Yi = fai Xfa2 Xfa3 ui
CHAPTER FOURTEEN: NONLINEAR REGRESSION MODELS 565
where a = ln fa. This model too is linear in the parameters. But now consider the following version of the C-D function:
As we just noted, C-D versions (14.1.2a) and (14.1.3a) are intrinsically linear (in the parameter) regression models, but there is no way to transform (14.1.4) so that the transformed model can be made linear in the parameters.2 Therefore, (14.1.4) is intrinsically a nonlinear regression model.
Another well-known but intrinsically nonlinear function is the constant elasticity of substitution (CES) production function of which the Cobb-Douglas production is a special case. The CES production takes the following form:
where Y = output, K = capital input, L = labor input, A = scale parameter, 5 = distribution parameter (0 < 5 < 1), and fa = substitution parameter (fa > — 1).3 No matter in what form you enter the stochastic error term ui in this production function, there is no way to make it a linear (in parameter) regression model. It is intrinsically a nonlinear regression model.
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