When learning econometrics I have usually seen stuff from the following perspective:
- Assume $Y_t = f(X_t) + e_t$, where f is some function of $X_t$ (typically linear). For example, assume $Y_t = X_t * \beta + e_t$. Then if $e_t$ satisfies certain properties the OLS estimator will converge to beta.
However I have also seen, but less frequently:
- Make no assumption on the function relationship between $Y_t$ and $X_t$. Without any assumptions we know there exists an optimal linear approximation of $E[Y_t|X_t]$ (the alpha such that $X_t*\alpha + e_t$ minimizes MSE, for example). Now if we assume that $(Y_t,X_t)$ is covariance stationary, the OLS estimator converges to alpha.
To me it seems like the perspective of 2. is more interesting because the analysis is not predicated on assuming that Y and X have a specific functional relationship. Instead, assumptions like "covariance stationary" seem more general than assuming that $Y = a + bX + e$.
Is there a reason why there seems to be more of a focus on 1.? Are the two perspectives related in some way?