Multiple linear regression linear relationship or not

Question

How should a multiple linear regression be interpreted that has statistically significant predictors, but an R Square value of $0.004$? Does that mean that there is a significant linear relationship (because statistically significant predictors), even though there is close to no linear relationship ($R^2$ of $0.004$ indicates close to no linear relationship).

Answer 1

Recall that $R^2 = 1 - \frac{\hat{\sigma}^2_{\epsilon|X}}{\hat{\sigma^2}_Y}$, namely if you have significant linear relationship between $y$ and your $(x_1,..,x_p)$, where the variance of $x_k$ is much smaller than that of $\epsilon$, (i.e., $\sigma^2_{x_k} < < \sigma^2_{\epsilon}$), you can get significant model with very low $R^2$. Take for example, $$ y_i = \beta_0 + x_{1i} + x_{2i} + \epsilon_i, $$ where $n=10,000$ and $\sigma^2_{\epsilon} = 200^2$ where $\sigma^2_{x_k} = 2$. In this case you have enough data points to get significance (convergence), however most of the conditional variability of $y$ is still "unexplained".