# Variable significance very sensitive to specification of non-correlated second variable

I´m doing research on a political science topic and my models leave me behind with a big questionmark at this point. I have a dataset containing 79 observations on a number of variables and trying and testing out different models I have specified one control variable in several different ways, to which my variables of interest have reacted by either being significant at the 5% level or not at all. I can´t really find an answer to this, as there is no (theoretical) link between the two variables so in my understanding it´s not a classical confounder. I have copied the regression outputs below, but for your better understanding some more information: While my dataset is originally in a panel data form, I "collapsed" the panel data into a cross-sectional dataset only including the maximum values of my dependent variable for each country. The independent variable that´s giving me nightmares since days is one, that in the original panel data is reported as a running value of the individuals in the respective country that experienced a certain event a, and thus never decreases. My other explanatory variables are time-invariant factors. So the original dataset looks something like this:

Country     Var1      Var2   Var3     region  Var4   IndepVar    Day
CountryA     0         4      6        1       2        34        1
CountryA     4         4      6        1       2        65        2
CountryA     9         4      6        1       2        65        3
CountryA     14        4      6        1       2        62        4
CountryA     17        4      6        1       2        82        5
CountryA     21        4      6        1       2        71        6


As described I changed this into a cross-sectional dataset, keeping only the first observation in which the Independent Variable reaches it´s maximum with respect to the country. So it looks something like this:

Country     Var1  Var2  Var3  region  Var4  IndepVar
CountryA     17    4     6     1       2       82


I further normalised Var1 on 100.000 citizens and did some additional transformations: A: the number of citizens first experiencing the event at the described time-point (on 100.000 citizens) B: the number of citizens first experiencing the event the week prior to the described time point. I estimated a linear regression with OLS, getting the following outputs:

For log of Var1 on 100.000 citizens

Residuals:
Min      1Q  Median      3Q     Max
-49.005  -4.818   1.881   7.794  18.363

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)               4.677e+01  1.333e+01   3.508 0.000838 ***
Var2                      7.619e-02  1.045e+00   0.073 0.942130
Var3                      1.816e+00  9.228e-01   1.968 0.053462 .
log(Var1 on 100.000 cit)  1.851e+00  9.262e-01   1.999 0.049957 *
Var4                     -3.334e-04  2.310e-04  -1.444 0.153815
Var5                      4.591e+00  2.040e+00   2.250 0.027936 *
East.Europe_Centr.Asia   -8.230e+00  6.089e+00  -1.352 0.181327
Sub.Sahara_Africa        -3.485e+00  5.115e+00  -0.681 0.498115
MiddleEast_N.Africa      -3.547e+00  5.970e+00  -0.594 0.554558
LA.America_Carribean     -1.050e+01  7.416e+00  -1.416 0.161631
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 13.36 on 63 degrees of freedom
(6 observations deleted due to missingness)
Multiple R-squared:  0.2673,    Adjusted R-squared:  0.1626
F-statistic: 2.554 on 9 and 63 DF,  p-value: 0.01435


For A:

Residuals:
Min      1Q  Median      3Q     Max
-43.786  -4.342   1.413   7.200  22.515

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)                                 6.377e+01  1.358e+01   4.694 1.46e-05 ***
Var2                                       -4.171e-02  9.977e-01  -0.042 0.966784
Var3                                        1.114e+00  8.872e-01   1.256 0.213858
log(A)                                      2.561e+00  6.857e-01   3.735 0.000402 ***
Var4                                       -3.921e-04  2.102e-04  -1.865 0.066735 .
Var5                                        4.293e+00  1.948e+00   2.203 0.031181 *
East.Europe_Centr.Asia                     -9.978e+00  5.271e+00  -1.893 0.062882 .
Sub.Sahara_Africa                          -4.915e+00  4.872e+00  -1.009 0.316783
MiddleEast_N.Africa                        -6.925e+00  5.713e+00  -1.212 0.229956
LA.America_Carribean                       -1.092e+01  7.073e+00  -1.544 0.127609

---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 12.77 on 64 degrees of freedom
(5 observations deleted due to missingness)
Multiple R-squared:  0.3597,    Adjusted R-squared:  0.2696
F-statistic: 3.994 on 9 and 64 DF,  p-value: 0.0004369


For B:

Residuals:
Min      1Q  Median      3Q     Max
-48.398  -5.018   0.807   8.299  25.956

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)                       4.831e+01  1.341e+01   3.601 0.000619 ***
Var2                             -1.905e-01  1.050e+00  -0.181 0.856602
Var3                              1.904e+00  9.334e-01   2.040 0.045520 *
log(B)                            1.945e+00  7.659e-01   2.539 0.013560 *
Var4                             -4.109e-04  2.281e-04  -1.801 0.076414 .
Var5                              5.530e+00  2.033e+00   2.720 0.008395 **
East.Europe_Centr.Asia           -1.111e+01  5.866e+00  -1.893 0.062875 .
Sub.Sahara_Africa                -4.147e+00  5.115e+00  -0.811 0.420498
MiddleEast_N.Africa              -5.264e+00  6.140e+00  -0.857 0.394446
LA.America_Carribean             -1.185e+01  7.556e+00  -1.569 0.121645
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 13.44 on 64 degrees of freedom
(5 observations deleted due to missingness)
Multiple R-squared:  0.2915,    Adjusted R-squared:  0.1919
F-statistic: 2.926 on 9 and 64 DF,  p-value: 0.005744


When omitting the variable at all, I get:

Residuals:
Min      1Q  Median      3Q     Max
-53.748  -4.453   1.743   7.763  18.564

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)            45.9891033 13.9324535   3.301  0.00157 **
Var2                   -0.0964768  1.0923946  -0.088  0.92990
Var3                    1.5724778  0.9621439   1.634  0.10702
Var4                   -0.0002251  0.0002250  -1.001  0.32072
Var5                    5.2805804  2.1139426   2.498  0.01503 *
East.Europe_Centr.Asia -5.1318216  5.5944169  -0.917  0.36237
Sub.Sahara_Africa      -3.5883081  5.3204603  -0.674  0.50243
MiddleEast_N.Africa     1.6412392  5.7302669   0.286  0.77547
LA.America_Carribean   -7.8610240  7.6936320  -1.022  0.31068
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 13.99 on 65 degrees of freedom
(5 observations deleted due to missingness)
Multiple R-squared:  0.2201,    Adjusted R-squared:  0.1241
F-statistic: 2.293 on 8 and 65 DF,  p-value: 0.03136


you can see, how especially Var3 (together with Var2 this is one of the factors I´m actually interested in) changes it´s significance level depending on how Var1 is operationalized. I really don´t get what is going on there and I would appreciate if someone could give me hint on how to deal with this. I have already tested for (multi)collinearity, but the coeffcients are not above 1.5/2 so in my understanding not critical.

Thank you very much!