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When we run a regression model (say OLS for simplicity) of y ~ x, we might have to use several control variables say z1 and z2. Now our model is y ~ x + z1 + z2, we may believe that z1 and z2 are not perfect controls then we introduce fixed effect dummies.

Now our model is y ~ x + z1 +z2 + fixed effects; but if the fixed effects take into account all control information then surely they must be seriously correlated with z1 and z2. Otherwise, by definition z1 and z2 are junk. Is this not going to introduce multi-collinearity?

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  • $\begingroup$ I am not sure I understand. Unless you are talking about multilevel models, all the variables on the right side of the equation are treated the same. Please elaborate $\endgroup$
    – Peter Flom
    Sep 9, 2014 at 20:09

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It's not possible to identify the parameters of a model when the fixed effects take into account all of the variation in the control variables. When you include fixed effects, the control variables that remain in the equations can only be variables that vary within each fixed effect group. Otherwise there will be perfect collinearity, and no variation with which to identify the coefficients of the control variables.

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  • $\begingroup$ Also, just to clarify terminology: statisticians use the term fixed effect to refer to anything that is not a random effect. So in your question a statistician may call Z1 and Z2 fixed effects. Economics and other social sciences use the term differently to refer only to variables that account for unobserved variation that is treated as non-random. $\endgroup$ Sep 9, 2014 at 20:13

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