Consider the model Y = B_0 + B_1X_1+B_2X_2+e, e ~ N(0, σ^2) where the ‘xs are all continuous variables. You also have data on a qualitative variable denoted d1 for each observation, where d1 = 1 if the observation belongs to group 1, and zero otherwise.
1) Assume you want to set an empirical model to see the expected value of Y was different for members of group 1. What model, null, and alternative hypothesis would you need to estimate and test, assuming identical continuous variable values? Show your work and explain. (Hint: use dummy variable)
2) What model, null, and alternative hypothesis would you need to estimate and test to allow for differences in both the intercept and the slope coefficient on X2? Show your work and explain. (Hint: you need an interaction term)
1. To test whether group 1 has an effect or no, consider the following model.
Y = β0 + β1x1 + β2x2 + β3x3 + e.
All terms are same as the original model, x3 is a dummy variable where x3 = 1 if the observation falls into group 1 and x3 = 0 otherwise.
What want to see if Y is different if the observation is in group 1 (i.e. x3 = 1) or not in group 1 (i.e. x3 = 0). That is, we want to see if β3 is significantly not zero.
H0: β3 = 0 (i.e. being in group 1 or not has no effect)
Ha: β3 ≠ 0 ...
Evaluate continuous variables.