Why Coefficient Interpretation Needs Diagnosis
An OLS table reports numbers, but a number is not yet an interpretation. A slope depends on how the outcome and regressor are measured, which other variables appear in the model, and—when interactions are included—the values of the interacting regressors. Reading a coefficient without checking those features can turn a correct calculation into an incorrect claim.
The aim is to describe the comparisons encoded by the fitted equation. This does not, by itself, establish that changing a regressor would cause the outcome to change. Causal interpretations require additional design or identification assumptions; a coefficient table alone does not supply them.
For example, in a model with one regressor, \(\widehat{y}_i=\hat{\beta}_0+\hat{\beta}_1x_i\), a one-unit increase in \(x_i\) changes the fitted value by \(\hat{\beta}_1\) outcome units. In a multiple regression, the same statement is conditional on holding the other included regressors fixed. These are statements about the fitted model, not automatically about conditional means, population effects, or causal effects.
Units and Rescaling
A coefficient's numerical value changes when a regressor's unit changes, even when the fitted predictions do not. If \(x^*=x/c\), then \(x=cx^*\). Substituting this into a fitted equation shows precisely how the coefficient must change.
Thus, the coefficient on \(x^*\) is \(c\hat{\beta}_1\). Since one unit of \(x^*\) represents \(c\) original units of \(x\), the new coefficient describes the fitted outcome change for that larger increment. The intercept and fitted values are unchanged. Always state the units rather than comparing coefficient magnitudes without considering scale.
Worked Example: Converting Experience From Years to Decades
Suppose a hypothetical OLS model predicts hourly earnings, in dollars, from education in years and work experience in years:
The experience coefficient is \(0.8\) dollars per hour for one additional year of experience, holding education fixed. Now define experience in decades as \(\text{experience}_{\text{decades}}=\text{experience}_{\text{years}}/10\). Since experience in years equals ten times experience in decades, the equivalent equation is:
The coefficient becomes \(10(0.8)=8\) dollars per hour per additional decade, again holding education fixed. For someone with 14 years of education and 20 years of experience, the first equation gives \(12+2.4(14)+0.8(20)=12+33.6+16=61.6\) dollars per hour. The second gives \(12+2.4(14)+8(2)=12+33.6+16=61.6\) dollars per hour. Rescaling changed the coefficient's numerical value and units, not the prediction.
The model is linear in these regressors as written, and this arithmetic checks the equivalence of the two parameterizations. The interpretation remains a model-based fitted comparison; it does not establish that an additional decade of experience causes earnings to rise by eight dollars per hour.
Rescaling the outcome has a corresponding effect: if the outcome is recorded in thousands of dollars instead of dollars, all coefficients, including the intercept, are divided by 1,000. A sound table interpretation therefore checks both the outcome scale and the regressor scale.
What Changes When Controls Are Added?
A coefficient in a multiple regression compares fitted outcomes for observations that differ in the regressor of interest while matching on the other included regressors. Adding controls changes which comparison is encoded. This is a property of the specification; it does not mean the coefficient must move in a particular direction, nor that the new coefficient is automatically causal.
A coefficient in a regression with education alone and a coefficient in a regression with education, experience, and region are not generally estimates of the same model comparison. The earlier material on controls and partialling out explains why estimates can change across specifications. For interpretation, name the regressors held fixed in each model and avoid claiming more than the model expresses.
Worked Example: Interpreting an Education Coefficient Across Specifications
Consider two hypothetical regressions where annual earnings are measured in thousands of dollars and education in years. The first includes education alone; the second also includes experience in years and a regional indicator \(R\), coded 1 for region B and 0 for region A.
In model 1, one additional year of education corresponds to a \(2.0\)-thousand-dollar difference in fitted annual earnings. In model 2, the corresponding fitted difference is \(1.3\) thousand dollars, comparing observations with the same experience and region. For example, at 16 years of education, 10 years of experience, and \(R=0\), model 2 predicts \(10+1.3(16)+0.4(10)+3(0)=10+20.8+4=34.8\) thousand dollars. At the same experience and region but 17 years of education, it predicts \(10+1.3(17)+0.4(10)+3(0)=36.1\) thousand dollars. The fitted difference is \(36.1-34.8=1.3\) thousand dollars.
The \(2.0\) and \(1.3\) figures answer questions about different fitted specifications. Neither comparison alone establishes a causal return to education. In particular, the model with controls still only supports a comparison conditional on the listed regressors; it does not show that all relevant sources of confounding have been addressed.
Interactions Make Slopes Conditional
When a model includes a product of two regressors, a coefficient on either regressor is not necessarily its universal slope. For a binary variable \(D\) and a quantitative variable \(x\), consider:
At \(D=0\), the fitted intercept is \(\hat{\beta}_0\) and the slope on \(x\) is \(\hat{\beta}_2\). At \(D=1\), the fitted intercept is \(\hat{\beta}_0+\hat{\beta}_1\) and the slope on \(x\) is \(\hat{\beta}_2+\hat{\beta}_3\). Equivalently, the fitted difference between \(D=1\) and \(D=0\) at a given \(x\) is \(\hat{\beta}_1+\hat{\beta}_3x\). These follow by substituting each value of \(D\) into the equation.
Worked Example: Reading an Interaction at a Specific Reference Score
Suppose a hypothetical model predicts a test score from a training indicator \(D\) and a baseline score centered at 70, denoted \(x_c=\text{baseline score}-70\):
Because \(x_c=0\) means a baseline score of 70, the coefficient 4 is the fitted training-versus-no-training difference for people with a baseline score of 70. For a person with a baseline score of 80, \(x_c=80-70=10\), so the fitted difference is \(4+0.2(10)=6\) test-score points.
The same result follows from predictions. At baseline score 80 with no training, the prediction is \(50+4(0)+0.6(10)+0.2(0)(10)=56\). With training, it is \(50+4(1)+0.6(10)+0.2(1)(10)=62\). The difference is \(62-56=6\) points. The baseline-score slope is \(0.6\) points per score point for \(D=0\), and \(0.6+0.2=0.8\) points per score point for \(D=1\). These are conditional fitted comparisons, not evidence that training causes the differences.
Centering matters for the main-effect interpretation: the coefficient 4 refers to baseline score 70, not score zero. If the uncentered baseline score were used instead, the coefficient on \(D\) would refer to a baseline score of zero, though fitted predictions would be unchanged after the coefficients were adjusted.
A Short Derivation: Why Centering Changes the Main Effect
The interaction example uses \(x_c=x-a\), where \(a\) is a chosen reference value. Write a model in the uncentered variable \(x\), then substitute \(x=x_c+a\):
The coefficient on \(D\) in the centered equation is \(\hat{\beta}_1+a\hat{\beta}_3\), exactly the fitted difference at \(x=a\) in the original equation. The centering changes the reference point for that coefficient; it does not change the fitted values when the parameters are transformed consistently.
Common Mistakes and a Practical Check
- Ignoring units. Calling a coefficient “the effect of \(x\)” without stating the increment and outcome units hides what the number measures. State, for example, “thousand dollars per additional year.”
- Reading a controlled coefficient as an unconditional comparison. A coefficient in a model with controls compares fitted values at the same values of those included controls.
- Treating an interaction main effect as universal. With \(D x\) in the model, the coefficient on \(D\) is the fitted difference at \(x=0\), or at the chosen reference value if \(x\) is centered.
- Calling a regression coefficient causal by default. The equation describes fitted associations under its specification. A causal claim needs additional justification beyond the coefficient and its standard error.
- Assuming a coefficient change proves bias or improvement. Different specifications encode different comparisons. A change alone does not establish which estimate is more credible.
Check Your Understanding
For each question, make the units and conditioning variables explicit.
- If a coefficient is 0.5 when \(x\) is measured in single units, what coefficient represents the same fitted relationship when \(x^*=x/100\)?
- In a model with education and experience, what must be held fixed when interpreting the education coefficient?
- In a model containing \(D\), \(x\), and \(Dx\), what comparison does the coefficient on \(D\) represent when \(x\) is not centered?
- Why can two education coefficients from regressions with different sets of controls represent different fitted comparisons?
- Does adding controls to an OLS model, by itself, establish that its coefficients are causal? Explain briefly.