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Thread: Change regression model ($x^*_i = x_i -10$)

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    Change regression model ($x^*_i = x_i -10$)




    Hi there.
    I'm solving an exercise on multiple linear regression. Near the end I will be asked for the same data as the previous model, it is the maximum likelihood estimates.

    The previous model:
    I have the matrix (X'X)^{-1} and the matrix X'y and the model is:

    Y_i = B_0 + B_1x_i + B_2x^2_i + e_i

    i= 1,...,10

    Now I have:

    Y_i = g_0 + g_1x^*_i + g_2(x^*_i)^2 + e_i

    x^*_i = x_i -10

    i=1,...,10
    To transform the model can I decrease the Matrix data for 10?

    The same thing can I make it also for the matrix X'y?
    I have many doubts.
    Publishing the text so it is more understandable.

    Consider the regression model linear:

    Y_i = B_0 + B_1x_i + B_2x^2_i + e_i

    with e_1, ...., e_n independent and identically distributed random variables and x_i, i = 1, ..., 0 constant fix.

    The only data I have are these:

    https://s32.postimg.org/o20r1i5et/Immagine.png

    I have to rewrite the whole thing ... I tried so:

    (X) = 
  \begin{bmatrix}
    1 & x_1 & x^2_1\\
    . & . & . \\
    . & . & . \\
    . & . & . \\
    1 & x_{10} & x^2_{10}\\
  \end{bmatrix}

    (X^*) = 
  \begin{bmatrix}
    1 & x_1-10 & (x_{1}-10)^2\\
    . & . & . \\
    . & . & . \\
    . & . & . \\
    1 & x_{10}-10 & (x_{10}-10)^2\\
  \end{bmatrix}

    X^{*'}X^* = 
  \begin{bmatrix}
    10 & \sum_{i=1}^{10} x_i-10 & \sum_{i=1}^{10}(x_{i}-10)^2\\
    \sum_{i=1}^{10} x_i-10 & \sum_{i=1}^{10} (x_{i}-10)^2 & \sum_{i=1}^{10}(x_{i}-10)^3\\
     \sum_{i=1}^{10} (x_{i}-10)^2 & \sum_{i=1}^{10}(x_{i}-10)^3 & \sum_{i=1}^{10}(x_{i}-10)^4\\
  \end{bmatrix}

    Now I have to calculate the inverse? I'm following proper solution?


    Excuse me , but does not write well in English
    Thanks.

  2. #2
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    Re: Change regression model ($x^*_i = x_i -10$)

    Are you just trying to find the estimates of the parameters for the model using the transformed data? I would just figure out what the new parameters are in terms of the old parameters and then use the invariance property of MLEs to get the estimates.
    I don't have emotions and sometimes that makes me very sad.

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    Re: Change regression model ($x^*_i = x_i -10$)

    The question is:
    Consider , for the same data , the model

    Y_i = g_0 + g_1x^*_i + g_2(x^*_i)^2 + e_i with x ^* _i = x_i - 10 , i = 1 , . . . , 10 , independent random variables and identical cally distributed N ( 0 , σ2 ) . It is the maximum likelihood estimates γ0 , γ1 , γ2.
    Last edited by valentina89; 06-24-2016 at 02:36 AM.

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    Re: Change regression model ($x^*_i = x_i -10$)

    Quote Originally Posted by Dason View Post
    and then use the invariance property of MLEs to get the estimates.

    In this case the estimate remains the same?

    B_0 = g_0 ecc..
    I do not have a real transformation of the initial parameters , but only a "data processing " . What I did is right ?

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    Re: Change regression model ($x^*_i = x_i -10$)

    Consider just a simple linear regression and the transformation you had. Our original model would be
    E[y] = \beta_0 + \beta_1x_i

    With x_i^* = x_i - 10 our model is

    E[y] = \beta_0^* + \beta_1^*x_i^*

    Replace with our definition of x_i^* to get

    E[y] = \beta_0^* + \beta_1^*(x_i - 10)

    Expand and rearrange and we have

    E[y] = \beta_0^* - 10\beta_1^* + \beta_1^*x_i

    Now since we already know E[y] = \beta_0 + \beta_1x_i

    we can now say that

    \beta_0 = \beta_0^* - 10\beta_1^*
    and
    \beta_1 = \beta_1^*

    So using this (since we're assuming we know the values for [math]\beta_0[math] and \beta_1 we can solve for the new coefficients. It's the process with your problem but since you have a quadratic the equations take a little bit more work.
    I don't have emotions and sometimes that makes me very sad.

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    Re: Change regression model ($x^*_i = x_i -10$)


    Wow!

    Thank you so much.
    I try now to do the exercise right!

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