Correlated Random Effects

Author
Affiliation

Mingze Gao, PhD

Macquarie University

Published

April 9, 2023

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,

(1)yit=β0+β1xit+β2ci+μi+εit

where,

  • yit is firm-year level outcome variable, e.g., firm ROA
  • xit is firm-year level independent variable, e.g., firm R&D expense
  • ci is an time-invariant firm-level variable, e.g., if the firm is VC-backed
  • μi is firm-level error and random intercept to capture the unobserved, time-invariant factors
  • εit is firm-year level error, assumed to be white noise and ignored in this post

We can estimate β0, β1, β2 and μi. Assuming that we’ve properly controlled for observable firm characteristics, β1 tells the relationship between R&D expenditure and firm performance. β2 tells the difference in firm performance between VC-backed and non-VC-backed firms.

However, we cannot rely on β2 to assert whether VC-back firms have better or worse performance. The drawback here is that we are unable to exhaustively control for all other time-invariant firm attributes that correlate with both firm performance and VC investment, thereby leading to biased β2 estimate due to omitted variables.

Similarly, our estimate of β1 may also be biased if some omitted firm-specific and time-invariant attributes correlate with both R&D expenditure and firm performance.

Fixed Effect Model

If we subtract the “between” model

(2)y¯i=β0+β1x¯it+β2ci+μi+ε¯i

from Equation , we have the fixed effect model in the demeaned form:

(3)(yity¯i)=β1(xitx¯i)+(εitε¯i)

The fixed effect model above removes the firm-level error μi so that the within effect (or fixed effect) estimate of β1 is unbiased even if E(μi|xit)0. This helps a lot and is why most of the time we control for firm fixed effects when estimating firm-year regressions.

However, the firm-level variable ci is also removed. It is now impossible to estimate β2 as in the random intercept model. In fact, we can no longer estimate the effect of any firm-level time-invariant attributes after controlling for firm fixed effects.

Hybrid Model

So, how to estimate both β2 when firm fixed effects are controlled for?

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 xit into two components, a between component x¯i and a cluster component (xitx¯i), so that

(4)yit=β0+β1(xitx¯i)+β2ci+β3x¯i+μi+εit

It is apparent that the β1 estimate gives the within effect as in the fixed effect model, identical to β1 in .

Moreover, the firm-level variable ci is kept in the model and we can estimate β2. The inclusion of cluster mean x¯i corrects the estimate of β2 for between-cluster differences in xit. Note that, however, for β2 estimate to be unbiased, we still require E(μi|xit,ci)=0 and μi|xit,ciN(0,σμ2).

Correlated Random Effect Model

A related model is correlated random effect model () that relaxes the assumption of zero correlation between the firm-level error μi and firm-year variable xit.

Specifically, it assumes that μi=πx¯i+vi, so becomes

yit=β0+β1xit+β2ci+μi+εit=β0+β1xit+β2ci+πx¯i+vi+εit

By including the cluster mean x¯i, we can account for the correlation between the random effects μi and the independent variable xit and obtain consistent estimates of the coefficients. The inclusion of x¯i in the random intercept (effect) model makes the estimate for β1 the same within effect (fixed-effect) estimate as in .

Of course, as the time-invariant firm-specific attribute ci remains in the model, we can estimate β2 as in the hybrid model.

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:

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References

Wooldridge, Jefrey M. 2010. Econometric Analysis of Cross Section and Panel Data. The MIT Press.