No, I understand. but I need to obtain an limit number, it look like the end formula is equal to f(1)/f(0). but what is the absolute value of it. I need to obtain the final function of S* with only the variable of T and theta

since the q and Cq is known, thus, in theory, it could derive the S* equation with only theta and sample size T. But I couldnt solve the limit sum term.

For specific model (e.g. AR, MA) , the term can be derived. For general model, the term is just what is to be estimated ( the long run variance). So it can be estimated by your method. So usually, you need to plug_in, that is, first assume a AR model, then derive the parameter, then derive the optimal truncation parameter. I think a paper of Don andrews discuss this issue but I forget the details.

The paper I mentioned is "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, 1991, 59: 817-858.  You can have a look at the paper to find how to use plu-in method to select the optimal truncation parameter.

I have read it, in that paper, he used QS kernel as optimal one not Bartlett. But the method is the same. I I only let residual subjected to AR(1) with autoregressive parameter equal to theta. but I dont know how to calculate this complicated formula. I have plug-in all the parameter but you know, my maths like rubbish. ....  Where are you right now? in US or UK?

你好，我现在在澳洲，办公室电脑没法打中文。。。对于AR MA模型 那一项是可以算的，对于AR模型 那一项由innovation的方差和自回归系数决定。如果我没记错的话 Andrews的文章有给出AR模型那一项的的具体值。当然你也可以自己计算，因为AR模型的不痛滞后阶的协方差是可以算的，将那些协方差代入那个公式，求无限个协方差的和。