Bias of a model that nests the DGP

Question

Consider model 1 and model 2 where the former is a special case of the latter. E.g. model 1 is $y=\beta_0+\beta_1 x+u$ while model 2 is $y=\gamma_0+\gamma_1 x+\gamma_2 x^2+v$. Suppose model 1 is the true data generating process. What is the bias (as in bias-variance decomposition) of model 2?

Answer 1

The bias of model 2 at a particular point $x_0$ is the difference between the expectation (taken over the distribution of the error term) of the fitted value and the deterministic part of $y_0$: $$ \text{Bias}(x_0) = (\mathbb{E}[\hat\gamma_0+\hat\gamma_1 x_0+\hat\gamma_2 x_0^2]) - (\beta_0+\beta_1 x_0). $$ It will be zero if the estimators of $\gamma$s are all unbiased, yielding $\mathbb{E}(\hat\gamma_0,\hat\gamma_1,\hat\gamma_2)=(\beta_0,\beta_1,0)$. This would be the case e.g. for OLS estimators under a classical set of assumptions. This would generally not be the case e.g. for LASSO or ridge estimators -- though it is technically possible that $x_0$ happens to be a point where the biases of $\hat\gamma_1^{LASSO}x_0$ and $\hat\gamma_2^{LASSO}x_0^2$ (or ridge) cancel out.
A closely related thread is "Bias-variance trade-off in case of biased estimators: is the bias zero?".