Is adding an interactive term to a regression analysis a valid way to examine time trend?

Question

I want to examine the effect of being a grandparent on work. My data spans from 2000 to 2020. The model is

$$\text{work}_{it}\sim \text{grandparent}_{it}+\text{covariates}_{it}+\delta_t+\varepsilon_{it}$$

where grandparent is an endogenous dummy variable instrumented by the dummy variable first_child_female ($=1$ is the individual's first child is female and 0 if otherwise). $\delta_t$ is time fixed effects and $\varepsilon_{it}$ is the error term.

Now I want to examine the time trend of the effect. Specifically, I want to examine the effect from 2000 to 2010, and from 2011 to 2020, to see how it changes. Instead of splitting the data into two halves, I'm thinking about constructing a dummy variable post2010 = 1 if the data was observed post-2010 and 0 if observed pre-2010.

Therefore, my model becomes

$$\text{work}_{it}\sim \text{grandparent}_{it} + \text{grandparent}_{it} \times \text{post-2010}+ \text{covariates}_{it}+\delta_t+\varepsilon_{it}$$

where $\text{grandparent}_{it}$ and $\text{grandparent}_{it}\times \text{post-2010}$ are endogenous variables instrumented by first_child_female and first_child_female * post2010.

Here are my two main concerns:

1. How to interpret the coefficients? I think the parameter of interest for 2000-2010 is the coefficient for $grandparent$, and the parameter of interest for 2011-2020 is the sum of the coefficients for $\text{grandparent}_{it}$ and $\text{grandparent}_{it}\times post2010$.
2. Is this a valid way to examine the time trend?

Thanks for the insights!

Answer 1

Happy to elaborate more as needed, but the answer to both questions is yes.

Briefly, I will use an example with a slightly different context, but the rationale is the same.

Let $y$ be your response variable, $x$ be a predictor (assume it to be a scalar variable) and let $d$ be a dichotomous indicator for group membership (1 in the group, 0 not in the group). Then the model $$\hat{y} = \beta_0 + \beta_1 · d + \gamma_0 · x + \gamma_1 · x·d$$ is the multiple regression model for the (multiplicative) moderation model of group membership on the relationship between $x$ and $y$.

If you are not in the group ($d=0$), then the equation relating $x$ and $y$ reduces to $$\hat{y} = \beta_0 + \gamma_0 · x$$ and if you are in the group ($d=1$), the equation reduces to $$\hat{y} = (\beta_0+\beta_1) + (\gamma_0+\gamma_1) · x$$

Finally, if you are doing a multiple regression, then you can use the p-value for the partial slope for the interaction term to determine if the slopes are statistically significantly different.

Hope this helps.