Change in the sign of Key independent variable in 3 different regressions

[lal mohan kumar]

Dear Stata Members
I ran 4 different regressions and my key variable of interest (Volatility_Index) changes its sign on a few occasions. The models include the same set of explanatory variables and the change I make is considering the lags. My Output is attached

 Model 1 All Independent variables are at time 't'
reghdfe Cash_to_Assets Volatility_Index SIZE_w LEV_w RD_DUM SG_w MB_w ROA_w PPE_w CFO_w CFO_VOL_w AGE_w, absorb (ind_dum year) cluster (id)
(MWFE estimator converged in 4 iterations)

HDFE Linear regression                            Number of obs   =    140,535
Absorbing 2 HDFE groups                           F(  11,  15652) =     566.01
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.1394
                                                  Adj R-squared   =     0.1391
                                                  Within R-sq.    =     0.1082
Number of clusters (id)      =     15,653         Root MSE        =     0.2398

                                    (Std. err. adjusted for 15,653 clusters in id)
----------------------------------------------------------------------------------
                 |               Robust
  Cash_to_Assets | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-----------------+----------------------------------------------------------------
Volatility_Index |  -.0097568   .0028779    -3.39   0.001    -.0153978   -.0041158
          SIZE_w |  -.0036144    .000836    -4.32   0.000     -.005253   -.0019758
           LEV_w |  -.0270098   .0083376    -3.24   0.001    -.0433525   -.0106672
          RD_DUM |  -.0059131   .0025762    -2.30   0.022    -.0109627   -.0008635
            SG_w |  -.0804705   .0024046   -33.47   0.000    -.0851837   -.0757572
            MB_w |  -.0142292   .0005879   -24.20   0.000    -.0153816   -.0130768
           ROA_w |  -.4183874   .0254141   -16.46   0.000    -.4682021   -.3685728
           PPE_w |  -.0443014   .0074621    -5.94   0.000     -.058928   -.0296749
           CFO_w |   .0424837    .014816     2.87   0.004     .0134426    .0715248
       CFO_VOL_w |  -.0925558   .0258393    -3.58   0.000    -.1432039   -.0419078
           AGE_w |   .0646821   .0016902    38.27   0.000     .0613691    .0679951
           _cons |   .1975041    .015831    12.48   0.000     .1664735    .2285346
----------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     ind_dum |        18           0          18     |
        year |        20           1          19     |
-----------------------------------------------------+

Model 2 Only Volatility_Index is lagged

reghdfe Cash_to_Assets l.Volatility_Index SIZE_w LEV_w RD_DUM SG_w MB_w ROA_w PPE_w CFO_w CFO_VOL_w AGE_w, absor
> b (ind_dum year) cluster (id)
(MWFE estimator converged in 4 iterations)

HDFE Linear regression                            Number of obs   =    140,406
Absorbing 2 HDFE groups                           F(  11,  15652) =     576.71
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.1409
                                                  Adj R-squared   =     0.1406
                                                  Within R-sq.    =     0.1097
Number of clusters (id)      =     15,653         Root MSE        =     0.2396

                                    (Std. err. adjusted for 15,653 clusters in id)
----------------------------------------------------------------------------------
                 |               Robust
  Cash_to_Assets | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-----------------+----------------------------------------------------------------
Volatility_Index |
             L1. |   .0358103   .0031561    11.35   0.000     .0296241    .0419965
                 |
          SIZE_w |  -.0046635   .0008355    -5.58   0.000    -.0063011   -.0030259
           LEV_w |  -.0264594   .0083334    -3.18   0.002    -.0427939    -.010125
          RD_DUM |  -.0098943   .0025574    -3.87   0.000    -.0149072   -.0048814
            SG_w |  -.0787333   .0024063   -32.72   0.000    -.0834499   -.0740168
            MB_w |  -.0136997    .000582   -23.54   0.000    -.0148404   -.0125589
           ROA_w |  -.4197649   .0253474   -16.56   0.000    -.4694487   -.3700811
           PPE_w |  -.0394118   .0074293    -5.30   0.000    -.0539741   -.0248495
           CFO_w |   .0333464   .0147884     2.25   0.024     .0043594    .0623334
       CFO_VOL_w |  -.1010976   .0257643    -3.92   0.000    -.1515985   -.0505966
           AGE_w |   .0657841   .0016723    39.34   0.000     .0625062     .069062
           _cons |   -.015407   .0167898    -0.92   0.359    -.0483169     .017503
----------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     ind_dum |        18           0          18     |
        year |        20           1          19     |
-----------------------------------------------------+

Model 3 All  Independent variables are lagged 

 reghdfe Cash_to_Assets  l.Volatility_Index  l.SIZE_w l.LEV_w l.RD_DUM l.SG_w l.MB_w l.ROA_w l.PPE_w l.CFO_w l.CF
> O_VOL_w l.AGE_w, absorb (ind_dum year) cluster (id)
(MWFE estimator converged in 4 iterations)

HDFE Linear regression                            Number of obs   =    128,955
Absorbing 2 HDFE groups                           F(  11,  14917) =     444.26
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.1358
                                                  Adj R-squared   =     0.1355
                                                  Within R-sq.    =     0.1070
Number of clusters (id)      =     14,918         Root MSE        =     0.2397

                                    (Std. err. adjusted for 14,918 clusters in id)
----------------------------------------------------------------------------------
                 |               Robust
  Cash_to_Assets | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-----------------+----------------------------------------------------------------
Volatility_Index |
             L1. |   .0186349   .0032806     5.68   0.000     .0122044    .0250654
                 |
          SIZE_w |
             L1. |  -.0044651   .0008571    -5.21   0.000     -.006145   -.0027851
                 |
           LEV_w |
             L1. |   .0326895   .0081656     4.00   0.000      .016684     .048695
                 |
          RD_DUM |
             L1. |   .0021194   .0025664     0.83   0.409    -.0029111    .0071499
                 |
            SG_w |
             L1. |  -.0172025    .002461    -6.99   0.000    -.0220264   -.0123786
                 |
            MB_w |
             L1. |  -.0210058   .0005584   -37.62   0.000    -.0221004   -.0199112
                 |
           ROA_w |
             L1. |   .3851511   .0149351    25.79   0.000     .3558764    .4144258
                 |
           PPE_w |
             L1. |  -.0527335   .0075056    -7.03   0.000    -.0674453   -.0380217
                 |
           CFO_w |
             L1. |   .1237895   .0145874     8.49   0.000     .0951964    .1523825
                 |
       CFO_VOL_w |
             L1. |  -.1812215   .0263877    -6.87   0.000    -.2329446   -.1294984
                 |
           AGE_w |
             L1. |   .0685172   .0017021    40.25   0.000     .0651808    .0718536
                 |
           _cons |  -.0033283   .0173806    -0.19   0.848    -.0373963    .0307398
----------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     ind_dum |        18           0          18     |
        year |        19           1          18     |
-----------------------------------------------------+

.

Model  4 Except variable of interest, all others are lagged. 

 reghdfe Cash_to_Assets  Volatility_Index  l.SIZE_w l.LEV_w l.RD_DUM l.SG_w l.MB_w l.ROA_w l.PPE_w l.CFO_w l.CFO_
> VOL_w l.AGE_w, absorb (ind_dum year) cluster (id)
(MWFE estimator converged in 4 iterations)

HDFE Linear regression                            Number of obs   =    129,050
Absorbing 2 HDFE groups                           F(  11,  14917) =     440.90
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.1358
                                                  Adj R-squared   =     0.1355
                                                  Within R-sq.    =     0.1071
Number of clusters (id)      =     14,918         Root MSE        =     0.2397

                                    (Std. err. adjusted for 14,918 clusters in id)
----------------------------------------------------------------------------------
                 |               Robust
  Cash_to_Assets | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-----------------+----------------------------------------------------------------
Volatility_Index |  -.0206103   .0029613    -6.96   0.000    -.0264149   -.0148057
                 |
          SIZE_w |
             L1. |  -.0034036   .0008562    -3.98   0.000    -.0050819   -.0017253
                 |
           LEV_w |
             L1. |   .0310474    .008152     3.81   0.000     .0150684    .0470263
                 |
          RD_DUM |
             L1. |   .0058614   .0025743     2.28   0.023     .0008154    .0109073
                 |
            SG_w |
             L1. |  -.0177188   .0024679    -7.18   0.000    -.0225563   -.0128814
                 |
            MB_w |
             L1. |  -.0214159    .000559   -38.31   0.000    -.0225116   -.0203201
                 |
           ROA_w |
             L1. |   .3780152   .0149347    25.31   0.000     .3487414     .407289
                 |
           PPE_w |
             L1. |  -.0573619   .0075174    -7.63   0.000    -.0720968   -.0426269
                 |
           CFO_w |
             L1. |   .1285699   .0146049     8.80   0.000     .0999425    .1571973
                 |
       CFO_VOL_w |
             L1. |  -.1735276   .0264111    -6.57   0.000    -.2252966   -.1217587
                 |
           AGE_w |
             L1. |   .0669816   .0017142    39.07   0.000     .0636215    .0703416
                 |
           _cons |   .1826648    .016266    11.23   0.000     .1507813    .2145482
----------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     ind_dum |        18           0          18     |
        year |        19           1          18     |
-----------------------------------------------------+

I am confused as there is no theory to help me to fixate on a particular model and how to test which model is correct?

[Carlo Lazzaro]

Ial:
if no theory can help you out, you may want to take a look at the Adj_Rsq to support your choice.
That said, both Adj_R-sq and within_Rsq do not vary that much across the 4 regressions.

[lal mohan kumar]

Thanks, Carlo for the helping hand. Can I ask you some follow up questions? If I choose one with say, positive association, then actually I am implying that the same kind of association is there in the population right? But given the alternative specifications, can I really hypothesize such a positive association? For presenting the results, should I show all outputs or is it okay to consider only convenient one. Finally, Can I put both, Volatility_Index at t and Volatility_Index at t-1 to get better interpretation as follows

. reghdfe Cash_to_Assets  l.Volatility_Index Volatility_Index  l.SIZE_w l.LEV_w l.RD_DUM l.SG_w l.MB_w l.ROA_w l.P
> PE_w l.CFO_w l.CFO_VOL_w l.AGE_w, absorb (ind_dum year) cluster (id)
(MWFE estimator converged in 4 iterations)

HDFE Linear regression                            Number of obs   =    128,955
Absorbing 2 HDFE groups                           F(  12,  14917) =     478.17
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.1415
                                                  Adj R-squared   =     0.1412
                                                  Within R-sq.    =     0.1129
Number of clusters (id)      =     14,918         Root MSE        =     0.2389

                                    (Std. err. adjusted for 14,918 clusters in id)
----------------------------------------------------------------------------------
                 |               Robust
  Cash_to_Assets | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-----------------+----------------------------------------------------------------
Volatility_Index |
             L1. |   .1293152   .0044349    29.16   0.000     .1206223    .1380081
             --. |  -.1208199   .0039855   -30.31   0.000     -.128632   -.1130078
                 |
          SIZE_w |
             L1. |  -.0034249    .000847    -4.04   0.000    -.0050851   -.0017647
                 |
           LEV_w |
             L1. |   .0282322   .0080696     3.50   0.000     .0124148    .0440496
                 |
          RD_DUM |
             L1. |   .0062318   .0025425     2.45   0.014     .0012482    .0112155
                 |
            SG_w |
             L1. |  -.0182055   .0024485    -7.44   0.000    -.0230048   -.0134061
                 |
            MB_w |
             L1. |  -.0205228   .0005493   -37.36   0.000    -.0215994   -.0194461
                 |
           ROA_w |
             L1. |   .3759544   .0148329    25.35   0.000     .3468801    .4050286
                 |
           PPE_w |
             L1. |  -.0531268   .0074179    -7.16   0.000    -.0676668   -.0385868
                 |
           CFO_w |
             L1. |   .1100924   .0144563     7.62   0.000     .0817563    .1384285
                 |
       CFO_VOL_w |
             L1. |   -.177259   .0261346    -6.78   0.000     -.228486    -.126032
                 |
           AGE_w |
             L1. |   .0652219   .0016863    38.68   0.000     .0619164    .0685273
                 |
           _cons |   .0547059   .0174951     3.13   0.002     .0204134    .0889984
----------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     ind_dum |        18           0          18     |
        year |        19           1          18     |
-----------------------------------------------------+

[Carlo Lazzaro]

Ial:
1) please consider that you are not dealing with a simple (i.e., one predictor only) regression. Therefore, each coefficient (statistically significant or not) is adjusted for the other ones. Provided that you gave a fair and true view of the data generating process, you could tell that, when adjusted for the other coefficient, -Volatility_Index- contributes to posituve variations in the regressand and statistically significantly so (however, the trivial criticism may be that with such a large sample size, it is not that difficult to reach statistical significance);
2) you may want to consider two model to show that lagging or not has a bearing on the sign of the predictor of interest.
3) Lagging or not it's a matter of methodological habits/theory of your research field. Unfortunately, I cannot advise you on that.

[lal mohan kumar]

Thanks, Carlo, the answers you gave, are enough to proceed with the results.