You regress a stock's return on your signal and get a coefficient of 0.8. Your signal has correlation 0.6 with market beta exposure, and beta exposure carries a true coefficient of 0.5 in the correct two-regressor model. Both regressors are standardised. Compute the omitted variable bias and the signal's true coefficient.

You regress a stock's return on your signal and get a coefficient of 0.8. Your signal has correlation 0.6 with market beta exposure, and beta exposure carries a true coefficient of 0.5 in the correct two-regressor model. Both regressors are standardised. Compute the omitted variable bias and the signal's true coefficient.

Approach: Write the short and long regressions, then use the identity relating the short coefficient to the long one through the auxiliary regression of the omitted variable on the included one.

0.5. Write the correct model as y = b1*s + b2*x + u with b2 = 0.5, and the short regression as y = c1*s + e. The omitted variable formula gives c1 = b1 + b2*delta, where delta is the slope from regressing the omitted x on the included s; with both regressors standardised, delta equals their correlation, 0.6. The bias is therefore 0.5*0.6 = 0.3, and since the short regression reported c1 = 0.8, the signal's own coefficient is b1 = 0.8 - 0.3 = 0.5. The omitted regressor is a confounder, and the sign of the bias is the product of two signs, so a positively priced factor that your signal loads on positively always inflates the estimate. Here 0.3 of the apparent 0.8, or 37.5% of the measured edge, is market beta you were being paid for taking risk rather than for the signal.

Follow-up: The omitted factor is measured with error, so you control for a noisy proxy. Does that remove the bias, and in which direction does the residual bias point?

Key concepts: omitted variable bias, confounding, standardised regressors, correlation.