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Correlated Random Effects

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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.

Table of Contents

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,

yit=β0+β1xit+β2ci+μi+εit\begin{equation} y_{it} = \beta_0 + \beta_1 x_{it} + \beta_2 c_i + \mu_i + \varepsilon_{it} \end{equation}


We can estimate β0\beta_0, β1\beta_1, β2\beta_2 and μi\mu_i. Assuming that we’ve properly controlled for observable firm characteristics, β1\beta_1 tells the relationship between R&D expenditure and firm performance. β2\beta_2 tells the difference in firm performance between VC-backed and non-VC-backed firms.

However, we cannot rely on β2\beta_2 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\beta_2 estimate due to omitted variables.

Similarly, our estimate of β1\beta_1 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

yˉi=β0+β1xˉit+β2ci+μi+εˉi\begin{equation} \bar{y}_{i} = \beta_0 + \beta_1 \bar{x}_{it} + \beta_2 c_i + \mu_i + \bar{\varepsilon}_{i} \end{equation}

from Equation (1), we have the fixed effect model in the demeaned form:

(yityˉi)=β1(xitxˉi)+(εitεˉi)\begin{equation} (y_{it} - \bar{y}_i) = \beta_1 (x_{it}-\bar{x}_i) + (\varepsilon_{it} - \bar{\varepsilon}_{i}) \end{equation}

The fixed effect model above removes the firm-level error μi\mu_i so that the within effect (or fixed effect) estimate of β1\beta_1 is unbiased even if E(μixit)0E(\mu_i|x_{it}) \ne 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 cic_i is also removed. It is now impossible to estimate β2\beta_2 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\beta_2 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 xitx_{it} into two components, a between component xˉi\bar{x}_i and a cluster component (xitxˉi)(x_{it}-\bar{x}_i), so that

yit=β0+β1(xitxˉi)+β2ci+β3xˉi+μi+εit\begin{equation} y_{it} = \beta_0 + \beta_1 (x_{it}-\bar{x}_i) + \beta_2 c_i + \beta_3 \bar{x}_i + \mu_i + \varepsilon_{it} \end{equation}

It is apparent that the β1\beta_1 estimate gives the within effect as in the fixed effect model, identical to β1\beta_1 in Equation (3).

Moreover, the firm-level variable cic_i is kept in the model and we can estimate β2\beta_2. The inclusion of cluster mean xˉi\bar{x}_i corrects the estimate of β2\beta_2 for between-cluster differences in xitx_{it}. Note that, however, for β2\beta_2 estimate to be unbiased, we still require E(μixit,ci)=0E(\mu_i|x_{it},c_i)=0 and μixit,ciN(0,σμ2)\mu_i|x_{it},c_i \sim N(0,\sigma^2_\mu).

Correlated Random Effect Model

A related model is correlated random effect model (Wooldridge 2010) that relaxes the assumption of zero correlation between the firm-level error μi\mu_i and firm-year variable xitx_{it}. Specifically, it assumes that μi=πxˉi+vi\mu_i=\pi\bar{x}_i + v_i, so Equation (1) becomes

yit=β0+β1xit+β2ci+μi+εit=β0+β1xit+β2ci+πxˉi+vi+εit\begin{align} y_{it} &= \beta_0 + \beta_1 x_{it} + \beta_2 c_i + \mu_i + \varepsilon_{it} \\ &= \beta_0 + \beta_1 x_{it} + \beta_2 c_i + \pi\bar{x}_i + v_i + \varepsilon_{it} \end{align}

By including the cluster mean xˉi\bar{x}_i, we can account for the correlation between the random effects μi\mu_i and the independent variable xitx_{it} and obtain consistent estimates of the coefficients. The inclusion of xˉi\bar{x}_i in the random intercept (effect) model makes the estimate for β1\beta_1 the same within effect (fixed-effect) estimate as in Equation (4).

Of course, as the time-invariant firm-specific attribute cic_i remains in the model, we can estimate β2\beta_2 as in the hybrid model.


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: