1.Consider a regression function of the form y = β0+ β1x + s. a. Formally state the homoskedasticity assumption (SLR5).
Consider a regression function of the form y = β0+ β1x + s.
a. Formally state the homoskedasticity assumption (SLR5).
b. What does this meanintuitively?
c. Draw some hypothetical data on an (x, y) scatterplot that would be consistent with this
d. On a second set of x and y axes, draw some hypothetical data that would exhibit heteroskedasticity, in violation of this
Formallystate the zero conditional mean assumption for a multiple linear regression (MLR 4). What might cause this assumption to be violated?
Supposeyou have data on the real (inflation-adjusted) incomes of men at age 40 (y) as well as the mens’ fathers’ real incomes at age 40 (x) (so each observation is a father-son pair of real incomes at age 40). Your estimated regression is yˆ = βˆ0 + βˆ1x.
a. Supposeyour estimated βˆ1 is equal to How would you interpret this?
b. Supposeyour estimated βˆ1 is equal to How would you interpret this?
c. Supposeyou wanted to allow βˆ1 to vary by race, with two racial How would you alter the above regression?
d. Supposeinstead there are R racial How would you alter the above regression?
Supposethe population regression function is y = β0 + β1x + s, where y is an indicator for whether or not a child eventually attends college and x is the child’s score on a 5th grade academic assessment (which ranges from 0 to 100, with mean µ and variance σ2). Assume the Gauss-Markov assumptions
a. Writethe expression for βˆ1 in the estimated regression of y on x.
b. Whatis the interpretation for βˆ1?
c. Supposethat, instead of using x in the regression, we use z, which is x standardized to have a mean of 0 and variance of 1. What is the expression for β1 now? How do we interpret its magnitude?
d. Showthat β1 = σβˆ1.
Supposethe population regression function is y = β0 + β1x1 + β2x2 + s, where β2 = 0, e., β2 has no linear relationship with y. Jorge suggests that you exclude x2 when you estimate your regression, as this will lead to better results. Torrance argues instead that it doesn’t matter whether you include x2 or not, as you will get the same results. Who is correct , and under what circumstances?
Suppose you wish to estimate a linear regression of Y on X1and X2. Write a model de- scribing each of the following scenarios, and fill in actual numbers which match the patterns described.
a. Y increases at an increasing rate with both X1and X2, and the coefficients on both continuous predictors can be interpreted as
b. Y increases with X1and X2, but the rate of change of Y with X1 is larger when X2 is larger.
c. Y increases at a constant rate with X1and X2, but X2 only matters for Y when Y is greater than 10,000.
Inthe potential outcomes model with a binary treatment D,
N AT E = AT E + E[Y 0 | D = 1] − E[Y 0 | D = 0] + (1 − P(D = 1))(AT T − AT U ).
a. Under what assumption on the relationship between D and (Y 1, Y 0) does N AT E=
AT E?
b. What does E[Y 0| D = 1] − E[Y 0 | D = 0] represent?
c. What is the formal definition ofAT T ?
d. What does AT T − AT U represent? Intuitively, why is thisimportant?
Below, I reiterate some of the formulas I presented in class and present some useful generalizations. Throughout the handout, let X, Y , and Z denote random variables while a and b denote arbitrary constants (numbers).