L1

Question 1

Slope of regression line = ?

Answer 1

Expected effect on Y of a unit change in X

Question 2

How does an OLS estimator work?

Answer 2

It minimises the avg square of differences between the actual Yi values and the predicted ones

Question 3

What is the interpretation of beta1(hat)?

Answer 3

For each extra unit of X, Y on avg. is beta1(hat) higher/lower

Question 4

What is the interpretation of beta0(hat)?

Answer 4

It is the intercept (ie. Y when X=0)

Question 5

Give an example when an interpretation of Beta0(hat) would not make sense?

Answer 5

If looking at test scores to student-teacher ratio, by extrapolating outside the data the line predicts a very high score, even though having 0-students to a teacher would not actually give a score!

Question 6

What does R^2 measure?

Answer 6

The fraction of the variance of Y that is explained by X (ranges from 0 (none) to 1 (complete explanation))

Question 7

What does it mean if R^2 is 0 or 1 for explained sum of squares?

Answer 7

If 0, means ESS=0

If 1, means TSS=ESS

Question 8

When does R^2=r^2?

Answer 8

When there is only one X variable

Question 9

What is r^2?

Answer 9

The square of the correlation coefficient between X and Y

Question 10

What does the standard error of the regression measure (SER)?

Answer 10

The magnitude of a typical regression residual in the units of Y

Question 11

What does the root mean squared error measure?

Answer 11

Same as the SER; in large samples the measures are very close

Question 12

Briefly, what are the least squares assumptions?

Answer 12

1) Error and X variable(s) are independent: E(u|X=x)=0
2) Observations are i.i.d. (Xi and Yi)
3) Large outliers in X and/or Y are rare
(not the same as the OLS assumptions!)

Question 13

When do we often encounter non i.i.d. sampling?

Answer 13

TS data and Panel data

Question 14

What is the sampling uncertainty of an OLS estimator?

Answer 14

When a different sample of data might lead to different beta1 estimate

Question 15

How is the variance of Beta1(hat) related to the sample size?

Answer 15

Inversely proportional; as sample size increases, variance decreases

Question 16

See: consistency, approximations and CLT

Answer 16

now

Question 17

How does the variance of X affect the variance of Beta1(hat)?

Answer 17

Larger variance of X, smaller variance of beta1(hat) (see notes)

if there is more variance in X, there is more information in the data therefore the regression is plotted with more confidence therefore less variance in coefficient

Question 18

State the CLT?

Answer 18

Suppose you have a set of i.i.d. observations with expected value 0 and variance=(sigma)^2, then when n is LARGE, 1/n(sum of obs.) is approximately distributed by N(0, Variance/n)

Question 19

Explain LSA 1: Error and X variable(s) are independent: E(u|X=x)=0?

Answer 19

For any given value of X, the mean of the error is zero!

Consider an ideal randomised controlled experiment: X is randomly assigned to people by a computer tf since X is randomly assigned, all other characteristics (present in u) are randomised so u and X are independent!

If an IRCE isn’t in action must consider if LSA 1 actually holds!

Question 20

Explain LSA 2: Observations are i.i.d. (Xi and Yi)? (ie. explain each i?)

Answer 20

Arises automatically if the entity is sampled by simple random sampling!

entities selected from same pop. so (Xi,Yi) are identically distributed for all i, and since they are selected at random the values (Xi,Yi) are also independently distributed tf i.i.d.

Question 21

Explain why a larger variance in the set of observations, X, leads to a better estimate of beta?

Answer 21

Eqn. in notes tells us that the variance of X is inversely proportional to the variance of beta(hat1)

if there is more variance in the X, this gives more information in the data therefore the regression will be plotted with more confidence and tf less variance in beta(hat1)