Breusch-Godfrey Test Assignment Help

Testing for autocorrelation - Breusch-Godfrey Test

Breusch-Godfrey Test:

A more powerful test that is also commonly used in empirical applications  is the Breusch-Godfrey (BG)  tyst, also known as the LM test. Using our earlier example of a single variable with intercept model the test proceeds in the following way. Note in this context that there can be many more variables as regressors, one  variable case  is  taken  for  simplicity only. Similarly,  lagged values of the dependent variable etc. can also be  there as a regressor. Assume that the error term follows the p-th order autoregressive,  AR(p) scheme (extension of the earlier AR(1) scheme), as follows  

990_Breusch-Godfrey Test.png

where E, is a stochastic error as before. The null hypothesis to be tested is:

889_Breusch-Godfrey Test1.png

In it is hypothesized that there is no autocorrelation of any order. The BG test involves the following steps:

1) Estimate by OLS and obtain  the residuals ti,

2)  Regress ui,  on the original  x, (as well as any other variables used as  regressors ealier)  and  2017_Breusch-Godfrey Test5.pngwhere  the latter are lagged values of the estimated residuals in step 1. Thus if p=4, we will  introduce four lagged values of the residuals as additional regressors in the model. In short, run  the following regression:

936_Breusch-Godfrey Test2.png

and obtain from this the auxiliary   egression.

3)  If the sample size is large then Breusch and Godfrey have shown that

401_Breusch-Godfrey Test3.png

where T is the total.number of observations. That is the BG test statistic is asymptotically  (as the sample size increases to infinity) distributed as a chi- squared distribution withp  degrees  of freedom. Therefore, once  the test statistic has been calculated this way it can be  compared to a standard chi-squared distribution  to accept or reject the null hypothesis of no autocorrelation of any order.

Note that unlike the Durbin-Waton test the BG test can test for both higher order schemes of  the  error process  AR(p) not  just AR(1).  It also  is  applicable when alternate time series for  the error term is specified such as  ,a Moving Average scheme.  MA(p), is defined as follows

232_Breusch-Godfrey Test4.png


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