Say that I fit some data with some model, for instance a linear function $y = mx+b$. What is the proper way to report the fitted coefficients and the goodness of fit? Specifically, if I do the fit in some numerical program like MATLAB, I might get:
$$a = 3.265093...$$ $$b = 5.782340...$$
perhaps even with 95% confidence bounds on the coefficients, e.g.,
$$a = 3.265093...\quad \left[3.021...,\ 3.5423...\right]$$ $$b = 5.782340...\quad \left[5.013...,\ 6.2541...\right]$$
And several measures of goodness of fit:
$$R^2 = 0.999999999978231...$$ $$SSE = 106.1...$$ $$RMSE = 0.813496...$$
What information should be reported, and how many significant digits should be reported (it seems a bit strange to report $R^2 = 0.999999999978231...$, but it also seems a bit strange to report $R^2 = 1$). How do I determine the appropriate number of significant digits (NOTE: the fit is to simulated data, so I can't get the significance from the precision of some measurement device and propagate it, the precision of the data is only limited by numerical precision, in this case double precision floating point numbers).
For the $R^2$, 2 is plenty. If your data is simulated, then getting a value of $1$ isn't that weird. Leave it at $0.99$ if you don't like $1$
For the coefficients, it depends on the context. The key question to ask is: Does adding a digit provide any useful information to your audience? Personally, I don't like having more than 3. If more are needed, then consider scaling the coefficients.