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问个econometrics问题（包子贴）

问个econometrics问题（包子贴）# Economics - 经济

s*w2009-03-10 07:03

1 楼

全部答上来的包子5个，否则答上来的前5位各包子一个。
weighted least square能不能确保给出consistent coefficient?
我记得wls estimator是BLUE，对吧？
另一个问题就是BLUE和consistency有啥必然联系么？
BLUE能不能推出consistency，why or why not?
怎么证明？

i*e2009-03-10 07:03

2 楼

my $0.02。
BLUE里面的U=unbiased，unbiasedness是个finite sample的概念，consistency是个
asymptotic的概念，两者没有必然关系。
计量教材里面有很多estimator在finite sample里面是biased但是consistent。另外一
方面，有的estimator是unbiased但是不consistent，例如我用样本里面的第一个观测
值X1作为sample mean的estimator，是unbiased，但通常不是consistent的，因为在样
本趋向无穷大时，lim( |X1-mean(x)|>epsilon )=常数，该常数一般不为0。
另外，WLS是不是BLUE或者consistent当然要看DGP的假设。

s*w2009-03-10 07:03

3 楼

Thank you for your reply.
I am ok with the difference between unbiasedness and consistency, but I am
not sure about the difference between the "Best" part (min var) and
consistency. I know consistent estimators are not necessarily "best", but is
"best" always consistent? why or why not? Can you prove?
Besides, WLS being "BLUE" is what I found from wikipedia. It seems that when
ppl refer to WLS they use 1/var as the weighted matrix. That is to say WLS
is referred to what we use to solve heterosca

f*r2009-03-10 07:03

4 楼

Not necessarily.
Not always.
Not really.
No. Unbiasedness does not mean consistency, vice versa.
Check Davidson and Mackinnon.
No need for baozi. hehe...

【在 s*****w 的大作中提到】

: 全部答上来的包子5个，否则答上来的前5位各包子一个。
: weighted least square能不能确保给出consistent coefficient?
: 我记得wls estimator是BLUE，对吧？
: 另一个问题就是BLUE和consistency有啥必然联系么？
: BLUE能不能推出consistency，why or why not?
: 怎么证明？

i*e2009-03-10 07:03

5 楼

My understanding is the "best" part in BLUE refers to a estimator having
minimum variance among the class of linear unbiased estimators (LUE), which
is similar to the concept of UMUE (uniformly minimum-variance unbiased
estimator). A consistent but biased estimator does not belong to the class
of LUE, thus can never be the best in the class. Meanwhile, a BLUE can be
consistent, for instance WLS under assumptions below. I am not sure a BLUE
or UMUE must be consistent. There may exist counter exam

【在 s*****w 的大作中提到】

: Thank you for your reply.
: I am ok with the difference between unbiasedness and consistency, but I am
: not sure about the difference between the "Best" part (min var) and
: consistency. I know consistent estimators are not necessarily "best", but is
: "best" always consistent? why or why not? Can you prove?
: Besides, WLS being "BLUE" is what I found from wikipedia. It seems that when
: ppl refer to WLS they use 1/var as the weighted matrix. That is to say WLS
: is referred to what we use to solve heterosca

s*w2009-03-10 07:03

6 楼

非常感谢两位！
包子是略表心意，请笑纳：）

which
consider
.
zero.

【在 i*******e 的大作中提到】

: My understanding is the "best" part in BLUE refers to a estimator having
: minimum variance among the class of linear unbiased estimators (LUE), which
: is similar to the concept of UMUE (uniformly minimum-variance unbiased
: estimator). A consistent but biased estimator does not belong to the class
: of LUE, thus can never be the best in the class. Meanwhile, a BLUE can be
: consistent, for instance WLS under assumptions below. I am not sure a BLUE
: or UMUE must be consistent. There may exist counter exam

s*w2009-03-10 07:03

7 楼

干脆说出我的问题吧，假设原regression是Y(t)=X(t)B+E(t).
我用WLS的时候weight是Y(t-1)，这样得到的的estimate：B(^)=(X'WY)/(X'WX)是不是
consistent呢？怎么证明？那本书好长，我看得头晕眼花的。
谢谢！

【在 s*****w 的大作中提到】

: 非常感谢两位！
: 包子是略表心意，请笑纳：）
:
: which
: consider
: .
: zero.

f*r2009-03-10 07:03

8 楼

I don't think it is "correct" to use Y(t-1) as weight. In theory, the
weight should be the inverse of each variance term. In reality, that's
never known. Therefore, some people use some variables which are believed
to determine variance as weights.
But I've never seen using Y(t-1) as weight. The problem with that is the
potential endogeneity problem introduced by X(t)/Y(t-1). If there is
autocorrelation for Y, then B hat won't be consistent.
The key point for consistency is that X is exogen

【在 s*****w 的大作中提到】

: 干脆说出我的问题吧，假设原regression是Y(t)=X(t)B+E(t).
: 我用WLS的时候weight是Y(t-1)，这样得到的的estimate：B(^)=(X'WY)/(X'WX)是不是
: consistent呢？怎么证明？那本书好长，我看得头晕眼花的。
: 谢谢！

k*g2009-03-10 07:03

9 楼

好post！
顶！

s*w2009-03-10 07:03

10 楼

回答得真好，谢谢！

【在 f*******r 的大作中提到】

: I don't think it is "correct" to use Y(t-1) as weight. In theory, the
: weight should be the inverse of each variance term. In reality, that's
: never known. Therefore, some people use some variables which are believed
: to determine variance as weights.
: But I've never seen using Y(t-1) as weight. The problem with that is the
: potential endogeneity problem introduced by X(t)/Y(t-1). If there is
: autocorrelation for Y, then B hat won't be consistent.
: The key point for consistency is that X is exogen

s*a2009-03-10 07:03

11 楼

关于这个，我觉得，如果存在一个consistent的estimator，则blue也consistent

【在 s*****w 的大作中提到】

a*n2009-03-10 07:03

12 楼

全都取决于你对error terms的假设。。

【在 s*****w 的大作中提到】

f*r2009-03-10 07:03

13 楼

We need to note that BLUE means "best"' linear unbiased estimator. It
guarantees unbiasedness, but not consistency. I agree that under certain
assumptions, WLS or GLS should be consistent, but that has nothing to
do with it being BLUE.
"Best" is sometimes over-emphasized. It just means it will have the
lowest standard error among all the unbiased estimators.
This does not come in without a price: you make
assumptions about the error structure to gain efficiency. In reality, we
don't care

【在 s*****a 的大作中提到】

:
: 关于这个，我觉得，如果存在一个consistent的estimator，则blue也consistent

h*e2009-03-10 07:03

14 楼

If the correct weight is known, then WLS is the BLUE. If not known, using a
weight matrix that is "closer" to truth than identity will reduce the
estimation error. If the variance can be well estimated somewhere else,
using the estimated variance for WLS is also good choice. The perhaps most
inefficient approach is huber sandwich estimator, but arguably robust.
I am also curious why econ people do not care efficiency. It seems you have
very large noise to signal ratio, and significance of regres

s*w2009-03-10 07:03

15 楼

再问一个问题，以上这些都是假设Y和X没有问题的。
如果知道观察到的Y是biased，因此E也是biased，而且知道Y(t-1)和Y(t)是负相关的，
那么用WLS的时候用Y(t-1)当weight是不是就能解决问题呢？得出来的B(^)还是不是
unbiased和consistent啊？因为现在E(E)<>0了。

【在 s*****w 的大作中提到】

a*n2009-03-10 07:03

16 楼

你真的需要看看书了。。。。

【在 s*****w 的大作中提到】

: 再问一个问题，以上这些都是假设Y和X没有问题的。
: 如果知道观察到的Y是biased，因此E也是biased，而且知道Y(t-1)和Y(t)是负相关的，
: 那么用WLS的时候用Y(t-1)当weight是不是就能解决问题呢？得出来的B(^)还是不是
: unbiased和consistent啊？因为现在E(E)<>0了。

s*w2009-03-10 07:03

17 楼

请具体指点，谢谢！如果你确认你很明白的话。
我以为我很明白WLS了，直到我遇到这个问题。
我看了好几本书，但都没有提到过这样的问题。

【在 a**n 的大作中提到】

: 你真的需要看看书了。。。。

v*o2009-03-10 07:03

18 楼

我猜alan说看书的意思是看BLUE, consistency, efficiency的定义。。。
然后照着定义去check就有答案了。没有书能把所有特例都解决

【在 s*****w 的大作中提到】

: 请具体指点，谢谢！如果你确认你很明白的话。
: 我以为我很明白WLS了，直到我遇到这个问题。
: 我看了好几本书，但都没有提到过这样的问题。

s*w2009-03-10 07:03

19 楼

这些定义我都懂，但是证明涉及到复杂的矩阵，我就懵了。

【在 v****o 的大作中提到】

: 我猜alan说看书的意思是看BLUE, consistency, efficiency的定义。。。
: 然后照着定义去check就有答案了。没有书能把所有特例都解决

a*n2009-03-10 07:03

20 楼

很奇怪为啥说Y是biased？你的X里面不包括constant term？
俺觉得你要看的是GLS
WLS只是一种特殊情况，按你的描述来看，好象是有serial correlation
那一般意义上的WLS不适用
GLS的关键是怎么知道error的variance-covariance matrix
如果你能够consistently estimate var(u_t)
那么fGLS就应该是consistent的
俺觉得用Y(t-1)不合理。。这个咋能估计var(u_t)呢。。

不是

【在 s*****w 的大作中提到】

s*w2009-03-10 07:03

21 楼

谢谢你，你confirm了我的想法。
我也觉得应该把bias扔到X里面的constant term，只不过那是一个我认为很不错的老师
说的，所以我先怀疑自己。
WLS不是也说即使var-cov matrx不consistent也可以给出consistent estimate吗？
应该是没有serial correlation的，否则就不能用WLS了。
Anyway,可能是题目本身有点问题。

【在 a**n 的大作中提到】

: 很奇怪为啥说Y是biased？你的X里面不包括constant term？
: 俺觉得你要看的是GLS
: WLS只是一种特殊情况，按你的描述来看，好象是有serial correlation
: 那一般意义上的WLS不适用
: GLS的关键是怎么知道error的variance-covariance matrix
: 如果你能够consistently estimate var(u_t)
: 那么fGLS就应该是consistent的
: 俺觉得用Y(t-1)不合理。。这个咋能估计var(u_t)呢。。
:
: 不是

f*r2009-03-10 07:03

22 楼

Measurement error in Y is not a big problem. It will increase noise,
therefore, standard errors will be inflated. But coefficient estimates are
correct, still. However, measurement error in X can be a big problem.