Our underlying statistical model can be expressed using the decomposition in AKM. Once this decomposition is estimated, we apply the formulae given therein to estimate the part of the inter‐industry wage differential due to person and firm effects. A summary of the methodology is given in this section.

### 3.1 The AKM decomposition

The linear statistical model, taken directly from AKM, is specified as:

\text{ln}\phantom{\rule{.3em}{0ex}}{w}_{it}={x}_{it}\beta +{\theta}_{i}+{\psi}_{\text{J}\phantom{\rule{2.77626pt}{0ex}}(i,t)}+{\u03f5}_{it}

(4)

where *x*_{
it
} denotes the time‐varying variables, *θ*_{
i
} the pure person effect, *ψ*_{J (i,t)} the pure firm effect, and *ϵ*_{
it
} the statistical residual. Note that the function J (*i*, *t*) gives the identifier for the dominant employer, *j*, of individual *i* at date *t*^{3}. In full matrix notation we have

y=X\beta +D\theta +F\psi +\u03f5

(5)

where *X* is the *N*^{∗} × *P* matrix of observable, time‐varying characteristics (in deviations from the grand means); *D* is the *N*^{∗} × *N* design matrix of indicators variables for the individual; *F* is the *N*^{∗} × *J* design matrix of firm indicators variables for the firm effects for the employer at which *i* works at date *t* (*J* firms total); *y* is the *N*^{∗} × 1 vector of dependent data (also in deviations from the grand mean); *ϵ* is the conformable vector of residuals; and{N}^{\ast}={\sum}_{i=1}^{N}{T}_{i}. We assume that *ϵ* has the following properties:

\text{E}\left[\u03f5\left|X,D,F\right.\right]=0

and

\text{Cov}\left[{\u03f5}_{i},{\u03f5}_{m}\left|{D}_{i},{D}_{m},{F}_{i},{F}_{m},{X}_{i},{X}_{m}\right.\right]=\left\{\begin{array}{c}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\left\{{\Sigma}_{{T}_{i}}\right\}}_{i}\text{,}\phantom{\rule{2.77626pt}{0ex}}i=m\\ 0\text{, otherwise.}\end{array}\right.

### 3.2 Industry effects^{4}

Industry is a characteristic of the employer. As such, the analysis of industry effects in the presence of person and firm effects can be accomplished by appropriate definition of the industry effect with respect to the firm Effects. We call the properly defined industry effect a “pure industry effect.” Denote the pure industry effect, conditional on the same information as in equations (4) and (5), as *κ*_{
k
} for some industry classification *k* = 1, … , *K*. Our definition of the pure industry effect is simply the correct aggregation of the pure firm effects within the industry. We define the pure industry effect as the one that corresponds to putting industry indicator variables in equation (5) and then defining what is left of the pure firm effect as a deviation from the industry effects. Hence, *κ*_{
k
} can be represented as an employment‐duration weighted average of the firm effects within the industry classification *k*:

{\kappa}_{k}\equiv \sum _{i=1}^{N}\sum _{t=1}^{T}\left[\frac{1\left(\text{K}\phantom{\rule{2.77626pt}{0ex}}\right(\text{J}(i,t))=k){\psi}_{\text{J}(i,t)}}{{N}_{k}}\right]

where

{N}_{k}\equiv \sum _{j=1}^{J}1\left(\text{K}\right(j)=k){N}_{j}

and the function K(*j*) denotes the industry classification of firm *j*. If we insert this pure industry effect, the appropriate aggregate of the firm effects, into equation (4), then the equation becomes

{y}_{it}={x}_{it}\beta +{\theta}_{i}+{\kappa}_{\text{K}\left(\text{J}\right(i,t\left)\right)}+({\psi}_{\text{J}(i,t)}-{\kappa}_{\text{K}\left(\text{J}\right(i,t\left)\right)})+{\u03f5}_{it}

or, in matrix notation as in equation (5),

y=X\beta +D\theta +FA\kappa +(F\psi -FA\kappa )+\u03f5

(6)

where the matrix *A*, *J* × *K*, classifies each of the *J* firms into one of the *K* industries; that is, *a*_{
jk
} = 1 if, and only if, K(*j*) = *k*. Algebraic manipulation of equation (6) reveals that the vector *κ*, *K* × 1, may be interpreted as the following weighted average of the pure firm effects:

\kappa \equiv {\left({A}^{\prime}{F}^{\prime}FA\right)}^{-1}{A}^{\prime}{F}^{\prime}F\psi .

(7)

The effect (*Fψ*−*FAκ*) may be re‐expressed as *M*_{
FA
}*Fψ*, where the column null space of an arbitrary matrix, *Z*, is denoted *M*_{
Z
} ≡ *I* − *Z* (*Z*^{′}*Z*)^{−}*Z*, and ()^{−} is a computable g‐inverse. Thus, the aggregation of *J* firm effects into *K* industry effects, weighted so as to be representative of individuals, can be accomplished directly by the specification of equation (6). Only rank(*F*^{′}*M*_{
FA
}*F*) firm effects can be separately identified using unrestricted fixed‐effects methods; however, there is neither an omitted variable nor an aggregation bias in the estimates of (6), using either of the class of methods discussed below. Equation (6) simply decomposes *Fψ* into two orthogonal components: the industry effects *FAκ*, and what is left of the firm effects after removing the industry effect, *M*_{
FA
}*Fψ*. While the decomposition is orthogonal, the presence of *X* and *D* in equation (6) greatly complicates the estimation using the fixed‐effects techniques discussed in Appendix A (Additional file1).

When the estimation of equation (6) excludes both person and firm effects, as most of the literature has done, the estimated industry effect,{\kappa}_{k}^{\ast \ast}, equals the pure industry effect, *κ*, plus the employment‐duration weighted average residual firm effect inside the industry, given *X*, and the employment‐duration weighted average person effect inside the industry, given the time‐varying personal characteristics *X*:

{\kappa}^{\ast \ast}=\kappa +{\left({A}^{\prime}{F}^{\prime}{M}_{X}FA\right)}^{-1}{A}^{\prime}{F}^{\prime}{M}_{X}({M}_{FA}F\psi +D\theta )

which can be restated as

{\kappa}^{\ast \ast}={\left({A}^{\prime}{F}^{\prime}{M}_{X}FA\right)}^{-1}{A}^{\prime}{F}^{\prime}{M}_{X}F\psi +{\left({A}^{\prime}{F}^{\prime}{M}_{X}FA\right)}^{-1}{A}^{\prime}{F}^{\prime}{M}_{X}D\theta .

(8)

Put simply, the raw industry effect, *κ*^{∗∗}, equals the true industry effect *κ* plus a bias that is exactly the aliasing bias from excluding person and firm effects from the original regression.

The exact decomposition is entirely parallel to our theoretical model: the inter‐industry wage differential is decomposed into two parts, a person and a firm component, both of which are properly adjusted for the presence of covariates. There are no ancillary, and unnecessary, orthogonality assumptions.

Notice that if industry effects, *FA*, were orthogonal to time‐varying personal characteristics, *X*, and to non‐time varying personal heterogeneity, *D*, so that *A*^{′}*F*^{′}*M*_{
X
}*FA* = *A*^{′}*F*^{′}*FA*, *A*^{′}*F*^{′}*M*_{
X
}*F* = *A*^{′}*F*^{′}*F*, and *A*^{′}*F*^{′}*M*_{
X
}*D* = *A*^{′}*F*^{′}*D*, the biased inter‐industry wage differentials, *κ*^{∗∗}, would simply equal the pure inter‐industry wage differentials, *κ*, plus the employment‐duration‐weighted, industry‐average pure person effect, (*A*^{′}*F*^{′}*FA*)^{−1}*A*^{′}*F*^{′}*Dθ*, or

{\kappa}_{k}^{\ast \ast}={\kappa}_{k}+\sum _{i=1}^{N}\sum _{t=1}^{T}\frac{\text{1}\left[K\right(J(i,t))=k]{\theta}_{i}}{{N}_{k}}

where{N}_{k}\equiv \sum _{i,t}1\left[\text{K}\right(\text{J}(i,t))=k].

### 3.3 Estimation of the fixed‐effects model by direct least squares

The estimation methods proposed by AKM have been improved so that the statistical model can now be solved exactly for the fixed‐effects case. The full solution and the associated identification analysis are reported in Abowd et al. (2002a), which is summarized in Additional file1: Appendix A to the present paper^{5}.

The nature of the identification of the firm effects can be more intuitively understood in terms of average treatment effects and local average treatment effects. The AKM decomposition identifies the average treatment effect of changing between two employers based on the contrast between the employer effects for those two employers, holding constant observables and individual heteogeneity. AKM uses the actual employment as the weights for this contrast, rather than just the movers, as would be the case for a local average treatment effect estimated by instrumental variables^{6}. Further analysis of the identification in terms of average treatment effects can be found in Card et al. (2012).