Q1.
Model 1: $Y=X_1\beta_1+\varepsilon$
Model 2: $Y=X_1\beta_1+X_2\beta_2+\varepsilon$
(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 $b_1=(X_1'M_2X_1)^{-1}X_1'M_2Y$ 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 $b_1=(X_1'X_1)^{-1}X_1'Y$) 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.
For Q1-b), take that formula $b_1 = (X_1'X_1)^-1(X_1'Y) $ and now plug in the true value for Y using model 2 and look what happens in the limit. You should end up with something like:
$E[b1]=\beta_1 + \beta_2 * b_{12} $ where $ b_{12} $ is the coefficient a regression of $X_2$ on $X_1 $ (Note, I'm taking that formula from memory, so it might not be exactly correct, but it's the general idea). That will give you cases of bias.
Q2:
I don't know how advanced the class is, but usually in cases of quantity and price, we're worried about endogeneity and simultaneity. However, it seems like in this case that is not the main point of the question. If we wanted the relation between quantity and price, one option is to simply regress quantity on price. $Q=\beta_1P + \epsilon $. However, here you're told that relationship differs across regions. One way to think about this model is as follows:
Suppose there were only 2 regions North and South, and I showed you a model:
$Q=\beta_1*P+\beta_{2}(P*N)+\epsilon $ where N is a dummy variable equal to 1 in the north. In this case, $\beta_1$ could be interpreted as the effect of price on quantity when N=0, i.e. in the south and $\beta_1+\beta_2$ is the effect of price on quantity when N=1; i.e. in the North. The case with 4 regions is similarly defined.