Partial Linear Model

A partially linear model requires the regression function to be a linear function of a subset of the variables and a nonparametric non-specified function of the rest of the variables. Suppose, for example, that one is interested in estimating the relationship between an outcome variable of interest y and a vector of variables (x, z). The economist is comfortable modelling the regression function as linear in x, but s hesitant in extending the linearity to z. One example, considered by Engle et al. (1986), is the effect of temperature on fuel consumption using a time series of cities. To do that, one can consider a regression of average fuel consumption in time t on average household characteristic and average temperature in time t. The analyst might be more comfortable with imposing linearity on the part of the regression function involving household characteristics but unwilling to require that fuel consumption varies linearly with temperature. This is natural since fuel consumption tends to be higher at extremes of the temperature scale, but lower at moderate temperatures. The regression function Engle et al. consider is: (1)<math display='block'> <mrow> <mi>y</mi><mo>=</mo><msup> <mi>x</mi> <mo>&#x2032;</mo> </msup> <mi>&#x03B2;</mi><mo>+</mo><mi>g</mi><mrow><mo>(</mo> <mi>z</mi> <mo>)</mo></mrow><mo>+</mo><mi>u</mi> </mrow> </math>$$y={x}^{\prime}\beta +g\left( z \right)+u$$ where x denotes a vector of household/city characteristics and z is temperature and u is a mean zero random variable such that is independent of (x, z). The function g(.) is unspecified except for smoothness assumptions. They term this the semiparametric regression model.