Exploiting results from the literature on non-parametric identification, we make three methodological contributions to the empirical literature estimating the matching function, commonly used to map unemployment and vacancies into hires. First, we show how to non-parametrically identify the matching function. Second, we estimate the matching function allowing for unobserved matching efficacy, without imposing the usual independence assumption between matching efficiency and search on either side of the labor market. Third, we allow for multiple types of jobseekers and consider an "augmented" Beveridge curve that includes them. Our estimated elasticity of hires with respect to vacancies is procyclical and varies between 0.15 and 0.3. This is substantially lower than common estimates suggesting that a significant bias stems from the commonly-used independence assumption. Moreover, variation in match efficiency accounts for much of the decline in hires during the Great Recession.