# Identification of a “modified” wage equation with firm-worker fixed effects

Consider the traditional AKM model where $$Y_{it}=\alpha_i+\psi_{j(i,t)}+\epsilon_{it}$$ for $$i=1,...,N$$ (individual index), $$t=1,...,T$$ (time index), $$j=1,...,J$$ (firm index), and $$j(i,t)$$ is the firm employing worker $$i$$ at time $$t$$. $$\alpha_i$$ is called worker $$i$$'s fixed effect. $$\psi_{j(i,t)}$$ is called firm $$j(i,t)$$'s fixed effect.

By stacking observations over individuals, the equation above can be rewritten as $$\underbrace{Y_i}_{T\times 1}=\underbrace{D_i}_{T\times N}\underbrace{\alpha}_{N\times 1} +\underbrace{F_i}_{T\times J} \underbrace{\psi}_{J\times 1}+\underbrace{\epsilon_i}_{T\times 1}$$

Assume that:

[A1] We have a sample of observations $$\{Y_i, F_i\}_{i=1}^N$$ i.i.d., for $$N$$ large and $$T$$ small.

[A2] $$E(\epsilon_i| D_i, F_i)=0$$ for each $$i=1,...,N$$.

As shown here, the identification of $$\alpha$$ and $$\psi$$ passes through dividing the sample into connected groups. In turn, under A1-A2 and for each of these groups $$g=1,...,G$$, $$\alpha_g$$ and $$\psi_g$$ are identified if and only if

• One element of the vector $$(\alpha_g, \psi_g)$$ is set equal to a known constant.

• $$N_g>1$$ and/or $$J_g>1$$, where $$N_g$$ (resp. $$J_g$$) are the numbers of individuals (resp. firms) in group $$g$$.

My question:

Consider the following "modified" AKM model $$Y_{it}=\alpha_i+X_{j(i),t}\psi_{j(i,t)}+\epsilon_{it}$$ where $$X_{j(i),t}$$ is an observed covariate with support in $$(0,1)$$, indexed by the time period $$t$$ and the entire history of employing firms $$j(i)\equiv \{j(i,t)\}_{t=1}^T$$. In other words, at a given time $$t$$, two workers are endowed with the same realisation of such covariate if and only if they face the same entire history of employing firms.

Note the purposely different subscripts of $$X_{j(i),t}$$ and $$\psi_{j(i,t)}$$.

My question is: does the presence of $$X_{j(i),t}$$ change the identification result of the traditional AKM model?

My intuition is that identification of the firm and worker fixed effects is still guaranteed only within connected groups. However,

(1) the normalisation of one among the firm-worker fixed effects within each group may not be needed anymore. Therefore, we can be able to make comparisons across groups.

(2) we may be able to identify firm-worker fixed effects also when $$N_g=J_g=1$$.

• Two points: 1.What exactly is the "firm-worker effect" here? If $\psi_{j(i,t)}$ in the first equation is a dummy taking the value 1 when worker $i$ is in firm $j$ at time $t$ than it is a firm-year fixed effect. 2. I find your explanation somewhat insufficient and confusing. You seem to rely on people having the time to read the articles you linked, or being already aware of this literature. This is unlikely if they aren't specializing in microeconometrics. It is not my impression that there are many such people here. – Grada Gukovic Feb 5 at 6:16
• I figured out some of what I didint understand. In the AKM equation $\psi_{j(i,t)}$ is a dummy variable. If $\psi_{j(i,t)}$ preserves its meaning in the modified version both $\psi_{j(i,t)}$ and $X_{j(i),t}$ are variables and $X_{j(i),t}\psi_{j(i,t)}$ is their interaction. In this case the regression equation should say $Y_{it} = \alpha_i + \beta_{i,j}X_{j(i),t}\psi_{j(i,t)} +\epsilon_{it}$. In which case I want to know why are you dropping $\psi_{j(i,t)}$ and keeping only the interaction. – Grada Gukovic Feb 10 at 14:48