When regression coefficient is nearly 0 (in fact in the real model it's exactly 0), what's the meaning of p value (<0.05) of the coefficient?
For example, I did a multiple variable regression with simulated data in R with lm().
Generate simulation data with the equation $$ y=2x_1^2+3x_2^2+3x_1+5 $$
The terms $x_1x_2$ and $x_2$ coefficients are zero. Using the data to do regression.
xmesh=mesh(seq(-4,4,0.1),seq(-4,4,0.1)) x1=as.vector(xmesh$x) x2=as.vector(xmesh$y) y=2*x1^2+3*x2^2+3*x1+5 model=lm(y~x1+x2+I(x1^2)+I(x2^2)+I(x1*x2)) summary(model)
The result is :
Call: lm(formula = y ~ x1 + x2 + I(x1^2) + I(x2^2) + I(x1 * x2)) Residuals: Min 1Q Median 3Q Max -8.871e-12 -4.500e-15 -7.000e-16 5.700e-15 4.194e-12 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 5.000e+00 3.301e-15 1.515e+15 < 2e-16 *** x1 3.000e+00 7.545e-16 3.976e+15 < 2e-16 *** x2 -3.348e-15 7.545e-16 -4.438e+00 9.22e-06 *** I(x1^2) 2.000e+00 3.609e-16 5.542e+15 < 2e-16 *** I(x2^2) 3.000e+00 3.609e-16 8.314e+15 < 2e-16 *** I(x1 * x2) -9.377e-16 3.227e-16 -2.906e+00 0.00367 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.429e-13 on 6555 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 2.313e+31 on 5 and 6555 DF, p-value: < 2.2e-16
We can see that the coefficients of term $x_2$ and $x_1x_2$ are nearly 0, and p-value<0.01. I think that lm() did the significance test of coefficient based on t-test with NULL hypothesis $\beta=0$. So p-value<0.05 should mean that the coefficient is significantly different to 0. However the coefficient should be 0 in my model. I am confused. How to interpret these two coefficients' significance?
Add a new test $y=2x_1^2+3x_1+0.001x_2+5$
> y2=2*x1^2+3*(x1)+5+0.001*x2 > model3=lm(y2~x1+x2+I(x1^2)+I(x2^2)+I(x1*x2)) > summary(model3) Call: lm(formula = y2 ~ x1 + x2 + I(x1^2) + I(x2^2) + I(x1 * x2)) Residuals: Min 1Q Median 3Q Max -9.237e-12 -1.700e-15 -1.000e-16 2.200e-15 2.757e-12 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 5.000e+00 2.840e-15 1.761e+15 <2e-16 *** x1 3.000e+00 6.492e-16 4.621e+15 <2e-16 *** x2 1.000e-03 6.492e-16 1.540e+12 <2e-16 *** I(x1^2) 2.000e+00 3.105e-16 6.441e+15 <2e-16 *** I(x2^2) -2.722e-16 3.105e-16 -8.770e-01 0.381 I(x1 * x2) -3.226e-16 2.776e-16 -1.162e+00 0.245 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.229e-13 on 6555 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.257e+31 on 5 and 6555 DF, p-value: < 2.2e-16
You can see that in the new test, the coefficients and std Error of term $x_1x_2$ and $x_2^2$ are essentially zero. Their p-value are large enough to accept the null hypothesis that $\beta=0$, it's a good result.
How to interpret the p-value of essentially zero coefficients in the two tests?