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  • #1

    AR(2) significance and missing AR(3) in xtabond2 with different laglimits

    Dear Statalist members
    I ran the following command:

    Code:
    xtabond2 p0 L.p0 rkfd rbbs yr2 yr4 yr5, gmm(p0, lag(2 .) collapse) gmm(rkfd, lag(2 .) collapse) gmm(rbbs, lag(2 .) collapse) iv(yr2 yr4 yr5) noleveleq twostep robust small
    The output shows AR(2) is significant (p < 0.05) and the Hansen test also indicates problems.

    Then I tried increasing the laglimit:

    xtabond2 p0 L.p0 rkfd rbbs yr2 yr4 yr5, gmm(p0, lag(3 .) collapse) gmm(rkfd, lag(3 .) collapse) gmm(rbbs, lag(3 .) collapse) iv(yr2 yr4 yr5) noleveleq twostep robust small artests(3)
    In this case, AR(2) remains significant (p = 0.012), while AR(3) does not appear in the output (“.”). The Hansen test, however, no longer shows issues.

    My questions are:
    1)     Is it normal that AR(3) does not appear in the xtabond2 output?
    2) Given that AR(2) remains significant (< 5%), does this mean the model results are invalid and cannot be used?

    Thank you very much for your guidance.

    Best regards,
    Hadi.
    Tags: None
  • #2
    1) This indicates that you only have a very small number of time periods and therefore insufficient observations to compute higher-order serial correlation tests.
    2) In principle, you could accept a significant test for second-order serial correlation in the first-differenced errors when you are only using higher-order serial lags. However, given the short number of time periods in your sample, you might not have enough instruments available. Furthermore, you would then require that higher-order tests are insignificant, which you apparently cannot test with your data. The issue thus remains inconclusive.

    For more details on dynamic panel data GMM estimation and serial correlation testing, you might find my two of my London Stata Conference presentations useful:
    https://www.kripfganz.de/stata/

    Comment

    • #3
      Originally posted by Sebastian Kripfganz View Post
      1) This indicates that you only have a very small number of time periods and therefore insufficient observations to compute higher-order serial correlation tests.
      2) In principle, you could accept a significant test for second-order serial correlation in the first-differenced errors when you are only using higher-order serial lags. However, given the short number of time periods in your sample, you might not have enough instruments available. Furthermore, you would then require that higher-order tests are insignificant, which you apparently cannot test with your data. The issue thus remains inconclusive.

      For more details on dynamic panel data GMM estimation and serial correlation testing, you might find my two of my London Stata Conference presentations useful:
      1) What if I run using lag (2 .) and lag (2 3), which produce AR(2) (0.000) < 5% and Hansen Test also (0.000) < 5%, but when using lag (3 3) it produces AR(2) = 0.221 > 5% while the Hansen Test & AR(3) has no value. Can the output from lag (3 3) be used or not?
      2) However, I also tried using system GMM two-step in a similar way. With lag (2 .) and lag (2 3), it produces AR(2) = 0.093 > 5%, AR(3) has no value, Hansen Test = 0.432, and Difference-in-Hansen tests (GMM instruments for levels) produce Difference (null H = exogenous) = 0.131. Meanwhile, with lag (3 3), it produces AR(2) = 0.224 > 5%, AR(3) has no value, Hansen Test = 0.148, and Difference-in-Hansen tests (GMM instruments for levels) produce Difference (null H = exogenous) = 0.148. Can these results also be used despite the absence of AR(3)?

      Thank you for your time and attention.
      Best regards,
      Hadi

      Comment

      • #4
        As per the FAQs, please provide full code and output, as otherwise it is hard to give advice.
        https://www.kripfganz.de/stata/

        Comment

        • #5
          Originally posted by Sebastian Kripfganz View Post
          As per the FAQs, please provide full code and output, as otherwise it is hard to give advice.
          Code:
          estimates table ols fe, keep(L.p0) b(%7.3f) se




----------------------------------

    Variable |   ols       fe    

-------------+--------------------

          p0 |

         L1. |   0.982     0.497  

             |   0.002     0.033  

----------------------------------

                      Legend: b/se

. xtabond2 p0 L.p0 rkfd rbbs yr2 yr4 yr5, gmmstyle(p0, laglimits(2 .) collapse) ///

> gmmstyle(rkfd, laglimits(2 .) collapse) gmmstyle(rbbs, laglimits(2 .) collapse) ///

> ivstyle(yr2 yr4 yr5) noleveleq twostep robust small

Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.




Dynamic panel-data estimation, two-step difference GMM

------------------------------------------------------------------------------

Group variable: id1                             Number of obs      =      1524

Time variable : year                            Number of groups   =       508

Number of instruments = 12                      Obs per group: min =         3

F(5, 508)     =    427.37                                      avg =      3.00

Prob > F      =     0.000                                      max =         3

------------------------------------------------------------------------------

             |              Corrected

          p0 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]

-------------+----------------------------------------------------------------

          p0 |

         L1. |   .9204626   .1160154     7.93   0.000     .6925335    1.148392

             |

        rkfd |  -.0839673   .0644258    -1.30   0.193    -.2105411    .0426065

        rbbs |  -.0230433   .0450035    -0.51   0.609    -.1114592    .0653725

         yr2 |   .7821115   .0594443    13.16   0.000     .6653247    .8988984

         yr4 |   .3105727    .069096     4.49   0.000     .1748236    .4463217

         yr5 |   .1415295   .0855534     1.65   0.099    -.0265525    .3096115

------------------------------------------------------------------------------

Instruments for first differences equation

  Standard

    D.(yr2 yr4 yr5)

  GMM-type (missing=0, separate instruments for each period unless collapsed)

    L(2/4).rbbs collapsed

    L(2/4).rkfd collapsed

    L(2/4).p0 collapsed

------------------------------------------------------------------------------

Arellano-Bond test for AR(1) in first differences: z =  -6.95  Pr > z =  0.000

Arellano-Bond test for AR(2) in first differences: z =   3.54  Pr > z =  0.000

------------------------------------------------------------------------------

Sargan test of overid. restrictions: chi2(6)    =  36.87  Prob > chi2 =  0.000

  (Not robust, but not weakened by many instruments.)

Hansen test of overid. restrictions: chi2(6)    =  31.22  Prob > chi2 =  0.000

  (Robust, but weakened by many instruments.)




Difference-in-Hansen tests of exogeneity of instrument subsets:

  gmm(p0, collapse lag(2 .))

    Hansen test excluding group:     chi2(3)    =   3.16  Prob > chi2 =  0.368

    Difference (null H = exogenous): chi2(3)    =  28.06  Prob > chi2 =  0.000

  gmm(rkfd, collapse lag(2 .))

    Hansen test excluding group:     chi2(3)    =   8.26  Prob > chi2 =  0.041

    Difference (null H = exogenous): chi2(3)    =  22.96  Prob > chi2 =  0.000

  gmm(rbbs, collapse lag(2 .))

    Hansen test excluding group:     chi2(3)    =  18.07  Prob > chi2 =  0.000

    Difference (null H = exogenous): chi2(3)    =  13.15  Prob > chi2 =  0.004

  iv(yr2 yr4 yr5)

    Hansen test excluding group:     chi2(3)    =   6.44  Prob > chi2 =  0.092

    Difference (null H = exogenous): chi2(3)    =  24.78  Prob > chi2 =  0.000
          1. From the output it is known that: AR(2) and the Hansen Test are < 5%. Therefore, I tried to extend the lag as presented below:
          Code:
          . xtabond2 p0 L.p0 rkfd rbbs yr2 yr4 yr5, gmmstyle(p0, laglimits(3 .) collapse) ///

> gmmstyle(rkfd, laglimits(3 .) collapse) gmmstyle(rbbs, laglimits(3 .) collapse) ///

> ivstyle(yr2 yr4 yr5) noleveleq twostep robust small artest(3)

Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.




Dynamic panel-data estimation, two-step difference GMM

------------------------------------------------------------------------------

Group variable: id1                             Number of obs      =      1524

Time variable : year                            Number of groups   =       508

Number of instruments = 9                       Obs per group: min =         3

F(5, 508)     =    354.74                                      avg =      3.00

Prob > F      =     0.000                                      max =         3

------------------------------------------------------------------------------

             |              Corrected

          p0 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]

-------------+----------------------------------------------------------------

          p0 |

         L1. |   .7924446   .2363788     3.35   0.001     .3280442    1.256845

             |

        rkfd |   -.301647   .3476644    -0.87   0.386    -.9846841    .3813901

        rbbs |   .2397314   .1290108     1.86   0.064    -.0137289    .4931918

         yr2 |   .6464067   .1646576     3.93   0.000     .3229131    .9699003

         yr4 |    .263106   .0873878     3.01   0.003       .09142    .4347921

         yr5 |    .093905   .1209952     0.78   0.438    -.1438076    .3316176

------------------------------------------------------------------------------

Instruments for first differences equation

  Standard

    D.(yr2 yr4 yr5)

  GMM-type (missing=0, separate instruments for each period unless collapsed)

    L(3/4).rbbs collapsed

    L(3/4).rkfd collapsed

    L(3/4).p0 collapsed

------------------------------------------------------------------------------

Arellano-Bond test for AR(1) in first differences: z =  -3.88  Pr > z =  0.000

Arellano-Bond test for AR(2) in first differences: z =   2.50  Pr > z =  0.012

Arellano-Bond test for AR(3) in first differences: z =      .  Pr > z =      .

------------------------------------------------------------------------------

Sargan test of overid. restrictions: chi2(3)    =   0.29  Prob > chi2 =  0.962

  (Not robust, but not weakened by many instruments.)

Hansen test of overid. restrictions: chi2(3)    =   0.41  Prob > chi2 =  0.938

  (Robust, but weakened by many instruments.)




Difference-in-Hansen tests of exogeneity of instrument subsets:

  gmm(p0, collapse lag(3 .))

    Hansen test excluding group:     chi2(1)    =   0.09  Prob > chi2 =  0.770

    Difference (null H = exogenous): chi2(2)    =   0.33  Prob > chi2 =  0.850

  gmm(rkfd, collapse lag(3 .))

    Hansen test excluding group:     chi2(1)    =   0.03  Prob > chi2 =  0.870

    Difference (null H = exogenous): chi2(2)    =   0.39  Prob > chi2 =  0.825

  gmm(rbbs, collapse lag(3 .))

    Hansen test excluding group:     chi2(1)    =   0.00  Prob > chi2 =  0.965

    Difference (null H = exogenous): chi2(2)    =   0.41  Prob > chi2 =  0.815
          2. From the output using lag (3 .), it can be seen that: AR(2) < 5% whereas it should be > 5%; AR(3) has no value because my T is very short = 5; then the Hansen Test is > 5%, but almost close to 1.000, whereas it should be '0.10<p<0.25', correct? Therefore, I tried to restrict the lag as shown below:

          Code:
          . xtabond2 p0 L.p0 rkfd rbbs yr2 yr4 yr5, gmmstyle(p0, laglimits(3 3) collapse) ///

> gmmstyle(rkfd, laglimits(3 3) collapse) gmmstyle(rbbs, laglimits(3 3) collapse) ///

> ivstyle(yr2 yr4 yr5) noleveleq twostep robust small artest(3)

Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.




Dynamic panel-data estimation, two-step difference GMM

------------------------------------------------------------------------------

Group variable: id1                             Number of obs      =      1524

Time variable : year                            Number of groups   =       508

Number of instruments = 6                       Obs per group: min =         3

F(5, 508)     =    157.10                                      avg =      3.00

Prob > F      =     0.000                                      max =         3

------------------------------------------------------------------------------

             |              Corrected

          p0 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]

-------------+----------------------------------------------------------------

          p0 |

         L1. |   .9364769   .3685193     2.54   0.011     .2124674    1.660486

             |

        rkfd |  -.6326397     .73555    -0.86   0.390    -2.077734    .8124548

        rbbs |   .3757894    .350604     1.07   0.284    -.3130229    1.064602

         yr2 |   .4704372    .384721     1.22   0.222    -.2854029    1.226277

         yr4 |   .2873559   .1046087     2.75   0.006     .0818369    .4928748

         yr5 |   .1608065   .1817469     0.88   0.377    -.1962616    .5178747

------------------------------------------------------------------------------

Instruments for first differences equation

  Standard

    D.(yr2 yr4 yr5)

  GMM-type (missing=0, separate instruments for each period unless collapsed)

    L3.rbbs collapsed

    L3.rkfd collapsed

    L3.p0 collapsed

------------------------------------------------------------------------------

Arellano-Bond test for AR(1) in first differences: z =  -2.16  Pr > z =  0.031

Arellano-Bond test for AR(2) in first differences: z =   1.22  Pr > z =  0.221

Arellano-Bond test for AR(3) in first differences: z =      .  Pr > z =      .

------------------------------------------------------------------------------

Sargan test of overid. restrictions: chi2(0)    =   0.00  Prob > chi2 =      .

  (Not robust, but not weakened by many instruments.)

Hansen test of overid. restrictions: chi2(0)    =   0.00  Prob > chi2 =      .

  (Robust, but weakened by many instruments.)
          3. From the output it can be seen that: AR(2) is already > 5%; AR(3) has no value because my T is very short = 5; then the Hansen Test also has no value (is this the same as AR(3) because T is short?). Then I also tried using System GMM two-step with lag (3 .):

          Code:
          . xtabond2 p0 L.p0 rkfd rbbs dgeo1 yr2 yr4 yr5, gmmstyle(p0, laglimits(3 .) collapse) ///

> gmmstyle(rkfd, laglimits(3 .) collapse) gmmstyle(rbbs, laglimits(3 .) collapse) ///

> ivstyle(dgeo1, equation(level)) ivstyle(yr2 yr4 yr5, equation(level)) twostep ///

> robust small artest(3)

Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.




Dynamic panel-data estimation, two-step system GMM

------------------------------------------------------------------------------

Group variable: id1                             Number of obs      =      2032

Time variable : year                            Number of groups   =       508

Number of instruments = 14                      Obs per group: min =         4

F(7, 507)     =  78686.32                                      avg =      4.00

Prob > F      =     0.000                                      max =         4

------------------------------------------------------------------------------

             |              Corrected

          p0 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]

-------------+----------------------------------------------------------------

          p0 |

         L1. |   .9548909   .0097411    98.03   0.000      .935753    .9740288

             |

        rkfd |  -.2373965   .1791103    -1.33   0.186    -.5892862    .1144931

        rbbs |   .3781727   .1527103     2.48   0.014     .0781497    .6781956

       dgeo1 |   .1059061   .0579206     1.83   0.068    -.0078879       .2197

         yr2 |   .7326321   .1107867     6.61   0.000     .5149745    .9502897

         yr4 |   .3770078   .0331926    11.36   0.000     .3117959    .4422197

         yr5 |   .2513513   .0355703     7.07   0.000     .1814679    .3212346

       _cons |  -.1532194   .2422266    -0.63   0.527     -.629111    .3226721

------------------------------------------------------------------------------

Instruments for first differences equation

  GMM-type (missing=0, separate instruments for each period unless collapsed)

    L(3/4).rbbs collapsed

    L(3/4).rkfd collapsed

    L(3/4).p0 collapsed

Instruments for levels equation

  Standard

    yr2 yr4 yr5

    dgeo1

    _cons

  GMM-type (missing=0, separate instruments for each period unless collapsed)

    DL2.rbbs collapsed

    DL2.rkfd collapsed

    DL2.p0 collapsed

------------------------------------------------------------------------------

Arellano-Bond test for AR(1) in first differences: z =  -7.63  Pr > z =  0.000

Arellano-Bond test for AR(2) in first differences: z =   1.68  Pr > z =  0.093

Arellano-Bond test for AR(3) in first differences: z =      .  Pr > z =      .

------------------------------------------------------------------------------

Sargan test of overid. restrictions: chi2(6)    =  10.55  Prob > chi2 =  0.103

  (Not robust, but not weakened by many instruments.)

Hansen test of overid. restrictions: chi2(6)    =   5.92  Prob > chi2 =  0.432

  (Robust, but weakened by many instruments.)




Difference-in-Hansen tests of exogeneity of instrument subsets:

  GMM instruments for levels

    Hansen test excluding group:     chi2(3)    =   0.29  Prob > chi2 =  0.962

    Difference (null H = exogenous): chi2(3)    =   5.63  Prob > chi2 =  0.131

  gmm(p0, collapse lag(3 .))

    Hansen test excluding group:     chi2(3)    =   1.98  Prob > chi2 =  0.576

    Difference (null H = exogenous): chi2(3)    =   3.94  Prob > chi2 =  0.268

  gmm(rkfd, collapse lag(3 .))

    Hansen test excluding group:     chi2(3)    =   0.62  Prob > chi2 =  0.893

    Difference (null H = exogenous): chi2(3)    =   5.31  Prob > chi2 =  0.151

  gmm(rbbs, collapse lag(3 .))

    Hansen test excluding group:     chi2(3)    =   3.98  Prob > chi2 =  0.264

    Difference (null H = exogenous): chi2(3)    =   1.95  Prob > chi2 =  0.584

  iv(dgeo1, eq(level))

    Hansen test excluding group:     chi2(5)    =   1.39  Prob > chi2 =  0.925

    Difference (null H = exogenous): chi2(1)    =   4.53  Prob > chi2 =  0.033

  iv(yr2 yr4 yr5, eq(level))

    Hansen test excluding group:     chi2(3)    =   0.11  Prob > chi2 =  0.991

    Difference (null H = exogenous): chi2(3)    =   5.81  Prob > chi2 =  0.121
          4. From the output it can be seen that: AR(2) is already > 5%; AR(3) has no value because my T is very short = 5; then the Hansen Test is > 5% and for GMM instruments for levels also > 5%. But I tried again by restricting the lag to (3 3) as shown below:

          Code:
          . xtabond2 p0 L.p0 rkfd rbbs dgeo1 yr2 yr4 yr5, gmmstyle(p0, laglimits(3 3) collapse) ///

> gmmstyle(rkfd, laglimits(3 3) collapse) gmmstyle(rbbs, laglimits(3 3) collapse) ///

> ivstyle(dgeo1, equation(level)) ivstyle(yr2 yr4 yr5, equation(level)) twostep ///

> robust small artest(3)

Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.




Dynamic panel-data estimation, two-step system GMM

------------------------------------------------------------------------------

Group variable: id1                             Number of obs      =      2032

Time variable : year                            Number of groups   =       508

Number of instruments = 11                      Obs per group: min =         4

F(7, 507)     =  81757.03                                      avg =      4.00

Prob > F      =     0.000                                      max =         4

------------------------------------------------------------------------------

             |              Corrected

          p0 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]

-------------+----------------------------------------------------------------

          p0 |

         L1. |   .9542529   .0119907    79.58   0.000     .9306954    .9778105

             |

        rkfd |  -.3022626     .35212    -0.86   0.391    -.9940567    .3895314

        rbbs |   .4529321   .4174496     1.08   0.278     -.367212    1.273076

       dgeo1 |   .1017565   .0557956     1.82   0.069    -.0078626    .2113756

         yr2 |   .6980213   .2294033     3.04   0.002     .2473233    1.148719

         yr4 |   .3841963   .0408712     9.40   0.000     .3038985    .4644941

         yr5 |   .2580085   .0486104     5.31   0.000     .1625059    .3535111

       _cons |  -.0970751   .3702215    -0.26   0.793    -.8244324    .6302821

------------------------------------------------------------------------------

Instruments for first differences equation

  GMM-type (missing=0, separate instruments for each period unless collapsed)

    L3.rbbs collapsed

    L3.rkfd collapsed

    L3.p0 collapsed

Instruments for levels equation

  Standard

    yr2 yr4 yr5

    dgeo1

    _cons

  GMM-type (missing=0, separate instruments for each period unless collapsed)

    DL2.rbbs collapsed

    DL2.rkfd collapsed

    DL2.p0 collapsed

------------------------------------------------------------------------------

Arellano-Bond test for AR(1) in first differences: z =  -5.42  Pr > z =  0.000

Arellano-Bond test for AR(2) in first differences: z =   1.22  Pr > z =  0.224

Arellano-Bond test for AR(3) in first differences: z =      .  Pr > z =      .

------------------------------------------------------------------------------

Sargan test of overid. restrictions: chi2(3)    =  10.50  Prob > chi2 =  0.015

  (Not robust, but not weakened by many instruments.)

Hansen test of overid. restrictions: chi2(3)    =   5.35  Prob > chi2 =  0.148

  (Robust, but weakened by many instruments.)




Difference-in-Hansen tests of exogeneity of instrument subsets:

  GMM instruments for levels

    Hansen test excluding group:     chi2(0)    =   0.00  Prob > chi2 =      .

    Difference (null H = exogenous): chi2(3)    =   5.35  Prob > chi2 =  0.148

  gmm(p0, collapse lag(3 3))

    Hansen test excluding group:     chi2(1)    =   1.77  Prob > chi2 =  0.184

    Difference (null H = exogenous): chi2(2)    =   3.58  Prob > chi2 =  0.167

  gmm(rkfd, collapse lag(3 3))

    Hansen test excluding group:     chi2(1)    =   0.01  Prob > chi2 =  0.913

    Difference (null H = exogenous): chi2(2)    =   5.34  Prob > chi2 =  0.069

  gmm(rbbs, collapse lag(3 3))

    Hansen test excluding group:     chi2(1)    =   3.46  Prob > chi2 =  0.063

    Difference (null H = exogenous): chi2(2)    =   1.88  Prob > chi2 =  0.390

  iv(dgeo1, eq(level))

    Hansen test excluding group:     chi2(2)    =   0.83  Prob > chi2 =  0.660

    Difference (null H = exogenous): chi2(1)    =   4.52  Prob > chi2 =  0.034

  iv(yr2 yr4 yr5, eq(level))

    Hansen test excluding group:     chi2(0)    =   0.00  Prob > chi2 =      .

    Difference (null H = exogenous): chi2(3)    =   5.35  Prob > chi2 =  0.148
          5. From the final output it can be seen that: AR(2) is already > 5%; AR(3) has no value because my T is very short = 5; then the Hansen Test is > 5% and for GMM instruments for levels also > 5%.

          My questions are:
          a) When faced with the condition that AR(3) has no value while AR(2) is already > 5%, can the output still be used, especially if the Hansen Test also has no value as in the output in point 3? I am thinking that if AR(2) is already clean (thus producing > 5%), even though AR(3) has no value (because T is short as you mentioned earlier), then if T were longer (say T = 6 so that AR(3) appears), wouldn’t it also produce AR(3) > 5%?

          b) From the five tests I conducted, the last one, 'system GMM (with lag (3 3)) in point 5', isn’t that the most robust and usable one (please correct me)? Or is it that none can be used, especially those where AR(3) has no value?

          Thank you for your time and attention.
          Best regards,
          Hadi
          Last edited by Redja Hadi; 24 Aug 2026, 05:16.

          Comment

          • #6
            2. Your number of instruments is very small (given the small number of time periods). There is no concern of instrument proliferation here. In this regard, a large p-value of the Hansen test is not a concern either. Taken at face value, it supports your specification. However, notice that overidentification tests rely on the untestable assumption that there are at least as many valid instruments as there are coefficients to be estimated. If there was higher order correlation, which you cannot test with the Arellano-Bond test, then potentially all of your instruments would be invalid. Thus, the Hansen test may not be reliable, irrespective of whether the p-value is 0.9 or 0.2.

            3. With 6 instruments and 6 coefficients, your model is just-identified. There are no overidentifying restrictions to be tested with the Hansen test. The logic is the same as mentioned above: You need at least as many valid instruments as there are coefficients - the validity of those instruments is not testable.

            4./5. Taken at face value, your specification test results are borderline acceptable. However, the same qualifications apply as above. System GMM introduces an additional assumption (mean stationarity). You are unable to assess all assumptions with your data. You can interpret the Hansen test as a test for validity of the instruments for the first-differenced model assuming that the instruments for the level model are valid, or vice versa. You cannot test both at the same time.

            Given these limitations, you can use the final results but should add a word of caution that the underlying assumptions are not all testable.
            https://www.kripfganz.de/stata/

            Comment

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