Is it meaningful to test equality of two coefficients, one significant, the other insignificant? My question is like this.
The regression specification is: 
$y=\delta_{1}D_{1}+\delta_{2}D_{2}+\delta_{3}D_{3}+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\epsilon$ 
where $D_{i}$ denotes dummy for each one of three categories, $x_{i}$ interaction between $D_{i}$ and independent variable $x$. 
I want to test $b_{1}=b_{2}=b_{3}$. $b_{i}$ is the estimated coefficient of $x_{i}$. After test, I will combine interaction terms for the categories not showing significantly different coefficients. 
Each category has different sample size. Category 1 have much more samples than category 2, and category 2 more than category 3.
Estimates show that $b_{1}$ and $b_{2}$ is significant (from zero), but $b_{3}$ not. 
I think the small sample size of category 3 could explain insignificance of $b_{3}$. 
My question: 
Is it meaningful (or right) to do joint test $b_{1}=b_{2}=b_{3}$ when $b_{3}$ is insignificant? 
If the test $b_{1}=b_{2}=b_{3}$ can not be rejected at e.g 5% level but $b_{1}$ and $b_{3}$ is significantly different, could I say no significant difference among $x_{1}$, $x_{2}$ and $x_{3}$? 
 A: Re: your first question "Is it meaningful (or right) to do joint test $b_1=b_2=b_3$ when $b_3$ is insignificant?"
The null hypotheses are different - in one case you're testing a heterogeneity of means and in another you're testing whether a particular coefficient is 0. The fact that two coefficients are not significantly different from 0 does not preclude them from being different from each other. For example, if $\hat{\beta}_{1} = -1$ and $\hat{\beta}_{2} = 1$ and ${\rm se}(\hat{\beta}_{1}) = {\rm se}(\hat{\beta}_{2})=1$. In that case, $H_0 : \beta_{1}=\beta_{2}$ would be rejected (using the normal approximation) at the 5% level but $H_0 : \beta_{1}=0$ and $H_0 : \beta_{2}=0$ both would not. 
Re: your second question - "If the test $b_1=b_2=b_3$ can not be rejected at e.g 5% level but $b_1$ and $b_3$ is significantly different, could I say no significant difference among $x_1$, $x_2$ and $x_3$?" - again the null hypotheses are different. In one case, you're testing a heterogeneity of means and in the second case you're testing for a pairwise difference. The two tests need not agree. This is a frequently discussed topic related to how ANOVA is different from pairwise tests and there are many threads on the subject. For example, see my answer here: In matlab, results from anova1 and multcompare disagree?
