Let's say we have two random variables Y and X used to form regression model Y = α + β X + μIt also holds that E ( μ ) = 0, Var ( μ ) = σ μ 2 , Var ( X ) = σ X 2 , Var ( Y ) = σ Y 2 , Corr ( X , Y ) = r and Corr ( X , μ ) = r X μ . Find β. I tried to solve this as follows:For simple linear regression β = Cov ( X , Y ) Var ( X ) and Corr ( X , Y ) = Cov ( X , Y ) σ X σ Y = r so that: β = Corr ( X , Y ) ⋅ σ X σ Y σ X 2 = r σ Y σ X Is this as simple as this?

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Let&#039;s say we have two random variables   Y and   X used to form regression model  Y  =  α  +  β  X  +  μIt also holds that   E  (  μ  )  =  0,   Var  (  μ  )  =      σ      μ    2  ,   Var  (  X  )  =      σ      X    2  ,   Var  (  Y  )  =      σ      Y    2  ,   Corr  (  X  ,  Y  )  =  r and   Corr  (  X  ,  μ  )  =      r          X      μ      . Find   β. I tried to solve this as follows:For simple linear regression   β  =                    Cov        (        X        ,        Y        )                    Var        (        X        )             and   Corr  (  X  ,  Y  )  =                    Cov        (        X        ,        Y        )                              σ          X                          σ          Y                      =  r so that:  β  =            Corr      (      X      ,      Y      )      ⋅              σ        X                    σ        Y                    σ      X      2        =  r            σ      Y              σ      X      Is this as simple as this?

Jake Mcpherson · Accepted Answer

Since you probably want the answer for the true   β given all moments let&#039;s be detailed.  E  (  Y  )  =  α  +  β  E  (  X  )    X  Y  =  X  α  +  β      X    2    +  X  μ    E  (  X  Y  )  =  E  (  X  )  α  +  β  E  (      X    2    )  +  E  (  X  μ  )    C  o  v  (  X  Y  )  =  E  (  X  Y  )  −  E  (  X  )  E  (  Y  )  =  E  (  X  )  α  +  β  E  (      X    2    )  +  E  (  X  μ  )  −  E  (  X  )  α    −  β  E  (  X      )    2    =  β  V  a  r  (  X  )  +  C  o  v  (  X  μ  )    S  o    r      σ    X        σ    Y    =  β      σ    X    2    +      r            X          μ            σ    X        σ    μ    β  =            r              σ        Y            −              r                        X                  μ                            σ        μ                    σ      X

Let's say we have two random variables Y and X used to form regression model Y =

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