Marginal effect with an interactive dummy variable

Problem

You have estimated the following model:

\[y_i= β_0+ β_1x_i+ β_2x_i^2+β_3x_id_i+ε_i\]

where \(x_i\) is a continuous variable and \(d_i\) is a dummy variable. You have the estimates \(b_0=10\) , \(b_1=4\) , \(b_2=-0.5\) , \(b_3=-2\) .

1. Estimate the marginal effect of \(x\) on \(y\) for an observation where \(x=2\) belonging to the group where \(d_i=0\) .
2. Estimate the marginal effect of \(x\) on \(y\) for an observation where \(x=2\) belonging to the group where \(d_i=1\) .
3. Find the \(x\) -value which would make the estimated marginal effect of \(x\) for group \(d_i=0\) equal to 0.