AR(1) and AR(2) test in dynamic panel GMM estimation?

[ram shrestha]

Hi,

I am using diff-GMM and sys-GMM for an unbalanced panel with time (T=5) and country (N=84). I am trying to get Maintained Statistical Model (MSM) following the guidelines given by
Kiviet 2020 (J. of Econometrics and Statistics). Across different model specifications, p-value of AR(1) test >0.1 and AR(2) test>0.1.

But, Kiviet(2020), suggested that p-value of AR(1) should be less than 0.05.

Will the MSM be invalid if p-value of AR(1)>0.05(0.1) ??

[Sebastian Kripfganz]

First of all, do not use p-values such as 0.05 as a hard threshold. The world does not suddently change if you jump from one side of the threshold to the other.

The aim of the Arellano-Bond tests is to check whether the idiosyncratic error term is serially correlated. The test is conducted for the first-differenced errors. If the error term in levels is serially uncorrelated, this implies that the error term in first differences has negative first-order serial correlation (with a correlation coefficient of -0.5) but no second-order or higher-order serial correlation. Thus, we should reject the null hypothesis of no first-order serial correlation in first differences (AR(1) test) but should not reject the null hypothesis of no higher-order serial correlation in first differences (AR(2), AR(3), ...).

If you do not reject the null hypothesis of the AR(1) test, this could indicate that your idiosyncratic error term in levels is highly serially correlated. In the extreme case, the error term in levels follows a random walk such that the first-differenced errors are serially uncorrelated. Such a situation would indeed invalidate the MSM.

[ram shrestha]

Thank you Sebastian Kripfganz for your response. This was a very useful reply to me.

What should be done to address this problem? Please provide me with some hints.
All my efforts with different lag lengths and instruments (collapse/curtail) and additional variables are proving futile.

Looking forward to hearing from you.
Thank you so much

[Sebastian Kripfganz]

I am afraid there is no cookbook recipe. With your small T, it is difficult to add further lags as regressors in an attempt to account for the serial correlation. If none of the usual strategies work (such as adding variables/lags or starting with deeper lags for the instruments), you might have to accept that your results are imperfect. You might still be able to draw tentative conclusions from your analysis.

Alternatively, you could think about directly specifying your dependent variable in first differences, if that still gives you a meaningful model.

[ram shrestha]

Dear Sebastian Kripfganz

Following your suggestion for I did quick check for the model (with first difference DV) as follows:
I created differeced dependent variable as:

gen dy=d.y

and

xtdpdgmm L(0/1).dy emig inv edu fd trade, model(diff) collapse ///
 gmm(dy, lag(2 .)) gmm( emig , lag(2 .)) gmm(inv edu fd trade, lag(2 .)) ///
 gmm(dy, lag (1 1) diff model(level) nocollapse) gmm(emig , lag (1 1) diff model(level) nocollapse) gmm( inv edu fd trade, lag(1 1) diff model (level) nocollapse) ///
 teffect small nocons two vce(r) overid
 estat serial, ar(1/3) // for serial correlation
estat overid

Group variable: iso3n                        Number of obs         =       213
Time variable: period                        Number of groups      =        83

Moment conditions:     linear =      37      Obs per group:    min =         1
                    nonlinear =       0                        avg =  2.566265
                        total =      37                        max =         3

                                 (Std. Err. adjusted for 83 clusters in iso3n)
------------------------------------------------------------------------------
             |              WC-Robust
          dy |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          dy |
         L1. |   .6174474   .1383448     4.46   0.000     .3422356    .8926592
             |
        emig |  -.0062989   .0081776    -0.77   0.443    -.0225668    .0099689
         inv |   .0021941   .0050445     0.43   0.665     -.007841    .0122292
         edu |   .0037634   .0025789     1.46   0.148    -.0013668    .0088937
          fd |  -.0026802   .0014388    -1.86   0.066    -.0055425     .000182
       trade |  -.0002736   .0005378    -0.51   0.612    -.0013435    .0007963
             |
      period |
          3  |  -.1014875   .1491103    -0.68   0.498    -.3981153    .1951403
          4  |  -.0602028   .1572571    -0.38   0.703    -.3730373    .2526318
          5  |  -.1328092   .1612046    -0.82   0.412    -.4534965    .1878781
------------------------------------------------------------------------------
Instruments corresponding to the linear moment conditions:
 1, model(diff):
   L2.dy L3.dy
 2, model(diff):
   L2.emig L3.emig L4.emig
 3, model(diff):
   L2.inv L3.inv L4.inv L2.edu L3.edu L4.edu L2.fd L3.fd L4.fd L2.trade
   L3.trade L4.trade
 4, model(level):
   4:L1.D.dy 5:L1.D.dy
 5, model(level):
   3:L1.D.emig 4:L1.D.emig 5:L1.D.emig
 6, model(level):
   3:L1.D.inv 4:L1.D.inv 5:L1.D.inv 6:L1.D.inv 3:L1.D.edu 4:L1.D.edu
   5:L1.D.edu 6:L1.D.edu 3:L1.D.fd 4:L1.D.fd 5:L1.D.fd 6:L1.D.fd
 7, model(level):
   3bn.period 4.period 5.period

.  estat serial, ar(1/3) // for serial correlation

Arellano-Bond test for autocorrelation of the first-differenced residuals
H0: no autocorrelation of order 1:     z =   -1.7673   Prob > |z|  =    0.0772
H0: no autocorrelation of order 2:     z =         .   Prob > |z|  =         .
H0: no autocorrelation of order 3:     z =         .   Prob > |z|  =         .

. estat overid // for overdi (after estimation of xtdpdgmm)

Sargan-Hansen test of the overidentifying restrictions
H0: overidentifying restrictions are valid

2-step moment functions, 2-step weighting matrix       chi2(28)    =   30.6127
                                                       Prob > chi2 =    0.3345

2-step moment functions, 3-step weighting matrix       chi2(28)    =   34.7022
                                                       Prob > chi2 =    0.1787

However, I could not get AR(2). Is it always the case or I did something wrong in estimation?
Please suggest me.
My sincere apology to bother you.

Best Regards
Ram

[Sebastian Kripfganz]

You lose one time period because of first differencing (and another because of the lagged dependent variable). Eventually, you do not have enough time periods anymore to conduct the AR(2) test.

The AR(1) test still has a quite low p-value. Taking the first-differenced dependent variable thus does not seem to have solved the problem. Instead, you could consider adding lags of the regressors to the model.