This paper presents finite-sample tests and confidence procedures for a multivariate linear regression system with cross-dependent possibly non-Gaussian disturbances, allowing for potentially missing explanatory variables. We consider a wide class of multivariate statis- tics based on the roots of a determinantal equation involving restricted and unrestricted least squares residuals. We characterize the exact distribution of these roots and identify important pivotal cases. General conditions for asymptotic validity are also given. To deal with missing explanatory variables, we introduce an identifiable ex-post parameter which embeds the impact of left-out variables, and show that usual tests remain valid from a conservative perspective. We derive simultaneous confidence bands for a vector (as well as linear combinations) of regression coefficients. The results are applied to asset pricing, with focus on risk-return analysis. We revisit the distinction between the alpha (risk-adjusted outperformance) and beta (risk component) of returns, from our ex-post parametrization perspective. Empirically, we analyze the risk-return profile of Catastrophe bond mutual funds.