# Multiple linear regression question

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?

• Thanks for the quick response, Peter. Just to add, time here is the year of data collection. I want to understand how Y changes just as a function $X_1$ in one part of the analysis (given the two other IVs), whereas in a second and separate analysis with a much larger dataset, I want to see how Y changed over time with $X_1$, $X_2$, and $X_3$ as independent variables, may be, with the variable 'time' as a classification variable. My objective in the first case is to plot Y vs. $X_1$ (given the two other IVs), and in the second case to is to see the progression of Y over time.
– Kris
Dec 7, 2013 at 15:10
• You can see Y as a function of $X_1$ by doing what I suggested. But, as I said, this will give a false picture if both variables are increasing or decreasing over time; any two variables that are linked to time will be linked to each other (e.g. attendance at NFL games and the value of the Dow Jones Industrial average). Dec 7, 2013 at 15:13
• That is what we want to know, how Y changes as a function of $X_1$ (e.g. whether Y decreases/increases as $X_1$ decreases/increases, after the effects of two other IVs are taken into account). The relationship of Y and $X_1$, with time, is not linear.
– Kris
Dec 7, 2013 at 15:36
• As long as both Y and $X_1$ have any relationship with time, any graph or equation that does not account for time will be misleading. Dec 7, 2013 at 16:16
• If there is any trend of Y and $X_1$ with time, then what would be best thing to do? Include time T as one of the independent variables in the regression equation to account for this? Thanks for your time.
– Kris
Dec 7, 2013 at 16:43