Linear regression model: module 2

Question 1

How is the LRM formulated?
What does all of the components stand for?

What is is we try to do with the LRM?

Answer 1

Formula: y(i)= β(1) + (2)*x(i) + ε(i)

β(1) and beta 2 are unkown parameters that are intercept and slope
x(i) is the independent/explanatoyu vriable
y(i) is the dependent/ explained variable
ε is the error term. It is an unobserved random variable

LRMs mainpoints:
observe data x and y (the only variables that are observable)
Make LRM assumptions
Find ESTIMATES of the two betas!

Question 2

What is an important assumption about ε?

Hint: it’s relation to x

Answer 2

ε and x can not be correlated! In other words: the x-varible is exogenous if E(ε|x)=0
Note: in whole it is =0

Question 3

What result of E(y|x) do we get if E(ε|x)=0?

Answer 3

Then E(y(i)|x(i))= β(1) + β(2)*x(i)

Question 4

What formula can we write error term epsilon as?

hint: expected value

Answer 4

Since we know that E(y(i) | x(i))= β(1) + β(2)*x(i)
We can write the LRM as: y(i) = E(y(i)|x(i)) + ε(i)
Which means that we can write
ε= y(i) - E(y(i)|x(i))

Question 5

If we have LRM + exogeneity, what can we estimate the LRM values to?

Answer 5

We can then use the OLS trendline
We estimate beta(1) to b(1) and beta(2) to b(2)
We estimate ε to e (residuals)
We estimate E(y | x) to the fitted values

Question 6

When is an estimator unbiased and cosistent? (both x-bar and b1, b2))

Answer 6

x-bar is unbiased estimator of μ (mean) if E(x-bar)=μ
x-bar is consistent if x-bar goes closer to μ when n goes to infinity

b is unbiased if E(b)= β
consistent if b becomes equal to as n goes to infinity

Question 7

When is an error term homoscedastic?

Answer 7

If Var(εi | xi) = σ (sqrd)
All error terms have the same variance (conditional on xi)

Question 8

When is an error term heteroscedastic? (ingen matte)

Answer 8

When each error term has its own variance which is contitional to xi

Question 9

Waht are the Gauss-Markov assumptions?

Answer 9

We have an LRM with random variables where the explanatory variable (x) is exogenous and the error terms are homoscedastic

Question 10

What are our OLS estimators?

Answer 10

b1,b2, fitted values, e

Question 11

Waht is the Gauss-markov theorem?

Answer 11

We have LRM with r. variables
OLS estimates are linear
In the LRM under GM-assumptions, the OLS estiamtes will be BLUE

Question 12

What does efficiency mean?

Answer 12

The smaller the variance - the more efficient

Question 13

What does BLUE mean?

Answer 13

Best linear unbiased estimator