Difference-in-difference unbalanced panel

[Elinor Eklund]

Hi!

I am doing a difference-in-difference analysis on how academic achievement (measured by a standardised test score) is affected by getting a diagnosis. My dataset is very big and I have information on background characteristics measured at birth, mental health and test scores for different school years. Individuals can be measured from 1 to 7 times depending on when they leave school.

To do my DiD I am comparing children who have a diagnosis before the 2nd grade to children who receive a diagnosis between the 2nd and 4th grade. I have test scores on their reading tests in the 2nd and 4th grade. I have reorganised my data to only include children who have both test scores. My model is:
𝑦𝑖𝑡=𝛼0+𝛼1Latediag𝑖+𝛼2Grade4𝑡+𝛿(Latediag*Grade4)𝑖𝑡+ 𝑥′𝑖𝑡𝛽+𝜀𝑖𝑡,

where y is a standardised test score, Latediag is 1 if the child receives a diagnosis between the 2nd and 4th grade (=0 otherwise), Grade4 is equal to one if the child is in grade 4 (0= otherwise) and X is a set of background characteristics and year dummies.

Right now I am estimating the model with OLS, but I am wondering if I should use the fixed effects instead. The reason for my wondering is that one child could have his measures of the 2nd and 4th grade tests in 2010 and 2012 and another child in 2013 and 2015. If I had measures from just two time periods e.g. 2010 and 2012 then I would use OLS and I know it would give me the same result as a fixed effects estimation. Is this also the case when I have two time periods with different years?

Best regards,
Elinor

[Clyde Schechter]

If I had measures from just two time periods e.g. 2010 and 2012 then I would use OLS and I know it would give me the same result as a fixed effects estimation.

That's not true.

You can always emulate fixed effects linear regression by adding a group of indicator variables (dummies) for the individual students to the OLS regression. But the model you specify doesn't have those, so it is not at all equivalent. If you do include them, then it is equivalent to doing a fixed effects regression regardless of the time periods corresponding to different years.

And just to emphasize the point: this emulation of fixed-effects regression by including indicators for the individual students only works for linear regression. If you try it with logit, poisson, etc., the results are not equivalent to a fixed-effects model.