DW can judge the first-order autocorrelation, which can generally be solved by difference method (first order)
The basic method to solve autocorrelation is to transform the original data through differential transformation to eliminate autocorrelation.
First, the difference method, first order.
Let the regression model from y to x be
yt =β 1+β 1xt+μt( 1)
μt =πμt- 1+vt
In the formula, vt satisfies all the assumptions of the least square method about the error term.
If the formula (1) lags behind for a period of time, there will be
yt- 1 =β0+β 1xt- 1+μt- 1(2)μt- 1 =πμt-2+vt- 1
Therefore, (1)-ρ×(2) gives yt-ρ yt-1= β 0 (1-ρ)+β1(XT-ρ XT-1)+957t.
Yt-ρYt- 1 =β 1(XT-XT- 1)+μt-μt- 1 =β 1(XT-XT- 1)+vt(4)
ρ is the autocorrelation coefficient
In other words, the first-order difference method is a special form of generalized difference method.
High-order autocorrelation is tested by BG, lm = t * r 2 obeys X 2 (p) (chi-square) distribution, t is the sample size, and p is the autocorrelation order you want to test. Chi-square distribution table, 95% confidence, that is, alpha =0.5. If t * r 2 > is the autocorrelation sequence you want to verify.
Generalized difference method (AR(p)) is used for correction.
Generalized difference method
For the model: YT = 0+ 1xt+ut-( 1), if ut has first-order autoregressive autocorrelation, that is, ut= u t- 1 +vt meets the usual assumptions.
Assuming that it is known, then: yt-1= 0+1XT-1+ut-1Multiply the two ends:
y t- 1 = 0+ 1 X t- 1+u t- 1-(2)
(1) minus (2):
yt-Y t- 1 = 0( 1-)+ 1X(Xt-X t- 1)+vt
Sequence: yt * = yt-yt- 1, XT * = (XT-XT- 1), 0 * = 0 (1-).
Then: Yt*= 0 *+ 1 Xt*+vt is called generalized difference model, and the random term satisfies the usual assumptions. The above equation can be solved by OLS estimation.
In order not to lose the sample points, let Y 1*= X 1*=
The above transformation for solving autocorrelation is called generalized difference transformation, and = 1 or =0, =- 1 is a special case.
Generalized difference transformation needs to know, and if it is not known, it needs to be estimated. The following methods are all developed according to the idea of finding the estimated value first and then carrying out differential transformation.
If the difference correction still doesn't work, it's your problem of regression variables. Some statistics have strong autocorrelation, such as GDP, which is inevitable. Some data must be castrated first and then co-integrated before they can be returned. I won't explain it in detail here. You should read the relevant chapters of econometrics textbooks carefully. You can ask me if you don't understand.
Mathematical Statistics and Management, Volume 27,No. 1 2008!
The linear correlation between components of n-dimensional random variables is studied, and the necessary and sufficient conditions for measuring the degree of linear correlation and having strict linear relationship (in the sense of probability 1) are given.