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After OLS, one can test the null hypothesis that all coefficients are 0 by calculating $N\cdot R^2$, which is distributed $\chi^2_{k}$ where $k$ is the number of $\beta$ (excluding the constant).

I have seen this called a Lagrange multiplier (LM) test. Why? I'm familiar with the LM test in MLE, but I am struggling to see the mathematical connection between these - although it must exist.

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The $NR^2$ test is a particular case of an LM test. You can refer to Davidson, R., & MacKinnon, J. G. (1993). Estimation and inference in econometrics, p. 90 for the exact derivation.

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