Gravity panel model: How to deal with high colinearity in important variables without having to remove them?

[Cynthia Hourizene]

Good morning,

I use Stata 13, I believe.

I have a gravity model, and I am estimating the effects of Free trade Agreements (FTAs) on trade/exports between the US states and the rest of the world (ROW). My independent variables include GDP in the exporting and importing countries, Population (both partners Exp and Imp), Production

(both partners Exp and Imp), farm income by state, Distance... and the FTAs (NAFTA, ASEAN...). There are two dummies for each of the FTAs where mm is a subscript for trade between members, and mn is for trade from a state to the ROW.

I am really interested in NAFTA signs here: Expected sign NAFTAmm is>0 and NAFTAmn is<0).

I am using both static (OLS, Fixed Effects, Two State gravity) and GMM. for the issue I am facing I am only going to show commands and output of OLS regression);
I have first run a model with only NAFTA and obtain the correct expected signs, then when I add all the other independent variables in every estimation from OLS to GMM, the signs of coefficients NAFTA are negative (see output below). . WHY?
Why does NAFTAmm goes from positive to Negative with other variables? Collinearity problem (continue below)

1- OLS with NAFTA only

reg lnStateExports NAFTAmm NAFTAmn, robust
Linear regression Number of obs = 232755
F( 2,232752) = 171.17
Prob > F = 0.0000
R-squared = 0.0019
Root MSE = 2.264
------------------------------------------------------------------------------
| Robust
lnStateExp~s | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
NAFTAmm | .0694875 .0399313 1.74 0.082 -.0087768 .1477517
NAFTAmn | -.4145053 .022455 -18.46 0.000 -.4585166 -.3704941
_cons | 13.52012 .0219329 616.43 0.000 13.47714 13.56311
------------------------------------------------------------------------------

2-OLS NAFTA and other independent vars

reg lnStateExports NAFTAmm NAFTAmn lnExpGDP lnImpGDP lnExpPop lnImpPop lnExpProd lnImpProd lnFarmIn
> c lnDistance BorderDummy, robust
Linear regression Number of obs = 46477
F( 11, 46465) = 1497.94
Prob > F = 0.0000
R-squared = 0.2393
Root MSE = 1.9851
------------------------------------------------------------------------------
| Robust
lnStateExp~s | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
NAFTAmm | -1.46886 .1077491 -13.63 0.000 -1.68005 -1.25767
NAFTAmn | -.0150688 .0482547 -0.31 0.755 -.1096486 .0795111
lnExpGDP | -.1744397 .0581043 -3.00 0.003 -.288325 -.0605544
lnImpGDP | .4816379 .0085027 56.65 0.000 .4649725 .4983034
lnExpPop | .877131 .0642196 13.66 0.000 .7512596 1.003002
lnImpPop | .1129621 .0124843 9.05 0.000 .0884926 .1374315
lnExpProd | -.3711653 .0177531 -20.91 0.000 -.4059615 -.336369
lnImpProd | -.0925821 .0098372 -9.41 0.000 -.1118632 -.0733009
lnFarmInc | .6124517 .0176502 34.70 0.000 .577857 .6470464
lnDistance | -.3987482 .0200808 -19.86 0.000 -.4381069 -.3593895
BorderDummy | 2.299835 .2741551 8.39 0.000 1.762487 2.837183
_cons | -10.49608 .3624592 -28.96 0.000 -11.2065 -9.785652
------------------------------------------------------------------------------

I have decided to run some multicolinearity diagnostics and below

vif
Variable | VIF 1/VIF
-------------+----------------------
lnExpPop | 37.17 0.026906 high multicolinearity
lnExpGDP | 35.37 0.028275 high multicolinearity
lnImpProd | 4.53 0.220672
lnImpPop | 3.98 0.251494
lnFarmInc | 3.72 0.268734
lnExpProd | 3.69 0.270983
lnImpGDP | 2.37 0.421065
lnDistance | 1.96 0.511224
ASEANm | 1.76 0.567868
NAFTAmn | 1.56 0.640402
NAFTAmm | 1.25 0.802299
MERCOSURmn | 1.13 0.881311
-------------+----------------------
Mean VIF | 8.21

Now I removed the variables above just to see what will happen (I have to say all these variables are really important here, hence I do not want to remove them.) the regression output seem ok. I also did the correlation matrix and it gives the output below.
I would like to know if there is a way to solve this issue without having to remove any of my variables?
Could you please tell me if there is another regression diagnostic I should do instead of focusing only on high collinearity? Thanks for you help.

running regression without EXpPOP and EXPGDP

. reg lnStateExports NAFTAmm NAFTAmn lnImpGDP lnImpPop lnExpProd lnImpProd lnFarmInc lnDistance ASEAN
> m MERCOSURmn, robust

Linear regression                                      Number of obs =   46477
                                                       F( 10, 46466) = 1005.83
                                                       Prob > F      =  0.0000
                                                       R-squared     =  0.1950
                                                       Root MSE      =   2.042

------------------------------------------------------------------------------
             |               Robust
lnStateExp~s |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     NAFTAmm |    -1.5044   .1081805   -13.91   0.000    -1.716436   -1.292365
     NAFTAmn |  -.3608108    .048148    -7.49   0.000    -.4551817     -.26644
    lnImpGDP |   .4777262   .0088465    54.00   0.000     .4603869    .4950655
    lnImpPop |   .0328538    .014152     2.32   0.020     .0051156     .060592
......


 vif

    Variable |       VIF       1/VIF
-------------+----------------------
   lnImpProd |      4.53    0.220731
    lnImpPop |      3.98    0.251540
   lnExpProd |      3.33    0.300253
   lnFarmInc |      3.01    0.331741
    lnImpGDP |      2.37    0.422257
  lnDistance |      1.95    0.513197
      ASEANm |      1.76    0.568594
     NAFTAmn |      1.45    0.688263
     NAFTAmm |      1.25    0.802492
  MERCOSURmn |      1.13    0.881512
-------------+----------------------
    Mean VIF |      2.48

. pwcorr lnStateExports NAFTAmm NAFTAmn lnExpGDP lnImpGDP lnExpPop lnImpPop lnExpProd lnImpProd lnFar
> mInc lnDistance ASEANm MERCOSURmn

             | lnStat..  NAFTAmm  NAFTAmn lnExpGDP lnImpGDP lnExpPop lnImpPop
-------------+---------------------------------------------------------------
lnStateExp~s |   1.0000
     NAFTAmm |   0.0029   1.0000
     NAFTAmn |  -0.0431   0.0497   1.0000
    lnExpGDP |   0.2308   0.0097   0.1563   1.0000
    lnImpGDP |  -0.0581  -0.0556  -0.3479  -0.0943   1.0000
    lnExpPop |   0.1712  -0.0143  -0.1683   0.4509  -0.0825   1.0000
    lnImpPop |   0.1491   0.1121  -0.0085  -0.0397   0.5758  -0.0197   1.0000
   lnExpProd |   0.1366  -0.0053   0.1189   0.5315   0.2612   0.1680  -0.0250
   lnImpProd |  -0.1147  -0.0573  -0.2563  -0.0496   0.9835  -0.0531   0.7958
   lnFarmInc |   0.2375  -0.0009  -0.0101   0.4262  -0.0502   0.2373  -0.0137
  lnDistance |   0.0508  -0.2313   0.0126  -0.0374   0.0361  -0.0254   0.3061
      ASEANm |   0.0590  -0.0601  -0.0097  -0.0306   0.1801  -0.0218   0.1661
  MERCOSURmn |  -0.0604  -0.0638   0.0002   0.0234   0.0070   0.0165  -0.0144

             | lnExpP~d lnImpP~d lnFarm~c lnDist~e   ASEANm MERCOS~n
-------------+------------------------------------------------------
   lnExpProd |   1.0000
   lnImpProd |   0.3098   1.0000
   lnFarmInc |   0.7343  -0.0186   1.0000
  lnDistance |   0.0124  -0.0258   0.0450   1.0000
      ASEANm |   0.0059   0.0586  -0.0119   0.3608   1.0000
  MERCOSURmn |   0.0094   0.0217   0.0126  -0.1854  -0.1162   1.0000

Thanks for help

[Clyde Schechter]

The way to solve this problem is to ignore the VIF results. In fact, you never should have done them in the first place. See Arthur Goldberger's textbook A Course in Econometrics. He has an entire chapter on why multicollinearity is, for the most part (and in your situation in particular), a bogus problem and why you should not look for it, and should ignore it if you find it.

[Cynthia Hourizene]

Thank you M. Clyde Schechter.
I have checked the textbook online and unfortunately I cannot afford it now. in the meantime, do you have any suggestion to solve the problem I am facing? what explains the change of signs in my regression as I add more variable? what can I do to solve it?

[Clyde Schechter]

what explains the change of signs in my regression as I add more variable? what can I do to solve it?

There are two separate issues. If you look at the N's for the two models, you will see that the sample sizes are very different. Due to missing values on variables, a lot of observations have been dropped from the model without when you go to the model with the extra variables. In fact, barely 1 of 6 observations survives the transition! It is possible that the resulting sample is very biased as a result. So the first step is to compare apples to apples: they have to be run on the same sample. The easiest way to do this is to run the model with the additional variables first. Any observation that is retained in this analysis will also be retainable when analyzing the smaller model. Next run the model omitting those variables, adding -if e(sample)- to the regression command (before the comma). You may well find that, when you run them on the same sample, the two models are in substantial agreement. The question, then, becomes whether this biased, reduced sample is suitable for answering your research questions.

Or perhaps the discrepancy between the models will persist. There is nothing unusual or surprising about estimates changing radically, including changing sign, when new variables are added to the model. This is known as Simpson's paradox, or the Yule-Simpson paradox, or, as applied to continuous variables, is sometimes called Lord's paradox. Wikipedia has a really good page on Simpson's paradox, which I recommend you read.

The change in signs, or other large difference, if it persists when running both models on the same sample, is not the problem you have to solve. It is what it is, and it is not a problem, nor does it require, nor admit, a solution. Your problem is that you need to decide whether it is the model with or without the additional variables that is appropriate to answering your research questions. This will depend on an in-depth understanding of your research questions and the roles the different variables play. It's not a statistical issue: it's a substantive one and you will benefit most from advice from other people in your discipline. In any case, one thing you should not do is choose a model because it gives you the results you like. Another thing you should not do is let a bogus problem like multicollinearity affect your choice of analysis.

Regarding Goldberger's textbook, it might be available in your institution's library. If you can borrow a copy (from there, or from a colleague) it's well worth a read. It's not only eye-opening it's written in a very entertaining style.

[Cynthia Hourizene]

Thank you sir, I will follow your instructions now