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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...

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  • $\begingroup$ maybe not normal (if you mean by normal "happens all the time"), but can sure happen $\endgroup$ – Christoph Hanck May 19 '15 at 12:11
  • $\begingroup$ 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). $\endgroup$ – Glen_b -Reinstate Monica May 20 '15 at 3:42
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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)

enter image description here

> 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  
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