Adding Noise Cannot Lower It

An OLS fit on 200 training rows has $\text{TSS} = \sum_i (y_i - \bar y)^2 = 800$ and $\text{RSS} = \sum_i (y_i - \hat y_i)^2 = 120$ , where $R^2 = 1 - \text{RSS}/\text{TSS}$ .

You then append one extra column of np.random.randn(200) to X and refit on the same 200 rows. What is the training $R^2$ before and after?

0.15 before, since $R^2$ is the ratio $\text{RSS}/\text{TSS}$ of error left in the model.

0.85 before; at least 0.85 after, and almost surely a little higher, since the old fit is still reachable with a zero coefficient on the new column.

0.85 before and 0.85 after, because least squares gives a pure-noise column a coefficient of exactly zero.

0.85 before and lower after, because an irrelevant feature adds variance and degrades the fit.