Correlated Random Effects
Can we estimate the coefficient of gender while controlling for individual fixed effects? This sounds impossible as an individual’s gender typically does not vary and hence would be absorbed by individual fixed effects. However, Correlated Random Effects (CRE) may actually help.
At last year’s FMA Annual Meeting, I learned this CRE estimation technique when discussing a paper titled “Gender Gap in Returns to Publications” by Piotr Spiewanowski, Ivan Stetsyuk and Oleksandr Talavera. Let me recollect my memory and summarize the technique in this post.
Random Intercept (Effect) Model
Consider a random intercept model for a firm-year regression, e.g., to examine the relationship between firm performance, R&D expense, and whether the firm is VC-backed,
where,
is firm-year level outcome variable, e.g., firm ROA is firm-year level independent variable, e.g., firm R&D expense is an time-invariant firm-level variable, e.g., if the firm is VC-backed is firm-level error and random intercept to capture the unobserved, time-invariant factors is firm-year level error, assumed to be white noise and ignored in this post
We can estimate
However, we cannot rely on
Similarly, our estimate of
Fixed Effect Model
If we subtract the “between” model
from Equation Equation 1, we have the fixed effect model in the demeaned form:
The fixed effect model above removes the firm-level error
However, the firm-level variable
Hybrid Model
So, how to estimate both
The same question, if paraphrased differently, is how to estimate the within effect in a random intercept model.
Interestingly, we can decompose the firm-year level variable
It is apparent that the
Moreover, the firm-level variable
Estimation
Note that there are many caveats for estimating CRE.
To be discussed.
Further Readings
This post is based on Within and between Estimates in Random-Effects Models: Advantages and Drawbacks of Correlated Random Effects and Hybrid Models.
Some other suggested readings include: