In a partially linear regression model via Robinson (1988), what is the meaning of conditional expectations of the errors being equal to zer

2018-02-13 01:30:36

From the work of Robinson (1988), he defines a partially linear regression (PLR) model through the following:

$$

Y = D\theta_0 + g_0(X) + U

$$

$$

D = m_0(x) + V

$$

where $E[U\mid X,D] = 0$ and $E[V\mid X]=0$. Here, $Y$ indicates an outcome variable, $D$ a treatment variable, and $X$ a vector of controls or confounders. $U,V$ are assumed to be disturbances or errors.

I am wondering what it means to have $E[U\mid X,D] = 0$ and $E[V\mid X]=0$. The way I take it is that since both are equal to $0$, then:

$$

E[U\mid X,D] = E[V\mid X]

$$

However, it seems like an awfully complex way to describe this. Additionally, conditional expectations usually are a function of the terms we condition on. Hence, is there something to be gained here from it being $0$? Thanks.