# Quadratic term has positive coefficient in simple regression, and negative in multiple regression?

Is it normal, that quadratic term has positive coefficient in simple regression, and negative in multiple regression? Linear term of the same variable is positive in both cases...

• maybe not normal (if you mean by normal "happens all the time"), but can sure happen – Christoph Hanck May 19 '15 at 12:11
• This is a particular example of a well-known phenomenon (see the picture, in particular). It has a variety of names under different situations, but the short outcome is, if you omit an important explanatory variable, your esitmates may be severely biased (including sign changes). – Glen_b -Reinstate Monica May 20 '15 at 3:42

Yes, for instance, an increasing trend with a decreasing rate of increase will fit your description:

set.seed(127)
x <- rnorm(300, 10, 2)
y <- 10*x - .3*x^2 + rnorm(300, 0, 2)

plot(x,y)

> lm(y ~ I(x^2))

Coefficients:
(Intercept)       I(x^2)
48.6200       0.1938

> lm(y ~ x)

Coefficients:
(Intercept)            x
28.09         4.07

> lm(y ~ x + I(x^2))

Coefficients:
(Intercept)            x       I(x^2)
-1.9046      10.3900      -0.3195