In a simple linear regression $y=a+ \beta X$, the ordinary least square or maximum likelihood (ML) estimation gives $\textrm{var}(\hat{\beta })=\sigma ^{2}(X{'}X)^{-1}$, where $\sigma ^{2}$ is the residual variance. In a case of one predictor x, $\textrm{var}(\hat{\beta })=\sigma^{2}/\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}$ (1).
In a general ML case, the asymptotic estimation of the standard error of a parameter is defined as the inverse of the Fisher information matrix: $\textrm{var}(\hat{\theta })=[\mathbf{\mathrm{I}}(\hat{\theta })]^{-1}$, where $\mathbf{\mathrm{I}}(\hat{\theta })$ refers to the Fisher information matrix.
1) Without going to the detail of the mathematical formulas, can someone give some intuitive explanations about how and why the standard error of a model coefficient is associated to the residual variance?
2) Given formula (1), it seems that if I have a wider range of predictor $x$ (relative to $\bar{x}$), or larger number of samples $n$, the estimated SE of $\beta $ decreases. Intuitively why?
3) If we look at my question from another perspective: if I randomly simulate a number of samples using a fixed relationship $y=3+2x+\epsilon$ where $\epsilon$ follows a normal distribution $N(0,\sigma)$, and apply a simple linear regression. How does the only randomness in my data generation process (i.e. $\sigma$ for $\epsilon$) end up in the uncertainty (or randomness,SE) of the coefficients? I mean when I simulate the data, there is no uncertainty in my model coefficients.