Worked Specification of an Explanatory Wage Model

From a Research Question to a Candidate Model

A question about education and earnings can lead to several different regression models, depending on what “earnings” means and which other characteristics the model includes. A useful specification makes these choices explicit before fitting the model. It also distinguishes a model for describing conditional associations from a claim that education causes a change in wages.

Suppose the research question is: Among employed adults in a defined population, how is educational attainment associated with hourly wages, comparing people with the same measured work experience and current occupation? This is an explanatory question about conditional associations. “Explanatory” here means that the model is organized to describe an association of interest and related variables; it does not mean that the model has established a causal effect.

For a coherent analysis, define who is included and what is measured. In the examples below, the target population is employed adults in a specified region and survey year, and the outcome is each person’s positive hourly wage in dollars at the survey date. The results would describe that population and period, not all workers at all times. A different question about annual earnings, for example, would require a different outcome definition and could have a different interpretation.

State the Outcome, Predictors, and Model Assumptions

Let $W_i$ be person $i$’s hourly wage in dollars, $E_i$ the number of completed years of education, $X_i$ a measure of work experience, and $O_i$ the person’s current occupation category. Let $B_i$ denote a set of measured background characteristics. Because wages must be positive for a logarithm, define the outcome as $Y_i=\log(W_i)$. Each candidate model below uses this same outcome and target population.

Definition: A specification states the outcome, the included predictor terms, and the form in which those terms enter a regression model. In a linear regression specification, the conditional mean of the outcome is represented as a linear combination of the included terms. The coefficients describe the fitted relationships associated with those terms, conditional on the other included terms.

For example, a candidate model with education, experience, current occupation, and background terms can be written as

Y_i=\beta_0+\beta_E E_i+\beta_X X_i+\beta_{X^2}X_i^2+\sum_{k=2}^{K}\gamma_k I(O_i=k)+\boldsymbol{\delta}^{\mathsf T}B_i+\varepsilon_i.

Here, $\beta_0$ is the intercept; $\beta_E,\beta_X,\beta_{X^2}$, the $\gamma_k$, and the entries of $\boldsymbol{\delta}$ are coefficients; and $\varepsilon_i$ is the error for person $i$. The indicator $I(O_i=k)$ equals 1 when the person is in occupation category $k$, and 0 otherwise. One category is omitted and serves as the reference category. $X_i^2$ is included only if a curved relationship between experience and log wage is part of the planned specification.

The model assumes that the conditional mean of $Y_i$, given the included predictors, has the stated form. It also assumes that the error has conditional mean zero: $E(\varepsilon_i\mid E_i,X_i,O_i,B_i)=0$. In the usual linear-regression setting, observations are treated as independent and the included terms must not be exact linear combinations of one another. These assumptions concern how the candidate model represents the data; they do not establish that changing education would change anyone’s wage. Standard small-sample $t$-based inference additionally relies on appropriate error-distribution conditions, such as constant conditional variance and, for exact normal-theory results, normally distributed errors. Those inference conditions do not determine which terms belong in the scientific specification.

Assign Each Predictor a Clear Role

Education is the main predictor of interest. In the model above, $\beta_E$ describes the difference in conditional mean log hourly wage associated with one additional year of education, for people who have the same included experience, occupation, and background values. Since the outcome is logarithmic, this is not a dollar-per-year coefficient. For a one-year difference, the corresponding wage ratio on the model’s conditional-log scale is $\exp(\beta_E)$; a percentage interpretation is approximately $100\beta_E\%$ when $\beta_E$ is small.

Experience represents time spent in work, which may be associated with wages independently of education. If its association is allowed to curve, both $X_i$ and $X_i^2$ appear. The derivative of the model’s conditional mean with respect to $X_i$ is $\beta_X+2\beta_{X^2}X_i$, so the experience association can vary with experience. This is a reason to include both terms when the research rationale supports curvature, not a reason to add a squared term automatically.

Current occupation represents the type of work a person does at the time wages are measured. Occupation is categorical, so indicators are used rather than treating labels such as “health,” “sales,” and “manufacturing” as numerical values. Background measures might include parental education or region of upbringing, if they are measured consistently and are relevant to the stated question. The role of each variable should be clear: a predictor can be included to describe an association, represent a substantive alternative account, or define the comparison being made.

These roles should not be confused with causal roles. Current occupation may itself be related to education, and education may influence occupational choice. Including occupation therefore changes the comparison: the education coefficient becomes an association among people with the same measured occupation, rather than an association that also includes differences connected with occupation. It does not automatically remove confounding, nor does it automatically estimate a direct causal effect.

Worked Example: Building a Core Specification

Suppose a hypothetical survey records hourly wages, completed education years, work experience, and current occupation for employed adults. The planned question concerns how education is associated with log hourly wage, conditional on experience and occupation. Use three occupation categories: service, technical, and administrative. Choose service as the reference category.

A core specification is

\log(W_i)=\beta_0+\beta_E E_i+\beta_X X_i+\beta_{X^2}X_i^2+\gamma_T I(O_i=\text{technical})+\gamma_A I(O_i=\text{administrative})+\varepsilon_i.

The model has two occupation indicators because there are three categories and service is the reference. For a hypothetical person with 16 years of education, 10 years of experience, and a technical occupation, the predictor terms are $E_i=16$, $X_i=10$, $X_i^2=10^2=100$, $I(O_i=\text{technical})=1$, and $I(O_i=\text{administrative})=0$. Substitution gives the model’s conditional mean:

E(Y_i\mid E_i,X_i,O_i)=\beta_0+16\beta_E+10\beta_X+100\beta_{X^2}+\gamma_T.

No coefficient values are calculated here because fitting has not been done. The expression shows exactly what the proposed comparison conditions on. If the person instead worked in service, both occupation indicators would equal 0; if the person worked in administration, only the administrative indicator would equal 1.

Measure Experience Rather Than Guessing It

The meaning of “experience” must be decided before choosing a term. Actual years in paid work would be preferable if measured reliably and available for the target population. When only age and education are available, a common proxy is potential experience:

X_i=\text{age}_i-E_i-6.

This proxy assumes that a person began formal schooling at about age 6 and entered work immediately after completing education. It is not the same as actual experience: time out of the labor force, repeated education, military service, and other pathways can create differences. If this proxy is used, the specification and interpretation should name it as potential experience, not actual years worked.

This construction also matters for the predictors in the model. Potential experience is calculated from age and education, so adding age, education, and potential experience together can create exact linear dependence. That violates the requirement that predictors not be exact linear combinations. The researcher must choose terms that represent the intended comparison without inserting redundant measures.

Worked Example: Calculating and Qualifying Potential Experience

In a hypothetical data set, a respondent is 34 years old and has completed 16 years of education. If actual work history was not collected, calculate the potential-experience proxy.

Using the stated definition,

X_i=34-16-6=12\text{ years}.

The specification may therefore use $X_i=12$, and if curvature was prespecified it also uses $X_i^2=12^2=144\text{ years}^2$. These values are arithmetic consequences of the proxy, not verified facts about the respondent’s work history. A model coefficient involving this $X_i$ describes an association with the proxy under its assumptions; it should not be described as the effect of one additional actual year of work.

The definition also makes clear why age should not be added alongside both education and this calculated $X_i$: $X_i=\text{age}_i-E_i-6$ means one of those three quantities is determined exactly by the other two. A model containing all three would be redundant.

Compare Core and Extended Specifications Fairly

A core model includes the terms needed to express the central question. An extended candidate can add measured background variables when the research rationale says they may help describe differences relevant to the education-wage association. For example, parental education and region of upbringing may be included if they are measured before a person’s own education and are available in the data. The outcome, population, and wage definition should remain the same when comparing such candidates; otherwise, a difference between models may reflect a changed analysis target rather than changed predictors.

The role of a background variable must still be interpreted cautiously. Measuring a variable before education can make it a plausible background measure, but timing alone does not prove that it controls all relevant differences. Nor does adding several background terms guarantee that the education coefficient has a causal interpretation. The models remain descriptions of conditional associations unless additional causal assumptions and a suitable study design justify a causal conclusion.

Worked Example: Extending the Model With Background Measures

Suppose the hypothetical survey also records parental education in years and region of upbringing, categorized as urban or rural. Keep log hourly wage as the outcome and extend the core specification by adding those measured background terms:

\log(W_i)=\beta_0+\beta_E E_i+\beta_X X_i+\beta_{X^2}X_i^2+\gamma_T I(O_i=\text{technical})+\gamma_A I(O_i=\text{administrative})+\delta_P P_i+\delta_R I(R_i=\text{urban})+\varepsilon_i.

Here $P_i$ is parental education in years, and rural upbringing is the reference category. Consider a hypothetical person with $E_i=16$, $X_i=12$, technical occupation, $P_i=14$, and urban upbringing. The squared experience term is $12^2=144$. Substituting the observed predictor values gives

E(Y_i\mid E_i,X_i,O_i,P_i,R_i)=\beta_0+16\beta_E+12\beta_X+144\beta_{X^2}+\gamma_T+14\delta_P+\delta_R.

If the fitted coefficient for education were, purely for illustration, $\beta_E=0.05$, then the education contribution in this linear predictor would be $16(0.05)=0.80$. That arithmetic does not give this person’s predicted wage by itself: all other terms, including the intercept and the person’s other predictors, also contribute. The coefficient value is illustrative, not an estimate from the hypothetical survey.

The extended specification changes the conditional comparison. Its education term compares people with the same included experience, occupation, parental education, and upbringing region. It does not assert that all such people are otherwise comparable in every causally relevant way.

Why the Distinction Matters

A specification makes the intended comparison visible. In the core model, education is compared among people with the same modeled experience and current occupation. In the extended model, it is additionally compared among people with the same measured parental education and region of upbringing. These are different conditional associations, even though the outcome is held fixed.

With a log outcome, a fitted conditional mean on the log scale can be exponentiated to obtain a wage-scale quantity. For example, if a model gives a conditional log mean of 3.34, then $\exp(3.34)\approx \$28.22$ per hour. This is the exponential of the modeled conditional log mean; it is not generally the conditional arithmetic mean wage.

A common mistake is to say, “We included occupation and background variables, so the education coefficient is the causal effect of education.” The reasoning is wrong because inclusion only conditions on the measured terms in the specified model. It does not prove that all relevant causes of both education and wages have been measured or modeled correctly, and occupation may itself follow from education. The correction is to describe the coefficient as a conditional association unless a separate causal argument supports stronger language.

Key takeaway: A defensible wage specification names the target population and fixed outcome, defines each predictor and its role, and states what the model conditions on. Adding experience, occupation, and background measures changes the comparison; it does not automatically establish causation.

Check Your Understanding

Answer each question using the model-specification choices described above.

Why should annual earnings and hourly wages not be substituted for one another across candidate specifications without changing the research question?
In a three-category occupation variable, why does a model with an intercept use two indicator variables rather than three?
A person is 40 years old and completed 18 years of education. Calculate potential experience using the stated proxy, and name one reason it might differ from actual experience.
What comparison does an education coefficient make when current occupation is included in the model?
Why does adding background measures not, by itself, justify calling the education coefficient a causal effect?