Lecture 3

Question 1

What is the systematic and stochastic component of the data in the population

Answer 1

Question 1

Recap of the PRF with diagram and explanation

Answer 1

Represents the general tendency that describes the relationship between y and x and

Question 2

What does B1represent

Answer 2

The ceteris paribus effect of a change in x on y

Question 3

What are estimators

Answer 3

Formulae that can be used in combination with data to generate estimates

Question 4

Recap of the OLS estimators

Answer 4

Question 5

How is the OLS estimator unbiased

Answer 5

The most likely estimates that the estimators produce are the actual population parameters, i.e., 𝐸(𝛽̂1 )=𝛽1 and 𝐸(𝛽̂0 )=𝛽0

Question 6

How is the OLS estimator efficient

Answer 6

Has the minimum variance of any linear unbiased estimators

Question 7

What is the unobservable model defining how the data is generated

Answer 7

𝑦=𝛽0+𝛽1𝑥+𝑢

Question 8

Show how the OLS estimator for B1=𝐶𝑜𝑣(𝑥,𝑦)/𝑉𝑎𝑟(𝑥) of is unbiased

Answer 8

Question 9

Explain the two parts of the OLS estimator for B1: 𝛽̂1=𝛽1+𝐶𝑜𝑣(𝑥, 𝑢)/𝑉𝑎𝑟(𝑥)

Answer 9

• There is a a non-random (deterministic) part that captures the true underlying relationship, i.e., that is equal to the population parameter, 𝛽1
• a random (stochastic) part responsible for variations around the population parameter

Question 10

Given that the OLS estimator 𝛽̂1 is a random variable, what does this mean

Answer 10

• The set of values it can take can be represented by a probability distribution

Question 11

What do we call this probability distribution of 𝛽̂1 and why

Answer 11

• The sampling distribution of 𝛽̂1
• If you drew lots of samples from the population to calculate an estimate of 𝛽̂1, that is what you would get

Question 12

What equation is required for the estimator to be unbiased

Answer 12

Question 13

What does this diagram show

Answer 13

That the estimator is biased given that 𝐸(𝛽̂1 )≠𝛽1

Question 14

What are the 4 assumptions that, if valid, mean the OLS estimators are unbiased

Answer 14

SLR1: The population model that describes how the data is generated is linear in parameters
SLR2: The sample of size n, taken from the pop and used to generate estimates is random
SLR3: Sample variation in the explanatory variable ( the variation in x)
SLR4: Zero conditional mean holds, 𝐸(𝑢|𝑥)= 0

Question 15

Explain how a model of the effect on wages holding other factors fixed such as innate ability ( 𝑤𝑎𝑔𝑒=𝛽0+𝛽1 𝑒𝑑𝑢𝑐𝑎𝑡𝑖𝑜𝑛+𝑢) leads to a bias in our estimator for 𝛽̂1

Answer 15

• 4th assumption is violated as the zero conditional mean does not hold because individuals. with higher innate ability (part of u) stay in education longer
• Given estimate is 𝛽̂1=𝛽1+𝐶𝑜𝑣(𝑒𝑑𝑢𝑐𝑎𝑡𝑖𝑜𝑛, 𝑢)/𝑉𝑎𝑟(𝑒𝑑𝑢𝑐𝑎𝑡𝑖𝑜𝑛), the cov(education,u) >0 as individuals with higher innate ability stay in education longer
• This means 𝐸(𝛽̂1 )>𝛽1 i.e., the expected value of our estimate of the effect of education on wages will be too high

Question 16

Why will the estimate be biased if one of the factors (within u) is positively correlated with x

Answer 16

• This means 𝐸(𝑢|𝑥)≠0 and, more specifically, 𝐶𝑜𝑣(𝑥, 𝑢)>0
• so 𝐸(𝛽̂1)≠𝛽1 as 𝛽̂1=𝐶𝑜𝑣(𝑥,𝑦)/𝑉𝑎𝑟(𝑥) =𝛽1+𝐶𝑜𝑣(𝑥, 𝑢)/𝑉𝑎𝑟(𝑥)
• the estimate will be biased upwards

Question 17

what does the diagram show

Answer 17

• The OLS estimator 𝛽̂1 is a random variable and the values it can take are represented by probability distribution called the sampling distribution of 𝛽̂1
• The OLS estimator is efficient

Question 18

Which of the 4 assumptions is most often invalid

Answer 18

SLR4- The zero conditional mean assumption

Question 19

Which assumption is pretty much guaranteed to be valid

Answer 19

SLR3- There is sample variation in the explanatory (X) variable