# With omitted variables is OLS estimator still the best linear predictor?

Suppose the true model is

$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon$$

where $$x_1$$ and $$x_2$$ are correlated and $$\epsilon$$ is white noise. I omit variable $$x_2$$ and apply OLS to estimate

$$y = \beta_0 + \beta_1 x_1 + \epsilon$$

Now my predictor $$x_1$$ is correlated with the noise ($$\epsilon + \beta_2x_2$$), thus the Gauss-Markov assumptions are not satisfied so I can't say the OLS estimator is the Best Linear Unbiased Estimator.

But what about predicting $$y$$ given only $$x_1$$? Is OLS still optimal in some sense?