One question regarding to the simple regression

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    Regression
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Discussion Overview

The discussion revolves around the relationships between parameters in a simple regression involving three variables: X, Y, and Z. Participants explore the conditions under which the relationship h = b/d holds, particularly in the context of estimated parameters versus actual parameters.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes the relationship Y = a + b*X + errorTerm and Z = c + d*X + errorTerm, leading to the question of how h relates to b and d.
  • Another participant suggests that if h, b, and d are actual parameter values, then h = b/d can be derived under certain conditions, but this may not hold if they are estimated parameters.
  • A third participant questions whether the parameters are known or estimated and asks about the specific distribution or analysis being considered.
  • The original poster clarifies that all parameters are estimated and expresses uncertainty about the validity of using h = b/d in this context, acknowledging the dependence on assumptions and estimations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether h = b/d can be applied when using estimated parameters, highlighting the complexity and uncertainty involved in the estimation process.

Contextual Notes

Participants note that the validity of the relationship h = b/d may depend on whether all assumptions of the regression are met, which remains unresolved in the discussion.

liujx80
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there are 3 variables. X,Y,Z.
Assuming
Y = a + b*X + errorTerm
Z = c + d*X + errorTerm,

Y = g + h*Z + errorTerm;

what can we say about the relationtion between h and (b,d)? under what
condition so we can have "h = b/d"?

thanks
 
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If h, b and d represent the actual parameter values then clearly you have:
Y = a + b*X + errorTermxy
Z = c + d*X + errorTermxz,

Y = g + h*(c + d*X + errorTermxy) + errorTermxz;
whence:
g = a - h*c
h = b/d
errorTermzy = h*errorTermxy + errorTermxz
But perhaps you mean b, d and h to be the estimated parameters. That gets more subtle.
 
Hey liujx80 and welcome to the forums.

Following on from haruspex's post above, are you trying to estimate them or do you know them already? If you are estimating them, do you have a specific distribution or analysis in mind? Do you have priors?
 
Thanks guys. The parameters are all estimated. I don't have distributions in mind. In general cases, can I use h=b/d using estimates? How wrong it could be? I had the same steps with haruspex's. However I'm not sure we then can say all assumptions are met. As you guys pointed out, it depends on estimations, which I have no clue .

Thanks
 

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