wootoo
如果一个真实的过程是：y(t)=A+B*sin(t)+y(t-1)+e(t);
应该如何做检验？
wootoo
DF检验、Perron检验都是基于如下假设：
即，真是过程要么是y(t)=y(t-1)+e(t);要么就是y(t)=A+y(t-1)+e(t)
可现实中，数据的生成过程是多种多样的，假如是如下两种情况:
DGP: y(t)=A+B*t+y(t-1)+e(t);
或者：y(t)=A+B*sin(t)+y(t-1)+e(t);
那么应该如何进行单位根检验呢？
虽然ADF检验要全面一些，但对于上述两例是否适用呢？（我觉得不行，不知大家有什么看法没有？）
谢谢指教
iamhappy
对于你提出的DGP:
y(t)=A+B*t+y(t-1)+e(t）
PP test, ADF 等都可以做。这种模型是再平常不过的了。
所以你说“DF检验、Perron检验都是基于如下假设：
即，真是过程要么是y(t)=y(t-1)+e(t);要么就是y(t)=A+y(t-1)+e(t)
”是不对的。只是你只知道这两种。
不过你提出的另一个非线性模型，说实话，我还从没见过，我还不觉得在经济学里会有人用这个模型。所以，对这个不感兴趣。
但按原则上讲，同样可以用pp test,和ADF.
只需使用一下FWL theorem,把limiting distribution中的布朗运动投影在那个空间上，取projection 就行了。
具体的，你可以参考相关书籍。如果有一定的理论知识，不难的。
wootoo
多谢了：）
其实我也不是非要做这么一个怪怪的模型不可。我问这个问题是因为：我感觉，在国内的期刊上发表的有关单位根检验的大部分文章都很“生硬”，都是拿过一个数据就套公式（情形1、情形2、情形3，DF、PP、ADF），然后就得出结论。可是有许多数据明显带有“周期成分”，比如GDP、农产品价格...
我想了解的是，当一个数据含有周期成分时，那些检验是否还有效（the power of test是否降低了？）
我搜索了一下处理周期数据的单位根检验的文章，发现都只是在做传统单位根检验之前做一下“季节调整”（或者进行“季节单位根检验”）。我觉得，如果数据是季度的、月度的，这样做还行，可如果是像股票或期货这种“每天的数据”，一旦周期很长（比如说356天一个周期），那怎么办？
所以，我开始设想的方法是：用“谱分析”方法找到主要的周期频率w，然后将这些“B*sin(wt)”进行剔除，之后再用传统单位根检验。但困难又出来了：要进行“谱分析”，数据就必需要求是平稳的，可怎么让数据平稳呢？又得依赖单位根检验...：（
后来，看了Hamilton的书后，想照葫芦画瓢，干脆推导一下“当真实过程是：y(t)=A+B*sin(t)+y(t-1)+e(t)时，y(t-1)前的系数的渐进分布”，但发现这要涉及到一堆“周期序列的级数收敛”问题，我又没那个实力，所以又走不通了。
希望再次得到iamhappy以及其他高手的帮助。再次感谢 Iamhappy
iamhappy
I take back some of what I said because I realized your question is actually about the methodology. Once again, I am telling you that what you suggested is definitely doable. It is NOT necessary to worry about the sin series or anything of this sort.
It can be done in the exact way that I told you.
The reason that no research has been done on your model is that it is too simple theoretically.
Here is why.
the limiting distribution is very easy to obtain.
As I have told you in my first answer post, simply project the brownion motion on the space spanned by the constant and the sin(.). Then replace brownion motion in the limiting distribution with no deterministic terms by this projection.
that is it!!!!!!
it won'''' t take anyone more than 2 minutes to do this. It is NOT necessary to worry about the sin series or anything of this sort.
Again, if you deal with some data displaying such cyclical behavior, using your specificaiton may give you high power.
by the way, Hamilton''''s book didn''''t mention this method (I guess, because I didn''''t use it as a textbook, only an occassional reference), because it is just a textbook and he didn''''''''''''''''t want to go too deep into the theory.
Better check out some papers by Philips, Perron, etc.
good luck
iamhappy
I was a little bit careless and didn''t answer your question.
sorry, man.
Cecil
This is what I think:
If y(t)=a+b*sin(t)+y(t-1)+e(t), then
y(t)-y(t-1)=a+b*sin(t)+e(t).
Using the ADF test framework, you may run the regression
y(t)-y(t-1)=a+b*sin(t)+c*y(t-1)+d*(y(t-1)-y(t-2))+...
and test for the significance of c.
However, unlike the case of linear time trend, you need to determine
the phrase and frequency of the sine function. That is, you are not
sure whether the time trend should be sin(t) or more generally sin(e+f*t).
You may be tempted to try running the regression
y(t)-y(t-1)=a+b*sin(e+f*t)+c*y(t-1)+d*(y(t-1)-y(t-2))+...
but the nonlinearity between parameters b, e and f will create a big problem in determining the critical value of c.
iamhappy
What you said is not an issue. I believe they all know how to construct the test statistic in this case.
the question is the limiting distribution. Without it, how can you do the test?
iamhappy
by the way, the projection of brownian motion on the deterministic space is just continuous time OLS
Cecil
(1) Will different e and f in sin(e+f*t) affect the shape of the limiting distribution?
iamhappy
yes, I think you have the point.
it will be different. I need to think about it more carefully.
参见讨论请到：http://bbs.efnchina.com/dispbbs.asp?boardid=33751&rootid=&id=40377
