# Variance Decomposition of Wage Equation

I am reading two recent papers studying between-firm and within-firm wage inequality, Barth et al 2016 (hereafter BBDF) and Song et al 2019 (hereafter SPGBV). I am confused by the different variance decomposition methods used in these two papers.

Both of these two paper first yield a simple log wage variance decomposition.

BBDF: $$V(\ln w)=V(s)+V(\varphi)+2 \operatorname{Cov}(s, \varphi)+V(u)$$

SPGBV: $$\operatorname{var}\left(y_{t}^{i, j}\right)=\operatorname{var}\left(\theta^{i}\right)+\operatorname{var}\left(\psi^{j}\right)+2 \operatorname{cov}\left(\theta^{i}, \psi^{j}\right)+\operatorname{var}\left(\epsilon_{t}^{i, j}\right)$$

Where the $$s$$ or $$\theta$$ is person effects, the $$\varphi$$ or $$\psi$$ is firm effects, and $$u$$ or $$\epsilon$$ is the match error.

However they then both rewrite this simple decomposition to a more complicated decomposition that distinguish the between-firm component and within-firm component, in somehow different ways.

BBDF: $$V(\ln w) = \underbrace{V(s)(1-\rho)+V(u)}_{\text {Within-firm component }} + \underbrace{V(s)\left(\rho+2 \rho_{\varphi}\right)+V(\varphi)}_{\text {Between-firm component }}$$ , where $$\rho=\operatorname{Cov}(s, S) / V(s)$$, $$\rho_{\varphi}=\operatorname{Cov}(s, \varphi) / V(s)$$, and $$S$$ is defined as the establishment's average level of the predicted wage from $$(s)$$.

SPGBV: \begin{aligned} \operatorname{var}\left(y_{t}^{i, j}\right)= \underbrace{\operatorname{var}\left(\theta^{i}-\bar{\theta}^{j}\right)+\operatorname{var}\left(\epsilon_{t}^{i, j}\right)}_{\text {Within-firm component }} +\underbrace{\operatorname{var}\left(\psi^{j}\right)+2 \operatorname{cov}\left(\bar{\theta}^{j}, \psi^{j}\right)+\operatorname{var}\left(\bar{\theta}^{j}\right)}_{\text {Between-firm component }}, \end{aligned}

Are these two decompositions the same thing but written in different ways? I try some calculations but fail to show that they are the same. Moreover while it is very clear how BBDF get their second decomposition (add and subtract one $$\operatorname{Cov}(s, S)$$ from the first decomposition), it is unclear to me where does the second formula in SPGBV come from? However in terms of interpretation, SPGBV seems to be a more intuitive way to explain the within- and between- components than the one in BBDF. I also wonder what is the principle behind a decomposition that separate the between- and within-firm effects?

For the BBDF expression, you can simply obtain the first from the second by substituting out $$\rho$$ and $$\rho_\varphi$$.
For the SPGBV expression things are a bit more tricky. Let $$i$$ represent the unit and $$g$$ the group variable (which is $$j$$ in your equation).
I'm guessing that $$\bar \theta^g = \dfrac{1}{n_g} \sum_{i \in g} \theta^i$$ where $$n_g$$ is the number of elements in group $$g$$. Let there be $$N ( = \sum_g n_g)$$ observations in total. Let $$p_g = \dfrac{n_g}{n}$$ be the probability of $$\bar \theta^g$$ (I take that $$\dfrac{1}{N}$$ is the probability of the unit $$\theta^i$$).
First look at the expression $$var(\theta^i - \bar \theta^g)$$​​. We can decompose it in the following way: $$var(\theta^i- \bar \theta^g) = var(\theta^i) + var(\bar \theta^g) - 2 cov(\theta^i, \bar \theta^g).$$ Now, we can expand the covariance term: \begin{align*} cov(\theta^i, \bar \theta^g) &= \frac{1}{N}\sum_{g}\sum_{i \in g} (\theta^i - \bar \theta)(\bar \theta^g - \bar \theta),\\ &= \frac{1}{N} \sum_g (\bar \theta^g - \bar \theta) \sum_{i \in g} (\theta^i - \bar \theta),\\ &= \frac{1}{N} \sum_g n_g (\bar \theta^g - \bar \theta)(\bar \theta^g - \bar \theta),\\ &= \sum_g p_g(\bar \theta^g - \bar \theta)^2,\\ &= var(\bar \theta^g). \end{align*} This gives: $$var(\theta^i - \bar \theta^g) = var(\theta^i) - var(\bar \theta^g)$$ Rewriting gives: $$var(\theta^i) = var(\theta^i - \bar \theta^g) + var(\bar \theta^g). \tag{1}$$ Next, let's have a look at the covariance term $$cov(\theta^i, \psi^g)$$. \begin{align*} cov(\theta^i, \psi^g) &= \frac{1}{N} \sum_g \sum_{i \in g} (\theta^i - \bar \theta)(\psi^g - \bar \psi),\\ &= \frac{1}{N} \sum_g (\psi^g - \bar \psi) \left( \sum_{i \in g} (\theta^i - \bar \theta)\right),\\ &= \frac{1}{N} \sum_g (\psi^g - \bar \psi) n_g(\bar \theta^g - \bar \theta),\\ &= \sum_g \frac{n_g}{N} (\bar \theta^g - \bar \theta) (\psi^g - \bar \psi),\\ &= \sum_g p_g (\bar \theta^g - \bar \theta)(\psi^g - \bar \psi),\\ &= cov(\bar \theta^g, \psi^g) \end{align*} So: $$cov(\theta^i, \psi^g) = cov(\bar \theta^g, \psi^g). \tag{2}$$ Subsituting $$(1)$$ and $$(2)$$ into the first SPGBV condition should give you the second one.