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Is it possible to test the following? Assume you have two linear regression models, one regressing $Y_1$ on $X_1$ the other $Y_2$ on $X_2$. This gives error variance $\sigma_1^2$ and $\sigma_2^2$. From a sample all variables are available. I would like to evaluate $$H_0: \sigma_1^2=\sigma_2^2,$$ or one-sided $$H_0: \sigma_1^2 \le \sigma_2^2.$$

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Assuming the two error distributions are normal (which you can also run diagnostics on), you can conduct an F-test on the ratio of the two variances.

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