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What can't be expressed as a non-linearlinear model?

Say I have outcome variablesvariable $$Y_i$$ 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 happenhappens 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 $$log(X_{i1} / X_{i2}^2)$$$$\log(X_{i1} / X_{i2}^2)$$ 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?

What can't be expressed as a non-linear model?

Say I have outcome variables $$Y_i$$ 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 happen 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?

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?

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