Answer-first
Heteroskedasticity breaks OLS efficiency because the variance of the error term is no longer constant. When errors have unequal variance, OLS still produces unbiased slope estimates, but they are no longer the best linear unbiased estimators (BLUE). Even worse, the usual standard errors become wrong, making hypothesis tests unreliable.
Warm intro (and where to find the “why” pages)
If you’re staring at a heteroskedasticity question at 11:47pm, feeling stuck, behind, or low‑key panicking because the “variance of the error term” explanation feels abstract, you’re not alone. These questions look simple—“use robust SEs”—but under exam pressure, students freeze when asked to explain why heteroskedasticity breaks efficiency and what actually goes wrong with inference.
If you’re rebuilding your foundation across topics, start at our Why Hub. If you need the full econometrics & time series roadmap for last‑minute studying or troubleshooting, see Econometrics & Time Series.
Problem setup
Consider the regression model:
\[ y_i = \beta_0 + \beta_1 x_i + u_i. \]
OLS assumes:
\[ \text{Var}(u_i \mid x_i) = \sigma^2. \]
Heteroskedasticity occurs when:
\[ \text{Var}(u_i \mid x_i) = \sigma_i^2, \]
meaning the variance depends on \(x_i\) or other factors.
Step-by-step solution (WordPress-safe MathJax)
Step 1: OLS remains unbiased
Even with heteroskedasticity:
\[ \mathbb{E}[\hat{\beta}_{OLS}] = \beta. \]
So the slope is still centered on the truth.
Step 2: But OLS is no longer efficient
OLS is BLUE only under homoskedasticity. When variances differ, OLS no longer minimizes the variance of the estimator.
Weighted Least Squares (WLS) becomes more efficient.
Step 3: Standard errors become wrong
The usual OLS standard error formula assumes constant variance. With heteroskedasticity, this formula underestimates or overestimates the true variability.
This breaks:
- t‑tests
- F‑tests
- confidence intervals
Step 4: Robust standard errors fix inference
Heteroskedasticity‑robust (White) standard errors correct the variance formula without changing the slope estimate.
Step 5: State the identifying assumption clearly
OLS requires homoskedasticity for efficiency and valid standard errors:
\[ \text{Var}(u_i \mid x_i) = \sigma^2. \]
When this fails, inference breaks.
Intuition
Heteroskedasticity means some observations are “noisier” than others. OLS treats all observations equally, so it gives too much weight to high‑variance points and too little weight to low‑variance points. This makes the estimator less precise and the standard errors misleading.
Common exam mistakes
- Claiming OLS becomes biased. It does not.
- Using non‑robust standard errors. This breaks inference.
- Confusing heteroskedasticity with autocorrelation.
- Thinking heteroskedasticity always inflates SEs. It can deflate them too.
- Ignoring model diagnostics. Residual plots matter.
Why this matters
Heteroskedasticity is everywhere in real data—income, firm size, prices, health outcomes. If you ignore it, your hypothesis tests become meaningless. Robust SEs and WLS are essential tools for credible empirical work.
Final summary
- Heteroskedasticity leaves OLS unbiased but inefficient.
- Standard errors become wrong under heteroskedasticity.
- Inference breaks unless you use robust SEs.
- WLS is more efficient when variances differ.
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