Multicollinearity Tests - Panel Data

[Ethan Kilmer]

Hello all,

I am currently working with a panel dataset on Stata18 and have some questions regarding multicollinearity. After reading various threads I am still slightly confused. I understand many are of the view that multicollinearity is not a huge issue, however, I am still hoping to explore and potentially mitigate the impact on my results (I have obtained very weird/large coefficients and st errors). For context, I am running a fixed effects model exploring the impact of ESG on the weighted average cost of capital (with various control variables).

'estat vif' does not seem to work on panel models. I tried running a simple 'pwcorr [variable list]'. However, after reading some suggestions, I tried the 'estat vce, corr' instead, which has given me mixed results. I am unsure whether to use robust standard errors in the xtreg before looking at correlations. When including robust standard errors, my correlations are much higher and indicate issues with multiple variables.

Could someone please help me understand which approach is most appropriate? Here are my differing results:

pwcorr test:

pwcorr ESG ZSCORE ASG CUR LEV INTCOV PTB ROA TQ BT SIGMA LEGALV INFL COVID ln_SIZE ln_SRT ln_GDPPC


             |      ESG   ZSCORE      ASG      CUR      LEV   INTCOV      PTB
-------------+---------------------------------------------------------------
         ESG |   1.0000
      ZSCORE |  -0.1697   1.0000
         ASG |   0.0097   0.0200   1.0000
         CUR |  -0.1018   0.4814  -0.0129   1.0000
         LEV |  -0.0713  -0.2297   0.0104  -0.3133   1.0000
      INTCOV |  -0.0131   0.0083  -0.0039   0.1752  -0.0908   1.0000
         PTB |  -0.2647   0.5836   0.0371   0.1336   0.0336  -0.0173   1.0000
         ROA |  -0.0258   0.3018   0.0936   0.4127  -0.3549   0.0726   0.2418
          TQ |  -0.1953   0.8527   0.0491   0.2905  -0.1539  -0.0097   0.7732
          BT |  -0.0720  -0.0847  -0.0459   0.0405   0.1231   0.0258   0.1270
       SIGMA |   0.1472   0.1000  -0.0148  -0.0030  -0.1160  -0.0442   0.1231
      LEGALV |  -0.3317  -0.0707   0.0071   0.1256   0.0534   0.0228  -0.1322
        INFL |   0.1222  -0.0579   0.0401  -0.0791   0.0117  -0.0257  -0.0463
       COVID |   0.0563  -0.0326   0.0156   0.0251  -0.0305   0.0445  -0.0435
     ln_SIZE |   0.5180  -0.0835   0.0220  -0.2159   0.0886  -0.0576  -0.0273
      ln_SRT |   0.4794  -0.1011   0.0193  -0.2684   0.0235  -0.0583  -0.0172
    ln_GDPPC |  -0.0885  -0.1063   0.0161   0.1304  -0.0574   0.0840  -0.1558

             |      ROA       TQ       BT    SIGMA   LEGALV     INFL    COVID
-------------+---------------------------------------------------------------
         ROA |   1.0000
          TQ |   0.3384   1.0000
          BT |  -0.1035  -0.1214   1.0000
       SIGMA |   0.0106   0.0847   0.1636   1.0000
      LEGALV |   0.0020  -0.0694   0.0161  -0.5173   1.0000
        INFL |   0.0791  -0.0463  -0.0466   0.1869   0.0038   1.0000
       COVID |   0.0339  -0.0328   0.0081  -0.0598   0.1672   0.2005   1.0000
     ln_SIZE |  -0.1353  -0.0866  -0.0251   0.2992  -0.7132   0.0543  -0.1201
      ln_SRT |  -0.1195  -0.1021  -0.0335   0.3013  -0.7447   0.0673  -0.1189
    ln_GDPPC |   0.1024  -0.1197  -0.0105  -0.3868   0.6927   0.3854   0.3054

             |  ln_SIZE   ln_SRT ln_GDPPC
-------------+---------------------------
     ln_SIZE |   1.0000
      ln_SRT |   0.9701   1.0000
    ln_GDPPC |  -0.4701  -0.4727   1.0000

estat vce, corr (Without robust standard errors) :

xtreg WACC ESG ZSCORE ASG CUR LEV INTCOV PTB ROA TQ BT SIGMA LEGALV INFL COVID ln_SIZE ln_SRT ln_GDPPC,fe

Fixed-effects (within) regression               Number of obs     =      1,019 Group variable: ID                              Number of groups  =         51
  
  R-squared:                                      Obs per group:
       Within  = 0.2454                                         min =         19
       Between = 0.1379                                         avg =       20.0
       Overall = 0.1392                                         max =         20
  
                                                  F(17, 951)        =      18.19
  corr(u_i, Xb) = -0.6622                         Prob > F          =     0.0000
  
  ------------------------------------------------------------------------------
          WACC | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
  -------------+----------------------------------------------------------------
           ESG |   .1266163   .0328428     3.86   0.000     .0621637     .191069
        ZSCORE |   .0468535    .041237     1.14   0.256    -.0340725    .1277794
           ASG |    -.00337   .0029078    -1.16   0.247    -.0090764    .0023365
           CUR |  -.9584892   .2990543    -3.21   0.001    -1.545372   -.3716065
           LEV |  -.2248722   .0287947    -7.81   0.000    -.2813807   -.1683637
        INTCOV |   .0000104   4.46e-06     2.34   0.020     1.68e-06    .0000192
           PTB |   .2069231   .1579241     1.31   0.190     -.102997    .5168431
           ROA |  -.0347309    .030474    -1.14   0.255     -.094535    .0250732
            TQ |   -.145192   .2926098    -0.50   0.620    -.7194275    .4290434
            BT |   .8209768    .331237     2.48   0.013      .170937    1.471017
         SIGMA |    13.7027   1.630689     8.40   0.000     10.50254    16.90287
        LEGALV |  -10.79952   3.204504    -3.37   0.001    -17.08824   -4.510804
          INFL |  -.0445178    .027403    -1.62   0.105    -.0982951    .0092596
         COVID |   -.000032    .000038    -0.84   0.400    -.0001067    .0000426
       ln_SIZE |   2.708554     1.0963     2.47   0.014     .5571075    4.860001
        ln_SRT |  -2.930164   .6583161    -4.45   0.000    -4.222084   -1.638244
      ln_GDPPC |   5.581056   2.122431     2.63   0.009     1.415865    9.746246
         _cons |  -32.63803    16.1497    -2.02   0.044    -64.33119   -.9448737
  -------------+----------------------------------------------------------------
       sigma_u |  4.5866087
       sigma_e |  4.1834419
           rho |  .54587392   (fraction of variance due to u_i)
  ------------------------------------------------------------------------------
  F test that all u_i=0: F(50, 951) = 7.78                     Prob > F = 0.0000

estat vce,corr
  Correlation matrix of coefficients of xtreg model
  
          e(V) |      ESG    ZSCORE       ASG       CUR       LEV    INTCOV       PTB       ROA        TQ        BT     SIGMA    LEGALV      INFL     COVID   ln_SIZE    ln_SRT  ln_GDPPC     _cons
  -------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
           ESG |   1.0000                                                                                                                                                                          
        ZSCORE |   0.0153    1.0000                                                                                                                                                                
           ASG |   0.0368   -0.0222    1.0000                                                                                                                                                      
           CUR |  -0.0303   -0.6145   -0.0054    1.0000                                                                                                                                            
           LEV |   0.0073    0.0909   -0.0281   -0.1710    1.0000                                                                                                                                  
        INTCOV |  -0.0260   -0.0679    0.0007    0.0988   -0.0126    1.0000                                                                                                                        
           PTB |  -0.0901    0.1738   -0.1125    0.1115   -0.2725   -0.0090    1.0000                                                                                                              
           ROA |   0.0448    0.0579   -0.0288   -0.1046    0.1375    0.0130   -0.2367    1.0000                                                                                                    
            TQ |   0.0981   -0.5633    0.0690    0.2545    0.1675    0.0415   -0.7312    0.1796    1.0000                                                                                          
            BT |  -0.0893   -0.0315    0.0206   -0.0663    0.0101    0.0107   -0.3408    0.1105    0.2357    1.0000                                                                                
         SIGMA |  -0.0251    0.0712    0.0314   -0.1232    0.0942    0.0498   -0.0799    0.0999    0.0483   -0.1128    1.0000                                                                      
        LEGALV |   0.0132   -0.0158    0.0199    0.0039   -0.0335    0.0522   -0.0213    0.0838   -0.0031    0.0165    0.1549    1.0000                                                            
          INFL |  -0.1078   -0.0585    0.0210   -0.0209    0.0569    0.0023   -0.1396   -0.0240    0.1069    0.0891   -0.2451    0.1852    1.0000                                                  
         COVID |  -0.1718   -0.0387    0.0022    0.0207    0.0109   -0.0194   -0.0793    0.0180    0.0526    0.0142   -0.0283   -0.0477    0.0303    1.0000                                        
       ln_SIZE |  -0.2173    0.1367   -0.1293    0.0919   -0.1666   -0.0042    0.6837   -0.1119   -0.4886   -0.0669   -0.1631   -0.0731   -0.1672   -0.0791    1.0000                              
        ln_SRT |   0.0400    0.0528   -0.0465   -0.0068    0.1136    0.0032   -0.0068   -0.2084   -0.0175    0.0147    0.1492    0.1539   -0.0217   -0.0014   -0.4483    1.0000                    
      ln_GDPPC |  -0.0808   -0.0383    0.0236    0.0481   -0.0039    0.0017   -0.1040    0.0639    0.0953   -0.0085    0.1299   -0.1527   -0.7622   -0.0764   -0.0045   -0.1644    1.0000          
         _cons |   0.1439   -0.0210    0.0476   -0.1186    0.0305   -0.0120   -0.2138    0.0201    0.1181    0.0154   -0.0973   -0.0265    0.7323    0.1398   -0.3645    0.1324   -0.8841    1.0000

estat vce, corr (with robust):

xtreg WACC ESG ZSCORE ASG CUR LEV INTCOV PTB ROA TQ BT SIGMA LEGALV INFL COVID ln_SIZE ln_SRT ln_GDPPC,fe vce(cluster ID)


Fixed-effects (within) regression               Number of obs     =      1,019
Group variable: ID                              Number of groups  =         51

R-squared:                                      Obs per group:
     Within  = 0.2454                                         min =         19
     Between = 0.1379                                         avg =       20.0
     Overall = 0.1392                                         max =         20

                                                F(17, 50)         =    1021.32
corr(u_i, Xb) = -0.6622                         Prob > F          =     0.0000

                                    (Std. err. adjusted for 51 clusters in ID)
------------------------------------------------------------------------------
             |               Robust
        WACC | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         ESG |   .1266163   .0478849     2.64   0.011     .0304367     .222796
      ZSCORE |   .0468535   .0729644     0.64   0.524    -.0996998    .1934067
         ASG |    -.00337   .0019584    -1.72   0.091    -.0073035    .0005635
         CUR |  -.9584892   .9860384    -0.97   0.336    -2.939006    1.022027
         LEV |  -.2248722    .118422    -1.90   0.063    -.4627298    .0129854
      INTCOV |   .0000104   7.78e-07    13.42   0.000     8.88e-06     .000012
         PTB |   .2069231   .1354552     1.53   0.133    -.0651467    .4789928
         ROA |  -.0347309   .0377041    -0.92   0.361    -.1104617        .041
          TQ |   -.145192    .353533    -0.41   0.683     -.855284    .5648999
          BT |   .8209768   .4355486     1.88   0.065    -.0538484    1.695802
       SIGMA |    13.7027   11.22082     1.22   0.228    -8.834986    36.24039
      LEGALV |  -10.79952   4.890098    -2.21   0.032    -20.62157   -.9774691
        INFL |  -.0445178   .0362501    -1.23   0.225    -.1173283    .0282928
       COVID |   -.000032   .0000444    -0.72   0.474    -.0001212    .0000571
     ln_SIZE |   2.708554   1.455806     1.86   0.069    -.2155177    5.632627
      ln_SRT |  -2.930164   2.054601    -1.43   0.160    -7.056952    1.196624
    ln_GDPPC |   5.581056   2.906683     1.92   0.061    -.2571898     11.4193
       _cons |  -32.63803   16.12961    -2.02   0.048     -65.0353   -.2407657
-------------+----------------------------------------------------------------
     sigma_u |  4.5866087
     sigma_e |  4.1834419
         rho |  .54587392   (fraction of variance due to u_i)
------------------------------------------------------------------------------

estat vce,corr

Correlation matrix of coefficients of xtreg model

        e(V) |      ESG    ZSCORE       ASG       CUR       LEV    INTCOV       PTB       ROA        TQ        BT     SIGMA    LEGALV      INFL     COVID   ln_SIZE    ln_SRT  ln_GDPPC     _cons
-------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
         ESG |   1.0000                                                                                                                                                                          
      ZSCORE |   0.2785    1.0000                                                                                                                                                                
         ASG |   0.2475    0.3166    1.0000                                                                                                                                                      
         CUR |  -0.3215   -0.9772   -0.3006    1.0000                                                                                                                                            
         LEV |  -0.4208   -0.8343   -0.2069    0.8798    1.0000                                                                                                                                  
      INTCOV |   0.0219   -0.0329   -0.1878   -0.0066   -0.2599    1.0000                                                                                                                        
         PTB |  -0.1889    0.1814    0.1335   -0.1129   -0.2361   -0.0513    1.0000                                                                                                              
         ROA |  -0.0417   -0.4253   -0.2902    0.5078    0.5498   -0.1301   -0.3822    1.0000                                                                                                    
          TQ |  -0.0772   -0.9000   -0.3320    0.8222    0.7310    0.0386   -0.4771    0.4502    1.0000                                                                                          
          BT |  -0.2923    0.1306   -0.0610   -0.1276   -0.1681    0.1829   -0.0683   -0.1792   -0.1104    1.0000                                                                                
       SIGMA |   0.3788    0.8802    0.2269   -0.9170   -0.9248    0.3610    0.1475   -0.5347   -0.7367    0.0814    1.0000                                                                      
      LEGALV |   0.1032   -0.2246   -0.2094    0.2271    0.2420    0.2724   -0.3723    0.2980    0.2843    0.1084   -0.2146    1.0000                                                            
        INFL |  -0.3218   -0.5684   -0.3296    0.5675    0.6453   -0.2035   -0.2911    0.5176    0.6128    0.0214   -0.6540    0.4680    1.0000                                                  
       COVID |  -0.3726   -0.7362   -0.3834    0.7686    0.8126   -0.2461   -0.2784    0.5972    0.6926   -0.0350   -0.8186    0.2869    0.7201    1.0000                                        
     ln_SIZE |  -0.1140    0.5756    0.1838   -0.5634   -0.5899    0.0483    0.7590   -0.5698   -0.6927    0.2380    0.5636   -0.4586   -0.4662   -0.5786    1.0000                              
      ln_SRT |  -0.4170   -0.8333   -0.2345    0.9028    0.8979   -0.1387   -0.1567    0.5095    0.6689   -0.0194   -0.9101    0.3034    0.5537    0.7789   -0.6391    1.0000                    
    ln_GDPPC |   0.4301    0.7509    0.2183   -0.8070   -0.8097    0.2233   -0.0020   -0.5016   -0.6140    0.0020    0.8468   -0.2595   -0.7824   -0.7546    0.4143   -0.8336    1.0000          
       _cons |  -0.2556   -0.6716   -0.1489    0.6927    0.7098   -0.3187   -0.2356    0.5420    0.6393   -0.1668   -0.7551    0.1001    0.7255    0.6905   -0.5823    0.7094   -0.8918    1.0000

[George Ford]

year fixed effect?

[George Ford]

you can convert your xtreg to just reg to use estat vif.

reg y x , absorb(id)
estat vif

[Ethan Kilmer]

I was unsure whether the year fixed effect was relevant for the multicollinearity test.

If running the 'estat vce, corr', should I include the robust standard error in the xtreg?

Thanks!

[George Ford]

Not sure, but my intuition tells me to use the cluster robust errors, because you will in the final model. I'd put year in there, and switch to reghdfe.

ssc install reghdfe
reghdfe WACC ESG ZSCORE ASG CUR LEV INTCOV PTB ROA TQ BT SIGMA LEGALV INFL COVID ln_SIZE ln_SRT ln_GDPPC, absorb(ID year) cluster(ID)

But run this to see what you get. Keep in mind, this will not detect multicollinearity of linear combinations.

 qui reg WACC ESG ZSCORE ASG CUR LEV INTCOV PTB ROA TQ BT SIGMA LEGALV INFL COVID ln_SIZE ln_SRT ln_GDPPC i.id i.year
estat vif

Also do this to exclude each variable to see how the results change. Might help you track down a problem.

global XMAIN ESG ZSCORE ASG CUR LEV INTCOV PTB ROA TQ BT SIGMA LEGALV INFL COVID ln_SIZE ln_SRT
local i = 1
foreach var in ESG ZSCORE ASG CUR LEV INTCOV PTB ROA TQ BT SIGMA LEGALV INFL COVID ln_SIZE ln_SRT {
    global XEX "`var'"
    global X : list global(XMAIN) - global(XEX)
    eststo e`i': reghdfe WACC $X , absorb(ID year) cluster(ID)
    local i = `i'  +1
}
esttab e1 e2 e3 e4 e5 e6 e7 e8
esttab e9 e10 e11 e12 e13 e14 e16 e16

[George Ford]

If you have a few main variables of interest, you can also include those in the regression and then add the other variables to determine when the problem starts.

[George Ford]

The reg option will not work. It makes a mess of it. Ignore that.

Can get VIFs this way.

global ALL ESG ZSCORE ASG CUR LEV INTCOV PTB ROA TQ BT SIGMA LEGALV INFL COVID ln_SIZE ln_SRT
foreach var in $ALL {
    quietly {
    global Y "`var'"
    global X : list global(ALL) - global(Y)
    reghdfe $Y $X , absorb(ID year) cluster(ID)
    }
    di "VIF_`var'" _col(20) %5.3f 1/(1-e(r2_within))
}

[Ethan Kilmer]

Okay I will try this out! Is there any particular reason to use reghdfe rather than xtreg y x, fe?

I really appreciate your input

[George Ford]

It absorbs multiple fixed effects, providing a cleaner output. (Though the latest version of Stata can do so as well, but I'm still trying to figure out why the SE are so different).

[Jeff Wooldridge]

estat vce, corr gives you the correlations among the parameter estimates, not the x. That’s very different. I can’t tell if the coefficients are large because you haven’t shown us units of your Y variable. Coefficients on logged vars often appear “large” because often it only makes sense to change the log by, say, 0.01 (a one percent increase in X).

[Ethan Kilmer]

Thank you for the clarification.

My dependent variable (ESG) is on a scale of 0-100.

"Coefficients on logged vars often appear “large” because often it only makes sense to change the log by, say, 0.01 (a one percent increase in X)" - so it makes sense to take the log by 0.01 rather than just "ln(variable)"? Could you provide me with the code for this?

[George Ford]

I'd start with TQ/ZSCORE and ln_SIZE/ln_SRT. Both have high correlations (the latter 0.97).

[Ethan Kilmer]

Yes, I am currently investigating the correlations and impacts of including these variables one by one.

In regards to my remaining data, does it make sense to take the regular natural logarithm of variables with large positive values (ln(variable))? For example, total assets of a firm, or GDP per capita? I am just slightly confused by Jeff's comment on 0.01.

Again, thanks for the input!!

[Ethan Kilmer]

George, I have some more questions regarding reghdfe. FYI, inn comparison to xtreg, it seems to provide me with more significant coefficients which I am happy with (thanks for the suggestion!!!).

1. Can I use interaction terms within this method?
2. Is it still considered FE regression? I am wondering how to explain this in my methodology.
3. Is interpretation of results the same?
4. for the 'absorb (year) portion, should I instead do 'absorb (Q_DATE) - my variables are measured quarterly.

Here is my output:

 
reghdfe WACC i.LEGAL#c.ESG TQ ASG CUR LEV ROA BT INFL ln_SIZE, absorb(Q_DATE) cluster(ID)

(MWFE estimator converged in 1 iterations)

HDFE Linear regression                            Number of obs   =      1,019
Absorbing 1 HDFE group                            F(  10,     50) =       7.31
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.2295
                                                  Adj R-squared   =     0.2069
                                                  Within R-sq.    =     0.1547
Number of clusters (ID)      =         51         Root MSE        =     5.0744

                                    (Std. err. adjusted for 51 clusters in ID)
------------------------------------------------------------------------------
             |               Robust
        WACC | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
 LEGAL#c.ESG |
          0  |   .0885032   .0465419     1.90   0.063    -.0049789    .1819853
          1  |   .0578929   .0377316     1.53   0.131    -.0178934    .1336791
             |
          TQ |   .2733341   .1296202     2.11   0.040     .0129842    .5336841
         ASG |  -.0006754   .0027836    -0.24   0.809    -.0062665    .0049157
         CUR |   .5938003   .2274227     2.61   0.012     .1370084    1.050592
         LEV |  -.0764465    .037516    -2.04   0.047    -.1517995   -.0010934
         ROA |    .039027   .0642866     0.61   0.547    -.0900964    .1681503
          BT |   1.418311    .508703     2.79   0.007     .3965507    2.440071
        INFL |   -.068703   .0985207    -0.70   0.489    -.2665876    .1291816
     ln_SIZE |  -.0954152   .2038758    -0.47   0.642    -.5049117    .3140813
       _cons |   12.35335   11.99887     1.03   0.308    -11.74709    36.45379
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
      Q_DATE |        20           0          20     |
-----------------------------------------------------+

. margins LEGAL,dydx(ESG)

Average marginal effects                                 Number of obs = 1,019
Model VCE: Robust

Expression: Linear prediction, predict()
dy/dx wrt:  ESG

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
ESG          |
       LEGAL |
          0  |   .0885032   .0465419     1.90   0.057    -.0027172    .1797236
          1  |   .0578929   .0377316     1.53   0.125    -.0160598    .1318455
------------------------------------------------------------------------------

[Jeff Wooldridge]

Ethan: But your use of reghdfe means you didn't include unit fixed effects. You only included time effects. You need to absorb(ID Q_DATE). Personally, in this sort of application I see no reason to use reghdfre. You can use xtreg with i.Q_Date with the fe option:

 
 xtreg WACC i.LEGAL#c.ESG TQ ASG CUR LEV ROA BT INFL ln_SIZE i.Q_DATE, fe vce(cluster ID)

This should reproduce reghdfe.

There's not code needed for what I was talking about. The point is that you are not going to increase ln(X) by one, as that means well over a 100% increase in X. You want to change ln(X) by a smaller amount when interpreting the coefficient.

Is there a reason you don't have LEGAL and ESG appearing by themselves? Unless they don't ever change across time, it's unusual to have interactions but not levels.