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 $\mathrm H_1: \beta_{\ln m} < 1$.