There are certain established ways to look at the problem for econometric purposes. Many studies pivot on how technology is diffused. The “Big-push industrialization” checks on jumps in the level of division of labor and related size of input-output relationships. You may choose big push from the market point of view examplified in Hong Kong and Taiwan, or from the central panning point of view as in China.
K. Murphy, A Shleifer, and R. Vishny (1989) “Industrialization and the Big Push,” J. of Political Economy, 97, pp. 1003-26 is one likely place to start. They show a general equilibrium framework to integrate the modern sectors with economies of scales, and the traditional sectors with constant returns to scale.
Paul Krugman and A. Venables (1995) “Globalization and the Inequality of Nations,” Quarterly Journal of Economics,” 110, pp. 857-880 looked for evidence that link division of labor and technical progress.
What 'upgrading' means to you in this context? It is related to diffusion of innovation the function to be estimated is typically non-linear. The logistic equation is often appropriate. But please clarify the question
Industrial upgrading can be defined as “the possibility for (developing countries) producers to move up the value chain, either by shifting to more rewarding functional positions or by making products that have more value added invested in them and that can provide better returns to producers” (Gibbon and Ponte, 2005).
Industrial upgrading is closely related to global production networks and global value chains.
Before looking into the empirics, you may like to give a look to endogenous growth models with quality changes, innovation, etc, such as in Chpaters 6-7-8 fo Barro and Sala-i-Martin. The empirics is mostly a matter of what data you have. As mentioned in my previous answer, I think S-shaped functions are often helpful.
I think you can use a production function "augmented", considering among the usual independent variables (capital and labor), also the R&D capital constituted by the R&D expenditures accumulated in the previous years with a suitable (well-timed) lag.
See on the topic an old paper of Goto e Suzuki in The Review of Economics and Statistics about which, sorry, I don't remember the year (1976 or 1986).
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