Why the magnitude of coefficient of interest decrease drastically when adding an interaction variable?

When examining whether the impact of laws on Y1 differently in developed countries is to add an interaction variable

pt is a variable of interest in a Differentce-in-Difference setting. pt_original is pt retrieved from this equation

Dependent_variables= pt + Independent_variables + fixed effects + error term

And pt along with developedpt is from this regression

Dependent_variables= pt + developed_dummy*pt + Independent_variables + fixed effects + error term

where developed_dummy equalling to 1 if this observation is in developed countries.

However, today, I faced this issue that the magnitude of coefficient of pt decreases drastically when controlling for the interaction variable. I am wondering what I should notice about this phenomenon? This is because the first coefficient estimate was estimated in the presence of omitted variable bias (OVB), and the effect of omitted interaction term just previously loaded onto the the beta coefficient of pt.

In presence of OVB coefficients can change substantially. For example If we assume that the true model is given by:

$$y= \beta_0 + \beta_1 pt +\beta_2 pt \cdot d + u$$

(where $$d$$ is developed dummy) but you will try to fit model that ignores the interaction term.

$$\hat{y} = \hat{\beta_0} + \hat{\beta_1} pt$$

In such conditions it can be shown that the expected estimated beta will be likely biased since the expectation of our beta will be given by (see Wooldridge Introduction to Econometrics):

$$E[\hat{\beta_1}]= \beta_1 + \beta_2 \frac{\sum (pt-\bar{pt})(pt \cdot d -\bar{pt \cdot d })}{\sum (pt-\bar{pt})^2}$$

Where the second term is the omitted variable bias. The intuition here is that if there is any covariance between $$pt$$ and $$pt \cdot d$$ and if $$y$$ actually depends also on $$pt \cdot d$$ (i.e. $$\beta_2 \neq 0$$) then if you omit the $$pt \cdot d$$ term the estimate of $$\beta_1$$ will be biased as it partially captures the relationship of $$y$$ and $$pt \cdot d$$. For example, if $$\beta_1=0.1$$, $$\beta_2= -2$$ and $$cov(pt,pt \cdot d )=0.5$$ and $$var(pt)=2$$ then if you estimate regression without $$pt \cdot d$$ the estimated $$\hat{\beta_1} = -0.4$$ whereas in regression where you correctly control for it would be $$\hat{\beta_1}=0.1$$ and $$\hat{\beta_2}=-2$$. So always if you omit important variables you will run into this problem.

Also, even though you are using fixed effects regression, remember, fixed effects only correct for firm invariant and time fixed effects only for time invariant unobservable. Presumably, $$pt$$ must be time variant (otherwise you could not run a fixed effects model in the first place), and also firm variant, so $$pt \cdot d$$ will be as well.

So your result just shows you omitted variable bias in action. If you ignore the fact that developed and developing countries are affected by laws differently, you find significant (at 10%) and negative coefficient but that was all driven by just developing countries. Once you properly control for the fact that some countries are developed you see that they were the ones driving the result since the $$pt$$ has significantly different effect there from the whole sample and the total effect of $$pt$$ there is quite large in absolute value ($$0.00000199-0.0199= -0.01989801$$).

• To use slightly different wording: -0.00506* is a sort of "average" effect for all countries. In the results with interaction, the coeff 0.00000199 (insignificant) on $pt$ is the effect of $pt$ for the countries with "developed=0". The effect of $pt$ for the countries with "developed=1" is, as 1muflon1 says, 0.00000199+(-0.0199). The estimated difference in the effects (between the developed & the underdeveloped) is -0.0199***. Jul 26 '21 at 1:48
• @1muflon1 "you find significant (at 10%) and negative coefficient but that was all driven by just developing countries", it should be the combination of developed and developing samples rather than being driven by developing countries, I deem? Jul 26 '21 at 5:03
• @BeautifulMindset no Chan is correct that this is an average effect but the results show that for developing countries there is simply no effect in your case so that average effect is driven by developed counties having negative effect and developing 0
– 1muflon1
Jul 26 '21 at 9:23