Week 3: Chapter 5&9

Question 1

GM assumptions 1-5, what’s the sample property

Answer 1

finite-sample property
- Holds for any sample size n, so long as n > k+1

Question 2

CLM assumptions 1-6

Answer 2

Exact sampling property
- Violation of MLR 6 invalidates our inference

Question 3

Two implications of a large sample (as n approaches infinity)

Answer 3

• Bj has an approximately normal distribution as n n approaches infinity)
• t and F statistics have approximately t and F distributions when as n approaches infinity

Question 4

Unbiased VS Consistency

Answer 4

Unbiasedness:
- On average B^ equals B
- The midpoint of the distribution of B^ is B
- Nothing to do with spread or distribution of B^

Consistency:
- As we add more observations, the distribution of B^ gets more tightly distributed around B
- Tells you something about the speed and distribution of B^

Question 5

Consistency definition

Answer 5

B^j is consistent when Var(B^j) and E(B^j) become more tightly distributed around Bj, until they collapse into a single value, as n tends to infinity

Question 6

Consistency formula P(Wn…)

Answer 6

notes

Question 7

Show how consistency follows from unbiasedness

Answer 7

Notes

Question 8

What happens when Cov(u,x) / Var ( x) = 0?

Answer 8

Plim B^1 = B1

Question 9

MLR4’

Answer 9

E(u) = 0 and Cov(xi,u) = 0.
- Zero Mean and Zero Correlation assumption
Requires that each xj is uncorrelated with u and E(u)
- Still consistent

Question 10

What does MLR1-4 Imply

Answer 10

OLS estimators are unbiased and consistent

Question 11

What does MLR1-4’ Imply

Answer 11

OLS estimators are consistent

Question 12

What does RESET test stand for and what does it test for?

Answer 12

Regression Estimation Specification Error Test
- Tests for non-linearities within the model
- RESET tests for functional form misspecification brought about by the exclusion of higher order polynomials of our x’s

Question 13

RESET equation and why there is no y^4

Answer 13

check notes for the equation.
- No y^4 as it uses up too many dofs

Question 14

What is the null of RESET test?
What does it mean when we fail to reject it?

Answer 14

H0: δ1 = δ2 = 0
If we fail to reject, means the original model was correct

Question 15

What are the drawbacks of the RESET test?

Answer 15

A drawback with RESET is that it provides no real direction on how to proceed if the model is rejected. THUS IT IS JUST A FUNCTIONAL FORM TEST

Question 16

Mizon-Richard Test

Answer 16

• focuses on log transformations
• y = g0 + g1x1 + g2x2 + g3log(x1) + g4log(x2) + u (g = a number symbol)

Question 17

Null and alternative hypothesis for MR test

Answer 17

H0: g3 = 0, g4 = 0 (original equation correctly specified)
H1: g1 = 0, g2 = 0

Question 18

David-MacKinnon Test

Answer 18

States that the fitted values from one model should be insignificant when added to another model, yˇ and y^ (fitted values)
1. y = b0 + b1x1 + b2x2 + d1yˇ + error
2. y = b0 + b1log(x1) + b2log(x2) + q1yˆ + error

H0 : d1 = 0 and H0 : q1 = 0

Question 19

How do you know if its a good proxy variable?

Answer 19

The closer Cor(proxy, variable) is to 1, the better the proxy

Question 20

The proxy variable final model

Answer 20

y = alpha0 + b1x1 + b2x2 + alpha3x3 + e

Question 21

What can we use instead of a proxy variable?

Answer 21

Lagged dependent variable
- accounts for historical factors that cause current differences

Question 22

Examples of measurement errors

Answer 22

Self-reported income, weight etc.
Always have errors, people misinterpret the information

Question 23

Proxy VS Measurement Error

Answer 23

• In the proxy case, the omitted variable is important to the extent to which it affects our other independent variables
• In the measurement error case, the mismeasurement in the independent variable is the issue

Question 24

Measurement error in the dependent variable:
- Model
- Implications

Answer 24

y* = 𝛽0 + 𝛽1x1 + . . . + 𝛽kxk + u
y = b0 + b1x1 + . . . + bkxk + v
V: e0 = y - y* (measurement error)

GM assumptions still hold, still consistent

Question 25

Measurement error in explanatory variables:

Answer 25

y = 𝛽0 + 𝛽1x1 + u
- CANNOT observe x1, have to observe x1
e1=x1- x1* and assume E(e1)=0, E(y|x1,x1)=E(y|x1)

Question 26

When does ME in dependent variables cause consistent estimators?

Answer 26

For consistent estimates of Bj, we require E(e0|x) = E(e0) = 0
- e0 is independent of x and has zero mean

Question 27

When does ME in independent variables cause consistent estimators?

Answer 27

1. e1 is uncorrelated with x1: Cov(x1, e1) = 0
2. e1 is uncorrelated with x1* : Cov(x1*, e1) = 0

Question 28

e1 is uncorrelated with x1: Cov(x1, e1) = 0, what is the model for this?

Answer 28

y=b0 +b1x1 + (u- b1e1)

Question 29

e1 is uncorrelated with x1: Cov(x1, e1) = 0, create the model and what is it called?
- show how plim bˆ1 does not = b1

Answer 29

Classical errors-in-variables (CEV assumption)
- plim bˆ1 = b1 + Cov(x1, u - b1e1)/Var(x1)
- Cov(x1, u - b1e1) = Cov(x1, - b1e1) = - b1Cov(x1, e1)
find rest in notes
GIVEN THIS, OLS PRODUCES BIASED AND INCONSISTENT ESTIMATORS

Question 30

Show what the Attentuation bias is

Answer 30

notes, always less than 1

Question 31

What are some examples of issues that will lead to bias in the Random Sampling (MLR2) assumption?

Answer 31

• Missing data
• Non-random samples
• Outliers

Question 32

What happens when data is missing at random?

Answer 32

Estimators are less precise (SSTx is lower in smaller samples), BUT they are still unbiased

Question 33

Exogenous Sample Selection

Answer 33

Sample selection based on the independent variables, can still be unbiased

Question 34

Endogenous Sample Selection

Answer 34

Sample selection based on the dependent variables, will lead to biased coefficients