This paper introduces a forward selection Fama-MacBeth (FS-FMB) procedure to identify higher-order terms and interactions of common linear factors that are priced in the cross-section of equity returns. Starting from six widely used factors (Fama-French 5 factors plus momentum), the method selects a small set of 7 higher-order factors (up to degree 3) from 57 candidates. The selected model substantially improves pricing performance, raising the adjusted cross-sectional R² from 31.2% (linear model) to 59%, and renders the intercept alpha insignificant. Moreover, it explains a large portion of the factor zoo: only 7 out of 148 zoo factors remain significant after controlling for the higher-order factors. The paper provides theoretical foundations for the FS-FMB estimator, including consistency and a debiasing procedure for inference. Out-of-sample tests confirm the model's robustness, and economic analysis via quantile regressions reveals that the higher-order factors capture non-linear exposure to macroeconomic tail risks, particularly financial intermediary distress, especially during recessions. Overall, the findings suggest that the pricing power of many zoo factors may stem from their exposure to higher-order manifestations of a few common factors.