# ECM with fixed effects in a long-run equation

I would like to make a simple estimate of regional GDP. Predicted values of GDP on country level are available, so I was thinking about using these values and create a simple model. Simple linear regression would assume that the reliationship between regional and national GDP will be the same all the time. So I tried ECM, it appears to me as a better choice. I want to make predictions for many regions, but I have only annual data from 2000, so it is just 15 years and for some regions even less.

I estimated long-run equation for every region and as I said, the problem is that there are only a few degrees of freedom and according to ADF test, there is no cointegration (probably because of lack of data). Now, I want to estimate 1 long-run relationship for all regions in the country and then I want to estimate ECM and make predictions. My long-run equations are better (logically) if I add fixed effects for every region. Is it possible to use such an equation in ECM model? Are the following formulas correct? Thanks for the advices.

$gdp_{ij,t} = a_{j} + b_{j}gdp_{j,t} + FixedEff(i) + \epsilon_{ij,t}\\ \Delta gdp_{ij,t}=\alpha_{j} + \beta_{j} \Delta gdp_{j,t} + \gamma_{j} (gdp_{ij,t-1}- a_{j} - FixedEff_{i} - b_{j}gdp_{j,t-1}) +u_{ij,t}$

where $gdp_{ij,t}$ is log GDP of region i in country j in time t.