How does one do a hypothesis test for elasticity?

Question

Given the regression output $$\widehat{\ln cons} = \underset{(0.6018)}{0.4054} + \underset{(0.0744)}{1.2739}\, \ln m - \underset{(0.1902)}{0.6666}\, \ln p_1 -\underset{(0.2645)}{1.6146}\, \ln p_2$$ where

• $\ln cons$ is the log of chocolate consumption,
• $\ln m$ is the log of income,
• $\ln p_1$ is the log of the price of chocolate, and
• $\ln p_2$ is the log of the price of sweets,

test whether chocolate is a luxury good.

Since $1.27 > 1$, it is logical to test whether $\beta_{\ln m}$ could be less than $1$. When I test elasticity, I base the null hypothesis on what is logical, as in this case if $\beta_{\ln m}$ is significantly greater than $1$, one shouldn't reject a (illogical) null hypothesis $\mathrm H_0: \beta_{\ln m} \geq 1$. So,

$\mathrm H_0: \beta_{\ln m} \leq 1$

$\mathrm H_1: \beta_{\ln m} > 1$

$\displaystyle t = \frac{1.2739-1}{0.0744}\approx 3.681$

Therefore, I reject the null hypothesis in favour of the alternative that chocolates are a luxury good.

Do you agree with the way I have set up this hypothesis test? If the estimate were less than 1, I would have stated $\mathrm H_0: \beta_{\ln m} \geq 1$ against the alternative $\mathrm H_1: \beta_{\ln m} < 1$.

Answer 1

Do you agree with the way I have set up this hypothesis test? If the estimate were less than 1, I would have stated $\mathrm H_0: \beta_{\ln m} \geq 1$ against the alternative $\mathrm H_1: \beta_{\ln m} < 1$.

Setting up null and alternative hypotheses is the first step in a t-test. You should not set them up after computing your estimates. In your specific example with income elasticity of demand, you set up the null and alternative correctly, meaning that you want the alternative hypothesis to be the statement about what you are trying to prove. You should use that null and alternative regardless of what the data then tells you.

P.S. You should report the one-sided p-value from your one-sided hypothesis test, not just the test statistic.