Justice -

For continuous data, I have used the "variance of the prediction error" to help determine the accuracy of prediction results. This can be found in econometrics books. For example:

Maddala, G.S.(2001), Introduction to Econometrics, 3rd ed., Wiley.

(There is a fourth edition with Lahiri.)

For finite populations, continuous data, we can look at the estimated variance of the estimated prediction error for both individual predicted values, and for estimated totals. (See first link below. The first few pages may be of interest.) If you are looking at such finite populations, and want to know what kind of sample sizes are needed, the second link might possibly be of interest. Your modeling may be far more complex, however, and you may want to look at answers to other questions on ResearchGate, related to yours for multiple regression of various complexities.

Also, note that if you try to collect too much data for your resources and circumstances, you may just increase nonsampling error, such as measurement error. That would mean increasing your sample size may just decrease your accuracy. The other extreme would be to not gather enough pilot data to even be able to test well which model to use.

I am not certain what, if anything, I've said applies to your situation, or to what degree, as I sometimes misinterpret and assume one thing is being done, when that may not be the case. I suppose that is a common problem with such Q&A when we may have very different focuses.

Best wishes on your project -

Jim 

Article Using Prediction-Oriented Software for Survey Estimation

Conference Paper Projected Variance for the Model-based Classical Ratio Estim...

It depends how efficient you need to be for your specific problem... You can have insight into this by simulating different sample size and see when the confidence interval(s) of your parameter(s) of interest are small enough for the aim you have.

Chapter 4 in Cochran, W.G.(1977), Sampling Techniques, 3rd ed., John Wiley & Sons, discusses sample sizes for random sampling. What that may have in common here is that in section 4.7 he discusses four ways to obtain the population variance estimates needed to estimate sample sizes. One would be a pilot study, and another includes "guesswork" regarding the "structure of the population," (here, instead, your regressors and corresponding y-values). That is, you need to use data that are reasonably representative of what you expect to obtain. This might help to supplement or feed into Emmanuel's simulations.