A partial solution to the dimensionality problem was also explored in Speckman (1988) and Robinson (1988). They considered the case where xi = (xi1, xi2) and m(x) = m(xi1, x12) = m1(xi1) + m2(xi2), but the researcher knows the functional form of m1(xi1) as x! ip, where xn is a dimensional and xt2 is q2 dimensional with no common elements. That is, they considered the partial linear models or semiparametric (SP) model of the following form
yi = x! ip + m(xi2) + ui,
where E(ui | x,) = 0. For example, in the earning functions log earning (y) may be an unknown function of age (xi2) but a linear function of education (xn). The estimation of P can be carried out by first eliminating m(xi2), and then using the following procedure of taking conditional expectations so that
E(у, | x!7) = E(x, i | x,2)P + m(xi2)
and y, – E(y, | xi2) = (x,1 – E(xn | xi2))P + u, or y* = x*Lp + u,. This can then be
estimated by the LS procedure as
X x*i y*.
For the implementation we need to know y* and x*, which can be obtained by estimating E( y,|xi2) and E(x,1 |xi2) using the LLS procedures in Section 2. After obtaining SSP one can proceed for the estimation of m(xi2) by writing
Уi – x, i Ssp = y * * = m(xa) + Щ = E( y* *| xa) + u,
and then again doing the LLS regression of y** on xi2.
While the estimator of m(xi2) achieves the nonparametric slow rate of convergence of (nhq/2)1/2, the remarkable point is that the 0SP achieves the parametric rate of convergence of n1/2. It is in this respect that this procedure is better than the univariate nonparametric convergence rates of generalized additive models above. However, in the partial linear model we need to be sure of the linearity of xnp. If m(xn, P) is a nonlinear function of xn and p, then it is not clear how one proceeds with the above estimation technique, though it seems that a nonlinear semiparametric LS procedure might be helpful. For the empirical applications of the above models, see Engle et al. (1986) for an electricity expenditure estimation, Anglin and Gencay (1996) for a hedonic price estimation of Canadian housing.
There are various extensions of the idea of the partial linear models. Fan and Li (1997) combine the partial linearity with the generalized additive models to consider
Vi = x aP + m2(xa) + m3(xs) + … + тц(хщ) + щ
and suggest the 4n convergent estimate of P and (nh)1/2 convergent estimates of ms(xis) for s = 2,…, q. This improves upon the (nhq)1/2 state of convergence of m(xi2) above.
The partially linear models have been extensively studied in the labor econometric literature on the selection models where m(xi2) = m(xi 2S) is an unknown function of single index xi2S and xi 2 and xi1 may have some common variables. For details on this literature, see Pagan and Ullah (1999, chs. 7-9). For the maximum likelihood estimation of the purely parametric model, yi = xiP + u, partial linear, and selection models without the assumption about the form of the density of ui, see the excellent work of Ai (1997). The estimation of panel data based partially linear models has been developed in Ullah and Roy (1998), Li and Ullah (1998) and Li and Stengos (1996), among others.