The regression specification is:
$y=\delta_{1}D_{1}+\delta_{2}D_{2}+\delta_{3}D_{3}+\beta_{1}D_{1} \cdot X+ \beta_{2}D_{2} \cdot X+\beta_{3}D_{3} \cdot X+\epsilon$
where $D_{i}$ denotes dummy for each one of three groups, $D_{i} \cdot X$ interaction between $D_{i}$ and independent variable $X$.
Each group has different sample size. Group 1 has much more samples than group 2, and group 2 more than group 3.
If I can incorporate some of these groups, I will remedy the problem of small sample size in group 2 and 3 to some extent.
So I want to test $\hat{\beta_{1}}=\hat{\beta_{2}}=\hat{\beta_{3}}$. $\hat{\beta_{i}}$ is the estimated coefficient of $D_{i} \cdot X$. After test, I will combine interaction terms for the groups not showing significantly different coefficients.
The questions are:
Should I use joint test $\hat{\beta_{1}}=\hat{\beta_{2}}=\hat{\beta_{3}}$? If this test cannot be rejected at e.g. 5% level, I will incorporate group 1, 2 and 3. Is that right?
If I should use joint test, since group 2 and 3 has very few samples, $\hat{\beta_{2}}$ and $\hat{\beta_{3}}$ may be insignificant even having wrong sign (I mean they are should be positive, but estimates are negative.) Is it right to use joint test $\hat{\beta_{1}}=\hat{\beta_{2}}=\hat{\beta_{3}}$ here?
Or should I use pairwise tests? If I use pairwise tests to find that $\hat{\beta_{1}}=\hat{\beta_{2}}$ and $\hat{\beta_{2}}=\hat{\beta_{3}}$ cannot be rejected but $\hat{\beta_{1}}=\hat{\beta_{3}}$ is rejected, how to incorporate these groups?
