Suppose the observation $(X_1, Y_1), \ldots, (X_n, Y_n)$ satisfies the following semi-parametric model $$Y_t = m(X_t, \alpha) + \sigma(X_t, \beta) U_t,$$ where $U_t$ is independent with $X_t$ with zero mean and unit variance, and $\alpha$, $\beta$ are finite-dimensional parameters with unknown link function $m(\cdot, \cdot)$ and $\sigma(\cdot, \cdot)$. Therefore, $(X_t, Y_t)$ follows a semi-parametric model. We have ample theories about the semi-parametric model, see Andrews (1994) and Newey (1994) for instance.
Now, I want to estimate the residual $U_t$, a natural thought is to estimate this semi-parametric model. Say we get the estimated parametric parts $\widehat{\alpha}$ and $\widehat{\beta}$, and estimated nonparametric parts $\widehat{m}$ and $\widehat{\sigma}^2$. So the estimated residual are $$\widehat{U}_t = \frac{Y_t - \widehat{m}(X_t, \widehat{\alpha})}{\widehat{\sigma}(X_t, \widehat{\beta})} \in \mathbb{R}$$. So the difference between the estimated residuals and the real residuals depends on $$\widehat{m}(X_t, \widehat{\alpha}) - m (X_t, \alpha) \quad \text{ and } \quad \widehat{\sigma}^2(X_t, \widehat{\beta}) - \sigma^2(X_t, \beta).$$ I want to know the general representations of the two differences above. I try my best to find some literature regarding it, but it is proved to be a fail. Does anyone know some papers about this semiparametric problem? Thanks so much!
