Difficulty: Intermediate | Prerequisites: OLS basics (Ch. 2), coefficient interpretation (Ch. 3 Q1–2)
This covers two pitfalls that trip students up every semester: perfect multicollinearity (when your variables are locked together by an identity so the regression cannot run) and omitted variable bias (when leaving out a relevant variable distorts the coefficient you care about). Both are testable and both require you to reason about the structure of your model before you ever touch a dataset. If you can handle these, you understand the "why" behind multiple regression, not just the mechanics.
Perfect multicollinearity means one explanatory variable can be written as an exact linear function of the others, which makes it impossible for OLS to separate their individual effects. Omitted variable bias arises when you leave a relevant variable out of the model and it is correlated with a variable you did include, causing that included variable's coefficient to pick up effects that belong to the omitted one. Both problems are about model structure, not data quality.
Perfect multicollinearity
A situation in which one explanatory variable is an exact linear combination of the other explanatory variables. When this holds, OLS cannot compute unique coefficient estimates.
In simple terms, the software literally cannot run the regression because the variables contain redundant information.
Assumption 3 (no perfect collinearity)
One of the Gauss-Markov assumptions. It requires that no explanatory variable is a constant or a perfect linear function of the others in the sample. Violating it means OLS breaks down entirely.
Think of it as the model needing each variable to bring at least some independent information to the table.
Omitted variable bias (OVB)
The distortion in an estimated coefficient that occurs when you leave a relevant variable out of the regression and that omitted variable is correlated with an included explanatory variable. The included variable's coefficient absorbs part of the omitted variable's effect.
In simple terms, you are blaming (or crediting) one variable for something that is partly caused by another variable you did not measure.
Gauss-Markov assumptions
The set of assumptions (linearity in parameters, random sampling, no perfect collinearity, zero conditional mean of the error) under which OLS is the best linear unbiased estimator (BLUE).
Identity constraint
An exact algebraic relationship among variables. In the time-allocation example, study + sleep + work + leisure = 168 for every student. This identity makes it impossible to include all four variables as independent regressors.
Dummy variable trap
A common instance of perfect multicollinearity that arises when you include dummy variables for every category plus a constant, so the dummies sum exactly to 1 (the constant). The fix is to drop one category.
The proposed model is:
GPA = \beta_0 + \beta_1 study + \beta_2 sleep + \beta_3 work + \beta_4 leisure + uEvery student's hours must sum to 168 (24 hours × 7 days), so:
study + sleep + work + leisure = 168
The question asks: does it make sense to hold sleep, work, and leisure constant while increasing study by one hour?
No. If three of the four are held fixed, the fourth is determined by the identity. You cannot add an hour of study without taking it from somewhere
The ceteris paribus interpretation of each coefficient is meaningless because the constraint forbids it
The identity means leisure = 168 - study - sleep - work
Leisure is an exact linear function of the other three variables plus a constant
Including all four alongside an intercept creates perfect multicollinearity
OLS cannot separate the effects because it cannot invert the matrix it needs to invert
Remove one of the four time-use variables (say, leisure) from the model:
GPA = \beta_0 + \beta_1 study + \beta_2 sleep + \beta_3 work + uNow each coefficient has a clear interpretation: the effect of one more hour of that activity, with the lost hour coming from the omitted category (leisure)
The choice of which variable to drop changes the interpretation but not the model's fit or predictive power
This is the same logic as the dummy variable trap: you always omit one category to avoid exact linear dependence
The true model is:
avgprod = \beta_0 + \beta_1 avgtrain + \beta_2 avgabil + uThis satisfies the Gauss-Markov assumptions. But suppose you only run the simple regression of avgprod on avgtrain, ignoring avgabil. You get a different slope estimate.
Grants have been given to firms whose workers have below-average ability
Firms that received grants have more training (higher avgtrain) but lower worker ability (lower avgabil)
So avgtrain and avgabil are negatively correlated: Corr(avgtrain, avgabil) < 0
The omitted variable bias formula says:
\tilde{\beta}_1 = \hat{\beta}_1 + \hat{\beta}_2 \cdot \tilde{\delta}_1where the tilde denotes the simple regression estimate and delta-tilde is the slope from regressing the omitted variable (avgabil) on the included variable (avgtrain).
The true effect of ability on productivity is positive: beta-2 > 0 (more able workers are more productive)
The correlation between avgtrain and avgabil is negative, so delta-tilde < 0
The bias term = (positive) × (negative) = negative
The simple regression coefficient on avgtrain is biased downward
You underestimate the true effect of training because the negative correlation with ability drags the estimate down
Firms with lots of training also happen to have less able workers (because of the grant programme). In the simple regression, some of the negative effect of low ability gets attributed to training, making training look less effective than it truly is.
study + sleep + work + leisure = 168This means any one variable equals 168 minus the other three. Including all four plus a constant creates perfect multicollinearity.
GPA = \beta_0^* + \beta_1 study + \beta_2 sleep + \beta_3 work + uNow beta-1 measures the effect of an additional hour of study at the expense of one hour of leisure (the dropped category). The intercept absorbs the leisure baseline.
\tilde{\beta}_1 = \hat{\beta}_1 + \hat{\beta}_2 \cdot \tilde{\delta}_1beta-tilde-1 = slope from the simple regression (omitting the relevant variable)
beta-hat-1 = slope from the multiple regression (including the relevant variable)
beta-hat-2 = coefficient on the omitted variable in the multiple regression
delta-tilde-1 = slope from regressing the omitted variable on the included variable
The bias is the product beta-hat-2 × delta-tilde-1. Its sign depends on the signs of the true effect (beta-2) and the correlation between the two explanatory variables.
Perfect multicollinearity shows up any time your data has a built-in constraint. Budget shares that must sum to 100%, time allocations that must sum to a fixed total, and sets of dummy variables that cover every category all produce the same problem. In policy evaluation, omitted variable bias is the reason observational studies can mislead: a programme evaluation that ignores baseline ability will misattribute outcomes to the programme itself.
Students sometimes think multicollinearity means two variables are merely correlated. Ordinary (imperfect) correlation between explanatory variables is fine and does not violate any assumption. Only exact linear dependence causes perfect multicollinearity.
Students often think omitted variable bias always makes a coefficient too large. The bias can go in either direction: it depends on the sign of the omitted variable's true effect and its correlation with the included variable.
A frequent mistake is thinking you can fix multicollinearity by collecting more data. If the linear relationship is an identity (like hours summing to 168), no amount of data breaks it. You must change the model.
Students sometimes confuse omitting a variable with dropping an observation. Omitting a variable changes the model specification. Dropping an observation changes the sample.
⚠️ Expect exam questions asking you to identify whether a model violates Assumption 3. Look for identities among the variables.
⚠️ "Which direction is the bias?" is a classic exam question. You need three things: the sign of beta-2 (the true effect of the omitted variable), the sign of the correlation between the omitted and included variables, and their product.
⚠️ Be ready to explain how to reformulate a model with an identity constraint. The answer is always: drop one variable so the remaining ones are not perfectly collinear.
⚠️ Understand the intuition behind OVB, not just the formula. Exams may ask you to explain the bias in words.
True or False: If two explanatory variables have a correlation of 0.85, the model violates Assumption 3. (False. Only perfect, exact linear dependence violates Assumption 3. High correlation is imperfect multicollinearity, which is allowed.)
Fill in the blank: In the time-allocation model, including study, sleep, work, leisure, and a constant creates perfect multicollinearity because the four time variables always sum to ____. (168)
True or False: When the omitted variable has a positive effect on y and is negatively correlated with the included variable, the simple regression coefficient on the included variable is biased upward. (False. Positive × negative = negative bias, so the coefficient is biased downward.)
Fill in the blank: To fix perfect multicollinearity caused by an identity constraint, you should ____ one of the constrained variables from the model. (drop / remove / omit)
True or False: Omitted variable bias disappears if the omitted variable is uncorrelated with all included variables. (True. If delta-tilde-1 = 0, the bias term is zero.)
Q: Why does including study, sleep, work, leisure, and a constant in a GPA regression violate Assumption 3?
A: Because the four time-use variables sum to exactly 168 for every student. This means any one of them is a perfect linear function of the other three plus a constant. OLS requires that no explanatory variable be an exact linear combination of the others, and this identity breaks that requirement.
Q: You drop leisure from the model. What does the coefficient on study now measure?
A: It measures the effect on GPA of one additional hour of study, with the hour taken from leisure (the omitted category), holding sleep and work constant.
Q: In the avgprod model, grants are given to firms with below-average worker ability. What is the direction of bias in the simple regression of avgprod on avgtrain?
A: Downward bias. The omitted variable (avgabil) has a positive effect on productivity (beta-2 > 0), and it is negatively correlated with avgtrain (firms with more training have lower ability because of the grant targeting). Bias = positive × negative = negative. The simple regression underestimates the true effect of training.
Q: If the grant programme instead targeted firms with above-average ability, how would the bias change?
A: Avgtrain and avgabil would now be positively correlated, so delta-tilde > 0. Bias = positive × positive = positive. The simple regression would overestimate the true effect of training.
Perfect multicollinearity connects to the dummy variable trap, which you will encounter whenever you work with categorical data (Chapter 7). Omitted variable bias is the conceptual foundation for instrumental variables (Chapter 15), where the entire point is to find a way around the bias when you cannot observe or control for the omitted variable. The bias formula here also foreshadows the Frisch-Waugh theorem, which formalises how adding or removing variables changes coefficient estimates.
perfect multicollinearity, Assumption 3, no perfect collinearity, identity constraint, time allocation, hours constraint, dummy variable trap, omitted variable bias, OVB, bias direction, Gauss-Markov, BLUE, model specification, avgtrain, avgabil, avgprod, simple vs multiple regression, Wooldridge Chapter 3, introductory econometrics