Lecture 7

Question 1

Example hypothesis: Demand Theory

Answer 1

• demand is affected by price, specifically, it is inversely related to price
• demand may be positively or negatively related to income
• individual characteristics affect demand in various ways

Question 2

Example MLR for demand theory

Answer 2

𝑄=𝛽_0+𝛽_1 𝑃+𝛽_2 𝑌+𝛽_3 𝑎𝑔𝑒+…+𝛽_𝑘 𝑥_𝑘+𝑢
where 𝑄 = demand, 𝑃 = price, 𝑌= income, 𝑎𝑔𝑒…𝑥_𝑘= individual characteristics

Question 3

What needs to be done to test the theory/hypotheses

Answer 3

• Reformulate the parameters

Eg price is inversely related to demand so:
𝛽1≠0 and 𝛽1<0

Question 4

How are hypotheses formed

Answer 4

• Set up in pairs
• The Null hypothesis, 𝐻0:
  the hypothesis we test
• The Alternative hypothesis, 𝐻1:
  the logical conclusion if the null hypothesis is false

Question 5

Hypothesis example for ‘Is demand affected by price?’

Answer 5

𝐻0: 𝛽1=0 𝐻1: 𝛽1≠0

Question 6

Hypothesis example for ‘Does demand fall when price increases?’

Answer 6

𝐻0: 𝛽1≥0 𝐻1: 𝛽1<0

Question 7

Which hypothesis is the theory captured in

Answer 7

The alternative hypothesis

Question 8

What should a null-alternative hypothesis pair encompass

Answer 8

All possibilities

Question 9

What two responses are there to the null hypothesis

Answer 9

We either ‘reject’ 𝐻0 (the null) or ‘fail to reject’ 𝐻0

Question 10

Theory of demand: 𝑄=𝛽0+𝛽1 𝑃+𝛽2 𝑌+𝛽3 𝑎𝑔𝑒+…+𝛽𝑘 𝑥𝑘+𝑢
For ‘Is demand affected by price?’
𝐻0: 𝛽1=0 𝐻1: 𝛽_1≠0
What does rejecting the null hypothesis imply and what type of test is required?

Answer 10

• That a relationship between price and demand exists
• Two-tailed test is requirement

Question 11

Theory of demand: 𝑄=𝛽0+𝛽1 𝑃+𝛽2 𝑌+𝛽3 𝑎𝑔𝑒+…+𝛽𝑘 𝑥𝑘+𝑢
For ‘Does demand fall when price increases?’
𝐻0: 𝛽_1≥0 𝐻1: 𝛽1<0
What does rejecting the null imply and what test do we undertake?

Answer 11

• Rejecting implies relationship is negative
• Undertake one tailed test

Question 12

What is assumption MLR6

Answer 12

We assume that the disturbances (𝑢) are normally distributed, because we know the means and variances of the sampling distributions of the 𝛽̂𝑗

Question 13

Why do we assume u is normally distributed

Answer 13

• We only know the means and variances of the sampling distributions of our estimators and a normal distribution assumption would mean we know everything and can draw perfect statistical inferences

Question 14

Full written assumption MLR6: Normality of u

Answer 14

The population disturbances (𝑢) are independent of the explanatory variables (𝑥1, 𝑥2, 𝑥3,…, 𝑥𝑘) and are normally distributed with zero mean and variance 𝜎^2, i.e., 𝑢~𝑁𝑜𝑟𝑚𝑎𝑙(0,𝜎^2)

Question 15

What two assumptions does MLR6 incorporate

Answer 15

MLR4: Zero conditional mean: 𝐸(𝑢|𝒙)=0
MLR5: Homoscedasticity: 𝑉𝑎𝑟(𝑢|𝒙)=𝜎^2

Question 16

Why is MLR6 stronger than MLR4 and MLR5

Answer 16

because it is about the overall shape of the distribution of 𝑢, not just its mean and variance

Question 17

What are MLR1 to MLR6 referred to as

Answer 17

the assumptions of the Classical Linear Model (CLM) for cross section data

Question 18

Under the assumptions of the CLM what can be written

Answer 18

𝑦|𝒙 ~ 𝑁𝑜𝑟𝑚𝑎𝑙(𝛽_0+𝛽_1 𝑥_1+…+𝛽_𝑘 𝑥_𝑘,𝜎^2)
i.e., conditional on 𝒙, 𝑦 has a normal distribution with a mean that is linear in 𝒙 and a variance that is constant irrespective of 𝒙

Question 19

Explain the decision rule: when to reject the null

Answer 19

Reject 𝑯𝟎 if it implies that the probability of getting the estimate that OLS yields is less than a small preselected probability (e.g. 5%)

Question 20

Example of the decision rule for ‘Is demand affected by price’

Answer 20

Question 21

What is the small pre-selected probability for the decision rule called

Answer 21

A significance level

Question 22

What does a significane level of 5% mean

Answer 22

we are willing to be wrong about rejecting the Null 5% of the time

Question 23

What is the process of applying the decision rule when 𝐻0: 𝛽𝑗=0

Answer 23

• Take the estimate 𝛽̂𝑗, - divide it by its standard deviation, 𝑠𝑑(𝛽̂𝑗),
• Then look at where the resulting number falls in the standard normal distribution, i.e., the distribution of 𝑍~𝑁𝑜𝑟𝑚𝑎𝑙[0,1]

Question 24

Decision rule summarised in an example

Answer 24

Question 25

Why can't we calculate the 𝑠𝑑(𝛽 ̂𝑗 )

Answer 25

Because it depends on 𝜎^2, which is unknown

Question 26

What is our estimate for the 𝑠𝑑(𝛽 ̂𝑗 )

Answer 26

the standard error of 𝛽 ̂𝑗

Question 27

Equation for standard error of 𝛽 ̂𝑗

Answer 27

Question 28

What distribution needs to be used to account for the fact we have used an estimate for 𝜎^2 to calculate the 𝑡 statistic

Answer 28

T distribution

Question 29

What does T distribution take in account for

Answer 29

- how much information we used in the estimation, i.e., the sample size, 𝑛, and - how much of that information we used up, i.e., the number of model parameters we estimated, 𝑘 slope parameters plus 1 constant - 𝑛−𝑘−1= the degrees of freedom

Question 30

So, under CLM assumptions what is the final test statistic

Answer 30