Initial per capita income is constant for all periods as you try to estimate the impact of initial income level on current growth level, hence giving you convergence coefficient. Time invariant variable would be dropped in within estimation.

Another trick is to convert the data into, say, 5-year averages and use the income  level of the first year of the five-year period to proxy for initial income.  

The estimate within eliminates the effects that do not vary over time. To keep these effects in the case of variables of interest, we must estimate the model by generalized least squares under the assumption of random effects. We can make this assumption verified by increasing the size and representativeness of the sample.

Hello, 

there is a method called fixed effects vector decomposition proposed by Plümper and Troeger (2007). Is allows to explain the fixed effects component of the model through non time varying variables.

There has, however, been some critique of this estimator by Greene (2010) who claims that the assumptions needed for the model essentially turn in into a random effects model. Plümper and Troeger (2011) refute this criticism.

Best

Makram

Thank you  to all for the invaluable information.