Appropriate way of calculating growth rates in a interval panel regression

[Marco Oldenhoff]

Dear statisticians,
I have the following problem. First of all i wasn't able to find anything related via the search function so i opened this topic. It is probably obvious for most of the people here in the forum, but as i want to be as accurate as possible i ask this question to verify my approach. As i want to establish a regression which is based on what is the effect of changes in population growth rates, labor growth rates, immigration inflow growth rates etc. on GDP/per capita growth. I want to conduct a simple OLS regression on this problem. So first of all i gathered all the variables as stock variables i.e. GDPper capita in year 1990, 1995, 2000, 2005, 2010 (data gathered in 5 year intervals) for all the variables mentioned above. But now i'm not 100 % sure which approach would be the right one to derive the growth rates. As there is on the one hand the classic approach by using the formula (end value - beginning value)/beginning value. And the other one is taking the log difference i.e. ln(x)-ln(xt-1) --> ln(gdp per capita 1995)-ln(gdp per capita 1990). If i established a variable i.e gdp growth with the ln method, is it then still appropriate to generate another variable i.e ln(gdpgrowth). Or would that alter the desired outcome? I would highly appreciate any opinions on this. I hope i explained it good enough to get a grasp of what i'm looking for if not i would gladly provide more information.

Best regards and thanks in advance for your attention

Marco

[Clyde Schechter]

So, by elementary properties of logarithms, ln x_t - ln x_t-1 = ln(x_t/x_t-1). The other approach calculates (xt - xt-1/xt-1), which is easily algebraically transformed to x_t/x_t-1 - 1. So denoting x_t/x_t-1 by r, one version uses ln r and the other uses r-1 as the variable. If you use the same representation on both sides of your regression equation, the ln r model is equivalent to fitting a power law regression: outcome growth = constant * predictor growth^{some power}, whereas the r-1 model says that outcome growth is linear in predictor growth. These are substantively different models and the choice between them should be governed by theory. (If there is no applicable theory, then graphical exploration is a good place to start, but for this type of data I suspect there is well developed theory to fall back on.)

It is also worth pointing out that if the growth rates are always small, say r between 0.9 and 1.1, then the two models are very close to equivalent. That's because for r close to 1, ln r is closely approximated by r-1.

[Marco Oldenhoff]

Thanks clyde for your fast response, i based my planned regression on the Solow model. But as im quite unfamiliar with the use of 5 year intervals i thought it might be a good thing to double check. So in essence it wouldn't make a difference if i first calculate the growth rates by (xt - xt-1/xt-1) and then take the ln of it. As in the solow model they use more or less the same procedure, and follow a log log regression. But with the little kink that i choose growth rates instead of stationary values.