Recently someone asked if I could provide a similarly motivating and intuitive example regarding random effects. Although I was not able to come up with a new example, I can definitely discuss random effects in the same context of the previous example. But first a little (less intuitive) background.

**Background**

To recap, the purpose of both fixed and random effects estimators is to model treatment effects in the face of unobserved individual specific effects.

y

_{it}=b x

_{it}+ α

_{i}+ u

_{it }(1)

In the model above this is represented by α

_{i . }In terms of estimation, the difference between fixed and random effects depends on how we choose to model this term. In the context of fixed effects it can be captured through a dummy variable estimation (this creates different intercepts or shifts capturing specific effects) or by transforming the data, subtracting group (fixed effects) means from individual observations within each group. In random effects models, individual specific effects are captured by a composite error term (α

_{i}+ u

_{it}) which assumes that individual intercepts are drawn from a random distribution of possible intercepts. The random component of the error term α

_{i}captures the individual specific effects in a different way from fixed effects models.

As noted in another post, Fixed, Mixed, and Random Effects, the random effects model is estimated using Generalized Least Squares (GLS) :

β

_{GLS}= (X’Ω^{-1}X)^{-1}(X’Ω^{-1}Y) where Ω = I ⊗ Σ (2)
Where Σ is the variance α

_{i}+ u_{it}. If Σ is unknown, it is estimated, producing a feasible generalized least squares estimate β_{FGLS}**Intuition for Random Effects**

In my post Intuition for Fixed Effects I noted:

*"Essentially using a dummy variable in a regression for each city (or group, or type to generalize beyond this example) holds constant or 'fixes' the effects across cities that we can't directly measure or observe. Controlling for these differences removes the 'cross-sectional' variation related to unobserved heterogeneity (like tastes, preferences, other unobserved individual specific effects). The remaining variation, or 'within' variation can then be used to 'identify' the causal relationships we are interested in."*

Lets look at the toy data I used in that example.

The crude ellipses in the plots above (motivated by the example given in Kennedy, 2008) indicate the data for each city and the the 'within' variation exploited by fixed effects models (that allowed us to correctly identify the correct price/quantity relationships expected in the previous post). The differences between the ellipses represents 'between variation.' As Kennedy discusses, random effects models differ from fixed effects models in that they are able to exploit both 'within' and 'between' variation, producing an estimate that is a weighted average of both kinds of variation (via Σ in equation 2 above). OLS, on the other hand exploits both kinds of variation as an unweighted average.

**More Details**

As Kennedy discusses, both FE and RE can be viewed as running OLS on different transformations of the data.

For fixed effects:

*"this transformation consists of subtracting from each observation the average of the values within its ellipse"*

For random effects:

*"the EGLS (or FGLS above) calculation is done by finding a transformation of the data that creates a spherical variance-covariance matrix and then performing OLS on the transformed data."*

As Kennedy notes, the increased information used by RE makes them more efficient estimators, but correlation between 'x' and the error term creates bias. i.e. RE assumes that α

_{i }is uncorrelated with (orthogonal to) regressors. Angrist and Pischke (2009) discuss (footnote, p. 223) that they prefer FE because the gains in efficiency are likely to be modest while the finite sample properties of RE may be worse. As noted on p.243 an important assumption for identification in FE is that the most important sources of variation are time invariant (because information from time varying regressors gets differenced out). Angrist and Pischke also have a nice discussion on page 244-245 discussing the choice between FE and lagged dependent variable models.

**References:**

A Guide to Econometrics. Peter Kennedy. 6th Edition. 2008

Mostly Harmless Econometrics. Angrist and Pischke. 2009

See also: ‘Metrics Monday: Fixed Effects, Random Effects, and (Lack of) External Validity (Marc Bellemare.

Marc notes:

*"Nowadays, in the wake of the Credibility Revolution, what we teach students is: “You should use RE when your variable of interest is orthogonal to the error term; if there is any doubt and you think your variable of interest is not orthogonal to the error term, use FE.” And since the variable can be argued to be orthogonal pretty much only in cases where it is randomly assigned in the context of an experiment, experimental work is pretty much the only time the RE estimator should be used."*