What Dropping a Regressor Does to the One You Keep

Implement two functions.

ols(X, y) - ordinary least squares with an intercept. Prepend a column of ones to X of shape (n, p) and return the (p + 1,) coefficient vector, intercept first. Use np.linalg.lstsq(..., rcond=None).

omitted_variable_bias(X, y, omit_idx) - for an X with exactly 2 columns, drop column omit_idx and return the tuple (full_kept, short, bias):

• full_kept - the coefficient on the kept regressor in the full two-regressor fit
• short - the coefficient on the kept regressor when the other is omitted
• bias - the predicted bias $\beta_{\text{omitted}} \cdot \delta$, where $\delta$ is the slope from regressing the omitted variable on the kept one

The omitted-variable bias identity says these satisfy short == full_kept + bias.

Hints

• np.column_stack([np.ones(len(X)), X]) builds the design matrix.
• ols is reused three times - full fit, short fit, and the auxiliary regression of omitted on kept.
• If the two regressors are uncorrelated, $\delta = 0$ and there is no bias.