Topic 3: Multiple Regression Model

Question 1

The OLS estimates are chosen to minimize what MLR model with k variables?

Answer 1

the sum of [(y_i) - (beta^0) - (beta^1xi1) - (beta^2xi2)-…-(beta^kxik)]^2

Question 2

In MLR, holding all other intendent variables equal except x, the change in y = ?

Answer 2

change in y^ = beta1 change in x^1

Question 3

What does it mean to say that a OLS estimator is unbiased?

Answer 3

That means that the expected value of the estimates is equal to the population coefficient.

Question 4

What is an important assumption for the unbiasedness of estimators?

Answer 4

the zero conditional mean assumption is what makes the expected value of the estimates to be equal to the parameter

Question 5

What is the bias in a simple regression from leaving out an important variable, beta2?

Answer 5

beta2 * the sum of [(xi1 - xbar)xi2] / the sum of [(xi1 - xbar)^2]

Question 6

When the correlation between x1 and 2x is greater than zero, and x2 is omitted from the estimated equation, when is the bias positive and negative?

Answer 6

When corr(x1,x2)>0 and beta2>0 the bias is positive (the expected value is greater than the parameter) and when corr(x1,x2)>0 and beta2<0 the bias is negative.

Question 7

When the correlation between x1 and 2x is less than zero, and x2 is omitted from the estimated equation, when is the bias positive and negative?

Answer 7

When corr(x1,x2)<0 and beta2>0 the bias is negative (the expected value is less than the parameter) and when corr(x1,x2)>0 and beta2<0 the bias is positive.

Question 8

How to remember the bias in beta~1, when x2 is omitted?

Answer 8

When the correlation and the beta2 have the same sign the bias is positive, when they have different signs, the bias is negative.

Question 9

Stata: How to leave how certain observations from a regression?

Answer 9

reg y x1 x2…. if var =

Question 10

What is multicollinearity?

Answer 10

Correlation among independent variables in a MLR model. An R-squared of 1 shows perfect collinearity, but there is no standard amount under 1 that always is too much.

Question 11

How do you interpret a the effect of a model on y where two variables are correlated (but not perfectly) with the variable of interest?

Answer 11

Combine the coefficients of the two variables in the regression to get the effect of the variable of interest on y.

Question 12

Why is the data in an experiment better than observational data when it comes to MLR assumptions?

Answer 12

In the experiment, it is given that the treatment is randomly assigned, thus the zero conditional mean assumption will hold.

Question 13

Would omitting a variable cause bias in your estimators?

Answer 13

Yes.

Question 14

Would including an irrelevant (uncorrelated) variable cause biased in your estimators?

Answer 14

No.

Question 15

Even though including uncorrelated variables does not cause bias, why would you not want to include such variables?

Answer 15

Because they could have undesirable effects on your variance.

Question 16

Why are OLS estimators used?

Answer 16

Because they are the Best Linear Unbiased Estimators. by MLR assumptions 1-5

Question 17

Why do we need a multiple regression analysis?

Answer 17

Because it is difficult to draw a ceteris paribus conclusion about x affecting y without other variables to explain more variation in y without being correlated with u.

Question 18

What is goodness of fit?

Answer 18

It is the proportion of the sample variation that is explained by the regression line, it is equal to SSE/SST. It usually increases as independent variables are added.

Question 19

What is the formula for the sampling variance of OLS estimators?

Answer 19

Var(beta^j) = (sigma^2) / SSTj*(1-Rjsquared), where SSTj is the sample variation in xj, and Rjsquared is from regression xj on all other xs.

Question 20

How could you find SSTj?

Answer 20

SSTj = the sum of (xij - xbarj)^2

Question 21

How could you find Rjsquared?

Answer 21

Run a regression of xj on the other interdependent variables, then look for the R-squared value in stata.

Question 22

What is the unbiased estimator of sigma^2 (the error variance)?

Answer 22

sigma^2 = [sum (u^i)^2] \ [n-k-1] = SSR/ (n-k-1)

Question 23

The variance of beta^j depends on what three factors?

Answer 23

The error variance (sigma^2), SSTj (total sample variance of xjs), and R^2

Question 24

What does larger R^2s do to Var(beta^j)?

Answer 24

var(beta^j) increases

Question 25

A small sample size (and SSTj) can lead to what kind of sampling variance?

Answer 25

large sampling variances

Question 26

What is Var(beta^j)?

Answer 26

Var(beta^j) = [sigma^2] / [SSTj(1-R^2)]

Question 27

What is the Gauss-Markov Theorem?

Answer 27

Theorem that states that under MLR 1-5, OLS estimators are BLUE, and have the smallest variance.