So, I have a linear-quadratic model that might not be good to do forecasting. So, I transform my model to log-linear etc but the transformed model is not good (violations of autocorrelation/heteroscedasticity),
But, when I transform my model into sqrt(regressors), then the transformed model is good and pass statistical assumptions. Why in this case sqrt-transformation is better than log transformations ?
If your data is best described by Poisson distribution, it is known that sqrt gives better approximation when you want it to be transformed for Gaussian-type modelling (see, e.g., http://tina.wiau.man.ac.uk/docs/memos/2001-010.pdf)
So, perhaps, you are dealing with count data where conditional PMFs are described by a distribution that is close to the Poisson distribution.
If you data has log-normal distribution (such as trees heights), it is better to use log transformed data.
A more general and flexible approach is to use Box-Cox transformations (may be good for negative binomial distributions which is a more general distribution compared to the Poisson distribution).
If the correct specification of the model is linear-quadratic (y=a+b*x+c*x²+random error), neither a log-linear nor a sqrt transformation of the regressors is feasible. But one should use binomial models (of which the linear-quadratic is a special form) only if there are really very good reasons for doing so. Otherwise, there may likely be problems for using such model for forcasts, because for growing x's, one reaches a point from which on y goes progressively upwords (c>0) or downword(c
@Andrey and Anton, Does logistic functions are suitable to replace my model ?
If one finds a certain function to be a correct specification according to theory or reasonable reflections, one cannot change to another function just to get better statistical results or forecasts. But bad results can be taken as a hint, that the assumed function was a bad choice and one should think over the problem again.
I do not think the logistic regression function is suitable in your situation.
Logistic regression relates to the case of binary dependent variables.
It would be great if you provided examples of your data, then it would be easier for people to advise you on what model would be most suitable.
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