TITLE: How reasonable is the linearity assumption in regression analysis?
QUESTION [5 upvotes]: As someone who moved from physics to economics, there are some things about regression analysis that have always bothered me. One of the most basic assumptions in econometrics is that for studying the effect of an independent variable $x$ on a dependent variable $y$, we can assume the following linear relationship between them:
$$y = \alpha + \beta x + e$$
Where $e$ is an error term. Coefficients $\alpha$ and $\beta$ are then estimated by minimising the least square errors. 
From a purely mathematical perspective, how reasonable is this assumption? If you have any function $y=f(x)$, if the domain of $f$ is restricted to a small number of real values (as it normally is in economics), when is it fine to assume linearity?
Also, is the nature of the variables in economics one of the reasons for this? I mean is it more appropriate to assume linearity when say $y$ is wage and $x$ is years of education than in the model of a physical system?

REPLY [2 votes]: Further to the points in the existing answer, note that any model of the form $y=\sum_i \alpha_i f_i(x)+e$ is amenable to these techniques, even if the $f_i$ are complicated, even if $x$ is multivariate, and even if the variable we call $y$ is obtained by transforming something else. For example, taking logarithms makes power laws amenable to this technique.
Power laws are the main focus of the rest of my answer. Calculus often legitimizes linearization in a "small" range of values for variables. But this is especially feasible for (exact or approximate) power laws, since then the plotted variables can be chosen as the logarithms of things we started with. That's what happens, for example, in stellar physics.
Using a vertical log-axis is part of why power laws in economics and econometrics are sometimes (e.g. here) described in terms of the ratio of percentage changes in variables; $y\propto x^n\implies\frac{dy}{y}=n\frac{dx}{x}$. (Okun's law is a little different in that it's more of a $\frac{dy}{y}\propto dx$ result.)
Power laws are often theoretically motivated by dimensional analysis, and in some cases theory even predicts an exponent. Even if neither economics nor econometrics found reasons to take this seriously (but they do!), they're not uniquely linearity-assuming fields, either for good or for ill. Just about any field with enough data to try regression will frequently be right to regress linearly for such reasons as discussed here (i.e. here on this page, not merely in my own answer).