Case 019Regression and model reviewCore

A signal's slope is 0.12 with an OLS standard error of 0.05, a t of 2.4, but the heteroskedasticity-robust standard error is 0.08. Recompute significance, say which to trust, and explain why the errors grow in volatile months.

The situation

Hiranyaka Capital's researcher regresses next-month stock returns on a value signal across ten years of monthly cross-sections. The pooled slope is 0.12 with an ordinary least squares standard error of 0.05, a t-statistic of 2.4, and the write-up calls the signal significant.

You rerun it with heteroskedasticity-robust (White) standard errors and get 0.08. A plot of the residuals against the month's market volatility shows them fanning out in volatile months, and the signal itself takes its most extreme values in the same months.

Your task

Recompute the t-statistic and the confidence interval, say which standard error to trust and why, and explain what makes the residuals grow in volatile months.

Quick check

What is the t-statistic with the robust standard error?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

With the robust standard error the t-statistic falls from 2.4 to 1.5, and the signal is no longer significant at 5%. The 95% interval widens from about 0.02 to 0.22 to about -0.04 to 0.28, which includes zero. Trust the robust one: the noise is larger in volatile months, and those are also the months with extreme signal values, which is exactly when ordinary standard errors are too small.

Step 1What changes when you switch standard errors?

Only the uncertainty, never the estimate. The slope stays 0.12. The t-statistic drops from 0.12 over 0.05, which is 2.4, to 0.12 over 0.08, which is 1.5, and its two-sided p-value rises from about 1.6% to about 13%. The signal goes from looking significant to being indistinguishable from noise at the usual 5% level. The researcher has not done anything wrong in the arithmetic; the formula behind the ordinary standard error assumed something about the data that is not true here.

Same slope, two intervals: only the honest one crosses zerozero: no signalOrdinary SE 0.05, t 2.40.0220.218Robust SE 0.08, t 1.5-0.0370.277-0.10.00.10.20.3
The slope of 0.12 has a 95% interval of 0.022 to 0.218 with the ordinary standard error, which excludes zero, but -0.037 to 0.277 with the robust standard error, which includes it, so the signal is not significant once the uneven noise is allowed for.
Step 2Why do the errors grow in volatile months?

Because everything moves more then. In a calm month, the returns the signal fails to explain are small; in a crash or a rally, every stock swings, and the misses are large. That is heteroskedasticityResidual variance that changes across observations, for example being larger in volatile months than in calm ones.. Ordinary standard errors assume one noise level for every month, so they average the calm and volatile months together and understate how uncertain the slope is when the signal's extreme values coincide with the noisiest months. A teacher who grades the same essay ten times on calm days and once during a fire drill should not trust the fire drill grade as much as the others; the ordinary formula trusts them equally.

The noise is not constant: residuals fan out in volatile months-20-10+10+200calm months: small missesvolatile months: large missesred: months with an extreme signal value, which sit where the noise is largest0.5x1.0x1.5x2.0x2.5xMarket volatility that month, relative to normal; residual in bps
In this illustration the regression residuals are small in calm months and fan out in volatile ones, and the months with extreme signal values sit in the volatile, noisy region, which is the pattern that makes ordinary standard errors too small.
Step 3Which do you trust, and what would you do next?

Trust the robust one. When the robust standard error is noticeably larger than the ordinary one, the gap is evidence of heteroskedasticity that matters, and the robust number is the honest measure. White's formula weights each observation's contribution by its own squared residual, so the volatile months count for what they are. Then try to do better than just widen the interval: divide returns and signal by each month's volatility, a weighted regression, so that calm months, which carry cleaner information, get more weight. That can tighten the estimate legitimately. Monthly cross-sections also share common shocks, so clustering standard errors by month is the next check.

Close with the decision. At t of 1.5 the signal is not dead, but it has not earned capital. Report both standard errors, explain the gap, and ask for the weighted version and more history before anyone sizes it.

Where candidates lose it

The common loss is defending the 2.4 because it was computed correctly. The arithmetic is fine; the assumption of constant noise is not, and the interviewer wants you to say which assumption failed and why it matters here.

The second is thinking robust standard errors change the slope. They change only its uncertainty, and a candidate who reports a different coefficient after switching has confused the estimator with its error bars.

What the interviewer asks next

  • When would robust standard errors be smaller than ordinary ones?
  • How would you adjust for both heteroskedasticity and correlation across stocks in the same month?
  • The weighted regression gives a slope of 0.09 with t of 2.6. Do you believe it more than the original?

Company names and figures are illustrative.