# Solved – What can’t be expressed as a linear model

Say I have outcome variable \$Y_i\$ and predictors \$X_{i1}\$ and \$X_{i2}\$ for some data point \$i\$. Wikipedia says that a model is linear when:

the mean of the response variable is a linear combination of the parameters (regression coefficients) and the predictor variables.

I thought this meant that a model can be no more complicated than: \$Y_i = beta_1 X_{i1} + beta_2 X_{i2}\$. However, upon further reading, I found out you could handle non-linear "interactions" of the predictors like in \$Y_i = beta_1 X_{i1} + beta_2 X_{i2} + beta_3 X_{i1} X_{i2}\$ by viewing \$X_{i1} X_{i2}\$ as just another predictor (which happens to be dependent on \$X_{i1}\$ and \$X_{i2}\$). This seems to mean that you can use any (linear or non-linear) function of the predictors like \$log(X_{i1} / X_{i2}^2)\$ or whatever. Conceptually, this type of "recoding" seems like it should work for the coefficients as well.

So: What exactly are the limits of linear regression, given you can do this kind of manipulation?

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