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I looked around online and couldn't find anything that was answering my question so I thought I would take to the stack!

I'm interested in knowing if there is a statistical or mathematical way of optimizing ones dependent variable in regression analysis by shifting the values of your independent variables. For instance in market mix modeling generally the regression equation looks something like:

$$\text{Sales = TV Spend + Print Spend + Online Spend + Radio Spend}$$

So once you have your model based on the data how do you optimize the spend levels to have the max sales without just randomly shifting spend from one category to another?

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Regression models are not created to do what you want. For example, if you had each of those 4 variables in the model like this: $$y=b0+b1*x1+b2*x2+b3*x3+b4*x4$$ then we can increase the predicted y to infinity by increasing any x variable that has a positive coefficient. Furthermore, we cannot fit the model in one range of observed x values and then use x values way out of that range to make predictions. The indep variables cannot be viewed in isolation -- if you advertise on radio, the impact of additional TV ads will have diminished impact so you need an interaction model. And something to include a decreasing marginal impact of ads in any one area. e.g, A(1-exp(-c*x)) If we created this highly non=linear model that actually described what was going on, then you have a math optimization model (nonlinear programming) to solve: maximize profit from advertizing (must take into account the cost of ads) subject to money < some number and maybe restrictions on min or max money on each medium.

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  • $\begingroup$ but in this case spend is a finite amount say each of the four x variables sums to equal 100% of the total ad spend. For example TV 45%, Print 20%, Radio 20% and Online 15% is the current mix but in fact the optimal mix is really TV 15% Print 30%, Radio 10% and Online 45%? Does this type of optimization still require an exponential model? $\endgroup$
    – moku
    Jul 19, 2014 at 15:06
  • $\begingroup$ yes of course you need a model with a budget constraint as I indicated. But with a linear model as in regression the solution with be 100% in whatever kind of ads have the best coefficient. Trivial solution. Need a more realistic economic model. Need to take into account that spending all money on tv, say, it not very effective. After watching 99 ads for a Big Mac i am already eating them or ignoring it. Putting that type of constraint in the model will keep the answer from being 100% for one variable. $\endgroup$
    – Mr.Spot
    Jul 19, 2014 at 21:46
  • $\begingroup$ Regression model can help estimate some costs and benefits involved but you need a different type of model. Are you familiar with Math Programming models? In this case, Linear Programming is too limited so you'll need a nonlinear optimization model. You need a lot of economic insight into the problem and some mathematical ability to choose the type of mathematical functions to be used. Anyone can build a simple model, but building a realistic model is part art and part science. I am sure models have been constructed for this exact case by experts. Look in academic journals. $\endgroup$
    – Mr.Spot
    Jul 19, 2014 at 21:58

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