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?

The parameter needs to enter linearly into the equation. So something like $E(Y)=beta_1 cos(beta_2 x_i + beta_3)$ would not qualify. But you can take functions of the independent variables as follows:

$E(Y)=beta_0 + beta_1X_i + beta_2X^2 + beta_3 e^{X_i}$

for example.

So the limits of linear regressions are: the mean of the $Y$ values is of the form parameter times (independent variable stuff) + parameter times (more independent variable stuff) … and so on.

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