Model with Independent lagged variables: FE, RE or POLS?

[Joern Sievering]

Dear all,

i am running a regression using panel data and looking at the effect of company financial data and characteristics on tax adjustments. The tax adjustments paid in period t are caused in periods before (t-1 oder t-2) and were found after tax audit. I don´t have many observations per group and i want to control for different company characteristics (dummy variables). The effects are primarily between the groups not within the time series. In my opinion the RE Model would be the logical consequence, but the hausman test rejects H0 so i can´t use RE.

This is how my xtreg, fe regression looks like:

Code:

xtreg LOG_STN l1.LOG_SALES l2.LOG_SALES l1.PROV l2.PROV l1.LEV l2.LEV l1.DEBT l2.DEBT l1.ROA l2.ROA l1.IVG_intense l2.IVG_intense l1.KAP_intense l2.KAP_intense l1.VOR_intense l
> 2.VOR_intense FamUN AUSL_KSTR y2012 y2013 y2014 y2015 y2016 y2017 bw bay hes sac nrw c f g m n, fe
note: FamUN omitted because of collinearity
note: AUSL_KSTR omitted because of collinearity
note: bw omitted because of collinearity
note: bay omitted because of collinearity
note: hes omitted because of collinearity
note: sac omitted because of collinearity
note: nrw omitted because of collinearity
note: c omitted because of collinearity
note: f omitted because of collinearity
note: g omitted because of collinearity
note: m omitted because of collinearity
note: n omitted because of collinearity

Fixed-effects (within) regression               Number of obs     =      3,518
Group variable: idc                             Number of groups  =      2,189

R-sq:                                           Obs per group:
     within  = 0.0216                                         min =          1
     between = 0.0001                                         avg =        1.6
     overall = 0.0002                                         max =          6

                                                F(22,1307)        =       1.31
corr(u_i, Xb)  = -0.1703                        Prob > F          =     0.1501

------------------------------------------------------------------------------
     LOG_STN |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   LOG_SALES |
         L1. |   .8047563   .3752725     2.14   0.032     .0685539    1.540959
         L2. |  -.7372711   .3916178    -1.88   0.060    -1.505539    .0309972
             |
        PROV |
         L1. |  -.2168305    1.86602    -0.12   0.908    -3.877553    3.443892
         L2. |   2.515237   1.810256     1.39   0.165    -1.036087    6.066562
             |
         LEV |
         L1. |   .0663668   .0781916     0.85   0.396     -.087028    .2197616
         L2. |   -.066027   .0374569    -1.76   0.078    -.1395093    .0074553
             |
        DEBT |
         L1. |  -.0293448   .6116295    -0.05   0.962    -1.229228    1.170538
         L2. |   .3254192   .5211656     0.62   0.532    -.6969934    1.347832
             |
         ROA |
         L1. |  -.6775415   1.158015    -0.59   0.559    -2.949314    1.594231
         L2. |  -2.684723   1.101758    -2.44   0.015     -4.84613   -.5233155
             |
 IVG_intense |
         L1. |   1.390094    2.55196     0.54   0.586    -3.616293     6.39648
         L2. |   -1.74212   2.698345    -0.65   0.519     -7.03568    3.551441
             |
 KAP_intense |
         L1. |  -.0716247   1.594696    -0.04   0.964    -3.200068    3.056818
         L2. |  -.6143406   1.284293    -0.48   0.632    -3.133843    1.905161
             |
 VOR_intense |
         L1. |   1.185455   .9368948     1.27   0.206     -.652527    3.023437
         L2. |    .800335   .8815385     0.91   0.364    -.9290501     2.52972
             |
       FamUN |          0  (omitted)
   AUSL_KSTR |          0  (omitted)
       y2012 |  -.1148903   .2005894    -0.57   0.567    -.5084027    .2786221
       y2013 |  -.2193175   .2076353    -1.06   0.291    -.6266523    .1880173
       y2014 |  -.3174528   .2204653    -1.44   0.150    -.7499573    .1150517
       y2015 |  -.2075198    .233365    -0.89   0.374    -.6653308    .2502912
       y2016 |  -.3926721   .2435855    -1.61   0.107    -.8705334    .0851892
       y2017 |  -.2898164   .3278856    -0.88   0.377    -.9330559    .3534232
          bw |          0  (omitted)
         bay |          0  (omitted)
         hes |          0  (omitted)
         sac |          0  (omitted)
         nrw |          0  (omitted)
           c |          0  (omitted)
           f |          0  (omitted)
           g |          0  (omitted)
           m |          0  (omitted)
           n |          0  (omitted)
       _cons |   1.903193    4.79805     0.40   0.692    -7.509529    11.31591
-------------+----------------------------------------------------------------
     sigma_u |  3.7078212
     sigma_e |  2.0251375
         rho |  .77023059   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(2188, 1307) = 2.12                  Prob > F = 0.0000

Hausman test:

 Test:  Ho:  difference in coefficients not systematic

                 chi2(22) = (b-B)'[(V_b-V_B)^(-1)](b-B)
                          =       76.77
                Prob>chi2 =      0.0000

xtreg, re:

 xtreg LOG_STN l1.LOG_SALES l2.LOG_SALES l1.PROV l2.PROV l1.LEV l2.LEV l1.DEBT l2.DEBT l1.ROA l2.ROA l1.IVG_intense l2.IVG_intense l1.KAP_intense l2.KAP_intense l1.VOR_intense l
> 2.VOR_intense FamUN AUSL_KSTR y2012 y2013 y2014 y2015 y2016 y2017 bw bay hes sac nrw c f g m n, re

Random-effects GLS regression                   Number of obs     =      3,518
Group variable: idc                             Number of groups  =      2,189

R-sq:                                           Obs per group:
     within  = 0.0006                                         min =          1
     between = 0.5251                                         avg =        1.6
     overall = 0.5054                                         max =          6

                                                Wald chi2(34)     =    2512.84
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

------------------------------------------------------------------------------
     LOG_STN |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   LOG_SALES |
         L1. |    .965822   .1913759     5.05   0.000     .5907322    1.340912
         L2. |   .1194994   .1908272     0.63   0.531    -.2545151    .4935139
             |
        PROV |
         L1. |   1.435207   1.127386     1.27   0.203    -.7744284    3.644843
         L2. |   -.734266   1.142288    -0.64   0.520    -2.973109    1.504578
             |
         LEV |
         L1. |  -.0132113    .027322    -0.48   0.629    -.0667613    .0403388
         L2. |   .0014444   .0193577     0.07   0.941    -.0364961    .0393849
             |
        DEBT |
         L1. |  -.1789497   .3023988    -0.59   0.554    -.7716405    .4137411
         L2. |   -.436075   .2741722    -1.59   0.112    -.9734427    .1012927
             |
         ROA |
         L1. |  -.1335986   .6398764    -0.21   0.835    -1.387733    1.120536
         L2. |   -1.52944   .6135618    -2.49   0.013    -2.731998   -.3268806
             |
 IVG_intense |
         L1. |   .0267801   1.624745     0.02   0.987    -3.157661    3.211221
         L2. |   1.663602   1.592992     1.04   0.296    -1.458605    4.785808
             |
 KAP_intense |
         L1. |   .2447735   .8247597     0.30   0.767    -1.371726    1.861273
         L2. |   1.489765   .7936084     1.88   0.060    -.0656792    3.045208
             |
 VOR_intense |
         L1. |  -.3348842   .4761082    -0.70   0.482    -1.268039    .5982708
         L2. |  -.3350098    .460146    -0.73   0.467    -1.236879    .5668598
             |
       FamUN |  -.5909862   .1461903    -4.04   0.000    -.8775138   -.3044586
   AUSL_KSTR |   .4322607     .13506     3.20   0.001     .1675479    .6969735
       y2012 |  -.0770948   .1694286    -0.46   0.649    -.4091688    .2549791
       y2013 |  -.3837821   .1674422    -2.29   0.022    -.7119628   -.0556015
       y2014 |  -.1691641   .1717899    -0.98   0.325     -.505866    .1675379
       y2015 |   .1384244   .1810782     0.76   0.445    -.2164823    .4933311
       y2016 |    .068587   .1829719     0.37   0.708    -.2900313    .4272053
       y2017 |  -.0408278   .2510171    -0.16   0.871    -.5328124    .4511567
          bw |  -.1638173   .1698556    -0.96   0.335    -.4967282    .1690935
         bay |  -.0336689    .161638    -0.21   0.835    -.3504736    .2831357
         hes |   .1426045   .2052941     0.69   0.487    -.2597646    .5449737
         sac |   .2499873   .2373464     1.05   0.292    -.2152031    .7151778
         nrw |  -.0404006   .1467679    -0.28   0.783    -.3280604    .2472591
           c |  -.5049047   .1654168    -3.05   0.002    -.8291157   -.1806938
           f |  -1.284322   .2200182    -5.84   0.000     -1.71555   -.8530946
           g |  -.8524794   .1796878    -4.74   0.000    -1.204661   -.5002978
           m |  -.2227419   .1809266    -1.23   0.218    -.5773516    .1318678
           n |  -.1955437   .3183816    -0.61   0.539    -.8195601    .4284727
       _cons |  -8.119061   .4488191   -18.09   0.000     -8.99873   -7.239392
-------------+----------------------------------------------------------------
     sigma_u |  1.7691576
     sigma_e |  2.0251375
         rho |  .43284126   (fraction of variance due to u_i)
------------------------------------------------------------------------------

Here xttest0 after xtreg, re:

Breusch and Pagan Lagrangian multiplier test for random effects

        LOG_STN[idc,t] = Xb + u[idc] + e[idc,t]

        Estimated results:
                         |       Var     sd = sqrt(Var)
                ---------+-----------------------------
                 LOG_STN |   13.63771       3.692926
                       e |   4.101182       2.025137
                       u |   3.129919       1.769158

        Test:   Var(u) = 0
                             chibar2(01) =    89.51
                          Prob > chibar2 =   0.0000

When my understanding for this tests is correct, i should use FE estimator, although the R-sq is very bad?

[Joern Sievering]

And is this regression structure with two time periods for each variable possible? Can i ignore multicollinearity between for example Sales(t-1) and Sales(t-2)?

[Carlo Lazzaro]

Joern:
according to -hausman- outcome, you should go -fe-.
The results of -xttest0- does not necessarily implies that the right specification is -re-; in fact, this test should be followed by -hausman- to decide whether -fe- or -re- is better for your data.
However, your problem seems more trivial: for both your models most of your coefficients do not show any evidence of an effect on the regressand.
You may want to consider a more parsimonious model, that gives a fair and true view of the data generating process.
Eventually, you may want to take a look at -estat vce, corr- after -xtreg- to investigate possible quasi-extreme multicollinearity issues among predictors.

[Joern Sievering]

Hello Carlo,
thank you for your advice.
I did -estat vce, corr- after xtreg, fe and saw values from 0.5 to 0.8 between the year dummies. After omitting these the results for the fixed effects regression changed totally. I thought the year dummies are always helpful to control for time-effects. Recently i have very high values between the Lag1 and the Lag2 of each variable. I looked at http://statisticalhorizons.com/multicollinearity but i´m not sure if i can ignore them.

here the results for -estat vce, vorr- after -xtreg, fe- and ommiting the year dummies:

Correlation matrix of coefficients of xtreg model

             |        L.       L2.        L.       L2.        L.       L2.        L.       L2.        L.       L2.        L.       L2.        L.       L2.        L.       L2.
        e(V) | LOG_SA~S  LOG_SA~S      PROV      PROV       LEV       LEV      DEBT      DEBT       ROA       ROA  IVG_in~e  IVG_in~e  KAP_in~e  KAP_in~e  VOR_in~e  VOR_in~e 
-------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------
 L.LOG_SALES |   1.0000                                                                                                                                                       
L2.LOG_SALES |  -0.9855    1.0000                                                                                                                                             
      L.PROV |  -0.0704    0.0635    1.0000                                                                                                                                   
     L2.PROV |   0.0645   -0.0654   -0.9386    1.0000                                                                                                                         
       L.LEV |   0.0453   -0.0524   -0.1013    0.1010    1.0000                                                                                                               
      L2.LEV |  -0.0470    0.0614   -0.0032   -0.0218   -0.4307    1.0000                                                                                                     
      L.DEBT |  -0.0323    0.0285    0.3462   -0.3108   -0.2584    0.0522    1.0000                                                                                           
     L2.DEBT |  -0.0277    0.0362   -0.2545    0.3040    0.0325   -0.1526   -0.6724    1.0000                                                                                 
       L.ROA |  -0.2565    0.2556   -0.0392    0.0372    0.0775   -0.0098    0.0257    0.0432    1.0000                                                                       
      L2.ROA |   0.1256   -0.1189    0.0817   -0.0886   -0.0222    0.0688    0.0706   -0.0857   -0.6536    1.0000                                                             
L.IVG_inte~e |  -0.1428    0.1354   -0.0741    0.0979    0.0325   -0.0194   -0.3372    0.3119   -0.0107    0.0377    1.0000                                                   
L2.IVG_int~e |   0.1240   -0.1234    0.0901   -0.0966   -0.0538    0.0245    0.3432   -0.3620    0.0154   -0.0178   -0.9320    1.0000                                         
L.KAP_inte~e |  -0.0519    0.0446    0.0337   -0.0255    0.0210   -0.0268   -0.1619    0.1997   -0.1168    0.0583   -0.0132    0.0090    1.0000                               
L2.KAP_int~e |   0.0540   -0.0528   -0.0540    0.0525   -0.0141    0.0566    0.1482   -0.2140    0.0901   -0.0463    0.0651   -0.0320   -0.8873    1.0000                     
L.VOR_inte~e |   0.1744   -0.1618   -0.0526    0.0664    0.0797    0.0127   -0.5767    0.5246   -0.0953    0.0535    0.1232   -0.1092    0.0886   -0.0539    1.0000           
L2.VOR_int~e |  -0.1780    0.1789    0.0860   -0.0938   -0.0379    0.0565    0.5175   -0.6347    0.0951   -0.0490   -0.0777    0.1083   -0.1103    0.1392   -0.7805    1.0000 
       FamUN |  -0.0104    0.0459    0.0297   -0.0033   -0.0144    0.0031   -0.0118   -0.0461   -0.0283    0.0010    0.0201   -0.0012   -0.0193    0.0116   -0.0335    0.0252 
   AUSL_KSTR |   0.0011   -0.0588    0.0529   -0.0426    0.0409    0.0111    0.0568   -0.0560   -0.0036   -0.0279   -0.1293    0.0339    0.0442   -0.0746   -0.0309   -0.0073 
          bw |   0.0346   -0.0249   -0.0668    0.0382    0.0136   -0.0392   -0.0234    0.0323    0.0483   -0.0663    0.0324   -0.0488   -0.0220   -0.0051   -0.0209   -0.0260 
         bay |  -0.0463    0.0396   -0.0276   -0.0183    0.0303   -0.0439   -0.0426    0.0705    0.0233   -0.0249    0.0181   -0.0656    0.0147   -0.0143    0.0437   -0.0263 
         hes |  -0.0148    0.0101   -0.0643    0.0213    0.0028   -0.0276   -0.0286    0.0345    0.0459   -0.0027    0.0154   -0.0303   -0.0468    0.0374    0.0070   -0.0296 
         sac |   0.0118    0.0009    0.0426   -0.0083    0.0195   -0.0636    0.0465    0.0205    0.0126    0.0259    0.0047    0.0069   -0.0027   -0.0020   -0.0179    0.0357 
         nrw |  -0.0192    0.0182    0.0103   -0.0674   -0.0295   -0.0319    0.0257   -0.0063    0.0503   -0.0076    0.0189   -0.0467   -0.0346    0.0114   -0.0384    0.0276 
           c |  -0.0281    0.0236   -0.0115   -0.0220   -0.0244   -0.0212    0.0670    0.0361    0.0009    0.0122   -0.0337    0.0342    0.0318   -0.0548   -0.1188   -0.1072 
           f |   0.0464   -0.0188   -0.0614    0.0138    0.0289   -0.0093   -0.0824   -0.0081   -0.0442    0.0192   -0.0287    0.0227    0.0334   -0.0297   -0.0664   -0.0820 
           g |  -0.0291    0.0200   -0.0270    0.0247    0.0402   -0.0135    0.0098    0.0063    0.0021   -0.0035   -0.0412    0.0393    0.0579   -0.0620   -0.1646   -0.1060 
           m |   0.0025   -0.0398   -0.0478    0.0284    0.0316   -0.0324   -0.0119    0.0091   -0.0170    0.0482   -0.0038   -0.0338    0.0047    0.0189   -0.0919   -0.0645 
           n |   0.0021    0.0092   -0.0403    0.0281    0.0206   -0.0289   -0.0326   -0.0311   -0.0162    0.0103   -0.0110    0.0037   -0.0491    0.0587    0.0036    0.0249 
       _cons |  -0.0345   -0.1177   -0.0294   -0.0021    0.0608   -0.0639   -0.1277   -0.1289   -0.0675   -0.0752    0.0324   -0.0058    0.0176   -0.0164   -0.0549   -0.0082 

             |                                                                                                                                  
        e(V) |    FamUN  AUSL_K~R        bw       bay       hes       sac       nrw         c         f         g         m         n     _cons 
-------------+----------------------------------------------------------------------------------------------------------------------------------
       FamUN |   1.0000                                                                                                                         
   AUSL_KSTR |   0.0838    1.0000                                                                                                               
          bw |   0.0111   -0.1671    1.0000                                                                                                     
         bay |  -0.0323   -0.0032    0.3048    1.0000                                                                                           
         hes |  -0.0469   -0.1833    0.2829    0.2779    1.0000                                                                                 
         sac |   0.0297    0.0014    0.1891    0.1880    0.1533    1.0000                                                                       
         nrw |   0.0255   -0.0775    0.3681    0.3640    0.3288    0.2239    1.0000                                                             
           c |  -0.1245   -0.2921   -0.0020   -0.0340    0.0561   -0.0127   -0.0133    1.0000                                                   
           f |  -0.1603   -0.0814    0.0817   -0.0333    0.0491   -0.0252   -0.0108    0.3604    1.0000                                         
           g |  -0.1519   -0.0979    0.0147   -0.1055   -0.0168   -0.0511   -0.0838    0.5368    0.3876    1.0000                               
           m |  -0.1492   -0.1127   -0.0258   -0.0330    0.0553   -0.0052   -0.0478    0.5834    0.3266    0.5161    1.0000                     
           n |  -0.0796   -0.0751   -0.0095   -0.0311   -0.0361   -0.0248   -0.0490    0.2123    0.1722    0.1963    0.2303    1.0000           
       _cons |  -0.1973    0.2835   -0.1208   -0.0466   -0.0409   -0.1897   -0.0840   -0.1092   -0.1790   -0.0495    0.0703   -0.0995    1.0000 

.

[Carlo Lazzaro]

Joern:
yoo may want to test whether your year dummies are jointly significant:

test y*

[Joern Sievering]

Carlo:
Thank you, that helped a lot. The year dummies are jointly not significant:

test y2012 y2013 y2014 y2015 y2016 y2017

 ( 1)  y2012 = 0
 ( 2)  y2013 = 0
 ( 3)  y2014 = 0
 ( 4)  y2015 = 0
 ( 5)  y2016 = 0
 ( 6)  y2017 = 0

       F(  6,  2188) =    0.71
            Prob > F =    0.6426

Regarding the lagged variables: There is for exapmle a value of 0.985 between l1.LOG_SALES and l2.LOG_SALES, similar for l1./l2.PROV and so on. Does it mean i can´t take both periods as explanatory variable or is this a special case where i can ignore this values?

[Carlo Lazzaro]

Joern:
you can safely omit -y*- from your set of predictors if this improves your model specification.
As far as your second question is concerned, it is not surprising to detect such a high correlation when lag is one year apart. If I were you, I would rethink my specification considering no lags altogehter (unless the literature in your research field proposes a different approach).

[Joern Sievering]

Carlo:
Thank you for your quick response!