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I am reading an article which uses a simple least squares model to measure the effect of a prevention campaign on methamphetamine use (http://www.ncbi.nlm.nih.gov/pubmed/20638737). In its second equation, it uses state fixed effects in an OLS model to capture the effect of living in each state.

Meth ~ constant + Betas*Attributes + State1Beta*State1 + ... + 
        StateZBeta * StateZ

I have two questions:

  1. Is the only difference between using random and fixed effects in this instance the distribution in which the values fall (ie, a random effect would have a mean of 0 and a normal distribution)? And,
  2. How does this jive with the interpretation that random effects are intercept shifts, and fixed effects are slope shifts?
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  • $\begingroup$ It is the first time I see the interpretation you mention in the second question. Could you give a reference? $\endgroup$ – mpiktas Oct 2 '12 at 2:42
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Since the author is an economist and from glancing at the tables in the working paper version of that paper, I would go with the standard interpretation that both the FEs and REs are state-specific intercepts. FEs are allowed to be correlated with other explanatory variables (but their magnitude is not restricted in any way), whereas REs are not. More details on the history in my answer to a similar question.

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  • $\begingroup$ The Wooldridge link will give you the econ history, the Gelman one will probably seem more familiar to you. $\endgroup$ – Dimitriy V. Masterov Oct 1 '12 at 20:18
  • $\begingroup$ Thanks very much -- is this interpretation of fixed effects as intercept shifts appropriate for other binary variables? Just trying to wrap my head around this, thanks. $\endgroup$ – mike Oct 1 '12 at 20:33
  • $\begingroup$ Strictly speaking, these are not quite fixed effects in a panel data setting. They're really just dummy variables for the states in a linear probability model and logged y regression applied to panel data. Other binary variables would have the same intercept shift interpretation. $\endgroup$ – Dimitriy V. Masterov Oct 1 '12 at 20:44
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  1. You will get different estimates of the beta coefficients using random vs fixed effects. The primary focus of the RE analysis is to estimate the variance of the distribution of the random effects - in this case, of the state effects. However you can also estimate the particular random effects that you get - most software implementing RE models allows you to do this. You will find that the RE coefficients are smaller in absolute value than the FE coefficients of the betas obtained through OLS. Random effects yield shrinkage estimators.
  2. I would not interpret the distinction between random and fixed effects in terms of slope and intercept changes. The distinction has to do with the error structure. Random effects introduce correlations in the response variable. You might want to check out some of the posts on this subject.
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