# interpretation of the main effect when interaction term is included and main effect changes sign

i have a panel data set where the N dimension is countries(127 countries) and T dimension is year(2010 to 2015). I use a fixed effects to estimate a model of the form:

Y= b1X1+b2X2+b3X3+b4X4+u.

I estimate the model and i find that the coefficient b4 is negative. All variables are continuous.

I am interested now to check the hypothesis that the effect of X4 to Y changes over time (that is if the effect of X4 to Y changes as the time dimension T=years increases) and if the effect increases/decreases.

In order to check the hypothesis i re-estimate the model including an interaction term as:

Y=b1X1+b2X2+b3X3+b4X4+b5years+b6X4*years+u

I treat the variable years as continuous. My first question is if this approach is correct in order to test the hypothesis.

After i estimate the second model i find that the coefficient of the interaction term b6 is negative and significant and the main effect b4 is positive and significant. I know that the coefficient b4 in the second model is not comparable to the coefficient b4 in the first model.

My second question is how can i interpretate this results in terms of my hypothesis.

Thank you for your time My Best Regards.

• In the first model was b4 statistically significant? In the second model which coefficients are statistically significant? The addition of year means the coefficients will change. Even if b4 is statistically in both models it could change signs because b4 no longer represents the full effect of X4 on Y. Some of it could be contained in the interaction term. – Michael R. Chernick Mar 17 '18 at 20:45
• in the first model b4 was negative and significant.You are correct that in the second model the coefficients change. Does that mean that i can not use this approach to test my hypothesis?. If this is the case which approach could i use to test the hypothesis? My best Regards – Thanos Mar 17 '18 at 21:16