Classical vs robust standard errors
The error spread grows with x (heteroskedasticity), yet the OLS slope stays unbiased. What breaks is the standard error: the classical SE is invalid here, while the robust (HC1) SE — Stata’s , robust — is asymptotically valid. The point estimate is fixed by construction, so only the SEs move.
SE(b₁)classical 0.158vsrobust 0.2141.36×
95% confidence interval for the slope b₁ (same centre, different width)
Slope b₁ (fixed by construction) 0.80Robust ÷ classical SE 1.36×
Heteroskedasticity6.0
The slope estimate is 0.80 at every setting, so heteroskedasticity has not biased it. But the classical SE (0.158) is invalid here, while the robust SE (0.214) is asymptotically valid. In this design robust is 1.36× the classical, so the classical confidence interval is too narrow and its t-test overstates significance.
Robust here is HC1 (finite-sample adjusted), the same estimator as Stata’s
regress y x, robust. It is asymptotically valid, not exactly correct, and need not be larger in general — it simply happens to be larger in this rising-variance design.The slope point estimate is 0.80, fixed by construction. The classical standard error is 0.158 and the heteroskedasticity-robust standard error is 0.214, a ratio of 1.36 to one. In this design the robust standard error is larger, so the classical confidence interval is too narrow.