Cross-sectional analysis using longtitudinal data, what is the best method?

Question

I have a small problem regarding the fact of doing a cross sectional analysis using a longitudinal data set. I have a set of countries (87 countries), with different observations measured in different years (e.g. [Country], [Year], [X]: Netherlands, 1990, 0.5; Netherlands 1991, 0.3; Germany 1991, 0.4).

Method 1

Now if I am doing a cross sectional analysis I would be picking all my observations from one year (i.e. $Y_{1990, i}=\alpha + \beta \cdot X_{1990, i}$), am I correct? However the problem is that I will only have about 24 observations, whereas the total amount of observations are 150, meaning that at each year only a subset of the 87 countries are measured.

Method 2

So to increase the amount of observations, I could just ignore the years. Giving me 150 observations, however I was wondering what kind of bias this would introduce. I am sure this will introduce some kind of bias, since some countries are sampled across multiple years.

Method 3

A different method would be taking the average of the countries that are sampled in multiple years, this would of course result in 87 observations. I am also wondering how this method would influence my estimations and what kind of biases this will introduce. So my questions are, given all these three “methods” which one should I use and what kind of biases do these methods introduce (theoretical, maybe some references). I was also wondering whether there are alternative “methods”.

Thanks in advance for your response!

Answer 1

Method 1
You are right. Cross-sectional analysis of this kind would mean to do the regression for a particular year for which you will only have 24 observations, so your estimates will be less precise. Otherwise there is nothing wrong with it but if you have the "luxury" to exploit also the time dimension of your data this might help (see below).

Method 2
You could use treat your data as pooled cross-sections and use your 150 but you should not "ignore" the years. In this case you will also need year dummies and cluster your standard errors on the country ID variable. The latter point is suggested by Cameron & Trivedi (2009) in order to correct the standard errors for serial correlation. You said correctly that some countries are sampled in multiple years, hence for country i the error in year t will be correlated with the error in year t-1.

The better alternative would be to declare your data an unbalanced panel and do a fixed effects estimation. This type of regression absorbs all unobserved time-invariant country characteristics into a constant such that these are not in the error term anymore. This is particularly useful if you suspect some of your explanatory variables to be correlated with such time-invariant characteristics. If you want a refresher or basic reference for fixed effects methods you can find one here.
The problem with fixed effects is that if you are interested in the effect of a time-invariant variable, you need to employ some tricks in order to do that. Otherwise it will just be absorbed in the fixed effect as well.

Method 3
Of the three offered alternatives this seems to be the least preferrable one. Suppose you have two countries and three time periods. You observe GDP of country A in year 1 and 2, and GDP of country B in year 3. You want to construct a cross-section for year 3, so now suppose that in year 1 there was a deep recession that affected both countries. Averaging country A such that it appears in the final period (the one you want to use as cross-section) might give you a much lower value for GDP that you would have obtained, had you observed GDP in country A also in year 3.
Actually, I haven't seen anyone using such averages to construct cross-sections but then I'm not a macro-guy either. Perhaps someone else knows more about your proposed method 3 but intuitively it seems less practical than the previous two. Still I hope this helps you.