<p>The <span class="math-container">$NR^2$</span> test is a particular case of an LM test. You can refer to Davidson, R., &amp; MacKinnon, J. G. (1993). Estimation and inference in econometrics, p. 90 for the exact derivation.</p>

CC BY-SA 4.0
2022-01-23T22:14:57.550Z
2022-01-23T22:14:57.550Z
LudwigNagasena
<p>After <a href="https://en.wikipedia.org/wiki/Ordinary_least_squares" rel="nofollow noreferrer">OLS</a>, one can test the null hypothesis that all coefficients are 0 by calculating <span class="math-container">$N\cdot R^2$</span>, which is distributed <span class="math-container">$\chi^2_{k}$</span> where <span class="math-container">$k$</span> is the number of <span class="math-container">$\beta$</span> (excluding the constant).</p>
<p>I have seen this called a <a href="https://en.wikipedia.org/wiki/Lagrange_multiplier" rel="nofollow noreferrer">Lagrange multiplier</a> (LM) test. Why? I'm familiar with the LM test in <a href="https://en.wikipedia.org/wiki/Maximum_likelihood_estimation" rel="nofollow noreferrer">MLE</a>, but I am struggling to see the mathematical connection between these - although it must exist.</p>

<econometrics><hypothesis-testing>
Why is N*R^2 called an "LM test"?
Michael Gmeiner
# Why is N*R^2 called an "LM test"?

(Asked by: username_0 on 2022-01-23)

After [OLS](https://en.wikipedia.org/wiki/Ordinary_least_squares), 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](https://en.wikipedia.org/wiki/Lagrange_multiplier) (LM) test. Why? I'm familiar with the LM test in [MLE](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation), but I am struggling to see the mathematical connection between these - although it must exist.

**Tags:** econometrics, hypothesis-testing


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## 1 answers

### Answer by username_1 on 2022-01-23. Score: 2 | Accepted answer

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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