# Time series regression - Lags of independent variable

I am trying to run a regression that has the (logged) values of GDP per capita (PPP) for the past 15 years in a given country (in my case Sudan and Rwanda, separatley of course) as dependent variable and an index that measures political stability (WGI) as independent variable. It seems to me that it would make sense to include lags of the independent variable as change in political stability does not have an immediate effect on GDP.

Taking the case of Rwanda if I run the regression without lags I get very good R squared (around 70%), if I include up to 5 lags the R squared goes up but the null hypothesis cannot be rejected for any of the variables except for the constant, however if I run a regression with the dependent variable and the lags taken separatley ex. GDP per capita and political stability (t-3) the R squared is very high and the variable is significant (the null can be rejected), I find that the lag for which I get the highest $R^2$ is t-5 (more than 80%) for both Sudan and Rwanda.

My question is: does it make sense to run a regression with the dependent variable and only the t-5 lag as dependent variable? Does any of this make sense? Am I doing something wrong?

P.S. please leave aside for now stationarity and cointegration