4- Hypothesis testing

Question 1

What does ^βols tell us?

Answer 1

Marginal effect on the dependent variable for a unit change in the covariate

Question 2

What do hats denote in regressions?

Answer 2

Estimated values, once we have estimated values of beta, the regression is now called the estimated regression equation

Question 3

What is the estimated error (û) in terms of y?

Answer 3

û = y - ŷ

Question 4

What is the explained sum of squares (ESS)?

Answer 4

Square of the difference between the estimated and the mean value Σ(ŷi - ȳ)²

Question 5

What is the residual sum of squares (RSS)?

Answer 5

Square of the difference between the observed and estimated value Σ(yi - ŷi)²

Question 6

What is the total sum of squares (TSS)?

Answer 6

Explained sum of squares + Residual sum of squares
Σ(yi - ȳ)²

Question 7

What is R squared?

Answer 7

Variation of Y that is explained by the regression

Question 8

What is R squared in terms of sum of squares?

Answer 8

ESS/TSS = 1 - (RSS/TSS)

Question 9

What is the main flaw of R squared?

Answer 9

It increases as the number of covariates in the model increases, even if they are nonsensical variables

Question 10

What is adjusted R squared?

Answer 10

It takes into account the number of parameters

Question 11

What is the formula for adjusted R squared?

Answer 11

1 - (n-1)(1 - R²)/n-K

Question 12

What is the null hypothesis?

Answer 12

Typically a statement that we do not expect (e.g. the coefficient is 0)

Question 13

What is the alternative hypothesis?

Answer 13

Typically a statement that we expect (e.g. coefficient is not equal to 0)

Question 14

What is the decision rule?

Answer 14

Method of deciding whether to reject the null hypothesis or not. This is done by comparing the sample statistic to the critical value

Question 15

What is the critical value?

Answer 15

Threshold value. If the sample statistic is greater than (less than) the critical value we reject (do not reject) the null hypothesis

Question 16

What is a Type 1 (false-positive) error?

Answer 16

When an investigator rejects a null hypothesis that is actually true in the population

Question 17

What is a Type 2 (false negative) error?

Answer 17

When an investigator fails to reject a null hypothesis that is actually false in the population

Question 18

What does the t-test check?

Answer 18

If the coefficients are significant i.e. non-zero

Question 19

What are degrees of freedom for a t test?

Answer 19

Observations minus parameters (n-K)

Question 20

What is the formula to run a t-test?

Answer 20

tₗ = ( ̂βₗ - H₀)/√( ̂vₗₗ)

Question 21

What is ̂vₗₗ in the t-test?

Answer 21

The relevant element from the variance matrix

Question 22

What are the 3 steps for a hypothesis test?

Answer 22

1. Get the critical value from t table
2. Plug given values into t test equation
3. If t value is greater than critical value, reject null

Question 23

What are the 3 steps for the global significance test?

Answer 23

1. Get critical value from fisher distribution
2. Plug numbers into formula
3. If result greater than critical, reject null

Question 24

What does the global significance test check?

Answer 24

Overall significance of model

Question 25

How can you find an upper and lower bound for an uncertain β based on a range of p values

Answer 25

-Figure out the lowest/highest p value the t stat must be greater/less than
-Form 2 inequalities using the t test formula
-Rearrange for an upper and lower value of β

Question 26

What is RSSᵣ & RSSᵤ?

Answer 26

RSS with restriction from null hypothesis & unrestricted original RSS

Question 27

What is the formula for an F-test?

Answer 27

[(RSSᵣ-RSSᵤ)/m] ÷ [RSSᵤ/(n-k)]

Question 28

What types of parameters can we not test using t or F tests?

Answer 28

Polynomial parameters; the tests only work on linear hypotheses

Question 29

When could you test regression coefficients using critical values from the normal distribution, not from the t distribution?

Answer 29

If n is extremely large, it might be appropriate to use the normal distribution, because the t(n-k) distribution converges on N(0,1) when n→∞

Question 30

What do you need to know to conclude whether a t-test is appropriate?

Answer 30

The distribution