I have my OLS regression: $y = \beta_1 + \beta_2 X_2 + \beta_3 X_3 +\beta_4 (X_3)^2$

Could anybody please explain to me the effect of a change in $X_3$ on the dependent variable?(Is the effect positive or negative). An example would be greatly appreciated.


As @Tomas already explained greatly in his answer you have to drive to $X_3$.

$$\frac{\partial y}{\partial X_3}= \beta_3 + 2 \times \beta_4 \times X_3$$.

As you want to now whether the derivative is positive you face the inquality:

$$0>\beta_3 + 2 \times \beta_4 \times X_3$$.

Depending on $X_3$ you can change the equation to

$$\frac{-\beta_3}{2 * \beta_4} > X_3$$

Let us look at an example in which you have real numbers for the Betas:

When you have the equation:

$\beta_1 = 5$

$\beta_2 = 3$

$\beta_3 = 4$

$\beta_4 = 2$

$y = 5 + 3X_2 + 4X_3 + 2(X_3)^2$

and $X_3$ will have a positive effect when:

$$\frac{-4}{2 \cdot 2} > X_3$$

which is simplified:

$$1 > X_3$$

my_df <- data.frame()
ggplot(df) + xlim(-10,10) + ylim(-10,10) + geom_abline(intercept=-4, slope=4) + geom_vline(xintercept=1, colour = "red")+ xlab("X_3") + ylab("derivative of X_3")

Everything left of the red line has a positive effect on the dependent variable

enter image description here

Note that the "change in the conditional mean of outcome y when regressors change by one unit" is also called marginal effect. You might look at the tag description and question tagged with this tag to get a broader understanding. In your case you want to know whether the marginal effect of $X_3$ is positive (or negative).

This question: Regression with Quadratic Term - Understanding Marginal Effect is closely related to your question as it also talks about the marginal effect with a quadratic term.


You just take the first derivative of the function with respect to $X_3$: $$\frac{\partial y}{\partial X_3}= \beta_3 + 2 \times \beta_4 \times X_3$$ You may see that the effect of changing $X_3$ on $y$ is not constant, but depends on the observed value of $X_3$ (for some, say, $i$-th observation). Also, ceteris paribus (other things fixed) interpretation holds only with respect to fixed $X_2$, but you cannot change $(X_3)^2$ while holding $X_3$ fixed.


The effect that a change in $X_3$ will have on the outcome depend on the coefficients of $\beta_3$ and $\beta_4$. If both $\beta_3$ and $\beta_4$ are positive, then any increase in $X_3$ will cause $Y$ to increase. If both $\beta_3$ and $\beta_4$ are negative, then the an increse in $X_3$ leads to a decrease in $Y$. (Granted, all this is given that $X_3$ is increased without changing the values of $X_2$.)

If the signs of $\beta_3$ and $\beta_4$ differ, then the net effect of an increase of $X_3$ on $Y$ will depend on the relative magnitude of $\beta_3$ and $\beta_4$, as well as the increase in $X_3$.


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