Say we have a linear regression model $$y_t = \mathbf{x}_t'\boldsymbol{\beta} + \epsilon_t, \qquad \epsilon_t \sim \mathcal{N}(0, \sigma^2)$$

Assuming $\boldsymbol{\beta}$ is random and that $\mathbf{x}_t$ is given, the variance of $y_t$ is given by $$\mbox{var}(y_t) = \boldsymbol{x}_t'\mbox{cov}(\boldsymbol{\beta})\boldsymbol{x}_t + \sigma^2$$

how do we know how much of the variance is purely random (from $\sigma^2$) and how much of it is due to variance in the regression coefficients $\boldsymbol{\beta}$? For example, if we estimate the variance of $y_t$ by the empirical variance $\hat{\mbox{var}}(y_t) = (1/T)\sum_{t=1}^T(y_t - \bar{y})^2$, how do we know how much of that number came from $\sigma^2$ and how much of it came from the variance of $\boldsymbol{\beta}$?


The question does not make much sense in that

  1. the variance of $y_t$ depends on $\mathbf{x}_t$, hence estimating it by $$(1/T)\sum_{t=1}^T(y_t - \bar{y})^2$$does not return the variance of $y_t$;
  2. in most linear regression settings, there is only one observation of $Y_t$ for a given $\mathbf{x}_t$. It is therefore impossible to estimate the variance of $y_t$ directly from $y_t$;
  3. despite the fact that the variance of $y_t$ decomposes as indicated when $\beta$ is random, the part $\sigma^2$ of this variance can be consistently estimated.

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