You should use stationary state of all variables in regression model. Some variables are stationary at level, use as it is. if some are stationary at first difference, use its first difference form and same as for second difference.

For example:

If Y is stationary at level (dependent)

X is stationary at first difference (independent)

Z is stationary at second difference (independent)

Then regression model will be:

Y = a + b1d(X) + b2 d(d(Z))

Dear Jawaid,

Thank you very much. I will consider that. The problem is the series are trend stationary, and I do know whether mixing them is a valid way.

Dear Beze, If some independent variables are stationary at level, you can use ARDL method, or common correlated effects mean group estimator.

I agree with Tesheen for panel regression. I dont know about cointegratiin.

To see possible data transformations on time series see (good summary):

http://www2.geog.ucl.ac.uk/~mdisney/teaching/GEOGG121/time_series/GEOGG121_5_TimeSeries_Wu.pdf

After transfomation(s), you can perform Dickey-Fuller test for stationarity:

- Dickey, D. A., and W. A. Fuller (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74: 427–431

- Becketti, S (2013). Introduction to Time Series Using Stata. College Station, TX: Stata Press

And/or use graphical analysis using time series plot + autocorrelation function ACF + partial autocorrelation function PACF.

All the best,

Rafa

Eyayaw Teka Beze , have you ever heard of DiPCA and DiPLS? Dynamic multivariate techniques will allow you to model system dynamics while separating autocorrelation effects from static.

Check these references:

Dong, Y., & Qin, S. J. (2018). A novel dynamic PCA algorithm for dynamic data modeling and process monitoring. Journal of Process Control, 67, 1-11.

Dong, Y., & Qin, S. J. (2015). Dynamic-inner partial least squares for dynamic data modeling. IFAC-PapersOnLine, 48(8), 117-122.

Ku, W., Storer, R. H., & Georgakis, C. (1995). Disturbance detection and isolation by dynamic principal component analysis. Chemometrics and intelligent laboratory systems, 30(1), 179-196.

Within the context of cointegration, when series are trend stationary they can either be detrended (be regressing on a linear trend and using the residuals) or a time trend can be added to the regression. In the former case the detrended series are stationary and not typically included in the cointegrating equation. In the latter case the time trend can be restricted (included) in the cointegrating equation with the trend stationary series (that has not been detrended). There can be a mix of I(1) and trend stationary series (as long as the trend stationary series is not the dependent variable). It should be OK to generalise this approach to panel data. Since there may be uncertainty over whether series are trend stationary rather than I(1), different tests could give different results, you might also consider the possibility that the trend stationary series is actually I(1) by estimating the cointegrating equation without the trend (and with the "trend stationary" series). You can compare the results of the two cointegrating equations and assess the need to include the trend in terms of whether the trend is significant as well as comparing how well determined and/or economically plausible the resulting estimates are.