Bins and Year Fixed Effects

Question

I am trying to understand the difference between two regression models. Before more complex models are run (ie. quantile) I split an key independent variable into bins to see if the relationship is non-linear. I do this in two ways.

Model (1): The independent variable x is grouped into 10 bins based on the decile. A regression is then run with y regressed on the 10 bins and year-fixed effects.

Model (2): The independent variable x is grouped into 10 bins based on the decile within each year. A regression is then run with y regressed on the 10 bins and year-fixed effects.

If Model (1) is run without year fixed effects, then it would seem that the 10 bins can be picking up some sort of time trend. Therefore, the method of calculating bins in Model (2) seems better. My question is if including year fixed effects in Model (1) removes the correlation between the bin variables and the time trend?

In my basic undergrad textbook (Introductory Econometrics: A Modern Approach) it states that "to reflect the fact that the population may have different distribution in different time periods, we allow the intercept to differ across periods, usually years." Then it talks about how year dummies accomplish this. So, it would seem that the time trend is controlled for. However, the distribution for bins in each year would differ between Model (1) and Model (2). For example, in Model (2) there would be an equal number of observations in each bin. This will not be true for Model (1).

Any advice? I get two very different results when I run these two models and am not sure which one is correct or how to interpret the differences.

Thank you!

Answer 1

Generally, if you treat the independent variables as non-random, you don't need the assumption of normality for them. Normality is necessary for the dependent variable in linear regression with hypothesis testing (actually, it's normality of the error terms, not of the variable itself, but in my experience a highly-non-normal dependent variable won't often yield a normal error term). This page gives a pretty layman's description. Without knowing more about your situation, you may not need to create the bins at all.

Related to the time trend, year fixed effects will take care of that for you, correct. But they may not be the ideal conception of time. If you have a small number of years in your panel, then fixed effects may be the best. But if the panel is fairly long, you may want to consider treating time as a continuous variable and add in polynomials as appropriate. You can generally explore that by graphing out the time series.

Getting to your two models, it makes perfect sense that they would give you different results. You are making a continuous variable ordinal, and the ordering of the observations is different in the two cases. Model 1 makes the ordering across all years, whereas Model 2 makes the ordering within year, and unless the observations are ordered the same each year, the results will be different. I don't intuitively see why you would want to take the Model 1 approach - if you really want an ordinal relationship, I would do it within the panels, not across them, as I think that will give you a cleaner interpretation of the coefficient.