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There's a more intuitive discussion in the context of simple linear regression here that may help motivate it for you.

There's a more intuitive discussion in the context of simple linear regression here that may help motivate it for you.

There's a more intuitive discussion in the context of simple linear regression here that may help motivate it for you.

The standard error is the square root of that estimated variance.

You can't. You can roughly estimate them by the residuals, but they don't even have the same variance, as we saw.

The standard error is the square root of that estimated variance up above.

Individually, you can't, at least not very well. You can roughly estimate them by the residuals, but they don't even have the same variance, as we saw. However, you can estimate their variance reasonably well from the residuals, if you appropriately adjust for the fact that the residuals are on average smaller than the errors.