# Regression with multiple (Y) values

I have several (x) variables which all come up to the total cost and in some way tell us what the profit margin (y) should be.

Some of the (x) variables are:
Weight
Gallons
# setups
# specifications
EAU
Inches
Sq ft
Run cost
Setup cost
Material cost
Paint cost
Purchase cost

I have about 100 items in my data set and know how to create a model based on multiple regression which will help me predict what the profit margin should be.  Problem we have is that we want to know how much margin we can charge for each of the following:
Run cost
Setup cost
Material cost
Paint cost
Purchase cost

We know run cost will be at 20% margin, and purchase will be at 30%.  Problem is that we don’t know what type of margin the other elements should be at given the response of the other x variables.  Is there any way I can calculate the result of multiple (y) variables, so I can see what my “subitems” of cost (ie. Run, setup, material, paint, and purchase) cost can be at for margin.  Maybe I am way off base with this or maybe I need to know some other factors to determine this.  We are trying to determine what the “market will bear” for these items.

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Commented:
what data do you have and what are the multiple y values?
100 data points is not a lot when you have 12 x variables to regress on.
Do you expect your y values to be linear functions of your x values?
Commented:
"Maybe I am way off base with this or maybe I need to know some other factors to determine this.  We are trying to determine what the “market will bear” for these items."
-
if  your desire to determine "what the market will bear" you do not need individual margins for each of your 100s of factors.
"The Market" does not care what margin you determine for something like inches. It is concerned ONLY with the final price. Markups in the detail for which you looking is but a crutch for estimators.
There IS no mark up for inches.
Find your costs so you do not lose money. Apply a company standard markup which you adjust for "What the Market will bear". Adjust the mark up depending on your experience.

Author Commented:
ozo - yes, we expect the y values to be linear functions of the x values.
aburr - based on what you said, I that we can scrap the Weight
Gallons
# setups
# specifications
EAU
Inches

We are still left with run, setup, material, paint, purchase cost.  Some of the purchase parts are highly specialized in the market and we know we can get 30, 35, 40% margin on those categories.  We need to know how to adjust profit margin % amongst run, setup, material, paint, and non-specialized purchase parts when some jobs have high setup content, or lots of inches, or weigh alot, etc.  Is this possible or what approach would you take?
Commented:
given a set of data points of the form
(x0,x1,x2,...,y)
a linear regression can find parameters a0,a1,...
that minimizes the sum of the errors
a0*x0 + a1*x1 +... - y

If there are few data points and many free parameters,
that minimum can often be a statistical artifact that is not a very robust
predictor of other points.
Applying domain specific knowledge, such as a model that predicts that certain x variables should be ignored, may allow the regression to give more meaningful results.
Multiple y variables can each be handled the same way
(unless you have reason to believe that the y variables are related to each other,
in which case you would need to incorporate that into your model)

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