# How to deal with unit roots in panel regression with fixed effects?

I am trying to figure out how to alter my panel regression in a case where fixed effects exists and one (or both) of the variables are I(1) processes (or in other words contain unit root). This is often the case in econometrics, when for example one variable is GDPGrowth(=I(0)) and the other is capital(=I(1)). For finding the unit root I am employing Levin–Lin–Chu(2002) test

The fixed effects are removed by taking the first difference(first difference estimator) or time demeaning (fixed effects estimator) all the cross sections independently. Both of those procedures affect the stationarity of the data series and do so differently, so should I perform the Levin–Lin–Chu unit root test on the demeaned or differenced data, or the data at level? Here is the formula for the fixed effects estimator model:

$y_{it} =\alpha_i+\beta_1 * x_{it}+u_{it}$

And likewise for the first difference estimator:

$Δy_{it} = \beta_1 * Δx_{it}+Δu_{it}$

Where $y_{it}$ is the dependent variable across time and cross sections(GDP growth), $\alpha_i$ is the cross section dependent fixed effect, $x_{it}$ is the independent variable (capital for example) at time t and cross section i. $\beta_1$ is the constant coefficient and $u_{it}$ is the idiosyncratic error term. An additional $δ_i*t$ could be added for trend (I am not sure if it should be added though). In the first difference model the $Δ$ stands for first difference.

How should the regression be modified if it's found out that one or both of the variables are not stationary and aren't cointegrated? Keeping in mind that long run effects are an important consideration, so differencing might not be advisable (and as such the first difference model should not be used). I am interested whether the first difference estimator takes care of the panel unit root by itself though, or if something more is required?

• Just curious, is GDP considered to be I(0)? I thought it is quite reasonable to assume it is a cumulative sum of random shocks plus a drift term; equivalently that GDP equals to its last value plus the newest random shock plus drift, which would make it I(1). – Richard Hardy Mar 18 '15 at 20:27
• @RichardHardy Yes, this is correct. I accidentally typed GDP when it should have been GDP growth. – Dole Mar 18 '15 at 20:54
• Would you perhaps be able to write out the model explicitly? – hejseb Mar 19 '15 at 7:59
• @hejseb Hopefully the edit works for you, although I must add that this problem is more general in nature and doesn't apply merely to basic panel regression models, but other panel models as well. – Dole Mar 21 '15 at 17:47