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Automated hiring systems are among the fastest-developing of all high-stakes AI systems. Among these are algorithmic personality tests that use insights from psychometric testing, and promise to surface personality traits indicative of future success based on job seekers' resumes or social media profiles. We interrogate the validity of such systems using stability of the outputs they produce, noting that reliability is a necessary, but not a sufficient, condition for validity. Our approach is to (a) develop a methodology for an external audit of stability of predictions made by algorithmic personality tests, and (b) instantiate this methodology in an audit of two systems, Humantic AI and Crystal. Crucially, rather than challenging or affirming the assumptions made in psychometric testing -- that personality is a meaningful and measurable construct, and that personality traits are indicative of future success on the job -- we frame our methodology around testing the underlying assumptions made by the vendors of the algorithmic personality tests themselves. Our main contribution is the development of a socio-technical framework for auditing the stability of algorithmic systems. This contribution is supplemented with an open-source software library that implements the technical components of the audit, and can be used to conduct similar stability audits of algorithmic systems. We instantiate our framework with the audit of two real-world personality prediction systems, namely Humantic AI and Crystal. The application of our audit framework demonstrates that both these systems show substantial instability with respect to key facets of measurement, and hence cannot be considered valid testing instruments.