What is a statistical model

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Structured variable selection and estimation (2009, background): “Interested readers are also referred to McCullagh (2002) for a rigorous discussion about what criteria a sensible statistical model should obey.”
On Some Principles of Statistical Inference (2015, background): “McCullagh (2002) emphasises the importance of careful delineation of design, covariate and treatment variables as an essential part of the correct specification of a statistical model.”
A theory of statistical models for Monte Carlo integration (2003, background): “These are fully fledged statistical models in the sense of the definition given by McCullagh (2002), no more or no less artificial than any other statistical model.”