Question

topic: model selection on applied linear regression

Exercise 5.5.3 LetY (6,8,9,4,4,4,4,4), X1 (3,0,6,2,4,7,0,0), X2 Consider the regression model Y-k) + Xi A +X2β2+ e, e ~ N (0 (3,0,6,2,4,7,7,0) , σ2 18). i) Find the VIFs for Xi and X2. ii) Estimate β1, β2 and find the variances of the estimates in terms of σ2 iii) Estimate σ2. iv) Find X3, which is a unit vector in the span of Xi,X2 but is orthogonal to X2 (Hints: consider (In-Ho)Xi for some Ho, and standardize.) v Repea-) by replacing X1 with X3.

Answer 1

1. Vif values of X1 and X2 are:

> vif (model) x1 x2 1.609938 1.609938

2. Estimates of B1 and B2 and estimated standard deviation

Coefficients: Estimate Std. Error t value Pr( ltI (Intercept) 5.4713 1.41743.860 0.0119 x1 x2 0.2420 -0.2102 0.4081 0.593 0.5789 0. 3923-0. 536 0. 6151

3. Estimated standard deviation: 2.352

Residual standard error: 2.352 on 5 degrees of freedom Multiple R-squared: 0.07385, Adjusted R-squared -0.2966 F-statistic: 0.1993 on 2 and 5 DF, p-value: 0.8255

# R codes:

library(car)

# Creating dataset

y = c(6,8,9,4,4,4,4,4)
x1 = c(3,0,6,2,4,7,0,0)
x2 = c(3,0,6,2,4,7,7,0)
data= data.frame(x1,x2,y)

# Building model
model = lm(y~.,data)
summary(model)
vif(model)