[有奖]The Consistency of beta??

九楼的出发点对了，就好像做数学考试最后一道大题一样，就是让大家讨论一下这个情况。

Hi, I worked in Mutual Fund and have been worked in these fields more than 4 years

In my experience, Beta is now good to describe market risk for the following reason:

it is not steady. for example the demension you used, 40 trading daily data? 5 years monthly data?

the second, moving Beta or Unique one?

Our Market Beta stays in one narrow range{0.7-1.2}.

Your Problem is that you used stable process, but the data are not, it is the KEY of your problem.

Care!

Read the problem again, please! Your beta is not the beta I mentioned.

There are two types of nonstationary time series:

1)trend-stationary process: yt=α+δt+rt, where rt is a stationary series, for example, a stationary AR(p) series;

2)unit root process: ARIMA process, a simplest example is random walk with a drift: yt= μ+yt-1+at, where at is a white noise series.

For type 1) process, we need to subtract δt (to produce a stationary process)before applying MLE to the data; for type 2) process, taking difference is indispensable (to achieve stationarity) before applying MLE.

Of course, in the first place we should test what kind of model fit the data best, and then follow the above method.

If the process is nonstationary, the traditional distributions (F or t) are no longer available. All kinds of estimators (ML, OLS, GMM, etc) will give rise to inconsistent estimates.

Y(t) = alpha + beta*Y(t-1) + u(t) ... AR(1)

assume u(t) follows a random walk

u(t) = u(t-1) + e(t), where e(t) is iid N(0, 1)

substitute u(t) to the AR(1) equation, you should find that e(t) are correlated with Y terms. As long as they are not orthognal, the estimates are not consistent. Large sample theory can not apply since the distribution is nonstandard. Similar proof can also be derived by converting the above AR(1) process to a MA(¥) .

14楼是正解，

We should first use ADF or other methods (usually ADF has low power to reject the null of unit root, so we can apply ,for instance, Doldado, Jenkinson, and Sosvilla-Rivero’s procedure(1990) instead ) to test for a unit root, that is , to test whether the original series is difference stationary or trend stationary . If the series is difference stationary , take difference and then apply MLE; if the series is trend stationary, detrend to get a stationary process and then employ MLE. Actually Macro economic series is often trend stationary under structural change, so employ Perron's unit root test under structural change and then do the similar things discussed above.

I would like to give some comments on the message provided by 16 楼. First, the type of nonstationarity is irrelevant to the original question. Second, time series analysis is based on the assumption that data generating process can be captured by using difference equations.

Basically, there is no way (do not trust the tests that show you these two can be distinguished) to identify whether a time series data is trend stationary or follows a stochastic process. A trend stationary process can be mimicked well by a random walk plus drift and vice versa.

107512.pdf (2.8 MB)

2007-4-13 06:28:00 上传

[有奖]The Consistency of beta??

不完全同意楼上观点，请参考这篇论文

Reply to 楼上,

I know (read it 10 year ago) this article. Before you make any judgements, try to do some simulation, and you can see my points.

Econometrics on time series is an "art" not pure science.

謝謝樓主的分享