Profile‐kernel versus backfitting in the partially linear models for longitudinal/clustered data

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Profile likelihood estimation of partially linear panel data models with fixed effects (2006, background): “A2 is assumed to simplify the proof and it is also assumed in Lin and Carroll (2001) and Hu et al. (2004).”
Semiparametric estimation in general repeated measures problems (2006, methodology): “However, Hu et al. (2004) also showed that in example 5 and equation (2) the use of the smoother that is defined in equation (5) leads to profiling and backfitting being asymptotically equivalent.”
Functional Data Analysis (2016, methodology): “A combined least squares method to estimate θ and β j simultaneously in a one step or proﬁle approach (Hu, Wang & Carroll 2004), where one estimates θ by proﬁling out the nonparametric components β j , is generally preferred over an alternative back-ﬁtting method.”