Depending of what you doing, but you may read the Arrelano-Bond approach

Dear Bogdan-Vasile Ileanu thank you for your good guidance 

estimators In partial adjustment model  are consistent, so we didn't need dynamic panel data estimators such arrelano-bond, IV, ... 

I use OLS estimators for that, and wanna just test autocorrelation 

the Arrelano-Bond test.

For the test of first-order autocorrelation absence H0 is not rejected.

The presence of correlation of first order in the differentiated waste does not imply that the estimates are inconsistent.

But the presence of second-order autocorrelation implies that if the estimates are inconsistent.

H0 in this case the absence of second-order autocorrelation is not rejected.

Arellano-Bond tests are valid for n tending to infinity. You should read the Econometrica paper by Alvarez-Arellano and references therein.

STATA software is a good statistical software for analysing Panel Data. Testing for autocorrelation is simply done by using the command xtserial y x1 x2 .......xn, when the statistic is significant it indicates the presence of autocorrelation.

Check the attached document by Chris Baum and the link below.

http://www.stata.com/support/faqs/statistics/panel-level-heteroskedasticity-and-autocorrelation/

 Thank you all, for your good answers

best regards 

 Hi, 

You can employ the Arellano-Bond test for AR(1)  and the Arellano-Bond test for AR(2).  If the absence of the second-order serial correlation in disturbances (which is the null hypothesis) is not rejected, you are safe. The first order serial correlation is expected (due to the lagged dependent term), and should not be a problem. 

You may have a look at this paper. 

http://link.springer.com/article/10.1007%2Fs10368-016-0341-7

Hi dear Olivier

I'm grateful because of your guidance 

Partial adjustment model (PAM) is a dynamic model, and so panel data PAM is a dynamic panel model; hence Arellano-Bond estimation is required. When the idiosyncratic errors in the panel are independently and identically distributed (i.i.d.), the first-differenced errors will become first-order auto-correlated. So, the expected result for the AR(1) test must present strong evidence against the null hypothesis of zero autocorrelation in the first-differenced errors at order 1. And there must be no autocorrelation in the first differenced errors at an order higher than 1; evidence against this implies that the moment conditions used in the estimation are not valid.

Arellano-Bond tests for AR(1) & AR(2)

Arellano–Bond test for first-order (AR1) and second-order (AR2) serial correlation.

panel data consists of several time series -- each tracking a different aspect of the individuals -- and each of these time series will tend to be autocorrelated, but there need not be any particular correlation between them.

Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them.

Check on this paper

https://www.mdpi.com/2227-7099/8/3/61/htm

You can use Arrellano- Bond tests AR(1) and AR(2). Due to the dynamic aspects, AR (1) should exist and hence AR(2) should indicate absence of second order serial correlation

panel data consists of several time series -- each tracking a different aspect of the individuals -- and each of these time series will tend to be autocorrelated, but there need not be any particular correlation between them.

You can use the Arellano-Bond estimator for datasets with many panels and few periods. This estimator is designed for datasets with no autocorrelation in the idiosyncratic errors1.