Difficulty: Introductory–Intermediate | Prerequisites: Simple regression (Ch. 2), basic algebra
This material sits at the heart of introductory econometrics. Once you can estimate a multiple regression, the next question is always: what do the numbers mean, and what can you do with them? These notes cover interpreting coefficient signs, computing predicted values, and comparing outcomes across individuals, all using the ceteris paribus logic that makes multiple regression useful. If you skipped Chapter 2, go back and get comfortable with simple OLS first.
Multiple regression lets you isolate the effect of one variable while holding others constant. To interpret a coefficient, ask: "If this variable increases by one unit and everything else stays the same, what happens to the predicted outcome?" A negative coefficient means the outcome falls as that variable rises, a positive one means it rises. Plugging specific values into the estimated equation gives you a predicted outcome for any combination of the explanatory variables.
OLS (Ordinary Least Squares)
The standard method for estimating the coefficients in a linear regression. It minimises the sum of squared differences between observed and predicted values.
In simple terms, OLS finds the line (or plane, in multiple regression) that fits your data as closely as possible.
Coefficient (slope parameter)
The number multiplying an explanatory variable in a regression equation. It tells you how much the predicted dependent variable changes when that explanatory variable increases by one unit, holding all other variables fixed.
Think of it as the "per-unit effect" of that variable, all else equal.
Ceteris paribus
"All other things equal." In multiple regression, each coefficient captures the effect of its variable while holding every other variable in the model constant.
In simple terms, this is the whole reason we run multiple regressions rather than simple ones: we want to isolate individual effects.
Predicted value (y-hat)
The value of the dependent variable you get by plugging specific numbers for the explanatory variables into the estimated equation. Written with a hat over the variable name.
Think of it as the model's best guess for a person or observation with those characteristics.
R-squared (R²)
The fraction of the total variation in the dependent variable that is explained by the explanatory variables in the model. Ranges from 0 to 1.
In simple terms, a higher R² means your model accounts for more of what is going on, but a low R² does not necessarily mean the model is useless.
hsperc (high school percentile)
The student's percentile rank in their high school graduating class. Defined so that hsperc = 5 means the student was in the top 5%. A higher number means a lower rank.
sat (SAT score)
The combined maths and verbal score on the student achievement test. Higher is better.
colgpa (college GPA)
College grade point average on a four-point scale. The dependent variable in the GPA2 regression.
The estimated equation is:
\widehat{colgpa} = 1.392 - 0.0135 \cdot hsperc + 0.00148 \cdot satn = 4,137, R² = 0.273
hsperc is defined so that a lower number means a better rank (hsperc = 5 means top 5%)
As hsperc increases (the student's rank gets worse), predicted college GPA falls
The negative sign is exactly what you would expect: students who ranked lower in high school tend to have lower college GPAs, holding SAT score fixed
To predict colgpa when hsperc = 20 and sat = 1,050:
Plug in: 1.392 - 0.0135(20) + 0.00148(1,050)
= 1.392 - 0.27 + 1.554
= 2.676
The predicted college GPA is about 2.68 on a four-point scale.
If Student A and Student B have the same hsperc but Student A's SAT is 140 points higher:
The hsperc terms cancel out (same value)
Predicted difference = 0.00148 × 140 = 0.2072
Student A's predicted GPA is about 0.207 points higher
This is a modest difference on a four-point scale, roughly one-fifth of a grade point
To find the SAT gap that produces a 0.50-point predicted difference in colgpa:
0.50 = 0.00148 × (difference in SAT)
Difference in SAT = 0.50 / 0.00148 ≈ 338 points
That is more than two standard deviations of the SAT distribution in this sample
The model says it takes a very large SAT gap to produce a half-grade-point difference, which suggests SAT alone is not a strong driver of college GPA once you account for high school rank
The estimated equation is:
\widehat{educ} = 10.36 - 0.094 \cdot sibs + 0.131 \cdot meduc + 0.210 \cdot feducn = 722, R² = 0.214
The coefficient on sibs is -0.094, which is negative as expected
More siblings tends to mean fewer resources per child (financial, parental attention), so each additional sibling is associated with about 0.094 fewer years of schooling, holding parental education fixed
To reduce predicted education by one full year: 1 / 0.094 ≈ 10.6 additional siblings would be needed
That is a large number, which tells you the per-sibling effect is fairly small in magnitude
The coefficient 0.131 means that one additional year of mother's education is associated with 0.131 more years of the son's education, holding sibs and feduc fixed
This is a partial effect: it isolates the mother's contribution from the father's
The father's education coefficient (0.210) is larger, but that does not necessarily mean fathers matter more, since the variables may be correlated and the model captures conditional associations, not causal effects
Man A: sibs = 0, meduc = 12, feduc = 12 Man B: sibs = 0, meduc = 16, feduc = 16
The sibs terms are identical (both zero), so they cancel
Difference = 0.131(16 - 12) + 0.210(16 - 12)
= 0.131(4) + 0.210(4)
= 0.524 + 0.840 = 1.364 years
Man B is predicted to have about 1.36 more years of education than Man A
This is a meaningful difference, roughly the gap between finishing a bachelor's degree and not quite completing one
\widehat{colgpa} = 1.392 - 0.0135 \cdot hsperc + 0.00148 \cdot sat\widehat{educ} = 10.36 - 0.094 \cdot sibs + 0.131 \cdot meduc + 0.210 \cdot feducTo compute a predicted value, substitute the values of each explanatory variable into the equation and do the arithmetic.
When comparing two individuals who differ on one variable but share the same value for all others, the predicted difference equals the coefficient on that variable multiplied by the difference in their values:
\Delta \hat{y} = \hat{\beta}_j \times \Delta x_jWhen they differ on multiple variables:
\Delta \hat{y} = \hat{\beta}_1 \Delta x_1 + \hat{\beta}_2 \Delta x_2 + \dotsUniversities use models like the GPA2 regression to predict student performance from admissions data. Admissions offices weigh high school rank and test scores precisely because they carry information about future GPA. The education model (WAGE2) reflects a longstanding finding in labour economics: parental education is one of the strongest predictors of a child's own educational attainment, even after controlling for family size.
Students often think a negative coefficient means something is wrong with the model. It does not. A negative coefficient simply means that variable and the outcome move in opposite directions, holding everything else fixed. Here, the sign on hsperc is negative because of how the variable is defined (higher number = worse rank).
Students sometimes plug in values and forget to apply the sign of the coefficient. If the coefficient is negative, the product will be subtracted, not added.
Students often confuse the coefficient on one variable with the "total effect" of that variable. In multiple regression, each coefficient is a partial effect, conditional on the other variables being held constant.
A common error is reading R² = 0.273 as "the model explains 27.3% of GPA," which is roughly right in spirit but imprecise. R² measures the share of variance explained, not the share of GPA itself.
⚠️ Exams frequently ask you to interpret the sign and magnitude of a coefficient. Always state the direction, the units, and the ceteris paribus condition.
⚠️ Prediction questions are common. Be methodical: write out the equation, substitute, and show your arithmetic.
⚠️ Questions comparing two individuals are a staple. Remember: if they share the same value of a variable, that variable's term drops out of the difference.
⚠️ Know how to invert the prediction formula. "What difference in X produces a 0.50 difference in Y?" means solving 0.50 = coefficient × ΔX for ΔX.
True or False: A negative coefficient always indicates a problem with the regression. (False. It means the variable and the dependent variable move in opposite directions, ceteris paribus.)
Fill in the blank: The predicted value of colgpa when hsperc = 20 and sat = 1,050 is approximately ____. (2.68)
True or False: If two students have the same hsperc, a 140-point SAT advantage translates into roughly a 0.50-point GPA advantage. (False. It translates into about 0.207 points.)
Fill in the blank: In the WAGE2 education regression, each additional sibling is associated with ____ fewer years of schooling, holding parental education constant. (0.094)
True or False: R² = 0.273 means the model is useless. (False. It means about 27.3% of the variation in colgpa is explained. Whether this is "good" depends on context.)
Q: In the college GPA regression, why is the coefficient on hsperc negative? Does this mean high school performance hurts college GPA?
A: No. The variable hsperc is defined so that a lower number is better (hsperc = 5 means top 5%). As hsperc rises, the student's rank is worse. So the negative coefficient correctly captures the idea that students who ranked lower in high school tend to have lower college GPAs, holding SAT fixed.
Q: Compute the predicted colgpa for a student at the 10th percentile with an SAT of 1,200.
A: colgpa = 1.392 - 0.0135(10) + 0.00148(1,200) = 1.392 - 0.135 + 1.776 = 3.033. The predicted GPA is about 3.03.
Q: In the education regression, what is the predicted difference in years of schooling between a person with 0 siblings whose parents each have 14 years of education and a person with 3 siblings whose parents each have 10 years of education?
A: Person A: 10.36 - 0.094(0) + 0.131(14) + 0.210(14) = 10.36 + 1.834 + 2.940 = 15.134. Person B: 10.36 - 0.094(3) + 0.131(10) + 0.210(10) = 10.36 - 0.282 + 1.310 + 2.100 = 13.488. Difference: 15.134 - 13.488 = 1.646 years.
Q: Holding hsperc constant, how large a difference in SAT scores is needed to produce a predicted colgpa difference of one full grade point?
A: 1.0 / 0.00148 ≈ 676 points. That is an enormous gap, well beyond the plausible range of SAT differences, which tells you the SAT coefficient, while positive, is small in practical terms.
This material connects directly to omitted variable bias (Chapter 3, later sections). The education regression shows how adding variables (meduc, feduc) changes the interpretation of other coefficients (sibs). In Chapter 2, you estimated simple regressions; here you see why multiple regression often gives different (and more useful) answers. The prediction skills here carry forward to hypothesis testing in Chapter 4, where you will ask whether a coefficient is statistically different from zero.
OLS, ordinary least squares, multiple regression, coefficient interpretation, predicted value, y-hat, ceteris paribus, all else equal, holding fixed, GPA2, WAGE2, college GPA, SAT score, high school percentile, parental education, siblings, R-squared, partial effect, slope coefficient, Wooldridge Chapter 3, introductory econometrics