Underfitting and Model Misspecification

Underfitting happens when a model is too simple, too heavily regularized, or trained on too few relevant features to capture the actual structure in the data — a straight line fit to a curved relationship, or a model missing an important predictor entirely. The tell-tale sign is that error stays high on both the training data and new, unseen data: unlike overfitting, where training error looks great but test error is much worse, underfitting produces a model that's failing everywhere, because it genuinely hasn't learned enough about the relationship to fit even the data it was trained on well.

Model misspecification is the closely related, broader idea: the model's functional form is fundamentally wrong for the process generating the data, regardless of how much data or regularization is applied. Fitting a linear regression to a relationship that's actually strongly non-linear will underfit no matter how many more rows of training data you add, because more data can't fix a model that's structurally unable to represent the true relationship.

For example, trying to predict next-day returns using only a linear function of yesterday's return, when the real relationship is that returns depend on a nonlinear combination of volatility, momentum, and macro regime, will show a persistently low $R^2$ on both the training sample and any holdout period — adding more historical rows to the training data won't help, because the model's linear form simply cannot represent the true, richer relationship.