Interpreting multiple linear regression Suppose $$E[y|\textbf{x}] = \beta_0+\beta_{1}x_{1}+\beta_{2}x_{1}^{2}+\beta_{3}x_{3}$$
Is there an easy interpretation of the coefficients? If there was no $\beta_{3}x_{3}$ term then we would have a quadratic and take the derivative to get an interpretation. So maybe we want take the partial derivative with respect to $x_1$?
 A: Just take partial derivatives.
If you increase $x_1$ by one unit, then the expected value of $y$ increases by $\beta_1 + 2\beta_2 x_1$ units. Note that this effect depends upon the level of $x_1$. 
If you increase $x_3$ by one unit, then the expected value of $y$ increases by $\beta_3$ units. Note that this effect does not depend upon the level of $x_3$.
As someone else mentioned, $\beta_0$ is the average value of $y$ when $x_1$ and $x_3$ are 0. I like to interpret the intercept as putting the regression  through the point of means, the combination of the average $x_1$, the average $x_1^2$ (not the (average of $x_1$)-quantity squared), the average $x_3$, and the average $y$.
A: There are two answers: Purely from the formula, you can say that $\beta_0$ is the expected value of y when $x_1$ and $x_3$ are both 0, that $\beta_1$ is the linear increase in the expected value of y holding $x_3$ constant, and so on. 
But to get a sense of what is going on, the best way is to plot some points and make some lines. Pick values of $x_1$ and $x_3$ that occur in your data and then compute y, then plot $x_1$ vs. y for various levels of $x_3$ and $x_3$ vs. y for various levels of $x_1$.
A: I still see a quadratic function $x_1$, adjusted by the $x_3$ term. You can think of $x_3$ as a second constant term, in addition to $\beta_0$. There is no multiplicative effect between $x_3$ and the other terms.
A: I guess you can consider $\beta_{1}x_{1}+ \beta_{2}x_{1}^{2}$ as a quadratic effect. Depending on the sign of $\beta_1$, for increasing values of $x_1$ the quadratic effect may increase or decrease. We can consider the effects of other covariates by holding the quadratic effect constant. 
