2. OLS - The simple regression model

Question 1

What assumption must be satisfied to argue that there is a causal relationship?

Answer 1

The conditional mean assumption

Question 2

When does B1 explain the relationship between x and y?

Answer 2

If everything else not accounted for in the model remains constant

Question 3

What is the conditional mean assumption?

Answer 3

The explanatory variable must not contain information about the mean of the unobserved factors

Question 4

Why is the conditional mean assumption unlikely to ever truly hold in the example of wage?

Answer 4

The conditional mean independence assumption is unlikely to hold because individuals with more education will also be more intelligent on average in this example (accounted for only in u)

Question 5

How do you ensure that B1 is the exclusive measure of change?

Answer 5

Expected value of all you can’t observe given x is 0, therefore you are holding everything else constant so B1 is the exclusive measure of change here

Question 6

Why is there a linear relationship between the explanatory variable and the dependent variable?

Answer 6

The population regression function tells us that as the mean of u given x is 0, we can express the average value of the dependent variable as a linear function of the explanatory variable (Slide 8 13/02/24)

Question 7

How is the distribution of y effected by x?

Answer 7

For any given x, the distribution of y is centred about E(y|x)

Question 8

What is the systematic part of the linear regression?

Answer 8

B0 and B1x: ie. the part of y explained by x

Question 9

What is the unsystematic part of y?

Answer 9

The part of y not explained by x

Question 10

How do we show that some variables are estimators?

Answer 10

We have a hat on the variable

Question 11

What are estimators dependent on?

Answer 11

The data sample used and the method used therefore there may be differences between samples but on average should reflect the true population

Question 12

What are regression residuals?

Answer 12

What we are unable to explain. Differences between what you predict and what you observe in the data. U hat also includes all the things that are not completely accurate with your data

Question 13

What does the linearity of the OLS model imply and why is it limited?

Answer 13

The linearity implies that a one-unit change in x has the same effect on y, regardless of the initial value of x. This is unrealistic for many economic applications.

Question 14

What is important to note about the population regression function?

Answer 14

The PRF tells us how the AVERAGE value of y changes with x, it does not say that y equals B0+B1x for all units of the population

Question 15

What is covariance?

Answer 15

Covariance measures how the deviation of one variable from it’s mean is related to the deviation of another variable from it’s mean

Question 16

What does the OLS say about the residual sum of squares?

Answer 16

Bo hat and B1 hat should be chosen such that the residual sum of squares are minimised

Question 17

Why do we aim to minimise the residual sum of squares?

Answer 17

Because if your residuals are too large, then your methods are not good so we aim to minimise them

Question 18

Why do we square the residuals rather than take the absolute values?

Answer 18

Because it makes the optimisation easier (we have to differentiate at some point)

Question 19

What is the Sample regression function?

Answer 19

The sample regression function (SRF) is the estimated version of the population regression function and forms the graph

Question 20

What is sample regression function also known by?

Answer 20

The OLS regression line

Question 21

How should you phrase talking about relationships between variables?

Answer 21

Instead of saying ‘caused by’ say associated with instead

Question 22

Under an OLS regression, what is true of deviations from the regression?

Answer 22

They sum to 0

Question 23

What is true of the covariance under an OLS model?

Answer 23

Covariance between deviations and regressors are 0

Question 24

What is true of the sample average of y and x?

Answer 24

They lie on the OLS regression line

Question 25

What are the three key algebraic properties of the regression model?

Answer 25

1. Deviations equal 0
2. Covariance equal 0
3. Sample averages on line