Teaching · applied econometrics
OLS assumptions
Seven simulations of classical linear-regression assumptions. Each changes one feature of the data-generating process and measures its effect on estimation and inference.
Bias, precision, and test size
Bias is the difference between the expected estimate and the target coefficient. Under the Gauss-Markov conditions, OLS has the smallest variance among linear unbiased estimators. The simulations compare coefficient bias, the reported standard error, and the frequency of rejecting a true null.
Each panel runs the same experiment. It builds a world where the true slope is exactly 1, draws 500 samples from it, estimates the slope in each, and tests a hypothesis that is true by construction. That gives three readings:
- Does the estimate still centre on the truth? The gap between the dashed line and the blue line is bias.
- Does the reported standard error match the real spread? The teal bar is how much the estimate actually moves between samples. The rust bar is how much the regression claims it moves.
- How often does a 5% test reject a true null? A correctly sized test rejects in 5% of repetitions; simulation estimates that frequency with sampling variation.
Every panel opens with its assumption satisfied. The first slider introduces the violation, and the three readings show which property of the estimate it affects.
Simulation results
| Assumption broken | Estimate still centred? | Standard error accurate? | Actual size of a 5% test |
|---|---|---|---|
| 1 · Specification | No, bias −0.5 | Roughly | 99% |
| 2 · Zero-mean error | Yes | Yes | 5% |
| 3 · Exogeneity | No, bias +1.2 | Roughly | 100% |
| 4 · Independence | Yes | No, understates 6× | ~70% |
| 5 · Homoskedasticity | Yes | No, understates 1.6× | ~22% |
| 6 · No collinearity | Yes | Yes | 5% |
| 7 · Normality | Yes | Understates on average | 5% or below |
Specification and exogeneity failures bias the coefficient in these examples. Serial correlation and heteroskedasticity affect inference without shifting the coefficient’s mean. The remaining examples concern intercept normalization, near-collinearity, and nonnormal errors.
Seven experiments at a glance
Each graph shows forty simulated 95% intervals at the maximum setting, ordered by their coefficient estimate. Highlighted intervals exclude the true coefficient.
Assumption 1
The model is linear, correctly specified, and has an additive error
Yᵢ = β₀ + β₁X₁ᵢ + ⋯ + βₖXₖᵢ + εᵢ
Protects Unbiasedness
The policy setting
Spending per pupil raises test scores, and the first thousand dollars does more than the tenth. A budget office fits a line and reports a single per-dollar effect.
What breaks
The omitted squared term pulls the fitted slope down. At full curvature the true relationship is y = 1 + x − 0.5x², so the marginal effect falls from 1 at the lowest spending level to 0 at the highest. The average estimate is 0.5, the average of that marginal effect over the sample, against a linear-term coefficient of 1. A single per-dollar figure understates the return to the first dollars and overstates the return to the last.
How you detect it
Plot residuals against fitted values and regressors. Systematic curvature can indicate a missing nonlinear term; RESET tests for certain specification errors.
What you do about it
Consider a squared term, a log specification, or a spline, according to the relationship being modeled. Changing standard errors leaves the misspecified coefficient unchanged.
Question
What does a straight line cost you when the policy actually has diminishing returns?
A quadratic spending relationship fitted with a straight line. The slider controls the omitted curvature.
01 Correct specification
Centre
On target
the estimate centres on the truth
Honesty
Honest
the reported error matches the real spread
The test
Test holds
6% across all 500 samples, against a promised 5%
All three readings are clean. The estimate centres on the truth, the reported uncertainty matches the real spread, and the test rejects the true null in 6% of the 500 samples, against a promised 5%. Of the hundred intervals drawn, 5 miss the truth; 95% coverage means five misses per hundred on average, and the count in any particular hundred varies around that figure.
Assumption 2
The error term has a population mean of zero
E(εᵢ) = 0
Protects Intercept normalization
The policy setting
A benefits survey collects self-reported spending. Respondents understate by roughly the same amount regardless of income, so every observation sits below its true value.
What breaks
The intercept absorbs the offset. Replacing the intercept by β₀ + 4 and the error by ε − 4 gives the same model with a zero-mean error.
How you detect it
OLS residuals sum to zero when a constant is included. That identity cannot establish the mean of the population error.
What you do about it
Include an intercept to absorb the common offset. Its interpretation depends on the outcome scale.
Question
If every household understates its spending by the same amount, what happens to the slope?
A common offset shifts the intercept and leaves the slope unchanged.
02 Zero-mean error
Centre
On target
the estimate centres on the truth
Honesty
Honest
the reported error matches the real spread
The test
Test holds
6% across all 500 samples, against a promised 5%
All three readings are clean. The estimate centres on the truth, the reported uncertainty matches the real spread, and the test rejects the true null in 6% of the 500 samples, against a promised 5%. Of the hundred intervals drawn, 5 miss the truth; 95% coverage means five misses per hundred on average, and the count in any particular hundred varies around that figure.
Assumption 3
No regressor is correlated with the error term
Cov(Xⱼᵢ, εᵢ) = 0 for every j
Protects Unbiasedness
The policy setting
A job-training programme is evaluated on earnings. Motivation is unmeasured, raises earnings on its own, and predicts who enrols in the first place.
What breaks
At full strength, the average estimate is 2.2 against a true effect of 1. The simulation rejects the true null in nearly every repetition.
How you detect it
Residual diagnostics alone cannot establish exogeneity. Examine how enrollment was determined and which common causes of enrollment and earnings were measured.
What you do about it
A randomized assignment, valid instrument, discontinuity, or credible difference-in-differences design may address selection. Each requires its own identifying assumptions.
Question
How does selection into training affect the estimated earnings effect?
An unmeasured driver raises earnings and also predicts who enrols. The slider is how strongly enrolment and that driver move together.
03 Exogeneity
Centre
On target
the estimate centres on the truth
Honesty
Honest
the reported error matches the real spread
The test
Test holds
6% across all 500 samples, against a promised 5%
All three readings are clean. The estimate centres on the truth, the reported uncertainty matches the real spread, and the test rejects the true null in 6% of the 500 samples, against a promised 5%. Of the hundred intervals drawn, 6 miss the truth; 95% coverage means five misses per hundred on average, and the count in any particular hundred varies around that figure.
Assumption 4
Observations of the error term are uncorrelated with each other
Corr(εᵢ, εⱼ) = 0 for i ≠ j
Protects Efficiency and valid standard errors
The policy setting
State unemployment is tracked quarterly after a stimulus. A bad quarter does not clear by the next one, so the shock persists.
What breaks
In this simulation, the slope remains unbiased. At full persistence, its sampling spread is about six times the reported standard error, and the nominal 5% test rejects roughly 70% of the time.
How you detect it
Plot residuals in time order and inspect their autocorrelation. Durbin-Watson and Breusch-Godfrey tests address specified forms of serial correlation.
What you do about it
Use a dependence-appropriate variance estimator, such as Newey-West. HC1 alone does not account for correlation across periods.
Question
How does serial correlation affect the standard error?
Errors follow an AR(1) process along a time trend. Total error variance is held fixed, so the only thing the slider changes is persistence.
04 Independent errors
Centre
On target
the estimate centres on the truth
Honesty
Honest
the reported error matches the real spread
The test
Test holds
5% across all 500 samples, against a promised 5%
All three readings are clean. The estimate centres on the truth, the reported uncertainty matches the real spread, and the test rejects the true null in 5% of the 500 samples, against a promised 5%. Of the hundred intervals drawn, 6 miss the truth; 95% coverage means five misses per hundred on average, and the count in any particular hundred varies around that figure.
Assumption 5
The error term has constant variance
Var(εᵢ) = σ² for every i
Protects Efficiency and valid standard errors
The policy setting
Municipal expenditure is regressed on resident income across cities. A town of four thousand and a city of four million are one observation each, and the large ones vary enormously.
What breaks
The slope stays unbiased and the reported standard error reports only about 60% of the true spread. A nominal 5% test rejects a true null roughly 22% of the time. The noisiest observations are also the highest-leverage ones, which is what makes the distortion this severe.
How you detect it
Plot residuals against fitted values and look for a fan opening to the right. The White and Breusch-Pagan tests formalise it, with White the more general and Breusch-Pagan the cheaper in degrees of freedom.
What you do about it
HC1 reduces the rejection rate from about 22% to about 9% in this example. Its finite-sample performance can remain imperfect; the choice of variance estimator and functional form needs further checking.
Question
Large cities are noisier than small ones. What does that do to a significance test?
Error spread grows with the regressor, and the regressor is skewed the way municipal data actually is: many small units, a few very large ones.
05 Constant variance
Centre
On target
the estimate centres on the truth
Honesty
Honest
the reported error matches the real spread
The test
Test holds
7% across all 500 samples, against a promised 5%
All three readings are clean. The estimate centres on the truth, the reported uncertainty matches the real spread, and the test rejects the true null in 7% of the 500 samples, against a promised 5%. Of the hundred intervals drawn, 4 miss the truth; 95% coverage means five misses per hundred on average, and the count in any particular hundred varies around that figure.
Assumption 6
No exact linear relationship between regressors
No Xⱼ is an exact linear combination of the others
Protects Identification, and then precision
The policy setting
Programme need is modelled on both the poverty rate and median household income across census tracts. The two are close to mirror images of each other.
What breaks
In this near-collinearity example, the estimate stays centered on the true coefficient while its spread grows about tenfold. The rejection rate remains near 5%.
How you detect it
Variance inflation factors, and the tell-tale pattern of a jointly significant F with no individually significant t. Wide confidence intervals on coefficients you expected to be sharp.
What you do about it
Additional independent variation can improve precision. Combining predictors changes the estimand; dropping one may change the specification. Exact collinearity prevents a unique coefficient estimate.
Question
If two predictors say almost the same thing, which one gets the credit?
Increasing the correlation between regressors widens the coefficient distribution and its reported standard error.
06 No exact collinearity
Centre
On target
the estimate centres on the truth
Honesty
Honest
the reported error matches the real spread
The test
Test holds
4% across all 500 samples, against a promised 5%
All three readings are clean. The estimate centres on the truth, the reported uncertainty matches the real spread, and the test rejects the true null in 4% of the 500 samples, against a promised 5%. Of the hundred intervals drawn, 0 miss the truth; 95% coverage means five misses per hundred on average, and the count in any particular hundred varies around that figure.
Assumption 7
The error term is normally distributed
εᵢ ~ N(0, σ²)
Protects Exact finite-sample inference
The policy setting
Disaster relief payouts are regressed on damage assessments. Most claims are modest and a few are enormous, so the errors are nothing like a bell curve.
What breaks
For the sample sizes and error distributions shown here, the two-sided rejection rate stays near or below 5%. Other nonnormal distributions or test statistics can behave differently.
How you detect it
A QQ plot of the residuals, or Jarque-Bera. Both will correctly tell you the errors are not normal, which is a different question from whether it matters.
What you do about it
Check the behavior of the statistic for the relevant sample size and error distribution. Normality also affects prediction intervals and other distribution-dependent calculations.
Question
Disaster payouts are wildly skewed. Does that invalidate the t-test?
Mean-zero lognormal errors vary in skewness and tail weight.
07 Normal errors
Centre
On target
the estimate centres on the truth
Honesty
Honest
the reported error matches the real spread
The test
Test holds
4% across all 500 samples, against a promised 5%
All three readings are clean. The estimate centres on the truth, the reported uncertainty matches the real spread, and the test rejects the true null in 4% of the 500 samples, against a promised 5%. Of the hundred intervals drawn, 2 miss the truth; 95% coverage means five misses per hundred on average, and the count in any particular hundred varies around that figure.
Implications for applied work
Specification and identification determine what a coefficient estimates. Variance assumptions determine whether the reported uncertainty is appropriate. These examples isolate one issue at a time; an empirical analysis may face several together.
Heteroskedasticity-robust standard errors address unequal error variance. They do not remove omitted-variable bias or establish a causal interpretation.