A pretty simple way to do it (since it is simply an equality constraint) is by rewriting the regression and estimate that through sm.OLS.

Y_t=c+β_1 X_t+β_2 Z_t+β_3 R_t \\
= c+(1 - β_2) X_t+β_2 Z_t+β_3 R_t \\
= c + β_2(Z_t - X_t) + X_t + β_3 R_t \\
Y_t - X_t = c + β_2(Z_t - X_t) + β_3 R_t

So create the variables Y_t - X_t and Z_t - X_t then regress Y_t - X_t on Z_t - X_t and R_t

If you want to test whether the restriction holds, a simple F test between the standard model you wrote in the question and the model above should work. If the F test is shows significant difference, the restriction doesn’t hold in the data