Back in 2013 I wrote a post that was titled, "Forecasting From Log-Linear Regressions". The basis for that post was the well-known result that if you estimate a linear regression model with the (natural) logarithm of y as the dependent variable, but you're actually interested in forecasting y itself, you don't just report the exponentials of the original forecasts. You need to add an adjustment that takes account of the connection between a Normal random variable and a log-Normal random variable, and the relationship between their means.
Today, I received a query from a blog-reader who asked how the results in that post would change if the dependent variable was the square root of y, but we wanted to forecast the y itself. I'm not sure why this particular transformation was of interest, but let's take a look at the question.
In this case we can exploit the relationship between a (standard) Normal distribution and a Chi-Square distribution in order to answer the question.