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...
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