Marginal effects in an AB/BA cross-over study

Question

I am currently working on an AB/BA cross-over study. There I have a treatment $T$, a time-varying covariate $X$ and a dummy variables $Z$. I wish to study the interaction between $X$ and $T, Z$. I also have some time-varying covariates $W$ that I wish to adjust for.

I assume the following linear model for the outcome $Y$:

$$Y = \beta_0 + \beta_1 T + \beta_2 X + \beta_3 XT+\beta_4W+\beta_5Z+ \beta_6XZ + \beta_7ZT + \beta_8XZT+\varepsilon $$

Denote $X_T$ and $X_C$ for the value $X$ takes at the treatment and control occasions respectively, with the corresponding definitions for $W_T$ and $W_C$ as well as for $Y_T$ and $Y_C$. I take the difference between the outcome at the treatment and control occasions and get $$Y_T - Y_C = \beta_1 + \beta_2(X_T - X_C) + \beta_3 X_T + \beta_4(W_T - W_C) + \beta_6 Z (X_T - X_C) + \beta_7 Z + \beta_8 ZX_T +\varepsilon^*$$

Thus linear regression of $Y_T - Y_C$ on $X_T-X_C$, $X_T$, $W_T-W_C$, $Z(X_T-X_C)$, $Z$ and $ZX_T$ yields identification of some of the parameters in the model.

Now, the coefficients of a model with interactions are always more complicated to interprete than those of a model without, so I want to also present the marginal effect of $X$ for different values of $Z$ and $T$. That is, I want to determine $\frac{\delta(Y_T-Y_C)}{\delta X}$ and get confidence intervals for those marginal effects of $X$ on $Y$ at different values of my dummy variables.

Had I had a more traditional linear regression model, I could have gotten this through the margins package in R. However, as $X$ is, in some sense, split into the $X_T$ and $X_C$ parts, said package can not (to my knowledge) be used for what I seek. I can get the manual point estimates by just adding the corresponding coefficients for the cases where $Z,T$ are 0 or 1, but the confidence intervals are proving tricky. Any suggestions would be much appreciated.

Answer 1

Turns out that the solution is pretty straight-forward. The point estimates, as I alluded to earlier, are gotten by simply taking the derivative and plugging in the corresponding value of $Z$ (and indirectly of $T$) in question. For instance, the marginal effect of $X$ on $Y$ when $T = 1$ and $Z = 0$ is simply $\beta_2 + \beta_3$, and the marginal effect when $T = 0$ and $Z = 1$ is $\beta_2 + \beta_6$. Replacing those parameters with their estimated counterparts from the regression model gives the point estimate of the marginal effect of $X$ on $Y$.

As for the standard errors, we get them by using rules for variance and covariance. For example: $Var\left(\hat{\beta}_2 + \hat{\beta}_3\right) = Var\left(\hat{\beta}_2\right) + Var\left(\hat{\beta}_3\right) + 2Cov\left(\hat{\beta}_2, \hat{\beta}_3\right)$.

From there we can construct confidence intervals via the normal or t-distributions as per usual.