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# Aufgabe 1 #
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MyRegression = function(y,X)
{
  # A beta = c
  A = t(X)%*%X
  Ainv = solve(A)
  c = t(X)%*%y
  betas = Ainv%*%c
  fittedvalues = X%*%betas
  errors = y - fittedvalues
  Rsquared = var(fittedvalues)/var(y)

  result = list(betas,fittedvalues,errors,Rsquared)
  names(result) = c("betas","fittedvalues","errors","Rsquared")

  plot(y,type="l")
  lines(fittedvalues,col="red")
  lines(errors,col="yellow")

  return(result)
}


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# Aufgabe 2 #
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x = (-10):10
eps = rnorm(21)
y = 5 - 0.5*x + eps
plot(x,y)

# 2a)
reslm = lm(y ~ x)
summary(reslm)
reslm$fit
reslm$res
plot(x,y,type="l")
lines(x,reslm$fit,col="red")
lines(x,reslm$res,col="yellow")

# 2b)
n = length(y)
x0 = rep(1,n)
X = cbind(x0,x)
resmy = MyRegression(y,X)
resmy$betas
resmy$fit
resmy$err
resmy$Rsq