Lecture 9

Question 1

What is the chance =? that we incorrectly reject a true null hypothesis

Answer 1

𝛼

Question 2

How many types of errors can be made in hypothesis testing

Answer 2

2

Question 3

What is a type I error

Answer 3

Reject a true null

Question 4

What is a type II error

Answer 4

Fail to reject a false null

Question 5

Table for type I and II errors

Answer 5

Question 6

What does type I error relate to

Answer 6

Significance level of a test =𝛼=𝑃𝑟𝑜𝑏(𝑇𝑦𝑝𝑒 𝐼)

Question 7

What does a type II error relate to

Answer 7

Power of a test =1−𝑃𝑟𝑜𝑏(𝑇𝑦𝑝𝑒 𝐼𝐼)

Question 8

How are the probabilities of making a type I and type II error related

Answer 8

They are inversely related

Question 9

Why do we tolerate a small type I probability

Answer 9

so that we don’t drive the Type II probability to unacceptably high levels

Question 10

What is the probability of making a type II error on a diagram showing the Distribution of 𝛽̂𝑗
under 𝐻0:𝛽𝑗=𝛽𝑗0 and the actual distribution of 𝛽̂𝑗

Answer 10

Question 11

If we select 𝛼 = 5% (rather than 𝛼 = 1%. what happens to the chances of commiting a type I and type II error

Answer 11

we make it a little more likely to commit a Type I error but much less likely to commit a Type II error

Question 12

How can we answer the question: ‘What range of values of 𝛽𝑗 is consistent with the sample estimate obtained using OLS?’

Answer 12

Using a confidence interval

Question 13

How to form a typical confidence interval for 𝛽𝑗

Answer 13

• we would not reject a two-tailed 𝐻0 if
  −𝑐<𝑡<𝑐
  −𝑐<(𝛽̂𝑗−𝛽𝑗)/(𝑠𝑒(𝛽̂𝑗))<𝑐
• Rearranging this expression around 𝛽𝑗 yields
  𝛽̂𝑗−[𝑐×𝑠𝑒(𝛽̂𝑗 )]<𝛽𝑗<𝛽̂𝑗+[𝑐×𝑠𝑒(𝛽̂𝑗 )]

Question 14

So what is the general confidence interval described as

Answer 14

Question 16

Key features of confidence intervals

Answer 16

• is symmetric around the estimate 𝛽̂𝑗
• depends on 𝑐, which is linked to 𝛼, the selected significance level:
  the smaller the significance level, the wider the interval
  95% confidence interval; 5% significance level; 𝑐 ~±1.96 (if 𝑛 large)
  99% confidence interval; 1% significance level; 𝑐 ~±2.58 (if 𝑛 larger
• depends on 𝑠𝑒(𝛽̂𝑗 ):

the more accurate our estimate, i.e., the smaller its 𝑠𝑒, the narrower the confidence interval

Question 17

What are confidence intervals also known as

Answer 17

Interval estimates

Question 18

Example of 95% confidence interval for 𝛽2 in this example:

Answer 18

Question 19

What test is used by econometricians to draw inferences about sets of coefficients as a group

Answer 19

the 𝑭 test

Question 20

What does regression analysis decompose each 𝑦𝑖 in the sample into

Answer 20

• a fitted value, 𝑦̂𝑖: the part that is
  explained by the estimated model (SRF)
• a linear function of the explanatory variables (𝑦̂𝑖=𝛽̂0+𝛽̂1 𝑥1𝑖+…+𝛽̂𝑘 𝑥𝑘𝑖)
• a residual, 𝑢̂𝑖: the part that is
  unexplained (𝑢̂_𝑖=𝑦_𝑖−𝑦̂_𝑖)

Question 21

How is the variation in 𝑦𝑖 decomposed

Answer 21

• The total sum of squares, SST which measures total variation in 𝑦𝑖
• the Explained Sum of Squares, SSE which measures the variation in 𝑦̂𝑖-
  the variation in 𝑦𝑖 that is explained by the model
• the Residual Sum of Squares, SSR
  measures the variation in 𝑢̂𝑖
  the variation in 𝑦𝑖 that is unexplained

Question 22

Equation for SST

Answer 22

Question 23

Equation for SSE

Answer 23

Question 24

Equation for SSR

Answer 24

Question 25

What is the coefficient of determination= 𝑅^2

Answer 25

Question 26

What does the coefficient of determination represent

Answer 26

- It is the ratio of the explained variation to total variation - It is the fraction of the sample variation in 𝑦 that is explained by 𝒙 0≤𝑅^2≤1

Question 27

If 𝑅^2= 0.10 what does that show

Answer 27

10% of variation in 𝑦 is explained by variation in the variables in 𝒙, the remainder (90%) by other factors (𝑢)

Question 28

What does a higher 𝑅^2 indicate for the model

Answer 28

That it is a better 'fit' to the data

Question 29

Diagrams to illustrate R^2 as a measure of model performance

Answer 29

Question 30

Ballentine Venn Diagram to show 𝑹^𝟐

Answer 30

Question 31

What is the problem with R^2

Answer 31

Including an additional regressor in a model never reduces and usually increases 𝑅^2

Question 32

What is used to overcome this problem

Answer 32

we adjust 𝑅^2 by a factor involving 𝑛 and 𝑘