What is doing the work in empirical growth models?

Most empirical growth models are not theoretically complicated. For example, one may be based on the assumption that output is

Output = A * Capital ^ a * Labour ^ (1-a)

or, dividing by labour,

Output per person = A * Capital per person ^ a

where A and a are constants. If one takes logs in the equation, then subtracts the same equation in the previous time period, one gets

ln output p.p.(t) - ln output p.p. (t-1) = a*(ln capital p.p. (t) - ln capital p.p. (t-1))

The left hand side is approximately equal to (output(t)- output(t-1))/(output(t)), which can be shown by using the Taylor expansion for ln (1+{output(t)/output(t-1) - 1}). The term is thus approximately equal to the growth in output, and we have a model

growth in output p.p. = a*growth in capital p.p.

The model has ignored many aspects of the economy, such as alternative production functions, differences across countries, aggregate demand, changes across time, other influences on growth, and influences of growth on changes in capital.

Yet the empirical testing of the model has been reasonably successful. One of the reasons is that the empirical testing does much of the clean-up work for the model. For example, if we test

growth in output p.p. =
a*growth in capital p.p. + country specific constant + time specific constant + error

the error will often be robustly estimated, so handling wide fluctuations in the quality of the fit and the direction of the model's bias. The country and time specific constants capture factors which may dwarf the explanatory power of the basic model. The estimation will frequently employ methods to handle endogeneity of variables, which allows for the possible omission of other relations between the variables in the model.

In short, if a growth model is assessed to be an empirical success, it is usually the model and the estimation method which are the successes.