AUTOCORRELATION Flashcards
What is AC?
One of the CLRM assumptions is that error terms are not correlated, that is there is no covariance between error terms Cov(Ut, ut-1)=0. Autocorrelation violates this assumption because the error term in time series t, is correlated with the error term in time series t-1. This occurs mostly in time series data and is due to the strong correlation in the shock in t and shock in time period t-1.
Non-formal detection of AC?
Graphical method:
- Plot the residuals, et, chronologically.
- Plot the residual et against et-1.
If a noticeable pattern occurs then AC may be present.
The two formal detections of AC?
- Durbin-Watson Test
2. Breusch-Godfrey Test
What is the DW test? What does it test for?
The Durbin-Watson test, tests for the first-order autocorrelation which means it tests if consecutive error terms are correlated.
D stat is defined as …
The sum of (et - et-1)^2/sum of et^2
If D>DU
No autocorrelation exists
If d less than DL
Have positive first-order autocorrelation - REJECT THE NULL
If D is between DL and DU
Inconclusive
If 4-DL less than D less than 4
Then negative first-order autocorrelation
What D test can we reject the null?
D less than DL
What are the assumptions of the Durbin-Watson Test?
1) Regression model must include an intercept in the OG model.
2) Variables are fixed in repeated sampling.
3) The error term, ut, follows an AR(1) scheme:
Ut = put-1 + Vt where P is the coefficient of AC and -1<=P<=1. This assumption means only the current error term and one period lagged error term is included.
4) Error terms follow a normal distribution
5) The regressors do not include lagged values of dependent variables. No Yt-1, Yt-2.
The null hypothesis of the Durbin-Watson Test
H0: RESIDUALS from OLS regression are not AC.
H1: Residuals have a first order (normally positive) AC
What is the BG test?
The BG test is a more general test for AC as it allows for more higher order AC schemes such as AR (2) and AR(3) so Ut-2 and Ut-3.
Ut = p1ut-1 + p2ut-2 + pput-p + Vt - current error term depends on the previous error term up to p-lags.
BG test also allows for lagged dependent variables.
How to run BG test?
- Run OLS and obtain residuals.
- Regress et on the regressors and the p auto-regressive terms:
et = A1 + A2lnDPIt + A3Dt + c1et-1 + C2et-2 - Obtain R2 from above.
What is the null hypothesis for the BG test?
Null H0:p1=p2=p3=pp=0 that is there is no AC
Alternative H1: Higher-order AC present
How to determine the BG hypothesis?
If Chi2?Critical Chi2 or F-value>critical F-value can reject the null also look at p-values of F and X2 if they are low might be AC present (cannot reject the null).
Solutions for AC?
The first difference transformation
How does it the solution for AC work?
If autocorrelation is of first-order then take first difference of dependent and all regressors. Want to remove the AC.
Ut - put-1 = Vt (now free from AC).
What if we know the value of p? What are the error term equation and the whole model equation
If we know the value of p, we can subtract p times the previous value of the error term from the current value.
Ut - Put-1 = Vt
Equation of the first difference transformation:
LnCt-pLnCt = B1(1-P) + B2(lnDt - plnDt-1) + b3(lnWt-plnWt-1) + (Ut-put-1) (which is now Vt and free from AC).
What id p=1?
Triangle Ct = B2triangleLnDPI + B3triangle lnWt + Vt
Removed B1
What is P(rho)
It is the correlation coefficient so by subtracting p we are removing first-order AC.
Consequence of AC
Not BLUE - large standard errors and small t-stats makes hypothesis testing suspect as cannot rely on T or F stats even in large samples
