Lecture 6

Question 1

What is assumption 1 for Multiple Linear Regression (MLR1)

Answer 1

The (true) population model is a linear function of the explanatory variables
𝑦=𝛽0+𝛽1𝑥1+𝛽2𝑥2+𝛽3𝑥3+…+𝛽𝑘𝑥𝑘+𝑢

Question 2

Assumption 2 for MLR (MLR2)

Answer 2

The sample of size 𝑛, {(𝑦𝑖, 𝑥𝑖1,𝑥𝑖2,𝑥𝑖3,…, 𝑥𝑖𝑘): 𝑖=1, 2, …, 𝑛}, taken from the population and used to generate the estimates, is random and 𝑛>𝑘+1

Question 3

Assumption MLR3- No perfect collinearity

Answer 3

• There is variation in all the variables
• There are no exact linear relationships among the independent variables

Question 4

MLR4- Zero conditional mean assumption

Answer 4

The disturbance term, 𝑢, has an expected value of zero given any values of the independent variables
𝐸(𝑢|𝑥_1,𝑥_2,𝑥_3,…, 𝑥_𝑘 )=0

Question 5

Why is the zero conditional mean assumption often not valid

Answer 5

Because factors affecting Y and correlated with x have been omitted from the model

Question 6

Why is zero conditional mean assumption more likely to be valid in the MLR model

Answer 6

Because more variable are explicitly included in the model

Question 7

If two variables are perfectly collinear, what is the solution for MLR

Answer 7

Exclude one of the explanatory variables from the model

Question 8

Assumption MLR5: Homoscedasticity

Answer 8

The errors have the same variance irrespective of the explanatory variables

𝑉𝑎𝑟(𝑢|𝒙)=𝜎^2

Question 9

Under the 5 assumptions, what are the variances of the OLS estimators (𝛽̂_1, 𝛽̂_2, 𝛽̂_3,…, 𝛽̂_𝑘)

Answer 9

Where 𝑅𝑗^2 is a measure of the correlation between 𝑥𝑗 and all of the other explanatory variables, i.e., all the other 𝑥s,

Question 10

How does the variance of the error, sample size and explanatory variables (x) effect the variances of the OLS estimators

Answer 10

-larger the larger 𝜎^2, the variance of the error (disturbance) terms

• smaller the larger 𝑛, the size of the sample used in the estimation
• smaller the greater the variance in the explanatory variables 𝒙

Question 11

How does the correlation between the explanatory variables (𝑅𝑗^2 ) impact the variances of the OLS estimators

Answer 11

larger the greater the correlation between the explanatory variables 𝒙

Question 12

In the model
𝑦=𝛽0+𝛽1𝑥1+𝛽2𝑥2+𝑢, how does a greater correlation between x1 and x2 impact the 𝑉𝑎𝑟(𝛽̂1 )

Answer 12

• The greater correlation between 𝑥1 and 𝑥2 the closer the 𝑅1^2 gets to its maximum value of 1
• The closer 𝑅1^2 gets to 1, the closer 𝑉𝑎𝑟(𝛽̂1 ) gets to ∞

Question 13

Which assumption is violated if 𝑅1^2=1

Answer 13

MLR3 would be violated as that means x1 and x2 are perfectly collinear and the assumption is that there is no perfect collinearity

Question 14

Diagram showing how multicollinearity increases the variance of the estimators

Answer 14

Question 15

How much of a problem of multicollinearity

Answer 15

• It does not cause bias in the estimators
• It only seriously inflates 𝑉𝑎𝑟(𝛽̂𝑗 ) when the correlation very high
  -It is tempting to remove one (or more) of the explanatory variables, but this will lead to bias in the estimators
  -Increasing the sample size, 𝑛, helps

Question 16

What is used, like in the SLR, as an estimator for 𝜎^2

Answer 16

the variance of the residuals, 𝜎̂^2=𝑉𝑎𝑟(𝑢̂𝑖 )

Question 17

What are the measures of the dispersion of the OLS estimators 𝛽̂1 to 𝛽̂𝑘, in the MLR

Answer 17

Question 18

What is the standard error of the regression

Answer 18

𝜎̂=√(𝜎̂^2 )

Question 19

What are the standard errors of the OLS estimators 𝛽̂1 to 𝛽̂𝑘, in the MLR

Answer 19

Question 20

Acronym for the Gauss-Markov Theorem (BLUE)

Answer 20

B est (smallest variance)
L inear (linear function of the data)
U nbiased (𝐸(𝛽̂𝑗 )=𝛽𝑗)
E stimators

Question 21

As long as the Gauss Markov assumptions are valid, is there any better estimator

Answer 21

NO