Heteroscedasticity

Homoscedasticity says Var(u_i | x) = σ² for every observation. Heteroscedasticity says the variance differs: Var(u_i | x) = σ_i². The usual picture is a fan: the points are tightly packed around the line at one end and spread out at the other. Typical causes are a variable that scales the noise (household income, firm size), learning (error rates fall as practice grows), outliers, and a wrong functional form or an omitted variable whose effect shows up as changing spread (lesson 7).

Real data show it clearly. In the World Bank data (life expectancy against ln GDP per head, 186 economies in 2019), the residuals of the straight-line regression have a standard deviation of 5.47 years among the poorest third of countries, 4.25 in the middle third and 2.78 among the richest third. Poor countries are far less predictable: some have high life expectancy for their income, some have been hit by conflict or disease. The invented survey, by contrast, was written homoscedastic, and its tests will say so.

Unbiasedness only needs E(u | X) = 0 (lesson 2): variances do not enter. So OLS stays unbiased and consistent. What fails is everything built on the assumption of constant variance:

• OLS is no longer best. It is not BLUE: there is a linear unbiased estimator with a smaller variance (weighted least squares), because OLS treats noisy observations as if they were as informative as precise ones.
• The usual standard errors are wrong, because the formula σ²(X′X)⁻¹ assumes constant σ². They can be too small or too large, depending on how the variance relates to the regressors. So t statistics, F statistics and confidence intervals are unreliable: they do not have their stated distributions, even in large samples.
• R² is not meaningfully affected as a descriptive measure.

One subtlety the Delhi School has asked: if the error variance depends on a variable z that is not in the model, the usual standard errors are wrong only if z is related to the regressors. What matters is Var(u | x), the variance given the regressors; if z were completely unrelated to x, the variance given x would be constant and the usual standard errors would be fine. In the textbook reading z is related to x, and the coefficients are unbiased while the estimated standard errors are biased.

Try it. Move γ from 0 to 1.5 and watch which numbers move. The slope stays centred on the truth. The usual standard error stops matching the real spread, and the 95% interval stops holding the truth 95% of the time. The robust standard error repairs the inference. Weighting gives a tighter slope. At small γ with x between 1 and 5, notice that the problem is mild: heteroscedasticity does the most harm when it is strong.

Start with a plot: the residuals (or their squares or absolute values) against a regressor or against the fitted values. A fan is the evidence. The tests put numbers on it. All have the null hypothesis of homoscedasticity.

For White’s test with the regression y = α₀ + α₁x₁ + α₂x₂ + u, the squared residuals are regressed on x₁, x₂, x₁², x₂² and x₁x₂ (five terms), and the test statistic is nR² from that regression with 5 degrees of freedom. Omitting the cross-product, or regressing on lagged residuals, gives a different test. (A regression on the lagged residual is a test for autocorrelation, the next lesson.)

On the World Bank regression, Breusch-Pagan gives LM = 8.54 (1 df, p = 0.0035), White gives 12.28 (2 df, p = 0.0022), and Goldfeld-Quandt, comparing the poorest 74 and the richest 74 countries after leaving out the middle 38, gives F = 4.01 with (72, 72) degrees of freedom (p < 0.001). All three reject. In the survey’s spending regression, where the variance is constant by construction, Breusch-Pagan gives p = 0.95 and White p = 0.97: no sign of a problem, as there should not be.

Two routes, with different goals. Weighted least squares (WLS) aims at the efficiency OLS lost. If Var(u_i) = σ²z_i², divide the equation by z_i and the new error has constant variance:

• WLS or GLS needs the form of the variances. If the form is known up to a constant, WLS is BLUE. If it is estimated first (feasible GLS), the result is efficient only in large samples and only if the form is right. “GLS can be used when the disturbances are autocorrelated or heteroscedastic”; it does not help with collinear regressors or an incorrect functional form.
• Robust (White, or HC) standard errors aim at valid inference without changing the estimate. They keep the OLS coefficients and replace the standard errors by ones that are consistent whatever the form of the variances. For the World Bank regression the usual standard error of the slope is 0.281 and the robust one is 0.355, about 26% larger: the usual formula understated the uncertainty. Nothing else changes: the coefficient is still 6.03.
• Transform the variables. Taking logs often steadies the variance, because proportional noise becomes constant noise (lesson 6).

Many applied economists now report robust standard errors by default, because they work whether or not the problem is present and need no model of the variance. WLS gains more when its model of the variance is right.

The Delhi School’s questions here (the case where the variance depends on a variable outside the model, and the form of White’s auxiliary regression) are in the hard-problems lesson at the end of the module. The matching of tests to problems is in the test atlas.