Conducting a joint test on signs of estimated coefficients after regression

[Gautam Sethi]

Dear all:
How do I conduct a joint test for testing the signs of two of the regression coefficients after running a regression?

If my model is Y = β₀ + β₁ X + β₂ Y + ...+ ε, I'd like to conduct the joint test (β₁ > 0) & (β₂ < 0).

Unfortunately, the test command does not allow inequalities. I'd love to hear your suggestions on how to conduct such a test.

[George Ford]

joint test the two are zero. look at the signs.

I suppose you could use the one-tailed test probability (1/2 the probability given). (look at ttest as an example.)

[George Ford]

https://www.stata.com/support/faqs/statistics/one-sided-tests-for-coefficients/#:~:text=If%20you%20wish%20to%20test,coefficient%20is%20equal%20to%20zero.&text=The%20Wald%20test%20given%20here,71%20denominator%20degrees%20of%20freedom.

[Gautam Sethi]

Thanks, George, for your response.

I'm not sure if I could run two t-tests, one for each coefficient, since the information I seek is about the joint distribution of b₁ and b₂, which are estimates of β₁ and β₂ respectively. Perhaps the picture below, which shows a confidence region for b₁ and b₂,will help?

Presumably, under the standard Gauss-Markov assumptions, the joint distribution of b₁ and b₂ is F-distributed. To do the computation myself, I need the conditional distribution of b₁ given b₂ (or the other way around) or the joint distribution itself. I was wondering if Stata has a built in command that yields the p-value for the joint "one-sided" test.

[Gautam Sethi]

Upon reflection, I think I might have answered my own question. Based on the figure below, if the one-sided p-value for b₁ is p₁ and the one-sided p-value for b₂ is p₂, then would the p-value for the joint test not simply be p₁ +p₂?

If this reasoning is correct, then the p-value I seek is merely the arithmetic mean of the p-values of b₁ and b₂, as reported by the regression output.