Topic 2: Simple Linear Regression

Question 1

In SLR model, if xbar = 0 then what does beta^_0=?

Answer 1

beta^0 = ybar

Question 2

Rearrange a SLR model to equal beta^_0

Answer 2

beta^_0 = ybar - beta_1*xbar

Question 3

Write the residual for observation i, u^_i, in terms of y_i’s

Answer 3

u^_i = y_i - y^_i

Question 4

What is the residual for observation i, u^_i equal to in terms of a SLR model?

Answer 4

u^_i = y_i - beta^_0 + beta^_1*x_i

Question 5

How to derive beta_1 from sums?

Answer 5

[the sum of (x_i - xbar)(y_i - ybar)] / [the sum of (x_i - xbar)^2] which is the sample covariance between x and y, divided by the sample variation.

Question 6

How would you find the coefficients of a model if given a table of the data?

Answer 6

find ybar and xbar, then use the formula for beta_1 (sample cov/sample var), then put these three into a formula to get beta_0

Question 7

What is the formula for fitted values, Y^_i , of observation i in terms of a SLR model?

Answer 7

(y^_i) = (beta^_0) + (beta^_1)*(x_i)

Question 8

What is the sum of squared residuals, SSR?

Answer 8

the sum of (u^_i^2) or the sum of (y_i - y^_i)^2

Question 9

What is a simple regression model used for?

Answer 9

Used to study the relationship between two variables as a tool for empirical analysis.

Question 10

Names for y variable in regressions?

Answer 10

dependent, explained, response, predicted, regressand

Question 11

Names for x variable in regressions?

Answer 11

independent, explanatory, control, predictor, regressor

Question 12

Describe the linear relationship in a simple regression model.

Answer 12

If the u term is held fixed at zero, x will have a linear affect on y, such that for every increase of a unit in x, there is a beta_1 change in y.

Question 13

Besides E(u)=0, how can you assure a ceteris paribus relationship in your SLR model?

Answer 13

The zero conditional mean assumption of E(u|x) = 0

Question 14

What is the conditional expectation function for the population regression function?

Answer 14

E(y|x) = (beta_0) + (beta_1)x, when the zero conditional mean assumption holds

Question 15

What is the minimizing function for the derivation of the OLS estimators in SLR?

Answer 15

the sum of [(y_i) - (beta_0) - (beta_1)*(x_i)]^2. This is the ordinary least squares we are trying to minimize (the coefficients)

Question 16

What is the residual of a SLR equal to in terms of parameters?

Answer 16

u = y - (beta_0) - (beta_1)(x)

Question 17

What are the first order conditions of the minimizing function for OLS estimators in SLR?

Answer 17

Take the partial derivative of the minimizing functions with respect to beta0 and beta1 to get two FOCs of: -2 sum of [(y_i) -(beta^_0) - (beta^_1)*(x_i)) = 0

Question 18

What is another way to write the formula for (beta^_1) = [the sum of (x_i - xbar)(y_i - ybar)] / [the sum of (x_i - xbar)^2]

Answer 18

(beta^_1) = [the sum of (y_i)(x_i - xbar)] / [the sum of (x_i - xbar)^2]

Question 19

What is homoskedasticity?

Answer 19

The errors in a regression have constant variance conditional on the explanatory variables, thus u has the same variance given any value of x.

Question 20

Under SLR1-4, do OLS estimates equal the true population parameters?

Answer 20

No, but under SLR1-4 the expected value of the estimates are equal to the parameter, making it unbiased.

Question 21

Would a violation of the zero conditional mean assumption make an estimator biased?

Answer 21

Yes.

Question 22

What is the expected value of residuals equal to?

Answer 22

0, both the sum and mean of residuals is 0.

Question 23

What is the bias in the estimator beta^1 if E(u_i) does not equal 0?

Answer 23

The bias in the estimator is equal to the sum of [(x_i - xbar)(u_i)] / the sum of (xi -xbar)^2

Question 24

What is the formula for the total sum of squares (SST)?

Answer 24

SST = the sum of (yi - ybar)^2 and is the measure of total sample variation in yi (thus the outcomes yi - ybar)

Question 25

What is the explained sum of squares (SSE)?

Answer 25

SSE = the sum of [(y^i - ybar)^2 and is the measure of sample variation in the fitted values (thus y^i)

Question 26

What is the residual sum of squares (SSR)?

Answer 26

SSR = the sum of u^i^2, the measure of variation in u^i

Question 27

What is the formula for total variation in y? (explained and unexplained?)

Answer 27

SST = SSE + SSR

Question 28

Stata: how do you get a frequency table of a variable?

Answer 28

tab var

Question 29

What is Var(beta^1)?

Answer 29

= sigma^2 / SSTx where SSTx = (sum xi- xbar)^2

Question 30

What is Var(y|x)?

Answer 30

= sigma^2, which is a constant and a measure of the variation in y given x.

Question 31

What is SLR 5?

Answer 31

Homoskedasticity, u has the same variance given any value of x, Var(u|x) = sigma^2 which is the error variance.

Question 32

What is Var(beta1)?

Answer 32

=0

Question 33

State the change in a level level model.

Answer 33

change in y = beta1 change in x

Question 34

State the change in a level log model.

Answer 34

change in y = (beta1/100)% change in x

Question 35

State the change in a log level model

Answer 35

% change in y = (100beta1) change in x

Question 36

State the change in a log log model.

Answer 36

% change in y = % change in x

Question 37

How can you remember the change of functional models?

Answer 37

A log with the variable makes it % change. For level-log beta1/100. For log-level beta1*100