求助推倒Bartlett kernel 的optimal bandwidth

搞论文做到NeweyWest 标准差的最后lag的确定问题， 研究了过去的paper啊，我想判断出来Barlett kernel的optimal bandwidth. 在文献中都是用spectral density function 推出来的，我的数学层次推不太出来，希望有人给指点一下 啊具体推倒公式见图片啊，退出S* 即可， Ut 是AR（1）模型，系数为theta。Cq=1.1447(l理论值已知)。先谢谢了啊

顶啊，顶啊，计量经济理论啊，求同道中人！！！！！

顶，别沉了

求指导

Notice that for Barlett kernel, q=1, thus in the right hand side formula, all the q can be replaced by 1. f^(0) corresponds to so-called spetral density evaluated at 0 ( the long run variance), the formula is given in your second equation by taking psai to 0, i.e., the |j|^(psai) terms are cancelled. So I think the three equations you provide have explaned everything.

cbqywl 发表于 2013-9-10 14:53
Notice that for Barlett kernel, q=1, thus in the right hand side formula, all the q can be replaced  ...

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

cbqywl 发表于 2013-9-10 14:53
Notice that for Barlett kernel, q=1, thus in the right hand side formula, all the q can be replaced  ...

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

cbqywl 发表于 2013-9-10 14:53
Notice that for Barlett kernel, q=1, thus in the right hand side formula, all the q can be replaced  ...

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.