Chapter 3 Brooks

Question 1

what is regression

Answer 1

evaluating the relationship between one variable and movements in one or more other variables

Question 2

What are we trying to do with regression

Answer 2

We are trying to explain movements in some dependent variable, y. We are trying to explain its movements by using explanatory variables.

Question 3

what is correlation

Answer 3

The degree of linear association between two variables

Question 4

what is the danger with correlation

Answer 4

1) Assume causaility
2) Misunderstand what linear association is

Correlation is given as:

cov(X,Y)/(sigma_X sigma_Y)

cov(X,Y) = E[(X - mu_x)(Y - mu_y)]

Question 5

what is the interpretation of covariance

Answer 5

The interpretation of covariance is “The expected difference from the mean of X multiplied by the difference from the mean of Y.” Meaning, the covariance of X and Y represent how much we expect X to move away from its mean in relation to how much Y moves away from its mean.

Question 6

elaborate on the role of correlation

Answer 6

The role of correlation is to provide an understanding of the linear relationship between two variables. If the linear relationship is perfect, it means that movement in one variable can perfectly explain the movement in the other.

HOWEVER: The correlation coefficeint is not interpretable for this. The coefficient only tell us the degree of the relationship.

We need to use linear methods to capture the linear relationship. This is where the linear regression come into play.

Question 7

first step to see if there is a relationship between two variables

Answer 7

Plot it visually

Question 8

elaborate on:

y = a + bx

Answer 8

This is an exact line. The problem with this is that it doesnt account for errors.

This is a model, a best fitting line, but it is not realistic

Question 9

What is a better model than

y = a + bx?

Answer 9

y_t = a + bx + u_t

This model assumes a relationship but there is a random disturbance term that always exist. Might be because of how it is impossible to catch everything, etc.

Question 10

generally speaking, how do we find values for alpha and beta

Answer 10

we need to find the alpha and beta that makes the sum of vertical distances between points y_t and y be as small as possible.

Question 11

why vertical and not horizontal distanves?

Answer 11

we make use of the assumption that the x-values are fixed in random samples. this means that there is no random element here. Therefore, this assumption reduce our task to finding only the y_t’s.

Question 12

Most common line fitting method

Answer 12

OLS

Question 13

general procedure of OLS

Answer 13

Square the distances between y_t and the exact line y. Sum together. find the y-line that makes this sum the samllest

Question 14

with correct notaiton how do we describe the method of OLS

Answer 14

Minimize the sum of squared differences between y_t and y_t-pred

Question 15

what is y_t and what is y_t_pred

Answer 15

y_t is the data point, as collected.

y_t_pred is the predicted data point.

Question 16

Other way of describing the method of OLS

Answer 16

Minimizing the sum of squared errors/residuals

Question 17

what is the residual mathematically?

Answer 17

The difference between y_t and y_t_pred

Question 18

what is PRF

Answer 18

Population regression function, the function that is thought to be producing the data

Question 19

give the PRF

Answer 19

y_t = a + bx_t + u_t

Question 20

Why does PRF contian error?

Answer 20

Because even though it is the true process, the true process can contian random elements.

Question 21

What is SRF?

Answer 21

Sample regression function

Question 22

Give the SRF

Answer 22

y_t_pred = a_pred + b_pred x_t

Question 23

why not error in SRF

Answer 23

Because it is the best fitting line that we have found to minimize the RSS.

Question 24

what is CLRM

Answer 24

the model of:

y_t = a + bx_t + u_t

along with the 5 assumptions

Question 25

what are the CLRM assumptions?

Answer 25

1) E[u_t] = 0 2) var(u_t) = sigma^2 < infinity 3) cov(u_i, u_j) = 0 4) cov(x_j, u_t) = 0 5) u_t ~n(0,sigma^2)

Question 26

do we need all the assumptions?

Answer 26

To use the model, no. we dont actually need number 5 to fit the model. but we need this assumption to make inference later.

Question 27

Elaborate on the properties of the OLS estimator

Answer 27

if assumptions 1-4 hold, OLS estimator gives estimates that are "BLUE". Best Linear Unbiased Estimator. Best refers to Gauss Markov theorem, stating that OLS estimator has the lowest variance in class. With the 1-4 assumptions in place, OLS estimator is unbiased, efficient and consistent.

Question 28

is it possible to find an estimator with lower variance than OLS estimator?

Answer 28

Yes. but the thing is that it would not be unbiased and linear.

Question 29

elaborate on the standard error of the OLS estimator

Answer 29

standard error is a measure of precision. Specifically, the standard error represent the standard deviation of the statistic. Theoretically, it is important to emphasize that standard error is an estimate of the standard deviation of the statistic. The standard errors represent a degree of variation in the statistic. Specifically, how much we should expect, on average, to see an estimate deviate from its mean. It is important to understand that standard error does not give any indication on the overall fit of the parameter. It only give us an answer in regards to "if we perform this experiment again, do we expect to see a very different result?".

Question 30

what informaiton is provided by standard error

Answer 30

Precision, not accuracy. This means that we will get some indication on whether our estimate is robust or not. But whether it is a good estimateo f the true populaiton parameter, it cannto say

Question 31

elaborate on t-ratio

Answer 31

special test where the hypothesis is that hte parameter we're testing for is 0. This gives us a ratio. We call it "t" because it follow student-t dist.