My Question concerns the significance of the main (independent) variable in a 2sls-regression with an interaction term. While the interaction term is not significant in neither the 2sls-regression, the reduced form, nor in the OLS-regression, the main (independent) variable stays significant only in the reduced form (with interaction) and in the OLS regression (with interaction). Surprisingly, the
main (independent) variable turns insiginifcant in the 2sls regression (with interaction). Why is the main (independent) variable insignificant in the 2sls regression (with interaction) but significant in the OLS and reduced form cases?

Being more specific, I have a confidential dataset, which is at the bank-firm level (stylized example follows).

Code:


bank firm Diff_log_loan (bank-firm) Dummy (bank) Instrument (bank) Dummy_Country (firm)




1 1 0.02 0 2.3 1


1 2 0.30 0 2.3 0


2 1 0.45 1 3.1 1


2 3 -0.34 1 3.1 0 3 2 -0.01 0 0.5 0 3 3 -0.02 0 0.5 1 3 4 0.04 0 0.5 0



I am running the following 2sls-regression (a) in STATA 15.0 and am interested in the coefficient estimate of the variable "Dummy" and in the coefficient estimate of the interaction between the variables "

Dummy" and "Dummy_Country". I use the variable "Instrument" as IV for "Dummy" and saturate my regression with firm fixed-effects (which are very important for identification in my case).






Code:


(a) reghdfe Diff_log_loan  $controls (Dummy c.Dummy#c.Dummy_Country =Instrument c.Instrument#c.Dummy_Country) , ///
absorb(firm) vce(cluster bank)

I obtain an insignificant coefficient estimate of the variable "Dummy" and an insignificant coefficient estimate of the interaction term "

c.Dummy#c.Dummy_Country."



Then, I run the following OLS-regression (b) and the following reduced form regression (c):






Code:


(b) reghdfe Diff_log_loan Dummy Dummy_Country c.Dummy#c.Dummy_Country $controls , ///
absorb(firm) vce(cluster bank)






Code:




(c)


reghdfe Diff_log_loan Instrument Dummy_Country c.Instrument#c.Dummy_Country $controls, ///


absorb(firm) vce(cluster bank)





Now,

the coefficeint estimate of the interaction term "


c.Dummy#c.


Dummy_Country


" stays insignificant in both cases (b) and (c), but the coefficient estimates of both (a) "Dummy" and (b) "Instrument" are significant, respectively.




In all three cases the used sample reported in the regression output is identical (49,494 observations).



Further details: When I run regressions (a), (b), and (c) without the interaction term "c.Dummy#c.Dummy_Country", then the coefficient estimate of "Dummy" is significant (5%-Level). Also, when I restrict the sample in the 2sls-regression (a) to observations for which the variable "Dummy_Country==0", then I get a significant coefficient estimate of "Dummy" (the regression includes 49,200 observations).




Why is the main (independent) variable insignificant in the 2sls regression (a) (with interaction) but significant in the OLS (b) and reduced form (c) cases (with interaction)?





Many thanks for your help.



PS: Please note that d

ue to confidentiality reasons I cannot use this dataset with a PC with internet connection. Therefore, unfortunately, I cannot post the regression output.