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I am building a fixed effect model:

$E(Y_{i,j}) = \beta_1 + \beta_2*Age_{i,j}+\alpha_i $

With $\alpha_i$ a fixed effect for each subject. Below is a plot of the data, and the measurements of three subjects highlighted. The green dot is the measurement of a subject with only one measurement.

I always learned that a fixed effect model looks at the longitudinal variation or the within subject variation, and that it is the same as first demeaning the measurements and fitting a regression model on the demeaned data (second plot).

But subjects with only one measurement do not have any longitudinal variation, if you demean their measurement you get 0. How is this measurement then used?

Or in other words, the fixed effect for these subjects matches the measurement perfectly, reducing the within subject variation to zero. How can this measurement than still contribute to the calculation of the time effect?

enter image description here

Kasper
  • 3,399

1 Answers1

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In a frequentist approach this fixed effect model has an unindentifiability problem (https://en.wikipedia.org/wiki/Identifiability) for $\beta_1$ and $\alpha_i$ for those subjects that have only one measurement. These subjects do not have a slope and do not contribute to the estimation of $\beta_2$.

In a Bayesian approach whether and how the subjects with one measurement contribute to the estimation of $\beta_2$ depends on the specifications of the priors.

winerd
  • 599