Week 1 and Week 2

Question 1

What is probability theory good for?

Answer 1

Learn about a large group:
large group = population
> Too large to look at everyone
• Look at small subgroups
> describe sample = descriptive statistics
• Infer properties of a population from a sample
> Inferential statistics

Question 2

How to choose a smart sample?

Answer 2

Choose a random sample

Question 3

How do we denote sample and realized sample?

Answer 3

Sample set: Ω
realized sample: ω_0

Question 4

Denote Expectation of X

Answer 4

E[X]

Question 5

How do we calculate variance? And why?

Answer 5

measures how accurately x is predicted by E[X]. Variance of x= var(x)=(x-E[X])^2

Question 6

If we calculate Var(x) and it’s LARGE, is E[X] a good prediction of x(ω_0)?

Answer 6

No, E[X] is not a good prediction of x(ω_0). Only if variance is small.

Question 7

How do we calculate standard deviation?

Answer 7

sd(x)=sqrt(var(x))

Question 8

Optimality of E[X]

Answer 8

E[X]=arg min E[(x-a)^2]

Question 9

What does it mean if:
Cov(Y_1,Y_2)<0
Cov(Y_1,Y_2)>0
Cov(Y_1,Y_2)=0

Answer 9

Cov(Y_1,Y_2)<0 = negative relationship, negatively correlated
Cov(Y_1,Y_2)>0= positive relationship, positively correlated
Cov(Y_1,Y_2)=0 = uncorrelated

Question 10

Does a random sample give a representative sample?

Answer 10

Yes

Question 11

What does k and K denote?

Answer 11

k = observed X’s
K = all of the X’s

Question 12

“we cannot predict the value of U by observing regressors” is denoted how?

Answer 12

E[U|X1,…,Xk]=0

Question 13

Assumption OLS-2 (exogeneity)

Answer 13

The linear regression model satisfies: E[U|X1,…,Xk]=0

The regressors are exogenous.

Question 14

If E[U|X1,…,Xk]=0 holds, what does that say about the covariance?

Answer 14

cov(U,Xj)=0
- each regressor uncorrelated with unobserved component
- find j with cov (U, Xj) ≠ 0 and exogeneity fails

Question 15

Assumption OLS-3

Answer 15

(Full rank, informal statement): The best linear prediction of Y is unique.
This assumption is often called the “no perfect collinearity assumption”

Question 16

B^_0 is equal to what?

Answer 16

B^_0=E^[Y]-B^_1E[X]

Question 17

How much does the score increase if the model is: score = β0 + β1 log(study time) + β2courses + U

Answer 17

We estimate βˆ1(ω0) = 16.74237. This means that the estimated effect of increasing study time by 1% is approximately 16.74237/100 points on the exam.

Question 18

How much does the score increase if the model is: log(score) = β0 + β1study time + β2courses + U.

Answer 18

• This implies that changing the variable X1 by 1 unit will change output by approximately 100β1 %.
• We estimate βˆ1(ω0) = .014173. This means that we estimate that studying one extra hour will increase your exam score by approximately 100 × .014173% = 1.4173%.

Question 19

How much does the score increase if the model is: log(score) = β0 + β1 log(study time) + β2courses + U.

Answer 19

• This implies that changing the variable X1 by 1% will change output by approximately β1%. In this case, the coefficient β1 is called the (approximate) elasticity of X1.
• We estimate an approximate elasticity of βˆ1(ω0) = .263274. This means that increasing your study time by 1% will boost your exam score by approximately .263274%.

Question 20

How much does the score increase if the model is: score = β0 + γ0stat + β1study time + U

Answer 20

• We estimate γˆ0(ω0) = 51.33396. The estimated effect can be interpreted as follows. Suppose we could re-write the past of someone who didn’t take a statistics course and make them attend at least one such course. It is estimated that this would boost their grade by approximately 51.33 points.

Question 21

How much does the score increase if the model is: score = β0 + β1study time + β2math + γ1(study time × math) + U.

Answer 21

• The marginal effect of study time is given by β1 + γ1math. If γ1 > 0 this means that the higher your mathematical ability the more efficiently you study.

Question 22

OLS assumption 1

Answer 22

Functional form

Question 23

Ols assumption 4

Answer 23

Random Sample

Question 24

If OLS 1-4 holds what does that mean?

Answer 24

That E[B^_1]=B_1 –> B^_1 is unbiased for B_1
- B^_1 is consistent for B_1
- B^_1 ≈ B_1 in large samples

Question 25

OLS 5

Answer 25

The conditional variance of the unobserved component U is constant. If OLS 5 is satisfied: the error term is HOMOSKEDASTIC.

Question 26

Can standard deviation replace standard error and vice versa?

Answer 26

Yes.

Question 27

What does large and small values of rj indicate?

Answer 27

Small values of rj indicate that Xj cannot be approximated well by the other regressors. Large values of rj indicate that Xj can be approximated quite well by the other regressors.

Question 28

What if
- σ decreases
- var(Xj) increases
- rj decreases

Answer 28

If σ decreases then the variance of βˆj will decrease.

If var(Xj) increases then the variance of βˆj will decrease.

If rj decreases then the variance of βˆj will decrease.

Question 29

Do we try to disprove the null hypothesis?

Answer 29

YES

Question 30

If H_0 is true and we reject the H_0, what is that called?

Answer 30

Type 1 error

Question 31

If H_0 is false and we don’t reject H_0, what is that called?

Answer 31

Type 2 error.

Question 32

What do we say if H_0 is rejected?

Answer 32

X is significant.

Question 33

When do we use F tests?

Answer 33

For testing of multiple hypothesis.

Question 34

When do we use t-test?

Answer 34

For testing of one hypothesis.

Question 35

F test:
Which one do we reject?

If p-value(ω0) ≥ α ⇒ Fˆ(ω0) ≤ cα

If p-value(ω0) < α ⇒ Fˆ(ω0) > cα

Answer 35

We reject if p-value(ω0) < α ⇒ Fˆ(ω0) > cα

Question 36

Can we say anything if we can’t reject the null hypothesis?

Answer 36

No. We learn nothing.

Question 37

Formula for R^2 when Var^ (Y) is unknown

Answer 37

R^2 = Var^ (Y^) / (Var^ (Y^) + Var^ (U^)) = 1 / (1 + Var^ (U^)/(Var^ (Y^))

Question 38

Formula for Var (X_1)

Answer 38

Var (X_1) = 1/n * (X - E(X) )^2

Question 39

Compute marginal effect of a square variable and another variable. Example m(x1) = β0 + β1x_1 + β2x_1^2

Answer 39

We derive and calculate as: m(x1) = β1 + 2*β2x_1.