Estimate of Coefficient Variance in multiple regression

Question

I'm trying to compute an estimate for the variance of the estimated coefficients in a non-linear regression using the formula described in link. I can't figure out how to build $F_{ij}$

Let's consider for simplicity a quadratic regression:

$f = \beta_0 + \beta_1 x + \beta_2 x^2$

and let's say we have computed the estimate $\hat{\beta}_0, \hat{\beta}_1, \hat{\beta}_2$ from a set of $m$ observations.

To me $$ F = \frac{\partial f(x_i)}{\partial \beta_j} = \left( \begin{array}{ccc} 1 & x_1 & x_1^2 \\ \vdots & \vdots &\vdots \\ 1 & x_m & x_m^2 \end{array} \right) $$

But of course it can't be cause it does not depend on $\hat{\beta}$ anymore...

Answer 1

The Matrix $F$ is actually correct and one can use it to build the covariance matrix of the regression coefficients: $$D(\hat{\beta}) = \sigma^2(F^T F)^{-1}$$ Now denoting with $d_{jj}$ the $j-th$ diagonal element of $D(\hat{\beta})$ we can specify the distribution of $\hat{\beta}_j$ as $$ \hat{\beta}_j \sim N(\beta_j, \sigma^2 d_{jj}) $$ where $\sigma^2$ can be estimated using the sum of squares of the residuals: $$ \hat{\sigma}^2 = \frac{1}{m - n} \sum_{i=1}^m \left( y_i - \hat{y}_i\right)$$