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Thread: Estimated Regression Equation

  1. #1
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    Canada
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    Estimated Regression Equation

    Hi,

    I like to get some help with the part c of the question below:

    X1 X2 Y
    30 12 94
    47 10 108
    25 17 112
    51 16 178
    40 5 94
    51 19 175
    74 7 170
    36 12 117
    59 13 142
    76 16 211

    For this question I had to the do the following:

    a.) Develop an estimated regression equation relating y to x1.

    I understand this part and got equation: y = 45.06 + 1.94x1

    b.) Develop an estimated regression equation relating y to x2.

    I also understand this part and got the equation: y = 85.22 + 4.32x2

    c.) Develop an estimated regression equation relating y to x1 and x2.

    I can't seem to figure out this part, As per the solution the equation is: y = -18.37 + 2.01x1 + 4.74x2

    I fail to understand how solution arrives at -18.37, 2.01 and 4.74

    Your help is greatly appreciated.

    Thanks
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  2. #2
    Senior Member
    Joined
    Mar 2008
    From
    Pennsylvania, USA
    Posts
    334
    Thanks
    45

    Re: Estimated Regression Equation

    Quote Originally Posted by scorpio2017 View Post
    Hi,

    I like to get some help with the part c of the question below:

    X1 X2 Y
    30 12 94
    47 10 108
    25 17 112
    51 16 178
    40 5 94
    51 19 175
    74 7 170
    36 12 117
    59 13 142
    76 16 211

    For this question I had to the do the following:

    a.) Develop an estimated regression equation relating y to x1.

    I understand this part and got equation: y = 45.06 + 1.94x1

    b.) Develop an estimated regression equation relating y to x2.

    I also understand this part and got the equation: y = 85.22 + 4.32x2

    c.) Develop an estimated regression equation relating y to x1 and x2.

    I can't seem to figure out this part, As per the solution the equation is: y = -18.37 + 2.01x1 + 4.74x2

    I fail to understand how solution arrives at -18.37, 2.01 and 4.74

    Your help is greatly appreciated.

    Thanks
    My favorite method for two independent variables is to solve for two unknowns in two equations using covariances.

    \begin{equation}Cov(Y,X_1) = b_1 Cov(X_1,X_1) + b_2 Cov(X_2,X_1) \end{equation}
    \begin{equation}Cov(Y,X_2) = b_1 Cov(X_1,X_2) + b_2 Cov(X_2,X_2) \end{equation}

    Remember that you can compute covariance using Cov(A,B)=E[AB]-E[A]E[B].

    Also note that Cov(A,A)=V(A), and make sure to use the population variance (varp in Excel).

    After computing b_1 and b_2, you can solve for b_0 using the regression line equation:
    \begin{equation} (Y-\bar{Y}) = b_1(X_1-\bar{X_1}) + b_2(X_2-\bar{X_2}) \end{equation}

    By hand, this is is rather cumbersome, but it works! Good luck!
    -Andy
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