Multiple regression problems (restricted regression, dummy variables)

Q1.

Model 1: $Y={X}_{1}{\beta}_{1}+\epsilon $

Model 2: $Y={X}_{1}{\beta}_{1}+{X}_{2}{\beta}_{2}+\epsilon $

(a) Suppose that Model 1 is true. If we estimates OLS estrimator ${b}_{1}$ for ${\beta}_{1}$ in Model 2, what will happen to the size and power properties of the test?

(b) Suppose that Model 2 is true. If we estimates OLS estrimator ${b}_{1}$ for ${\beta}_{1}$ in Model 1, what will happen to the size and power properties of the test?

-> Here is my guess.

(a) ${b}_{1}$ is unbiased, inefficient estimator. (I calculated it using formula for "inclusion of irrelevant variable" and ${b}_{1}=({X}_{1}^{\prime}{M}_{2}{X}_{1}{)}^{-1}{X}_{1}^{\prime}{M}_{2}Y$ where ${M}_{2}$ is symmetric and idempotent matrix) Inefficient means that it has larger variance thus size increases and power increases too.

(b) ${b}_{1}$ is biased, efficient estimator. (I use formular for "exclusion of relevant variable" and ${b}_{1}=({X}_{1}^{\prime}{X}_{1}{)}^{-1}{X}_{1}^{\prime}Y$) Um... I stuck here. What should I say using that information?

Q2.

Let $Q$ and $P$ be the quantity and price. Relation between them is different across reions of east, west, south and north, and as well, for different 4 seasons. Construct a model.

-> Actually, I don't know well about dummy variables. So any please solve this problem to help me.