I was puzzled by the problems as follows when trying to solve the DSGE model. Would you please give any advice? Thank you for your time.

(1) When I use averages of empirical data(e.g. C/Y, I/Y) to do the calibration in DSGE model with no trend, do I need to eliminate the growth rate of these data series(e.g. Y, C, I ) respectively? I divide raw quarterly data series by quarter-on-quarter growth, is this correct?

And also, before I deal with these data series for the bayesian estimation, do I also need to eliminate the growth rate ?

(2) When comparing the second moments of variables between model and empirical data to check the model fit, do I need to confirm all major variables fit well with empirical data? I read in some papers that not all the moments of empirical data perfectly lie in the interval that the model implies. Does this mean that it is acceptable to allow the error in between? Will this impact the reliability of simulated results ?

(3) I want to add investment shock into capital accumulation function, and I read the formation like or . How to decide which one should be chosen?
And what about the function like ? Is it correct to add investment shock like ?

Thank you for reading the post. Any reply will be appreciated.

(1) I read in a book that I should eliminate the trend term of data series before using them to calibrate the parameters in a dsge model with no trend. Is this correct ? And how should I realize this ?

(2) Sorry but I did not find the answer in that post. Maybe I should make it more clear. For instance, the std of output variable Y that the model implied is between 1.5 and 2.5 while the std of empirical data Y is 2.8 or sth out of the interval that model implied. Do I need to confirm all the stds or cross correlations of major variables fit well with empirical data?
I read in some papers that not all these features of empirical data perfectly lie in the model-implied interval. Does this mean that it is acceptable to allow the error in between ? And to what extend will this impact the reliability of simulated results ?

(3) Sorry and I rewrite the question:
I want to add investment shock into capital accumulation function, and I read the formation like {{k}_{t}}=\left( 1-\delta \right){{k}_{t-1}}+\varepsilon _{t}^{i}\left[ 1-\frac{{{\kappa }_{i}}}{2}{{\left( \frac{{{i}_{t}}}{{{i}_{t-1}}}-1 \right)}^{2}} \right]{{i}_{t}}
or {{k}_{t}}=\left( 1-\delta \right){{k}_{t-1}}+\left[ 1-\frac{{{\kappa }_{i}}}{2}{{\left( \frac{\varepsilon _{t}^{i}{{i}_{t}}}{{{i}_{t-1}}}-1 \right)}^{2}} \right]{{i}_{t}}.
How to decide which one should be chosen?

And what about the accumulation function like {{k}_{t}}=\left( 1-\delta \right){{k}_{t-1}}+{{i}_{t}}-\frac{{{\kappa }_{i}}}{2}{{\left( \frac{{{i}_{t}}}{{{k}_{t-1}}}-\delta \right)}^{2}}{{k}_{t-1}} ?
Is it correct to add investment shock as {{k}_{t}}=\left( 1-\delta \right){{k}_{t-1}}+{{i}_{t}}-\frac{{{\kappa }_{i}}}{2}{{\left( \frac{\varepsilon _{t}^{i}{{i}_{t}}}{{{k}_{t-1}}}-\delta \right)}^{2}}{{k}_{t-1}} ?

Without details or context it is impossible to know. If the calibration only requires ratios and there is cointegation, then there is no point in detrending. There may be cases where detrending is necessary to take care of things not in the model, e.g. capital deepening as one form of unbalanced growth.

A model will never perfectly fit the data in all regards. After all, it’s a stylized representation of reality. You should check whether the fit for the relevant variables is acceptable. Of course, it’s up to you to judge and argue what is acceptable. If your model is completely off in a dimension that is important for your result then you cannot trust your results.

The first two are investment adjustment costs, the last two capital adjustment costs. They imply different dynamics and there is a large debate which formulation performs better. Where did you get the second formulation from? It very much looks like a depreciation shock.