One question regarding to the simple regression

[liujx80]

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

 

[haruspex]

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

Y = g + h*(c + d*X + errorTerm_xy) + errorTerm_xz;
whence:
g = a - h*c
h = b/d
errorTerm_zy = h*errorTerm_xy + errorTerm_xz
But perhaps you mean b, d and h to be the estimated parameters. That gets more subtle.

 

[chiro]

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?

 

[liujx80]

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