gemini69: for example

Tse 2000 J of Econometrics, his LM test has an empirical power of 50% for sample size 300, and 90% for sample size 1000. it over-rejects the H0 for 300 and is well-sized for 1000.

Of course the sample size should be reasonably large since the test statistic is based on CLT. well you can say that 30 is infinity. But without enough number of obs, no usful information is available.

[此贴子已经被作者于2006-2-13 3:12:57编辑过]

gemini69:

maybe you can recommand me a paper in the core English Econometric journal that deals with cointegration using only 40 obs.

Well，

1、"渐近" ：随着样本渐大的性质，并不等同样本就要很大才会有的性质，它不是恒等式；何况，多少的样本数才是大？多少的样本数才是小，或是以多少样本数区分有限与渐近？ 你也拜托一下，这是质的观念，不是量的标准！

2、这跟收敛速度有关吧！

3、 40个年资料，两个变量，每一个就有20年的资料，严不严谨是一回事，叁考价值多不多是另一回事。

4、不需要给 paper吧，这根本是基本观念。不然你翻一下任何一本基础的概率或是计量教科书，有没有那个作者，给你拍胸保证，样本个数40的估计式，不管什麽状况，钢定、铁定不符合渐近性质。

随便摘录几段话：

幼稚园级： (Basic Econometrics 4E, Gujarati, pp903)

" It often happens that an estimator does not satisfy one or more of the desirable statistical properties

in small samples. But as the sample size increases indefinitely, the estimator possesses several desirable

statistical properties. These properties are known as the large-sample, or asymptotic, properties."

幼稚园级： (Introductory Econometrics 1st Edition, Wooldridge, pp68)

" Asymptotic analysis involves approximating the features of the sampling distribution of an estimator.

These approximations depend on the size of the sample. Unfortunately, we are necessarily limited in

what we can say about how “large” a sample size is needed for asymptotic analysis to be appropriate;

this depends on the underlying population distribution. But large sample approximations have been

known to work well for sample sizes as small as n ＝ 20. "

初级： (Econometric Theory and Methods_Davidson & MacKinnon, pp145)

"Asymptotic theory is concerned with the distributions of estimators and test statistics as the sample size n

tends to infinity. It often allows us to obtain simple results which provide useful approximations even when the

sample size is far from infinite."

计量学者： 台湾中研院院士， PhD, UCSD (Prof. White 的学生)

" 许多书都强调样本规模必须大过 30 (或 50) 才够，但事实上这个问题是没有答案的。

因为：有时样本规模小于 30 时，参数与估计值便非常接近。有时即使样本规模大到 3000，

估计值仍不会接近参数。 所以我们只能说 n 越大，估计值愈有可能非常接近真实参数。

(请参阅课本214页，图7.1)"

Usually depend on your sample, if your sample distribution is consistent with population distribution,

that is no problem, just as above, work well for small sample

if the sample distribution is not consistent with population distribution,

unfortunately, you have to use large sample to approximate it

the only problem is how we know whehter the sample distribution is consistent or not

usually we choose large sample to get convergence in case there is consistence problem

半年不来大家越来越高深了

幼稚园级： (Introductory Econometrics 1st Edition, Wooldridge, pp68)

" Asymptotic analysis involves approximating the features of the sampling distribution of an estimator.

These approximations depend on the size of the sample. Unfortunately, we are necessarily limited in

what we can say about how “large” a sample size is needed for asymptotic analysis to be appropriate;

this depends on the underlying population distribution. But large sample approximations have been

known to work well for sample sizes as small as n ＝ 20. "

That is for undergraduate Econometrics, but not for modern Time series.

If you are really concerned with cointegration analysis, nonstationary time series analysis, do you really would like analyse a data set of sample size 40.

计量学者： 台湾中研院院士， PhD, UCSD (Prof. White 的学生)

" 许多书都强调样本规模必须大过 30 (或 50) 才够，但事实上这个问题是没有答案的。

因为：有时样本规模小于 30 时，参数与估计值便非常接近。有时即使样本规模大到 3000，

估计值仍不会接近参数。 所以我们只能说 n 越大，估计值愈有可能非常接近真实参数。

So that is why people will check the small sample property if the test statistics is asymptotic.

I think 30 is more concerned with statistics than with time series analysis. If your derivation starts from alpha-mixing, rho-mixing, functional Brownian motion, and then you deal with a 30-sample size data in your application section, will that be sensible?

Prof. White 的学生: I know White's students are all around the world. But this is not the point.

有时样本规模小于 30 时，参数与估计值便非常接近: the problem is that in nonstationary time series analysis this is not the case.

But large sample approximations have been

known to work well for sample sizes as small as n ＝ 20. ------------- that is also again not for cointegration, unit root. -----

[此贴子已经被作者于2006-2-13 19:56:54编辑过]

If you think about PPP, in general people find that in the long run, there is cointegration. Small sample size does not lead this conclusion in most cases. Well recently, people devise threshold cointegration which I think is much more intuitive. there is a growing literature of this kind. If you believe in such non-linear model,of course then large sample is welcome since normally  there are more parameters in the model. On the other hand if you think of the ADF, in order to get rid of the autocorrelation in the residual, quite a few lagged dependent variable need to be included(for example, sometimes, 15th lag turns out to be useful[OK, one can also delete the insignificant intermediate lagged variables]). Then with 40 obs, how many degrees of freedom can be left rendering the analysis really informative.

协整滞后期一般等于var最优滞后期-1

而Sims（1980）提出了决定VAR模型最优滞后阶数的似然比检验法。根据模型选择的似然比统计量LR的计算方程式为：LR（p，p+1）=-2（logLp-logLp+1）。似然比统计量服从卡方分布。

是的. 我用的就是这种方法, 就是找AIC和SC同时达到最小阶的VAR, 如果二者冲突, 则用对数似然比LR检验.

But there are some statistics can evalute how long the lag terms should be.you can find them in eviews and so on.

Maybe the software's suggestion is not like what you think, but it really can give you some guide. Now that you use the software,you should believe it.

Sometime the theory is not the same with the fact.

事实上，选择协整检验的滞后阶数，估计连johansen本人也没有确定的答案。好像有论文建议，通过增加滞后阶数，检验残差的白噪声性或正态性，滞后期越大，这些检验越容易通过，但需要估计的参数越多，自由度也就越少，具体操作中需要在两者间做出取舍。当然，有时也会遇到滞后期已经很长，残差仍然难以通过上述检验，此时就应该考虑模型构建问题，可能需要在Var模型中包含其他有解释能力的变量了！！