The residual is where a growth model puts everything it cannot name, so the useful research programme of the last forty years has been to name pieces of it and pull them out.
Human capital came out first. Mankiw, Romer and Weil augmented the production function with a stock of skills accumulated by schooling, and found that the augmented model explains far more of the cross-country variation in income and predicts a convergence speed that matches the data. Practically, this means a country's education stock belongs in your growth build as an input rather than as a hope, and it means the schooling-to-skills conversion rate matters more than the enrolment ratio. India's enrolment gains over two decades were very large; its learning outcomes, measured by the annual education surveys, moved far less, and the growth arithmetic responds to the second.
How do institutions affect growth?
Institutions came out next and are harder to measure. Daron Acemoglu, Simon Johnson and James Robinson argued in 2001 that the institutions colonial powers established, extractive where settlers could not survive and inclusive where they could, still explain a large share of income differences today. Their identification strategy used settler mortality as an instrument, which is why the paper is famous and also why it is contested, and Robert Hall and Charles Jones made a related argument in 1999 under the name social infrastructure.
The mechanism in both is the same. Property rights, contract enforcement and constraints on expropriation determine whether it is rational to invest and to innovate, and therefore determine both s and A.
For an analyst that literature converts into three questions you can actually answer about a country before forecasting its growth. How long does it take to enforce a contract through the courts. What share of activity sits outside the tax and regulatory perimeter. And how often does the rule governing an industry change without notice. None of those numbers is precise, and all of them move the residual more than any plausible change in the saving rate.
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Common mistake: charging for weak institutions twice
The seam to keep clean is the one between a growth input and a discount-rate input. Weak institutions lower growth and raise the risk premium, and an analyst who reflects the same weakness in both a lower g and a higher country risk premium has charged for it twice. Pick the channel deliberately, say which one you used, and leave the other alone.