I want to use only 1 independent variable - GDP per capita - and to control for time effects (6 dummy variables for each wave).
I suggest that you read these two recent articles:
How many countries for multilevel modelling? A comparison of Frequentist and Bayesian approaches Daniel Stegmueller, 2013 American Journal of Political Science,Volume 57, Issue 3, pages 748–761, July 2013
Two Multilevel Modeling Techniques for Analyzing Comparative Longitudinal Survey Datasets Malcolm Fairbrother, Political Science Research and Methods 10/2013; DOI:10.1017/psrm.2013.24
Both of these report simulations studies for the sort of problem that you are interested in, so you can make your own judgement.
I don't want to do multilevel modeling though. I am not sure if the same logic is applicable to FE and RE models.
I am a bit lost by your reply - multilevel modelling is a name for models that have both fixed and random effects and there are often good reasons to use this formulation as by adaptive pooling of information it is very efficient
see: Explaining Fixed Effects: Random Effects modelling of Time-Series Cross-Sectional and Panel Data; Andrew Bell, Kelvyn Jones
Political Science Research and Methods 12/2013; forthcoming.
https://www.researchgate.net/publication
/233756428_Explaining_Fixed_Effects_Random_Effects_modelling_of_Time-Series_Cross-Sectional_and_Panel_Data
I think the confusion comes from the problem that FE and RE have two completely different meanings. One has nothing to do with multilevel modelling. I am interested in analyzing longitudinal data with FE and RE at only one level - that of countries.
Plamen - there are different meanings of RE, but they both have something to do with multilevel modelling. A RE model in your case is a model that has country-level random effects, and *is* a multilevel model, where the levels are occasions (level 1) and countries (level 2).
If you include country fixed effects (ie a dummy for each country) in your model, then you will not be able to identify country random effects (and fit a RE/multilevel model) because all the country-level degrees of freedom will have been consumed by the fixed effects.
Returning to your original question, if I understand rightly your dataset is very small - 6 countries measured over 6 occasions but on average only 3-4 times per country? If you include 6 country FEs, and 6 time dummys, you are immediately knocking out most of your degrees of freedom.
You could save degrees of freedom by using a RE (aka mutlilevel) model instead (with only 6 countries you would need MCMC - see the Stegmueller paper Kelvyn linked to above), and/or by specifying time as a linear/quadratic function rather than dummies (if it is appropriate to do so). But even then you are asking a lot of a very small dataset.
Thank you for the explanation, Andrew. I think the idea to use linear/quadratic time variable instead of dummies is great and could be feasible.
As far as I know, the downside of RE is that it doesn't control for unobserved heterogeneity. In your paper you argue that this is solvable by using Mundlak's formulation? Even if that's the case, don't FE and RE require the same degrees of freedom?
With such a small data set, can I just be more relaxed with the statistical significance of the coefficients?
If you mean RE, yes you are right, and yes the Mundlak formulation will help if there are unobserved country-level factors linked to GDP that you want to control for. But this also uses up degrees of freedom. Without Mundlak, RE uses less degrees of freedom (because the random effects are assumed part of a distribution, only the variance of that distribution needs to be estimated, and you don't need a parameter for each RE).
Sadly, no, statistical significance can't just be ignored - if you don't have evidence for something, then you don't have evidence for it, regardless of the sample size.
