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Suppose I have the model

$y_i= a+ b_1X_{i}D_{1,i} + b_2X_{i}D_{2,i} + b_3X_{i}D_{3,i} + b_4D_{2,i} + b_5D_{3,i} + e_i$

where $D_{j,i}=1$ if observation $i$ belongs to group $j=1,2,3$. There are only three groups so that each observation must belong to one of the three groups.

Suppose I find $b_1$ is significant at $5\%$ level and that $b_2$ and $b_3$ are not significant even at $10\%$. This would mean that $X$ only affects $y$ for group $1$. However, I also find that $b_1$ is not statistically different from $b_2$ and $b_3$. In practice, $b_1=0.2$ and $b_2=b_3=0.1$.

If $b_1$ is significant and the $b_2$ and $b_3$ are not, it ought to mean that $b_1$ is larger than $b_2$ and $b_3$. Yet, I do not find a statistically significant difference between the coefficients. In this case, is it still fine to interpret the results as $X$ only affecting $y$ for group $1$?

Daniel Pinto
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  • Isn't there the risk of model misspecification here since you include the interaction terms but not the main effects? See this [question](https://stats.stackexchange.com/questions/11009/including-the-interaction-but-not-the-main-effects-in-a-model). In addition, whether a coefficient is statistically significant or not only provides some information on the uncertainty of the estimate; not the magnitude of the estimated effect. – horseoftheyear Jun 06 '20 at 12:11
  • Thanks for the reply. The main effect is there because I include all possible groups. I can also write $y_i= \tilde{\alpha} + \tilde{b}_1X_{i} + \tilde{b}_2X_{i}D_{2,i} + \tilde{b}_3X_{i}D_{3,i} + \tilde{b}_4D_{2,i} + \tilde{b}_5D_{3,i} + \epsilon_i$ . In this case $b_1= \tilde{b}_1$ but $b_2 = \tilde{b}_1+\tilde{b}_2$. The information is the same. – Daniel Pinto Jun 07 '20 at 13:39

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