran Dickey–Fuller test on all variables and found only oil prices are stationary. Therefore, I 1st or 2nd differenced all non-stationary variables and specified the estimation model in eq(1):

eq(1): DLOG(operating expenses)= constant + LOG(oil price)+DLOG(GDP,2) + DLOG(CPI in air transportation)+ DLOG(operating expenses (-1)) + arma (1-3) + error

I have two questions that would very much appreciate your insights.
1) How to interpret the coefficients of differenced variables, such as DLOG(GDP,2) and DLOG(CPI in air transportation) in eq(1)?
2) Is there another way to address non-stationary time series other than differencing? For instance, can I seasonally adjusted all variables and add a trend term to eq(1)—the rationale behind is that seasonality and trends cause spurious regressions, so we can fix/address the issue by eliminating seasonality in the time series and capturing the effect of trends. What I just describe is in eq(2) below:

eq(2): LOG(operating expenses)= constant+ LOG(oil price)+LOG(GDP) + LOG(CPI in air transportation)+ LOG(operating expenses (-1)) + arma (1-3) + @trend + error
I wondered if this is a legit way to address spurious regressions. Thank you very much!