# One question regarding to the simple regression

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

## Answers and Replies

haruspex
Science Advisor
Homework Helper
Gold Member
2020 Award
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.

chiro
Science Advisor
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