# Sign of Estimate (Coefficient) of Interaction Terms

I create a linear regression model with interaction term in the model say: $$y=a_0+a_1x_1+a_2x_2+a_3x_1*x_2+e$$ where $x_1$ is continuous and $x_2$ is binary. Now, I have couple of questions:

1. If $x_1$ and $x_2$ both are significant and have positive coefficients, that means that if $x_1$ and $x_2$ increase, then $y$ increases. What can I say if $x_1*x_2$ is statistically insignificant? Does it mean that increase in. Does it mean that for $x_2=1$, the effect of $x_1$ on $y$ is not changed, i.e. increase in $x_1$ still leads to increase in $y$? What if the interaction is statistically significant?

2. What can I say if the interaction term is statistically insignificant and negative? Does it indicate that if $x_2=1$, increase in $x_1$ actually leads to decrease in $y$?

Thanks.

• "you discuss the interaction ... in terms of what happens to ... a1 and a2" in a model without an interaction. Indeed the interaction model changes the interpretation of a1 and a2, to the point it's probably better to give them different notation: $E_I(y | x_1, x_2) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \gamma x_1 \cdot x_2$ versus $E_M(y | x_1, x_2) = \alpha_0 + \alpha_1 x_1 + \alpha_2 x_2$ – AdamO May 14 '15 at 0:53