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Math Help - Linear Regression Models (3)

  1. #1
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    Linear Regression Models (3)







    3) "A linear regression model is of the form:
    Y = β0 + (β1)(X1) + (β2)(X2) + ... + (βk)(Xk) + ε
    If there is more than one independent variable, then the model is called a MULTIPLE linear regression model."

    This idea doesn't seem too clear to me. Here, what can the Xi's be? What are some specific examples of mutliple linear model? Does a linear model always have to be a straight line or a plane?

    Thanks for clearing my doubts!

    β
    ε

    note: also under discussion in "Talk Stats Forum"
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  2. #2
    MHF Contributor matheagle's Avatar
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    LINEAR in BETAs, not x's

    Y=\beta x^5+\epsilon is linear in BETA

    Y=\beta_0\beta_1 x+\epsilon is NOT linear in BETA

    NOR is Y=\beta^2_0 +\beta_1 x+\epsilon

    but here I would let \beta_0^*=\beta^2_0 to make this linear in beta's.


    AND USE MATRICES TO SOLVE THESE. Wackerly does a decent job, except at the end with the F tests.
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  3. #3
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    OK, I think the trickiest point to notice when I first read through the definition of linear regression model is that it is linear in β's while in calculus, when we talk about linear, we are usually trying to say that the function is linear, i.e. straight line, plane.

    I have some more questions regarding MULTIPLE linear regression models...

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  4. #4
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    3) "A linear regression model is of the form:
    Y = β0 + β1*X1 + β2*X2 + ... + βk*Xk + ε "


    (i) Y = β0 + β1*X + β2*exp(X) + ε
    (ii) Y = β0 + β1*X1 + β2*X2 + β3*(X1*X2) + β4*(X1)^2 + β5*(X2)^2 + ε

    For (i), X1=X, X2=exp(X)
    For (ii), X3=X1*X2, X4=X1^2, X5=X2^2
    The latter X's depends on the previous X's. In particular, X3 depends on TWO of the previous X's: X1 AND X2, which looks a bit funny to me? Are those allowed? Somehow I am having a lot of troubles understanding this...I understand the general form of a multiple linear regression model, but I don't seem to understand the specific examples of it like (i) and (ii).
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