lower order term has positive prediction in simple regression but negative prediction in quadratic regression I am using SPSS to conduct a quadratic regression. The IV is positively and significantly related to the DV in a simple linear model. However, after the squared IV is put in the model, the lower order of IV becomes insignificant and the direction shifts (i.e., it is now negatively related to the DV); the squared IV is not significant either. Why's that and how can I interpret the data? 
P.S. only one IV and one DV are in the model and both the IV and the DV are continuous data. No anomalous data point is found.
 A: There's nothing weird going on and it need have nothing whatever to do with errors. It has to do with where the minimum of the best-fitting quadratic is.
Consider the following quadratic relationship, which has no error (aside perhaps some miniscule floating point error):

Clearly if I fit a straight line, the fitted line has a positive slope.
But the actual quadratic is $y=x^2-4x$.
There's nothing "weird" about this. Indeed, it's true of adding other predictors, even ones that are not squared terms -- as long as they're correlated.
If you had data from a relationship like $y=\beta_0+\beta_1x_1+\beta_2x_2+e$ but you only fitted $\hat{y}=b_0+b_1x_1$, then you could see the same thing ... that the coefficient of $b_1$ in the fitted equation could flip around from the one you get if you fitted $\hat{y}=b_0+b_1x_1+b_2x_2$.
Indeed, in the quadratic case, the negative coefficient in the linear term (when the coefficient of the quadratic is positive) corresponds to the minimum of the quadratic occurring at a positive $x$-value:

If that minimum is right of 0, the linear term must have a negative coefficient. Another way to look at it is that the negative coefficient tells you about the slope of the quadratic at 0 (see how it slopes down there at the far left of the blue curve? The slope there is -4). 
Now, neither of those two related facts (that the minimum of the fitted quadratic occurs at a positive x, or that the slope at 0 is negative) tell you much that's very useful about what the fit is way over there where the data are.
Which is to say, don't try to interpret the linear coefficient (at least not the way you have been!) when you add a quadratic term.
It might pay you to play around a bit with quadratic equations and see what a negative term does.
[If you do want to be able to interpret the sign of the linear term the way you want to even after fitting a quadratic term, use orthogonal polynomials. The coefficients of those will stay the same as you add higher order terms.]
