Lecture 5

Question 1

What does the mutiple regression model allow us to obtain

Answer 1

Estimates of the relationships between Y and many factors, rather than just between Y and X

Question 2

Example of a multiple regression model for effect of education and experience on wages

Answer 2

𝑤𝑎𝑔𝑒𝑖=𝛽0+𝛽1𝑒𝑑𝑢𝑐𝑖+𝛽2𝑒𝑥𝑝𝑒𝑟𝑖+𝑢𝑖

Question 3

In the education/experience example what can we investigate using multiple regression

Answer 3

• investigate the relationship between experience and wages
• investigate the relationship between education and wages, while controlling for the effect of experience on wages

Question 4

𝑤𝑎𝑔𝑒𝑖=−3.39+0.64𝑒𝑑𝑢𝑐𝑖+0.07 𝑒𝑥𝑝𝑒r𝑖

for this example,what is the predicted wage rate for when educ=0 and exper=0

Answer 4

-3.39

Question 5

𝑤𝑎𝑔𝑒𝑖=−3.39+0.64𝑒𝑑𝑢𝑐𝑖+0.07 𝑒𝑥𝑝𝑒r𝑖
for this example, what is the marginal effect of an additional year of education on wages

Answer 5

Holding experience fixed, ∆𝑒𝑥𝑝𝑒𝑟=0, an additional year of education is predicted to increase wages by $0.64 per hour, (𝜕𝑤𝑎𝑔𝑒𝑠)̂/𝜕𝑒𝑑𝑢𝑐=0.64

Question 6

𝑤𝑎𝑔𝑒𝑖=−3.39+0.64𝑒𝑑𝑢𝑐𝑖+0.07 𝑒𝑥𝑝𝑒r𝑖
for this example, what is the marginal effect of an additional year of workplace experience on wages

Answer 6

Holding education fixed, ∆𝑒𝑑𝑢𝑐=0, an additional year of workplace experience is predicted to increase wages by $0.07 per hour, (𝜕𝑤𝑎𝑔𝑒𝑠)̂/𝜕𝑒𝑥𝑝𝑒𝑟=0.07

Question 7

For the estimators to be unbiased in the multiple regression model, what needs to hold using the education/experience example

Answer 7

For our estimators to be unbiased, the zero conditional mean assumption
𝐸[𝑢|𝑥1, 𝑥2]=0, must be valid
i.e., for any values of education and experience in the population, the average value of 𝑢 is zero

Question 8

What is the general multiple regression model

Answer 8

𝑦=𝛽0+𝛽1𝑥1+𝛽2𝑥2+𝛽3𝑥3+…+𝛽𝑘𝑥𝑘+𝑢

Question 9

For this general model, what assumption is still necessary

Answer 9

The zero conditional mean assumption is still necessary
For this general model with 𝑘 regressors the assumption is: 𝐸(𝑢|𝑥1, 𝑥2, 𝑥3, . . . 𝑥𝑘)=0

Question 10

If any one of the explanatory variables is correlated with 𝑢, then no matter how large 𝑘 is, what will this mean for the OLS estimators

Answer 10

It will be biased

Question 11

What is the sample regression function (SRF) for multiple regression

Answer 11

𝑦̂=𝛽̂0+𝛽1𝑥1+𝛽2𝑥2+𝛽3𝑥3+…+𝛽𝑘𝑥𝑘

Question 12

What does OLS do for multiple regression

Answer 12

OLS finds the values for 𝛽̂0,𝛽̂1, 𝛽̂2, 𝛽3,…𝛽̂𝑘 that minimise the sum of squared residuals (𝑆𝑆𝑅)

Question 13

When OLS does the minimisation, what will be the result

Answer 13

we get 𝑘+1 first order conditions called ‘normal equations’ that we need to solve simultaneously…

Question 14

Solving these ‘normal equations’ simultaneously what is the result

Answer 14

We get the OLS estimators 𝛽̂0,𝛽̂1, 𝛽̂2, 𝛽̂3,…𝛽̂𝑘

Question 15

Visual representation of the simple regression model:𝑦=𝛽0+𝛽1𝑥1+𝑢, SRF: 𝑦̂=𝛽̂0+𝛽̂1𝑥1,
𝛽̂1= 𝐶𝑜𝑣(𝑥,𝑦)/𝑉𝑎𝑟(𝑥)

Answer 15

Question 16

Labelled Visual representation of the multiple regression model: The multiple regression model: 𝑦=𝛽0+𝛽1𝑥1+𝛽2𝑥2+𝑢

Answer 16

Question 17

What are the three important implications of moving from simple to multiple regression analysis

Answer 17

• The estimators change: In general, the estimators from simple & multiple regressions are not equal

-A ceteris paribus (causal) interpretation of the estimators (and the estimates they generate) is more likely to be correct

• The estimators (and the estimates they generate) have a “partialling out” interpretation

Question 18

In what cases are the estimators not different for multiple regression model

Answer 18

• 𝑥1 and 𝑥2 are uncorrelated, i.e., 𝐶𝑜𝑣(𝑥1𝑥2)=0, then the SLR and MLR estimators are the same
• the ceteris paribus effect of a change in 𝑥2 on 𝑦̂ is zero, i.e., 𝛽̂2=0

Question 19

Why is a causal interpretation of the multiple linear regression estimators more likely to be correct than simple linear regression

Answer 19

SLR: 𝑦̂=𝛽̂_0+𝛽̂1 𝑥1
MLR: 𝑦̂=𝛽̂_0+𝛽̂1 𝑥1+𝛽̂2 𝑥2

• In the MLR, we have controlled for the effect of 𝑥2 on 𝑦. In the SLR we have not.

Question 20

Explain the ‘partialling out’ process for MLR

Answer 20

• In the MLR we “partial out” that part of the variation in 𝑦 that could be explained by either 𝑥1 or 𝑥2 or a mixture of both