Omitted-variable bias: trading frequency and returns

Question

You regress each account's monthly return on its number of trades per month and get a positive coefficient, suggesting frequent trading pays. A colleague points out you omitted leverage (risk-taking), which in your sample is positively correlated with both trading frequency and returns.

Is your estimated coefficient on trading frequency biased upward or downward? State the reasoning.

Accepted Answer

The coefficient is biased upward. The omitted-variable-bias rule. If the true model is Y = \beta1 X + \beta2 Z + \varepsilon but you omit Z, the coefficient you estimate on X is approximately \hat\beta1 \approx \beta1 + \beta2 \, \delta, \qquad 	ext{where } \delta = \frac{\operatorname{Cov}(X, Z)}{\operatorname{Var}(X)} is the slope from regressing the omitted Z on the included X. The bias is \beta2 \delta, and its sign is the product of two signs: the effect of the omitted variable on the outcome, and the direction of its correlation with the included variable. Apply it. Here X is trading f…

Omitted-variable bias: trading frequency and returns

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