Regression Model with even function?

Question

Is there any method to test if the mean function, $f(x)$, of a regression model $y=f(x)+\epsilon$ is even or not?

Answer 1

$\newcommand{\m}{\begin{bmatrix}}\newcommand{\em}{\end{bmatrix}}$ You're not saying what your data look like, so any answer must be fairly limited.

Suppose you fit a polynomial $y=a+bx+cx^2$ by least squares. You could then consider the null hypothesis $b=0$ versus the alternative hypothesis $b\ne 0$. You have $$ \m y_1 \\ \vdots \\ y_n \em = \m 1 & x_1^2 & x_1 \\ \vdots & \vdots & \vdots \\ 1 & x_n & x_n^2 \em \m a \\ c \\ b \em + \m \varepsilon_1 \\ \vdots \\ \varepsilon_n \em, $$ and write this as $$ Y = X A + \varepsilon. $$ The least-squares estimates $\hat A=\m \hat a \\ \hat c \\ \hat b \em$ are given by $$ \hat A = (X^\top X)^{-1}X^\top Y $$ Now write $X$ as $$ X = \m X_1 & X_2 \em \text{ where }X_1 = \m 1 & x_1^2 \\ \vdots & \vdots \\ 1 & x_n \em $$ and $X_1$ is the last column.

Take the residuals from regression of $Y$ on these two "even" columns, and regress that vector on the third column $X_2$. Do the usual two-sided $t$-test of whether the slope $b$ is $0$, with $n-3$ degrees of freedom. (Or, since the alternative hypothesis $b\ne0$ is two-sides, do the $F$-test with one degree of freedom in the numerator and $n-3$ in the denominator, and it will be equivalent.)

I won't go into higher-degree terms before knowing more about the problem.