I am running a multiple regression of the form Y = $\beta_0$ + $\beta_1$*$X_1$ + $\beta_2$*$X^2_1$ + $\beta_3$*$X_2$ + $\beta_4$*$X_3$ on a time-series dataset. I want to plot the relationship between Y and $X_1$ based on this equation such that I get a curve after taking into account the effects of $X_2$ and $X_3$ (or in other words, after the effects of $X_2$ and $X_3$ are removed from Y). What is the most appropriate way to do this?
You can plot the predicted value of Y from the equation at various levels of $X_1$, but you have to choose values of $X_2$ and $X_3$ for those lines. One choice would be the median value of each.
Taking a step back: 1) Are you sure you want multiple regression on a time series? This can cause some problems (e.g. if Y and any of the X are both increasing over time) 2) Where is 'time' in your formula? Is it one of the X's?