Week 3

Question 1

What is a population bivariate linear regression?

Answer 1

A tool that allows to decompose random variable Y into a linear function of another variable X and a regression residual.

Y= a + bX + e

[a + bX] is population regression line

Question 2

What is the best linear predictor of Y given X?

Answer 2

Population regression line

Question 3

Why is the population regression line the best linear predictor of Y?

Answer 3

It minimizes mean squared error. Minimizes e = Y - a - bX

Question 4

Minimise the mean squared error by calculus and show how to get (b) and (a).

Answer 4

Question 5

What is the relation with regression residuals and regressors?

Answer 5

Regression residuals are uncorrelated with regressors by construction. Since b is defined by the property that Cov(X, e) = 0

Question 6

What are a and b in the population bivariate regression?

Answer 6

a is the intercept and b is the slope of the the regression line.

Question 7

An economist has an economic model of earnings where education (X) has a positive causal effect on earnings (Y). What is (u) in Y = a + bX + u

Answer 7

(u) is any other determinant of earnings other than education.

Question 8

What are the 3 assumptions we make that (X) is not affected by (u).

Answer 8

1) E(u|X) = 0

Average error term for any given amount of X is zero

2) E(u|X) = E(u)

Average error term is same for any given amount of X

3) Cov(u, X)

Error term is uncorrelated with X

Question 9

What is zero conditional mean assumption?

Answer 9

Implies E(u) = 0 hence, E(u|X) = E(u), which also implies uncorrelatedness

Question 10

Show that b and a are unbiased estimators in population regression function.

Answer 10

Question 11

How to show omitted variable bias in both cases? compare when X and u and correlated and uncorrelated.

Answer 11

Question 12

How to describe the assumption Cov(X,u) = 0 makes sense using years of schooling and average ability?

Answer 12

If years of schooling were randomly assigned to individuals across a population. Average ability would be same across different years of schooling. In that case schooling and ability would be uncorrelated.

Question 13

In reality we do not have all data in population we use a sample. what are the formulas for the estimand (b) and estimator (b hat).

Answer 13

Question 14

Under which 2 assumptions is b hat a desirable estimator of b.

Answer 14

1) Large random sample

2) Large outliers in X and Y are unlikely

Question 15

What is an assumption of b hat that allows us to contract confidence intervals and conduct hypotheses tests.

Answer 15

b hat is asymptotically normally distributed

Question 16

What are the 3 assumptions that b hat is a desirable estimator for b in bivariate econometric model.

Answer 16

1) Large Random Sample

2) Large outliers in X and Y are unlikely

3) Zero conditional mean E(u|X) = 0