# How to include off-diagonal elements in covariance matrix in uncertainty of variable

I am performing a linear regression (y = a + bx) and calculate the correlation matrix:

$$\begin{bmatrix} 1 & -0.84 \\ -0.84 & 1 \\ \end{bmatrix}$$

and covariance matrix:

$$\begin{bmatrix} 0.003 & -0.0054 \\ -0.0054 & 0.01 \\ \end{bmatrix}$$

The diagonal elements of the covariance matrix are the variances of my variables and I get the standard deviation or uncertainty by applying the square root:

$$a = 0.99 \pm \sqrt{0.003} = 0.99 \pm 0.0547$$

$$b = 1.94 \pm 0.1$$

However, the variables are highly anti-correlated and the off-diagonal elements are not 0, which means I have to include them in the uncertainty of a and b. I have tried to search the internet and didn't find an answer. In principle, a and b have also different units, so I really don't know how to include these numbers in the uncertainties.

• By "uncertainty of $a$ and $b$, do you mean the standard estimation error? In that case, here you have some formulas (denoting standard errors for $a$ and $b$ as $SE(\hat{\beta_0})$ and $SE(\hat{\beta_1})$ stats.stackexchange.com/questions/289457/… Commented May 29, 2019 at 11:47
• Or see this page for this single-predictor case and the the links from it for the more general multiple regression case.
– EdM
Commented May 29, 2019 at 12:22