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When there is a variable dependent only on one independent variable, we can use a least-squares method to find a function that best fits the data. What if our dependent variable is dependent now on two or more independent variables? Let's say that I am running a company and have two expenses: employee pay & utilities. My dependent variable is "total expenses". Suppose, though, that I rename this variable and send my expense data off to a statistics company just to see if they can figure out what this variable represents. How might they figure it out?

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  • $\begingroup$ You can also use least squares if you have more than one independent variable (multivariate regression mode). A simple model should give two coefficients of 1, even though I'm not sure it's possible to compute the model, since given the independent variables, you can perfectly predict the dependent variable. $\endgroup$ – tho_mi Dec 8 '15 at 1:51
  • $\begingroup$ I'll adjust my answer. The coefficients won't ve exactly one, since it depends on the distribution of the spending categories across the observations. It could be the case that the estimates are suspiciously precise. $\endgroup$ – tho_mi Dec 8 '15 at 1:55
  • $\begingroup$ Note that an equation like "total expense = a + b×employee pay + c×utilities + noise" is linear. Are you really asking about nonlinear relationships? $\endgroup$ – Glen_b Dec 8 '15 at 2:40
  • $\begingroup$ @glen_b Yeah, I guess. $\endgroup$ – moonman239 Dec 9 '15 at 7:10
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You don't need a non-linear method here: We can just use the least squares method with more than one predictor, so you could have:

$ Y=\beta_0+\beta_1\times\mathrm{Employee\ Pay}+\beta_2\times\mathrm{Utilities} $.

We could try a slightly more flexible least squares regression by testing to see if there is a synergy between Employee Pay and Utilities:

$ Y=\beta_0+\beta_1\times\mathrm{Employee\ Pay}+\beta_2\times\mathrm{Utilities}+\beta_3\times\mathrm{Employee\ Pay}\times\mathrm{Utilities}$

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