4- Hypothesis testing Flashcards
What does ^βols tell us?
Marginal effect on the dependent variable for a unit change in the covariate
What do hats denote in regressions?
Estimated values, once we have estimated values of beta, the regression is now called the estimated regression equation
What is the estimated error (û) in terms of y?
û = y - ŷ
What is the explained sum of squares (ESS)?
Square of the difference between the estimated and the mean value Σ(ŷi - ȳ)²
What is the residual sum of squares (RSS)?
Square of the difference between the observed and estimated value Σ(yi - ŷi)²
What is the total sum of squares (TSS)?
Explained sum of squares + Residual sum of squares
Σ(yi - ȳ)²
What is R squared?
Variation of Y that is explained by the regression
What is R squared in terms of sum of squares?
ESS/TSS = 1 - (RSS/TSS)
What is the main flaw of R squared?
It increases as the number of covariates in the model increases, even if they are nonsensical variables
What is adjusted R squared?
It takes into account the number of parameters
What is the formula for adjusted R squared?
1 - (n-1)(1 - R²)/n-K
What is the null hypothesis?
Typically a statement that we do not expect (e.g. the coefficient is 0)
What is the alternative hypothesis?
Typically a statement that we expect (e.g. coefficient is not equal to 0)
What is the decision rule?
Method of deciding whether to reject the null hypothesis or not. This is done by comparing the sample statistic to the critical value
What is the critical value?
Threshold value. If the sample statistic is greater than (less than) the critical value we reject (do not reject) the null hypothesis
What is a Type 1 (false-positive) error?
When an investigator rejects a null hypothesis that is actually true in the population
What is a Type 2 (false negative) error?
When an investigator fails to reject a null hypothesis that is actually false in the population
What does the t-test check?
If the coefficients are significant i.e. non-zero
What are degrees of freedom for a t test?
Observations minus parameters (n-K)
What is the formula to run a t-test?
tₗ = ( ̂βₗ - H₀)/√( ̂vₗₗ)
What is ̂vₗₗ in the t-test?
The relevant element from the variance matrix
What are the 3 steps for a hypothesis test?
- Get the critical value from t table
- Plug given values into t test equation
- If t value is greater than critical value, reject null
What are the 3 steps for the global significance test?
- Get critical value from fisher distribution
- Plug numbers into formula
- If result greater than critical, reject null
What does the global significance test check?
Overall significance of model
How can you find an upper and lower bound for an uncertain β based on a range of p values
-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
What is RSSᵣ & RSSᵤ?
RSS with restriction from null hypothesis & unrestricted original RSS
What is the formula for an F-test?
[(RSSᵣ-RSSᵤ)/m] ÷ [RSSᵤ/(n-k)]
What types of parameters can we not test using t or F tests?
Polynomial parameters; the tests only work on linear hypotheses
When could you test regression coefficients using critical values from the normal distribution, not from the t distribution?
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→∞
What do you need to know to conclude whether a t-test is appropriate?
The distribution
