# How to calculate prediction intervals using best-fit parameters when parameters covary?

I'm doing a nonlinear least-squares regression to find best-fit values for two parameters. I then want to use these best-fit parameters and their variances to extrapolate to a predicted value.

It's easy enough to plug the best-fit values in and get the corresponding best-fit value for the extrapolation, but I'm getting confused about how to put prediction intervals on this prediction. My first impulse was to just calculate the 95% prediction interval as I would normally, i.e. by using the variances for my two parameters that comes out of the covariance matrix of the regression. But the covariance from the covariance matrix (ie the off-diagonal entry) is nonzero (usually between 0.1 and 0.2), so I'm wondering if this is kosher. How do I deal with the fact that the uncertainty in my parameters covaries when making a prediction interval estimate?

Sorry if some of this isn't clear; I'm pretty new to this.

Thanks much!

• Google "delta method" for how to calculate the variance of a nonlinear function of random variables (the estimate parameters). – Dimitriy V. Masterov May 16 '18 at 0:25