The Gauss-Markov theorem tells us that the ordinary least-squares (OLS) estimator is the best linear unbiased estimator (BLUE) for the coefficients in a linear regression (given some conditions on the errors). I can understand why we want an unbiased and minimum-variance ("best") estimator, but why linear? Why not an estimator that depends on any other power (square, square root, etc) of the data?
More specifically, for an $n\times m$ data matrix $X$ predicting an $n \times 1$ response vector $y$ in the model $y = \beta X + \epsilon$, the OLS estimator for the coefficients $\beta$ is,
$$\hat\beta = (X^TX)^{-1}X^Ty = Cy.$$
Thus each $\hat\beta_j$ can be defined linearly in terms of $y_i$, as
$$\hat\beta_j = c_{j0} y_0 + c_{j1} y_1 + c_{j2} y_2 + \cdots,$$
and is therefore a linear estimator. Is there a particular reason we don't consider non-linear estimator, for example, of the form,
$$ \tilde\beta_j = Cy^a = c_{j0} y_0^a + c_{j1} y_1^a + c_{j2} y_2^a + \cdots $$