[Abdul Sazali]

Hi JanDitzen,

Thank you very much for your response. Could you please suggest me which estimator and command that might be suitable for my panel data while addressing the issues?

Many thanks,
Abdul

[JanDitzen]

I would use standard panel estimator such as fixed effects and IV+fixed effects. Do not use the mean group estimator, it is not possible to estimate the unit specific equations.

Given the short time dimension, tests for/estimation of strong cross-section dependence will be biased or over powered and hence I would not worry about it. Adding time fixed effects is a better solution.

[Abdul Sazali]

Dear JanDitzen,

Thank you very much for your suggestion.

I have previously used 2SLS FE by including the IV with fixed effects into the model to address endogeneity issue using the following command:

My endogenous variable is lnpax and my instrumental variables are lnpop and lntrade.

. xtset routeid year

Panel variable: routeid (strongly balanced)
 Time variable: year, 2012 to 2019
         Delta: 1 unit

. xtivreg lnfare (lnpax = lnpop lntrade) asam totalcarriers desigcarriers fscper lngdp countryfuel atii i.year, fe vce(robust)

Fixed-effects (within) IV regression            Number of obs     =      5,032
Group variable: routeid                         Number of groups  =        629

R-squared:                                      Obs per group:
     Within  =      .                                         min =          8
     Between = 0.0121                                         avg =        8.0
     Overall = 0.0015                                         max =          8


                                                Wald chi2(15)     =   33619.84
corr(u_i, Xb) = -0.6653                         Prob > chi2       =     0.0000

                               (Std. err. adjusted for 629 clusters in routeid)
-------------------------------------------------------------------------------
              |               Robust
       lnfare | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
        lnpax |   .5087595    .322941     1.58   0.115    -.1241932    1.141712
         asam |  -.0898262   .0396187    -2.27   0.023    -.1674775   -.0121749
totalcarriers |  -.0583593    .031977    -1.83   0.068     -.121033    .0043143
desigcarriers |  -.0613801   .0325806    -1.88   0.060     -.125237    .0024767
       fscper |   .0023645   .0011204     2.11   0.035     .0001686    .0045604
        lngdp |  -.2661736   .1708156    -1.56   0.119    -.6009659    .0686188
  countryfuel |  -.1371823   .0428347    -3.20   0.001    -.2211367   -.0532279
         atii |  -.0119196   .0058801    -2.03   0.043    -.0234443   -.0003948
              |
         year |
        2013  |  -.1701968   .0547684    -3.11   0.002    -.2775408   -.0628527
        2014  |  -.2884796   .0704503    -4.09   0.000    -.4265597   -.1503994
        2015  |  -.5909791   .1349744    -4.38   0.000    -.8555241   -.3264342
        2016  |  -.6658244   .1440267    -4.62   0.000    -.9481116   -.3835373
        2017  |   -.631819   .1415059    -4.46   0.000    -.9091655   -.3544725
        2018  |  -.5676029   .1322243    -4.29   0.000    -.8267579    -.308448
        2019  |  -.6477493    .137719    -4.70   0.000    -.9176736    -.377825
              |
        _cons |   3.165659   2.092094     1.51   0.130    -.9347709    7.266089
--------------+----------------------------------------------------------------
      sigma_u |  .68559818
      sigma_e |  .24511511
          rho |  .88666584   (fraction of variance due to u_i)
-------------------------------------------------------------------------------
Endogenous: lnpax
Exogenous:  asam totalcarriers desigcarriers fscper lngdp countryfuel atii
            2013.year 2014.year 2015.year 2016.year 2017.year 2018.year
            2019.year lnpop lntrade

.
end of do-file

As I assume that 2SLS FE cannot address the cross-sectional dependence issue, I try to combine the IV FE with DK Standard Errors using the following command:

. reg lnpax lnpop lntrade

      Source |       SS           df       MS      Number of obs   =     5,032
-------------+----------------------------------   F(2, 5029)      =    398.67
       Model |  1028.50044         2  514.250218   Prob > F        =    0.0000
    Residual |  6487.01765     5,029  1.28992198   R-squared       =    0.1369
-------------+----------------------------------   Adj R-squared   =    0.1365
       Total |  7515.51809     5,031   1.4938418   Root MSE        =    1.1357

------------------------------------------------------------------------------
       lnpax | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       lnpop |   .2790867    .027533    10.14   0.000       .22511    .3330634
     lntrade |   .1553236   .0083099    18.69   0.000     .1390326    .1716145
       _cons |   5.081224   .4201694    12.09   0.000     4.257509    5.904939
------------------------------------------------------------------------------

.
end of do-file

. predict lnpax_hat, xb

.
end of do-file


. xtscc lnfare lnpax_hat asam totalcarriers desigcarriers fscper lngdp countryfuel atii i.year, fe lag(7)

Regression with Driscoll-Kraay standard errors   Number of obs     =      5032
Method: Fixed-effects regression                 Number of groups  =       629
Group variable (i): routeid                      F( 15,     7)     =   1748.71
maximum lag: 7                                   Prob > F          =    0.0000
                                                 within R-squared  =    0.3672

-------------------------------------------------------------------------------
              |             Drisc/Kraay
       lnfare | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
    lnpax_hat |   .3029089   .0770061     3.93   0.006     .1208183    .4849995
         asam |  -.0466783   .0067542    -6.91   0.000    -.0626496   -.0307071
totalcarriers |  -.0113627    .004257    -2.67   0.032    -.0214288   -.0012966
desigcarriers |  -.0193062   .0026888    -7.18   0.000    -.0256642   -.0129482
       fscper |   .0007941   .0002361     3.36   0.012     .0002357    .0013524
        lngdp |  -.0804757   .0450378    -1.79   0.117    -.1869733    .0260219
  countryfuel |  -.0989573    .017999    -5.50   0.001    -.1415182   -.0563963
         atii |  -.0074234   .0036861    -2.01   0.084    -.0161396    .0012928
              |
         year |
        2012  |          0  (empty)
        2013  |  -.0847815   .0019525   -43.42   0.000    -.0893983   -.0801646
        2014  |   -.180373   .0050292   -35.86   0.000    -.1922653   -.1684808
        2015  |  -.3930609     .02954   -13.31   0.000    -.4629118   -.3232099
        2016  |  -.4575731    .036477   -12.54   0.000    -.5438275   -.3713186
        2017  |  -.4227544   .0308447   -13.71   0.000    -.4956905   -.3498183
        2018  |  -.3692728    .021621   -17.08   0.000    -.4203982   -.3181473
        2019  |  -.4439532    .024419   -18.18   0.000     -.501695   -.3862113
              |
        _cons |   3.213833   .7129194     4.51   0.003     1.528046    4.899619
-------------------------------------------------------------------------------
.
end of do-file

I am not confident at all with the solution that I have used in combining 2SLS FE with DK standard errors. As per your suggestion that "the short time dimension, tests for/estimation of strong cross-section dependence will be biased or over powered," please kindly advice:

1. Should I just ignore the potential of cross-sectional dependence issue in my model given that my panel data has large N and small T?
2. Is my first model -xtivreg lnfare (lnpax = lnpop lntrade) asam totalcarriers desigcarriers fscper lngdp countryfuel atii i.year, fe vce(robust)- already correct and robust to be applied by focusing only on endogeneity, heteroskedasticity, and autocorrelation issues?

Thank you very much for your helpful advice. I really appreciate it.

Best regards,
Abdul

[Abdul Sazali]

I am also not confident with my 2SLS FE model as the R-squared is very small.

Thanks,
Abdul

[JanDitzen]

To the best of my knowledge, there is no theory on IV + DK SE, so I would be very careful with it. Also note that your first stage is missing any FE and the other exogenous variables.

On your questions:
1) Correct. Strong CSD in the Pesaran (2006) context/interactive fixed effects will be impossible to accommodate. Hence I would add the time fixed effects.
2) I am not an expert in this, but I think robust accounts for heteroskedasticity, not for autocorrelation. Instead of xtivreg I would use xtivreg2 which is based in ivreg2 and has some variance estimator for the presence of autocorrelation.

[Abdul Sazali]

Dear Jan,

Thank you very much your answers and explanation. They are very helpful.

Best,
Abdul

[Abdul Sazali]

Hi JanDitzen,

I have tried running the model using -xtivreg2- to address the issues on endogeneity and heteroskedasticity in the presence of autocorrelation. I prefixed the code with -xi:- following the advice in this post using the following command:

 . xi: xtivreg2 lnpax (lnfare = lncountryfuel atii) totalcarriers desigcarriers lnpop lngdp lntrade visa ttci i.year, fe cluster(routeid)
i.year            _Iyear_2012-2019    (naturally coded; _Iyear_2012 omitted)

FIXED EFFECTS ESTIMATION
------------------------
Number of groups =       629                    Obs per group: min =         8
                                                               avg =       8.0
                                                               max =         8

IV (2SLS) estimation
--------------------

Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity and clustering on routeid

Number of clusters (routeid) =     629                Number of obs =     5032
                                                      F( 15,   628) =    29.53
                                                      Prob > F      =   0.0000
Total (centered) SS     =  730.4952556                Centered R2   =   0.1046
Total (uncentered) SS   =  730.4952556                Uncentered R2 =   0.1046
Residual SS             =  654.0980561                Root MSE      =    .3854

-------------------------------------------------------------------------------
              |               Robust
        lnpax | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
       lnfare |  -1.217123   .9949571    -1.22   0.221    -3.167203    .7329569
totalcarriers |   .0934962   .0212316     4.40   0.000      .051883    .1351093
desigcarriers |   .0565155    .035087     1.61   0.107    -.0122538    .1252848
        lnpop |   1.391638   .8241165     1.69   0.091    -.2236003    3.006877
        lngdp |   .1782343   .1908032     0.93   0.350    -.1957331    .5522017
      lntrade |   .1357787   .0618093     2.20   0.028     .0146347    .2569226
     visafree |   .1968128   .1221559     1.61   0.107    -.0426084     .436234
         ttci |  -.0890813   .1397518    -0.64   0.524    -.3629897    .1848272
  _Iyear_2013 |   .0434494    .093443     0.46   0.642    -.1396955    .2265943
  _Iyear_2014 |  -.0530545   .1617121    -0.33   0.743    -.3700043    .2638953
  _Iyear_2015 |  -.0928531   .2440553    -0.38   0.704    -.5711927    .3854864
  _Iyear_2016 |  -.1418022   .3160252    -0.45   0.654    -.7612003    .4775958
  _Iyear_2017 |  -.1281309    .332412    -0.39   0.700    -.7796465    .5233846
  _Iyear_2018 |  -.1143521   .3646413    -0.31   0.754    -.8290358    .6003316
  _Iyear_2019 |  -.2141635   .4398561    -0.49   0.626    -1.076266    .6479386
-------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic):              4.554
                                                   Chi-sq(2) P-val =    0.1026
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic):                3.101
                         (Kleibergen-Paap rk Wald F statistic):          2.276
Stock-Yogo weak ID test critical values: 10% maximal IV size             19.93
                                         15% maximal IV size             11.59
                                         20% maximal IV size              8.75
                                         25% maximal IV size              7.25
Source: Stock-Yogo (2005).  Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments):         0.184
                                                   Chi-sq(1) P-val =    0.6677
------------------------------------------------------------------------------
Instrumented:         lnfare
Included instruments: totalcarriers desigcarriers lnpop lngdp lntrade visafree
                      ttci _Iyear_2013 _Iyear_2014 _Iyear_2015 _Iyear_2016
                      _Iyear_2017 _Iyear_2018 _Iyear_2019
Excluded instruments: lncountryfuel atii
------------------------------------------------------------------------------

Could you please help me check whether the steps that I have taken are already correct or not as most of the variables are not statistically significant and the R-squared is still small?

Many thanks,
Abdul