Heteroskedasticity

Heteroskedasticity, error variance that changes across observations, is the normal state of financial data, not the exception. The Gauss–Markov assumption of spherical errors requires $\operatorname{Var}(\varepsilon_i) = \sigma^2$ for every $i$; markets flatly refuse. Volatility clusters, larger firms have noisier returns, and cross-sectional regressions mix calm and turbulent observations. The crucial thing to understand is what breaks and what doesn't: your coefficients are fine, your standard errors are lies, and the fix is a one-line change to the covariance estimator.

The problem, precisely

Keep the linear model $y = X\beta + \varepsilon$ with strict exogeneity $\mathbb{E}[\varepsilon\mid X] = 0$, but let the conditional variance vary:

$\operatorname{Var}(\varepsilon \mid X) = \Omega = \operatorname{diag}(\sigma_1^2, \dots, \sigma_n^2), \qquad \sigma_i^2 \ne \sigma_j^2.$

The errors are still uncorrelated (off-diagonals zero) but no longer identically scaled. Because strict exogeneity survives, the unbiasedness proof is untouched:

$\hat\beta = \beta + (X^\top X)^{-1}X^\top\varepsilon, \qquad \mathbb{E}[\hat\beta\mid X] = \beta.$

So OLS remains unbiased and consistent, this is the single most important fact and the most common thing candidates get wrong. What fails is efficiency (OLS is no longer BLUE; see The Gauss–Markov Theorem) and, more damagingly, the variance formula.

Why the standard errors are wrong

The textbook OLS covariance $\sigma^2(X^\top X)^{-1}$ was derived assuming $\operatorname{Var}(\varepsilon) = \sigma^2 I$. With general $\Omega$, the correct sandwich is

$\operatorname{Var}(\hat\beta\mid X) = (X^\top X)^{-1}\,\big(X^\top \Omega X\big)\,(X^\top X)^{-1}.$

The naive formula replaces $X^\top\Omega X$ with $\sigma^2 X^\top X$, which is simply wrong when $\Omega \ne \sigma^2 I$. The reported standard errors can be too small or too large, often too small when high-variance observations coincide with extreme regressor values, so $t$-statistics are inflated and you declare significance that isn't there. Every hypothesis test and confidence interval built on the naive formula is invalid.

The White / robust fix

Halbert White's insight (1980): you do not need to know $\Omega$ to estimate the sandwich consistently. You cannot estimate $n$ separate $\sigma_i^2$ from $n$ points, but you can estimate the $k\times k$ matrix $X^\top\Omega X$ by plugging in squared residuals:

$\widehat{X^\top\Omega X} = \sum_{i=1}^n \hat\varepsilon_i^2\, x_i x_i^\top,$

giving the heteroskedasticity-consistent (HC0) covariance

$\widehat{\operatorname{Var}}(\hat\beta) = (X^\top X)^{-1}\Big(\sum_i \hat\varepsilon_i^2\, x_i x_i^\top\Big)(X^\top X)^{-1}.$

These are White standard errors (robust, HC standard errors). Finite-sample refinements HC1–HC3 rescale $\hat\varepsilon_i^2$ (e.g. HC3 divides by $(1-h_{ii})^2$ using the hat-matrix leverage) and are preferred in small samples. The practical rule on a desk: use robust standard errors by default, they are correct under homoskedasticity too (they just cost a little efficiency), so there is little reason not to.

Detecting it

• Breusch–Pagan test, regress squared residuals $\hat\varepsilon_i^2$ on the regressors (or their determinants of variance); a significant fit means variance depends on $X$. It is a score/LM test: $nR^2 \sim \chi^2_p$ under homoskedasticity.
• White test, the same idea but including squares and cross-products of regressors, catching general (nonlinear) heteroskedasticity at the cost of many degrees of freedom.
• Eyeball, plot residuals against fitted values or against time; a fanning "megaphone" shape or visible volatility clustering is heteroskedasticity you can see.

Worked example: pooled cross-sectional return regression

Regress monthly stock returns on a value signal across all stocks and months, $r_{i,t} = \alpha + \beta\,\text{value}_{i,t} + \varepsilon_{i,t}$. Residual variance is far larger in crisis months (Oct 2008) than calm ones, and larger for small caps than mega caps, textbook heteroskedasticity. The OLS slope $\hat\beta$ (the value premium estimate) is still unbiased. But the naive $t$-stat might read 3.5 and the White $t$-stat only 2.0: the naive standard error understated uncertainty because the high-variance crisis observations also had extreme value spreads. Reporting the naive $t$ would overstate how sure you are the value premium exists. (In panels you would go further and cluster by month and/or firm.)

Failure modes in financial data

• Volatility clustering is heteroskedasticity in time series form; modeling it explicitly (rather than just robustifying standard errors) is the entire subject of GARCH Volatility Models.
• Robust SEs don't restore efficiency. White fixes inference but leaves OLS inefficient. If the variance structure is known or modelable, weighted/generalized least squares recovers the lost efficiency and is BLUE.
• Heteroskedasticity plus autocorrelation. Financial errors usually have both. White handles only the variance; you need the HAC (Newey–West) estimator to also handle serial correlation. See Autocorrelation and Serial Correlation.
• Leverage points. A few high-leverage observations dominate $\sum \hat\varepsilon_i^2 x_i x_i^\top$; HC3 guards against this, plain HC0 does not.

In interviews

The clean two-part answer: heteroskedasticity leaves OLS unbiased and consistent but not efficient, and, the part that actually matters, makes the standard errors wrong, so all $t$-tests and confidence intervals are invalid until fixed. Be able to write the sandwich covariance $(X^\top X)^{-1}(X^\top\Omega X)(X^\top X)^{-1}$ and explain that White's trick estimates the middle with $\sum \hat\varepsilon_i^2 x_i x_i^\top$, you cannot estimate every $\sigma_i^2$ but you can estimate the $k\times k$ sandwich meat. Name a detection test (Breusch–Pagan / White) and know the practical default: always report robust standard errors. A strong closer connects it to GARCH as the time-series counterpart.