3/4- Linear regression with one regressor

Question 1

What 3 main things does linear regression allow us to do?

Answer 1

• Quantify different relationships (through correlations)
• Estimate causal effects under assumptions
• Carry out a predictive analysis

Question 2

What does an error term summarise?

Answer 2

All the determinants of y besides x

Question 3

What is the causal/treatment effect?

Answer 3

E(y|x=1) - E(y|x=0) = β1

Question 4

Why will an experiment not give us the true value of β1 unless we can sample the entire population?

Answer 4

Randomisation ensures E(y|x=1) = E(y|x=0) in samples as n goes to infinity. But, in any sample, the sample average value of u may differ between small and large classes

Question 5

What are the 2 main properties of ^β1 as an estimator of β1?

Answer 5

• ^β1 is an unbiased estimator of β1, if we did the experiment repeatedly on different samples, the average value of ^β1 would equal β1: E(^β1)=β1
• ^β1 is a consistent estimator of β1, as the sample size goes to infinity ^β1 gets closer to β1

Question 6

What is the selection bias?

Answer 6

The difference between E(u|x=1) - E(u|x=0)

Question 7

What does the slope of the population regression line (β1) give us?

Answer 7

The average change in Y for a unit change in x

Question 8

What do beta coefficients with hats (^β) refer to, and what do those without (β) refer to?

Answer 8

• ^β refers to a sample

- β refers to population parameters

Question 9

What is the line of best fit?

Answer 9

The best fit line is the closest to all the sample points, so to find it we have to minimize the distance between the line and the data points

Question 10

What does the intercept (β0) of the OLS estimator tell us?

Answer 10

The predicted value when X = 0

Question 11

What does the Total Sum of Squares (TSS) tell us?

Answer 11

TSS measures the variation in the dependent variable

Question 12

What does the Explained Sum of Squares (ESS) tell us?

Answer 12

ESS represents variation explained by the regression

Question 13

What does the Residual Sum of Squares (RSS) tell us?

Answer 13

RSS represents variation not explained by the regression

Question 14

What does R^2 measure?

Answer 14

The fraction of the variance of Y that is explained by the regression. It measures how well the regression line fits the data

Question 15

How does TSS relate to ESS and RSS?

Answer 15

TSS = ESS + RSS

Question 16

What are the range of values R^2 can take?

Answer 16

Between 0 and 1, 0 implies no fit and 1 implies perfect fit

Question 17

How does the interpretation of β1 change when regression is in semi-logarithmic form?

Answer 17

A one unit change in x produces a β1*100 percent change in Y

Question 18

How can you calculate the error term/residual (u)?

Answer 18

Find the difference between the actual results for a given parameter and the result the model would predict given that parameter

Question 19

What is a key feature of OLS residuals?

Answer 19

They all add up to 0

Question 20

What is the relationship between the predicted/fitted value of y and the residual?

Answer 20

The sample covariance between them is 0

Question 21

What is an advantage of using a semi-logarithmic regression?

Answer 21

If we just had the dependent variable then the value of an extra unit of the independent variable would be constant

Question 22

What is a constant elasticity model?

Answer 22

When both the dependent and independent variable are logarithmic

Question 23

What is the relationship between y and β1 in the constant elasticity model?

Answer 23

a 1% increase in y leads to a β1% increase in x

Question 24

What does the expected value of the OLS estimator tell us?

Answer 24

Tells us if the estimator gets it right on average, it is the average outcome across all possible random samples

Question 25

What does the standard error of the OLS estimator tell us?

Answer 25

Tells us how reliable our results are

Question 26

What are the 4 main assumptions for the unbiasedness of SLR?

Answer 26

• Linear in parameters: can have log(y) but not log(β)
• Random sampling
• Sample variation in the explanatory variable: sample outcomes on xi are not all the same value
• Zero conditional mean: E(u|x)=0

Question 27

What happens to the parameters if the OLS is unbiased? i.e all 4 assumptions hold

Answer 27

E(^β1)=β1