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### Detecting multicollinearity with Longley Data (Table 10.8)
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## Load the data
longley <- read.table("Table_10_8.csv", header = TRUE)

## Simple regression
longleyLM <- lm(Y~X1+X2+X3+X4+X5, data = longley)
summary(longleyLM) # Hight R2 but few variables are significant,  what was the
                   # problem?


## Check what went wrong with the covariates
X <- as.matrix(longley[, 3:7]) # The covariates matrix
n <- nrow(longley)
k <- ncol(X) + 1

## Check the correlations
cor(X) # Correlation
pairs(X) # Pairwise plot

## Check the eigenvalues
longleyEigen <- eigen(t(X)%*%X)
longleyEigenVec <- longleyEigen$values

## The conditional number
max(longleyEigenVec)/min(longleyEigenVec) # what is your conclusion?


## Study the VIF
betaVCOV <- vcov(longleyLM) # variance covariance matrix
betaVAR <- diag(betaVCOV)
sigma2hat <- sum((longleyLM$residuals)^2)/(n-k)

VIF1 <- betaVAR[2]*sum((X[, 1]-mean(X[, 1]))^2)/sigma2hat
VIF2 <- betaVAR[3]*sum((X[, 2]-mean(X[, 2]))^2)/sigma2hat
VIF3 <- betaVAR[4]*sum((X[, 3]-mean(X[, 3]))^2)/sigma2hat
VIF4 <- betaVAR[5]*sum((X[, 4]-mean(X[, 4]))^2)/sigma2hat
VIF5 <- betaVAR[6]*sum((X[, 5]-mean(X[, 5]))^2)/sigma2hat