QA8 - Regression with Multiple Explanatory Variables

Question 1

Distinguish between the relative assumptions of single and multiple regression

Answer 1

Multiple regression assumes that explanatory variables are not perfectly linearly dependent

Question 2

Interpret regression coefficients in a multiple regression

Answer 2

Have to hold all other variables constant then standard interpretation

Question 3

Interpret goodness-of-fit measures for single and multiple regressions, including R^2 and adjusted R^2

Answer 3

TSS = ESS + RSS
sum(yi - ybar)^2 = sum(yhat_i - ybar)^2) + sum(errors^2)

R^2 = ESS / TSS = 1 - (RSS / TSS)
Percentage of variation in the data explained by the model

Adjusted R^2 = 1 - ((n-1)/(n-k-1)) * (1 - R^2)
Penalises additional parameters in the model

Question 4

Construct, apply, and interpret joint hypothesis tests and confidence intervals for multiple coefficients in a regression

Answer 4

F = ((RSSr - RSSu)/q) / (RSSu /(n - k_u - 1)) ~ Fq,(n -k_u -1)

Where RSS is the restricted model and u is the unrestricted model