The idea of combining statistical and economic models dates to mid-20th century and work of the Cowles Commission.
For example, a regression is often called a reduced-form equation even when no standard economic model would generate it as the reduced form relationship between variables.
A structural model often involves sequential decision-making under uncertainty or strategic environments where beliefs about other agents' actions matter.
Parameters of such models are estimated not with regression analysis but non-linear techniques such as generalized method of moments, maximum likelihood, and indirect inference.
The reduced-form of such models may result in a regression relationship but often only for special or trivial cases of the structural parameters.