I was curious about: Why do we write errors (or disturbance term) $\varepsilon$ as a additive term in a regression model? To elaborate, whether we consider a paramteric or non-paramteric regression model, given by: $$Y_i = m(x_i)+\varepsilon_ i \ \ \ldots \ i=1,\ldots,n$$ where, m(.) is the unknown regression function and the errors are given by $(\varepsilon_1,\ldots,\varepsilon_n)$
Why does this error term come as an additive random variable in a regression model. Couldn't error be written as a multiplicative, like $$Y_i = m(x_i)\varepsilon_ i \ \ \ldots \ i=1,\ldots,n$$
or as an exponent:
$$Y_i = m(x_i)^{\varepsilon_ i} \ \ \ldots \ i=1,\ldots,n$$
Or in any other non-linear form? Is there a particular reason we always think that the true response ($Y_i$) is a sum of some value plus some noise?