x <- c(16,20,24,28,32,36,40,44,48,52)
y <- c(14,13,18,19,22,23,24,28,30,29)

model <- lm(y~x)   #  linear model  (linear regression, OLS)
summary(model)

plot(x,y,xlab="Income",ylab="Consumption")
abline(model)

# Comment-out
# OLS via Linear Algebra
y <- cbind(y)
y

X <- cbind(rep(1,10),x)   # column combine #rep replicate  # t transpose
X                         # %*%  matrix multiplication # solve matrix inverse

bOLS <- (solve(t(X)%*%X))  %*%  (t(X)%*%y)
bOLS


t(X)%*%X
solve(t(X)%*%X)
t(X)%*%y


> t(X)%*%X
          x
   10   340
x 340 12880

> solve(t(X)%*%X)
                          x
   0.97575758 -0.0257575758
x -0.02575758  0.0007575758

> t(X)%*%y
     y
   220
x 8120

> bOLS
          y
  5.5151515
x 0.4848485


# Example of a LOOP with iterations
sumy <- 0
for(i in 1:length(y)){
  sumy <- sumy + y[i]
}
sumy