Polynomial regression

Question 1

New linearity assumption for polynomial regression

Answer 1

E[Yi|Zi] = f(zi|β₀,…β_l) = Σ_j=0^l β_j z_i^j = Σ_j=0^l β_j Bj(zi) BASES

Question 2

ML estimation of polynomial regression

Answer 2

β^= (Z’Z)^-1 Z’Y
Since columns highly correlated, better use orthogonal polynomials that will not however change the conclusions nor the fitted model

Question 3

Hypothesis testing for nested polynomial regression models

Answer 3

F = ΔSSE/SSE * (n-d)/r| H0 ∼ Fr,n-d
d = l+1

Question 4

Polynomial regression in R

Answer 4

p = poly(z, degree=l, raw=F) #If =T, not orthogonal
fit = lm(y~p) #l+1 parameters
anova(fitmin, fitmax)
lht(fitmax, C, d, test=”F”)