ECON2515 · Intermediate Applied Econometrics Ii
The Multiple Linear Regression Model
Multiple regression fits y = β₀ + β₁x₁ + … + βₖxₖ + u so that each slope measures the effect of its own variable holding all the other regressors constant — the ceteris-paribus, or partialled-out, slope that turns regression into a tool for causal questions. This ECON 2515 Week 4 topic is where you learn to interpret a coefficient other-things-equal, sign the bias from a variable you leave out (omitted-variable bias), and judge model fit with adjusted R² rather than raw R². It underpins every t-test, F-test and diagnostic in the rest of the course, so the interpretation habits you build here decide most of the final-exam Part B marks.
What this chapter covers
- 011. The MLR model — y = β₀ + β₁x₁ + … + βₖxₖ + u; why we add regressors (isolate an effect, cut bias)
- 022. The ceteris-paribus slope — every βⱼ is an 'other things equal' effect, not a raw correlation
- 033. Partialling-out — βⱼ is the slope of y on the part of xⱼ the other regressors cannot explain
- 044. MLR.1–6 assumptions — linearity, random sampling, no perfect collinearity, E[u|x]=0, homoskedasticity, normality
- 055. MLR.4 (zero conditional mean) — the exogeneity hinge that makes OLS unbiased and the slope causal
- 066. Omitted-variable bias — Bias = β₂ × δ; sign it with the effect on y AND the correlation with the included x
- 077. Marginal effects & turning points — for a β₃x + β₄x² pair, ME = β₃ + 2β₄x and the peak is −β₃/(2β₄)
- 088. Goodness of fit — R² only ever rises with k, so compare models with adjusted R²; df = n − k − 1
Sign an omitted-variable bias — is the return to experience over- or under-stated?
- +3(a) The bias is the omitted variable's effect on y times its relationship with the included regressor: Bias(α̂₁) = β₂ × δ, where δ is the slope from regressing the omitted ABILITY on EXPER (δ = Cov(EXPER, ABILITY) / Var(EXPER)). This is the amount ABILITY's channel leaks into α̂₁ when it is left in the error.
- +3(b) Sign each piece: β₂ > 0 (ability raises salary) and δ > 0 (experience and ability are positively correlated). The product (+)(+) is positive, so the bias is upward — α̂₁ over-states the true return to experience, because it also picks up the salary gains that were really due to the higher ability that tends to come with more experience.
- +2(c) Leaving ABILITY out pushes it into the error term v, and because ability is correlated with the included EXPER, the error is now correlated with a regressor: E[v|EXPER] ≠ 0. That is a failure of MLR.4, the zero-conditional-mean (exogeneity) assumption — exactly the condition that makes OLS unbiased, which is why the estimate is biased.
Key terms
- Multiple linear regression (MLR)
- A model y = β₀ + β₁x₁ + … + βₖxₖ + u with two or more regressors, still linear in the parameters, estimated by OLS just like simple regression.
- Ceteris paribus slope
- The interpretation of any βⱼ in an MLR: the effect of a one-unit change in xⱼ on the mean of y while holding all the other regressors constant ('other things equal').
- Partialling-out
- The mechanism behind the ceteris-paribus slope: β̂ⱼ equals the slope of y on the residual part of xⱼ that the other regressors cannot explain — OLS purges the other variables out of xⱼ first.
- Omitted variable bias (OVB)
- The bias in an OLS coefficient when a variable that belongs in the model and is correlated with an included regressor is left out; Bias = β₂ × δ, so its sign needs both channels.
- MLR.4 (zero conditional mean)
- The exogeneity assumption E[u|x₁…xₖ] = 0 — the error is unrelated to every regressor. It is what makes OLS unbiased and a slope causal, and it is exactly what an omitted variable breaks.
- Perfect collinearity
- A regressor being an exact linear function of the others (e.g. months and years=months/12). MLR.3 forbids only this exact case; ordinary high correlation is the separate multicollinearity problem.
- Adjusted R²
- R̄² = 1 − (1 − R²)(n − 1)/(n − k − 1): R² penalised for the number of regressors. Unlike raw R² it can fall (or go negative) when a useless variable is added, so it is used to compare models of different size.
- Marginal effect & turning point
- When a variable enters as x and x² its effect is not constant: the marginal effect is β₃ + 2β₄x (plug in the stated x) and earnings/etc peak where it is zero, at x* = −β₃/(2β₄).
The Multiple Linear Regression Model FAQ
What does 'holding the other variables constant' actually mean?
It means each slope is a partialled-out effect: OLS first strips the influence of the other regressors out of xⱼ (and of y), then measures xⱼ's effect only on the variation the others cannot explain. In an exam always write the phrase 'ceteris paribus' or 'holding the other regressors fixed' when you interpret a coefficient — leaving it out usually costs a mark on every interpretation part.
How do I get the direction of an omitted-variable bias right?
You need two signs, not one. Bias = β₂ × δ, where β₂ is the omitted variable's effect on y and δ is its correlation with the included regressor. Same signs give an upward bias (the coefficient over-states the true effect); opposite signs give a downward bias. Saying only 'the omitted variable raises y' does not fix the direction — trace both channels.
Why can't I just compare models using R²?
Because R² can only rise as you add regressors — even a column of random noise nudges it up — so it always 'prefers' the bigger model. Use adjusted R², which subtracts a penalty for extra variables and can fall when a variable does not earn its place, or run an F-test to compare a restricted and unrestricted model formally.
What degrees of freedom does multiple regression use?
df = n − k − 1, where k is the number of slope coefficients and the extra −1 is for the intercept. This df drives σ̂², adjusted R², and every t- and F-test that follows. Forgetting the −1 (using n − k) is one of the most common and avoidable slips in the course.
Is a multiple-regression coefficient automatically causal?
No. A slope is causal only if the exogeneity assumption MLR.4 (E[u|x] = 0) holds — no relevant, correlated variable left in the error, no reverse causality, no serious measurement error. Adding good controls moves you toward it, but a significant coefficient is still just an association unless you can defend MLR.4.
Exam move
Drill interpretation until it is automatic, because that is where the Part B marks live. For every coefficient practise saying the ceteris-paribus sentence out loud — 'a one-unit rise in xⱼ is associated with a β̂ⱼ change in mean y, holding the other regressors fixed' — in the variables' real units, and for any squared term compute both the marginal effect β₃ + 2β₄x at the stated point and the turning point −β₃/(2β₄). Build a one-page reference with the MLR.1–6 assumptions and what each buys, the omitted-variable-bias sign grid (Bias = β₂ × δ, same signs → up, opposite → down), and the adjusted-R² formula with df = n − k − 1, then test yourself by covering it and reconstructing each row. Finally, work past-style questions from R output end to end: interpret every slope, sign any bias and name the assumption it breaks, judge fit with adjusted R² rather than raw R², and close with a sentence separating statistical significance from economic magnitude and causal validity.
