Factor models based on Arbitrage Pricing Theory (APT) characterize key parameters jointly and nonlinearly, which complicates identification. We propose simultaneous inference methods which preserve equilibrium relations between all model parameters including ex-post sample-dependent ones, without assuming identification. Confidence sets based on inverting joint tests are derived, and tractable analytical solutions are supplied. These allow one to assess whether traded and nontraded factors are priced risk-drivers, and to take account of cross-sectional intercepts. A formal test for traded factor assumptions is proposed. Simulation and empirical analyses are conducted with Fama-French factors. Simulation results underscore the information content of cross-sectional intercept and traded factor restrictions. Three empirical results are especially noteworthy: (1) the Fama-French three factors are priced before 1970; thereafter, we find no evidence favoring any factor relative to the market; (2) heterogeneity is not sufficient to distinguish priced momentum from profitability or investment risk; (3) after the 1970s, factors are rejected or appear to be weak, depending on intercept restrictions or test portfolios.