Hello,
I am running regressions on fertility rates on yearly data for T = 15 and 34 countries. During this period, fertility has been rather stable for few countries, for the others it has increased and for a last group it has decreased. But trends are not really linear since in many countries they have changed after the onset of the great recession.
I have run two-ways fixed regressions using xtpcse, hetonly to account potential autocorrelation of residuals, but I am wondering whether I should also include country specific time trends in my model, in order to de-trend the data and to capture the potential effect of unboserved characteristics (for instance, changes in norms or preference shifting fertility down in some countries).

my dependent variable is fertility rates, and my independent variables are female employment rates, unemployment rates, the share of temporary contracts, the proportion of workers working more than 50 hours, the share of people with very short job tenure (450 observations); and in another model I add information on the share of self-employment and the incidence of low pay (and my number of observations falls to 215). I also add the "mean age at births" to somehow capture the effect of changing "preferences" in fertility behaviours.

I have tested two ways to include time trends.
- country specific time trends for the 34 countries, but I think I have too few degrees of freedom to do so and by including such trends I introduce more problems than it helps to solve them. These trends seems to be highly correlated with some of my explanatory variables, and I face multicollinearity problems.

- group specific time trends, as suggested by Clyde Schechter in a previous post; I have made 3 groups of countries (one where trends are pretty stable, one where fertility is increasing, the other where it is decreasing), but again, when I look at the data, the trends do not look very linear.

My concern is that the three models give very different results for few variables. For instance, the incidence of self-employment is positive and highly significant in the model without linear time trends, positive but not statiscally significant in the model with "group" time trends, and strongly negative with the country specific time trends.

I am tempted to stick with the results of the model that do not include time trends, but at the same time, I know that I miss unobserved factors if I don't control for any time trends. What would you recommend, please? Anybody to advise?
Many thanks in advance,
Olivier