# Testing for ARCH Process

I have sales data that I think would be best modeled with an ARCH model.

I believe the following is true, but I'm not sure:

If the residuals are regressed against the sales data using the equation $e_t=b_0+b_1*X_t$, where $e$ represents the residual terms from the original regression and $X$ represents the sales data, if $b_1$ is statistically different from zero, then the regression model contains an ARCH process.

Is what I'm saying true?

You could test for ARCH patterns up to order $m$ by checking the joint statistical significance of all slope coefficients in the model $$e^2_t=\alpha_0+\alpha_1 e^2_{t−1}+\dots+\alpha_m e^2_{t−m}+u$$ in the Lagrange multiplier form: take $R^2$ from the regression and multiply it by the number of observations $T$; this will be distributed as $\chi^2(m)$ under the null of no ARCH effects. This is known as ARCH-LM test. See more in Wikipedia.