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?


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